the geometry of world - springer978-94-015-9303... · 2017-08-27 · the differential geometry of...

Appendix A

The Geometry of World Lines

In this appendix to the main work, we present a supplementary discussion of the differential geometric properties of world lines, including a complete derivation of the results that were summarized in Chapter 3. For purposes of completeness, some of the definitions and equations in Chapters 2 and 3 are repeated here. The reader should also be aware of a change in notation, in that the discussion in this final chapter is limited to the case of a single particle in arbitrary motion, completely without reference to any dynamical law. Thus the subscripts i and j here refer to spatial components of covariant vectors, rather than particles of an n-body system. Likewise, in discussing frame-dependent quantities, the dot refers to differentiation with respect to the frame time t, and not to the dynamical time r, which does not appear here.

A.l The Geometry of 1-d Curves

A.l.l Curves in the Space En

The differential geometry of curves1 is the study of the properties of onedimensional objects which can be considered as regular curves, which will be defined below.

1M. Lipschutz, Theory and Problems of Differential Geometry (McGraw-Hill, N. Y., 1969); M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. II (Publish or Perish, Berkeley, 2nd ed. 1975).

279

280 APPENDIX A. THE GEOMETRY OF WORLD LINES

Consider an arbitrary one-dimensional curve in n-dimensional Euclidean space En. Let the curve be continuously differentiable up to the nth derivative, i.e., of class en.

In a coordinate basis in the space, let the curve C be represented as a position vector function xc (A) of some parameter A over some interval I. Then over that interval, the vector function is xc = xc(A) is a regular parametric representation of the parameter A over I provided that the function xc = xc(A) is continuously differentiable over I (of class C 1)

and the derivative of the function xc = xc(A) with respect to A is nonvanishing everywhere along the interval.

The parameter A in general is not unique. An allowable change of parameter A -+ 0 over an interval I is a function A = A ( 0) such that the function A = .\( 0) is continuously differentiable over the interval Io and the derivative of .\( 0) with respect to 0 is nowhere vanishing along the interval. Without loss of generalization, let C be an oriented curve such that an arbitrary allowable parameter A is a monotonically increasing function along the forward direction of C.

For a given curve C, the allowable changes of parameter define an equivalence class. Let a regular curve be defined as an equivalence class of regular parametric representations.

In a given metric space, one may define the arc length function s(.X.) along the curve. Since s(.X.) monotonically increases with any parameter, it is possible, and often desirable, to use the arc length s as the parameter of the representation of the curve. A regular parametric representation of the form xc = xc(s) is known as a representation in terms of arc length or more simply as a natural representation of the curve.

The unique specification of a given curve C in En is described by the following theorem:2

Theorem 1 Let xc (A) be a regular parametric representation of a regular curve C in Euclidean space of n dimensions, where n 2:: 2. Then C is specified uniquely (up to background transformations of the coordinate basis) by n - 1 intrinsic curvature coordinates as functions of the parameter A.

Moreover, it is possible to uniquely characterize the curve by a set of n mutually orthogonal unit vectors which are functions of A along the curve.

2 For a proof, see Spivak, Vol. II, op. cit., footnote 1 on p. 279.

A.l. THE GEOMETRY OF 1-D CURVES 281

The intrinsic curvature coordinates can be thought of as describing the local rates and directions of the bending and twisting of the curve. By convention, the intrinsic parameters and differential equations of local curve geometry are given as functions of arc length, although it is always possible to represent them as functions of any other arbitrary parameter A.

A.1.2 Curves in the Space E 3

Consider a regular curve C in three-dimensional Euclidean space E 3 •

Then C is characterized by two intrinsic curvature parameters and the corresponding co-moving basis set consists of three mutually orthogonal unit vectors.3

Let x = x(£) be a natural representation of the curve, where d£ is the distance

d£= (dx 2 +dy2 +dz2 ) 112 .

The unit tangent vector u = u( £) is then

dx u =de"

(A.l)

(A.2)

The second derivative, i.e. the derivative of the unit tangent vector u = u(£) with respect to arc length s, is called the curvature vector kat the point x(£)

du k = d£" (A.3)

Because u = u(£) is a unit vector, the curvature vector k = k(£) is everywhere orthogonal to u along the curve, with k pointing instantaneously in the direction into which the curve is turning. The magnitude of the curvature vector is the absolute curvature lk1 1 at the point x( £) and is given by

(A.4)

The absolute curvature lk1l = lki(£)1 is the first of the two intrinsic curvature parameters in E 3 . The inverse of the absolute curvature k1

is known as the radius of curvature of the curve at the point x( £). Any point x( £) at which the absolute curvature lk11 vanishes is known as a point of inflection of the curve.

3 M. Lipschutz, op cit., footnote 1 on p. 279.


The second vector in the set of three co-moving basis vectors in E 3

is known as the principal unit normal n = n(£). At any point that is not a point of inflection, n can be defined as the curvature vector k = k(£) at the point x(£) divided by its magnitude,

k k n= Tkf = TkJ" (A.5)

The principal unit normal n = n(£) is therefore a unit vector which points instantaneously in the direction into which the curve is turning. The definition eq. (A.5) is a bit unsatisfactory in that n is not continuous through points of inflection. The remedy for this 4 is to define n instead as the vector such that

(A.6)

where k1 = k1 (£) is simply called the curvature and is a positive or negative quantity whose absolute value is identical the absolute curvature jk1 j. The vector n is then continuous through points of inflection and points either parallel or antiparallel to the curvature vector k depending on the sign of k1. Although this technique restores the continuity of n through points of inflection, it must be remembered that only the absolute value of the curvature k1 (i.e. the absolute curvature jk1 1) has any intrinsic meaning in the local geometry.

The third unit vector in E 3 , called the unit binormal h = h( £), is defined to be

h = u x n, (A.7)

and is therefore a unit vector orthogonal to both u and n, such that the three vectors form the desired orthonormal set, i.e.,

U·U

u·n

n · n = h · h = 1,

u ·h = n · h = 0. (A.8)

Because all three vectors are functions of arc length e (or any other parameter A) along the curve, they define a co-moving trihedron { u n h} which spans the space at any given parameter value A (or £) along the curve. The derivatives of these vectors with respect to any parameter must therefore be linear functions of the vectors themselves. In three-dimensional Euclidean space, if the parameter is chosen to be the



Euclidean arc length s, then the derivatives of these co-moving basis vectors can be represented as a system of linear first-order differential equations in matrix form

.!!._ [:] = [-~1 dC h 0

(A.9)

where k2 = k2 (C) is the second intrinsic curvature parameter for E 3

defined by the equation

(A.lO)

The intrinsic parameter k2 is known as the torsion and is a measure of the local rate at which the curve is emerging from the plane containing the unit tangent u and the principal unit normal n. At any point where k1 # 0, it can be shown that the torsion is given by

k _ (dxjdC · (d2xjdC2 x d3xjdC3))

2 - JdxjdC X d2x/dC2 J2 (A.ll)

Because the torsion is defined by eq. (A.lO) and not by eq. (A.ll), the torsion is a continuous function along C which vanishes at points of inflection. A similar situation will arise in the Minkowski space intrinsic parameters below.

The curvature k1 and torsion k2 as functions of Euclidean arc length C therefore uniquely determine a regular curve in E 3 up to transformations of the coordinate basis (i.e. the Galilean group).

The linear equations (A.9) are known as the Serret-Frenet Equations for a one-dimensional curve in three-dimensional Euclidean space and are the basic equations for the study of regular curves in E 3 .

A.1.3 Applications to Nonrelativistic Motion

Although the Serret-Frenet equations provide a complete geometric description of an arbitrary differentiable curve in E 3 , the intrinsic curvature coordinates as functions of arc length do not furnish a general representation for particle trajectories in nonrelativistic mechanics, although they remain useful in certain situations. The primary reasons for this lack of general applicability to kinematics are two-fold. The


first difficulty is that the conditions of differentiability can be violated at certain positions at which the particle is instantaneous at rest. In the simple example of an object thrown directly upwards on the surface of the earth, the differential geometrical picture breaks down at the turning point of the trajectory. Although it would be possible in this particular example to choose a moving reference frame to restore the differentiability of the trajectory, the exclusion of the stationary frame is contrary to the usual formalism of mechanics. Likewise a piecewise analysis of the problem is not satisfactory, as there is nothing special about the turning point from the standpoint of acceleration.

The second difficulty concerning the use of differential geometry for nonrelativistic trajectories is more fundamental and problematic, arising from the fact that differential geometry treats the trajectory as an onedimensional object unto itself without any need to specify a particular parameterization of the curve. Although the arc length parameterization is the conventional choice to represent the intrinsic curvature coordinates and the Serret-Frenet equations, any other parameter A (satisfying the allowability conditions) may be substituted in place of f. The arc length f is the simplest choice, but beyond its identification with the metric, as a parameter it plays no intrinsically preferred role.

In non relativistic kinematics, however, there does in fact exist a preferred parameterization of the trajectory, namely the absolute time t at which the particle was found at a particular point on the curve. The differential geometric description of a Newtonian trajectory as a function of an arbitrary parameter A would contain no information about when the particle was located at any point along the curve (i.e. the situation which arises in the Kepler problem where the quadrature solutions to the Lagrangian equations of motion at first yield only orbit equations for r as a function of</>, instead of r and </>as functions of time t. In nonrelativistic kinematics,5 the necessity of choosing the timet as the preferred parameter introduces an additional intrinsic coordinate arising from the physical kinematics but independent of the local curve geometry. This additional coordinate is specifically the chain rule derivative v = df/ dt and is identical to the speed of the particle along its trajectory. In other words, the unit tangent vector u = u(s) in eq. (A.2) is not identical to the particle velocity vector v(t) but is rather a unit vector parallel to

5 H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 2nd ed. 1980).


v(t). Moreover, the curvature vector k = k(s) is not parallel to the acceleration a(t) but is a time-dependent linear combination of v(t) and a( t). The resulting "kinematical" representations of the Serret-Frenet equations contain additional "nongeometric" terms arising from derivatives of the speed v with respect to arc length £. Although it is possible (and at times useful) to employ such a formalism, 6 its cumbersome nature is usually contrary to the simplicity motivations which would lead one to seek a. geometric picture of the kinematics in the first place.

Because the intrinsic curvature coordinates in Euclidean space of n dimensions are in genera.! functions of the nth order derivatives, the use of intrinsic curvature coordinates for nonrela.tivistic motion is often avoided on the grounds that it makes reference to the third derivative b(t) = x (t). This objection is purely aesthetic, however, and should not be seen in any way a.s a. prohibition against intrinsic curvature representations of motion. Likewise, in the spacetime M 4 , one should expect the eventual appearance of fourth-order derivatives in the derivation of the intrinsic coordinates.

A.1.4 Applications to Relativistic Motion

Although the differential geometry of curves fails as a. complete kinematical description of the nonrelativistic particle due to the preferred role of timet a.s a. parameter, the situation in regard to relativistic kinematics is quite different. The first difficulty, regarding the pathology of a. particle at rest, is alleviated by the fact that a particle is never instantaneously at rest in Minkowski space. More significantly, the difficulty regarding the problem of the preferred parameterization does not arise because in Minkowski space the world line trajectory furnishes a complete kinematical description of the particle independent of the parameterization, i.e the geometric relationship of neighboring events along the world line completely specifies all the information which can be recorded by any particular inertial observer. The world line may be therefore treated as a one-dimensional object unto itself. The need for the extrinsic parameter v = v(t) has been removed at the cost of the introduction of an additional Euclidean dimension.

Although the kinematics in Minkowski space is independent of the parameterization, one may nevertheless choose by convention to repre-

6 See, e. g., G. N. Plass, Rev. Mod. Phys. 33, 37 (1961).


sent the intrinsic curvature coordinates as functions of Minkowski arc length s, which in a given coordinate basis is

(A.l2)

and which is invariant, by construction, under the frame transformations.7

The Minkowski arc length representation has the additional advantage in that it is measured explicitly by a co-moving proper· clock which is affixed to the particle. As is the case in E 3 , the conventional use of the arc length representation must in no way be seen to preclude the use of any other arbitrary parameter A beyond the allowability condition that ds / dA be a positive definite function along the world line. In the following sections, the label s will refer unambiguously to both the Minkowski arc length along the world line and the proper time of the moving particle.

It is often stated that because of the form of the metric function eq. (A.l2), the proper time parameters is invariant under background transformations of the coordinate basis, where the coordinate transformations are the transformations of the inhomogeneous Lorentz group. It must be seen, however, that any allowable parameterization A is invariant under the coordinate transformations, in that the values of A are identified with particular events along the world line, each of which is held stationary under frame transformations. That s is the arc length simply means that it demarcates the world line in even metrical segments whereas A does not. Although different segments of equal A are in general unequal in metrical length within any given reference, so that in general the segment dx J.L ( Ai+2 - Ai+d is unequal in length to the segment dxJ.L (Ai+l - Ai), each of these segments is separately invariant in length under a frame transformation, i.e. the segment dxJ.L (Ai+l- Ai) is identical in length to the segment dx~ (Ai+l - Ai)· Whether a particular A is directly measurable by a set of co-moving instruments would depend on the particular choice but in no way would affect the issue of allowability.

From the indefinite form of the Minkowski metric eq. (A.l2), the space M 4 is not the Euclidean space E 4 , and one must use the generalization if Theorem 1 to M 4 with caution. In fact, an arbitrary curve in M 4 cannot be studied using the tools pertinent to En because an

7 W. Pauli, The Theory of Relativity (Pergamon, N. Y., 1958).

A.2. SPACETIME CURVES 287

infinitesimal lightlike separation along a curve (i.e. ds = 0 for two neighboring points) would violate the conditions for regularity of the curve. For world lines of classical material particles, however, this situation is never encountered8 and therefore one may treat the Minkowski metric as if it were a definite metric. Moreover, the intrinsic curvature coordinates for a world line in M 4 will be derived "from scratch" and Theorem 1 will not explicitly appear in the derivations but will serve as a motivation for the final result.

A.2 Spacetime Curves

A.2.1 Special Relativistic Kinematics

Consider a point particle of arbitrary mass moving under the influence of an arbitrary nongravitational action-at-a-distance potential. In a given coordinate basis (t, x, y, z) the particle will be observed to have a trajectory given by some vector function x = x(t). Likewise in that basis the particle will be observed to have a velocity, acceleration, etc. given by the derivatives of position x(t) with respect to the timet of that frame:

X X (t),

v :X (t), a X: (t), (A.l3)

b x (t) , d x(4 ) (t),

where the dot indicates differentiation with respect to the coordinate time t of the frame.

The frame-dependent kinematic observables of eq. (A.l4) have magnitudes given by the Euclidean metric, i.e.,

8 Cf. Section 1.7 of this work; E. C. G. Stckelberg, Helv. Phys. Acta 14, 372, 588 (1941).


b (A.l4)

d

as well as frame-dependent scalar products, i.e.

(A.l5)

Consider a Lorentz transformation to a second coordinate basis with axes parallel to the first but with origin moving at velocity /1 with respect to the origin of the first frame, where f3 = j/Jj < 1. Without loss of

generality let /1 lie along the x axis of both frames. The position x and time t of any event will transform according to the equations

t "'( ;3 ( t' + f3 x') , X 'Y/3 (x' + f3 t') , y y', (A.l6) z z'

' where the primes indicate the "old" coordinates and where 'Y/3 = 'Y/3 ((3)) is defined to be the function

( 2) -1/2 'Y/3 = 1 - (3 . (A.17)

Dividing the infinitesimal form of the Lorentz Transformations for dx, dy, and dz ofeq. (A.17) by the transformation equation for dt yields

dx dx' + f3dt' -dt dt' + f3dx''

dy dy' (A.l8)

dt "'((J (dt' + f3dx')'

dz dz' -dt 'Y/3 (dt' + f3dx')'

Dividing the top and bottom of all three expressions by dt' gives

dx dx' I dt' + f3 dt 1 + f3 ( dx' I dt') '

A.2. SPACETIME CURVES

dy dt

dy1 1 dt1

r/3 (1 + (3 (dx 1jdt1))'

dz dz1 jdt1

dt rf3(1+f3(dx1jdt1))'

289

(A.19)

which generalize to the transformations (originally due to Poincare) of the components of velocity parallel and perpendicular to the boost vector !1,

V.L vi

.L

(A.20)

Extending the process in eqs. (A.18 -A.19) by dividing the velocity transformations eq.(A.20) by dt1 yields the transformations for the components of the acceleration,9

( ~ )3' r3 1 + (3. yl

(A.21)

1 [ 1 ~ ( I I)] ----~-------,:-3 a.L + (3 x a x v .

~~ ( 1 + (3 · v 1)

Likewise the components of the third derivative b =X. transform as

(A.22) 1

9 J. D. Jackson, Classical Electrodynamics (Wiley, N. Y., 2nd ed. 1975).


[b/ (3- (b' ') 3 (/1. a') ( I (3- ( I '))] 1. + X XV - ( _ ) al. + X a XV . 1 + f3 · v'

Likewise the components of the fourth derivative d = x( 4 ) transform as

1

rE(1+/1·v') 5

[d' _ 7(/J·a') b' + ( 15(/l·a'f _ 3(/J·b')) '] II ( 1 + iJ. v') II ( 1 + iJ. v'r ( 1 + iJ. v') all ,

(A.23)

1 ------=-X

r4 ( 1 + /1 · v' f [ d ~ + /1 X ( d' X v') + /1 X (b' X a')

7 (/J ·a') _ (b~ +/Jx (b' x v')) ( 1 + f3 · v')

( (- )2 (- ) ) l 15 f3 · a' 3 f3 · b' + _ 2 - _ (a~ + /1 x (a' x v')) .

( 1 + f3 . v') ( 1 + f3 · v') The basic Lorentz transformations for x and t in eq. (A.17) may be

generalized in that any four quantities (qo, q1, q2 , q3) that transform in the same fashion as (t, x, y, z) are said to comprise a covariant fourvector. The coordinates (t, x, y, z) of any point on a particle world line are said to define the covariant position vector of the event, labeled xf.L, where f-L runs over the indices 0-3, i.e.,

(A.24)

and where by convention, the top line gives the "time" or {0} component and the bottom line gives the "space" or { i = 1, 2, 3} components of the covariant vector, i.e., for an arbitrary covariant four-vector


{ qo ql-' = q' (A.25)

where q = (qx, qy, qz)· .] ust as the magnitude p of position vector x~-', given by

(A.26)

is invariant under frame transformations eq. (A.17), the magnitude of any covariant four-vector q~-', given by

(A.27)

is likewise invariant under frame transformations. The scalar product between between any two covariant vectors ql-'

and P~-t is the Lorentz invariant expression

(A.28)

Einstein 10 showed from the velocity transformation equations (A.20) that the components ofv could be assembled into a covariant four-vector of the form

ul-' = { "'!

'{V

(A.29)

where"'!= 'f(v) is the frame-dependent function of speed

"'! = ( 1- v2) - 12 (A.30)

and must not be confused with the quantity "'/(3 in eq. (A.17). Although they have the same functional form, '{ = "'!(v) is a quantity that is measured within a given reference frame, whereas "'/(3 is defined only in the transformation between two reference frames.

By eq. ( A.20), the speed v transforms under the Lorentz transformations by

( 2) 1/2 v'2 + {3 2 + 2j] · v' -liJ X v'l

v= ~ ' 1 + f3 · v'

(A.31)

10 A. Einstein, Bull. Am. Math. Soc., April 1935, p. 223.


so that 1 = 1(v) transforms under a Lorentz transformation of magnitude (3 by the equation

I ( v) = lt311 ( v) ( 1 + ,B · v') . (A.32)

By experience, 11 the quantity 1 = 1(v) appearing in u~" is equal to the dilation rate of the coordinate time t with respect to the proper time s of a particle moving instantaneously with speed v in the frame, i.e.,

dt ds = l(v), (A.33)

from which it can be shown that Minkowski arc length (A.l2) is equal to proper time.

From the dilation equation (A.33), the covariant vector u~" can be written as

{ dtjds

u~" = dxjdt · dtjds (A.34)

From the chain rule, therefore, the Einstein covariant velocity u~" = u~" ( s) is identical to the derivative of the covariant position vector x ~~ = x~"(s) with respect to proper time, i.e.,

dx~" UM=-.

ds (A.35)

Because "Lorentz invariance" refers to equality of measured value among reference frames and not to constancy in time within a given frame, the magnitude of the covariant position vector lx~"x ~-"I of a moving particle in general depends on the proper time s, although its value is the same for all observers at any given value of proper time. The magnitude of the covariant velocity uf..l at any times, however, is not only invariant among frames but is also identically a constant equal to unity for all speeds v, i.e.,

uf..luf-l = 1. (A.36)

One may interpret this condition as the result of the fact that no independent fourth component of v exists, and therefore one creates the

11 H. E. Ives and C. R. Stilwell, J. Opt. Soc. Amer. 28, 215 (1938); 31, 369 (1941); H. P. Robertson, Rev. Mod. Phys. 21, 378 (1949).


{0} component of u'" by "stretching" the three independent components of v into four dependent components. The cost of the extra component is borne by a loss of freedom in magnitude.

A.2.2 World Lines as Regular Curves

Let the particle have a world line in M 4 that is continuously differentiable up to the fourth derivative (i.e., of class C 4 ) and oriented such that its forward angle with respect to the t axis of any coordinate basis is everywhere less than 45 degrees. Let A be an arbitrary (frame-invariant) monotonically increasing parameter along the world line. Then the position four-vector x'" = x'"(.X) of a particle is a regular parametric representation and the world line can be considered a regular curve.

Let x'" = x'" ( s) be a natural representation of the world line. Then the derivative dx '"/ ds of x'" with respect to arc length s defines the unit tangent vector to the world line at the point x'" ( s).

Since arc length in M 4 is experimentally measured by the proper time clock affixed to the particle, then by eq.(A.35) the unit tangent vector dx '"/ ds is exactly identical to the Einstein covariant velocity four-vector u,, = u'"(s) defined in eq.(A.29). One may therefore interpret the constant unit magnitude of u'" additionally as a result of the experimental fact that proper time is identical to the arc length of spacetime M 4 . Let u'" = u'" ( s) therefore be identified unambiguously as both the covariant velocity and as the unit tangent vector at the point x'" ( s). Note that u{, is a timelike unit four-vector, as it always lies within the forward light cone around the t axis of any coordinate basis. At any point x'" ( s), the unit tangent u'" therefore defines a spacelike subspace perpendicular to u'", known as the normal subspace at the point x'"(s).

Since X I" = X I" ( s) is of class C4 ' one may define the curvature fourvector a'" to be the derivative of the unit tangent vector with respect to arc length s

(A.37)

Because u'" is a unit vector, it follows that a'" is everywhere orthogonal to u'" and lies within the normal subspace. It is tempting to envision a'" as pointing instantaneously in the direction in M 4 into which the curve is turning, and although this is true in the metrical sense, it must


be remembered that orthogonality in M 4 does not in general correspond to perpendicularity in a Euclidean depiction.

Because uil is both the covariant velocity and the unit tangent vector, the vector ail = ail ( s) can be unambiguously labeled both the curvature four-vector and the covariant acceleration. To derive the components of ail in a given coordinate basis, one uses eq. (A.29), as well as the chain rule

d dt d d ds = ds dt = 1 dt' (A.38)

in order to obtain the derivative of 1 = 1 ( v) with respect to proper time s. The chain rule gives

d1 d ( 1 ) d1 dv ds = ds ~ = dv ds'

(A.39)

where the derivative of 1 with respect to speed v is

v 3

( 2)3/2 = vI ' 1- v (A.40)

and where the derivative of speed v = lvl with respect to proper time s

lS

dv dv dt dv -=--=-1· ds dt ds dt

The derivative of speed v with respect to time is

which by the chain rule yields

dv v ·a -=-1· ds v

(A.41)

(A.42)

(A.43)

The derivative of 1 = 1 ( s) with respect to proper time s is therefore

d1 4 - = (v ·a) 1 . ds

(A.44)


Taking the derivative of the components of ull in eq. (A.29) with respect to sand using eq. (A.44) gives the the components of the curvature four-vector all= dulljds as

{ (v ·a) 1 4

all = a'/'2 + v (v. a) 1'4 . (A.45)

From this it can be verified explicitly that all and ull are everywhere orthogonal,

allu - ( (v. a) 1'4 ) . ( 'Y ) ll - a 1 2 + v (v ·a) 1 4 V'Y '

(A.46)

allull ((v·a)'Y4) 'Y-(a'Y2 +v(v·a)'Y4) V'/',

allull ( 1- v2) (v ·a) 1 5 - (v ·a) 1 3

allull (v ·a) 1 3 - (v ·a) 1 3

allull 0. (A.47)

The Lorentz invariant magnitude of the curvature vector is

which gives

( 1 - v2) (v · a) 2 1 8 - 2 (v · a) 2 1 6 - a21\

(v. a)2 1'6- 2 (v. a)2 1'6- a2'/'4,

-a2'Y4 _ (v. a)2 1'6·

(A.48)

(A.49)

The magnitude of all = all(s) is negative because all is a spacelike four-vector. Let the positive quantity a= a(s) be defined by

(A. 50)

such that

(A.51)


Because a = a( s) is the (positive) magnitude of the curvature fourvector all, let a be called the intrinsic curvature of the world line at the point xll(s). For the nonrelativistic particle, v--+ 0 and a reduces to the nonrelativistic acceleration a. The curvature a is therefore a measure of the Lorentz invariant proper acceleration of the particle, i.e., the instantaneous acceleration of the particle relative to its local inertial rest frame defined by the unit tangent uw

A point on the world line at which all = 0 is known as a point of inflection of the world line, so that at a point of inflection, the invariant proper acceleration a vanishes. In dynamics, one would expect points of inflection to occur at points where the particle is instantaneously moving as a free particle.

Let a =/= 0 along some segment of the world line. The vector nil = nll(s), defined by

1 (A.52)

is a spacelike unit vector that is everywhere orthogonal to the unit tangent vector Uw

This definition of nil suffers from the same problem as the definition of n in E 3 in that it is not continuous through points of inflection, since it is not defined where a = 0. Using the identical "inverted definition" technique as in eq. (A.6), however, the vector nil can be defined to be continuous through points of inflection. Define the principal unit normal at the point x 1,(s) along the world line to be vector nil= nll(s) such that either

(A.53)

everywhere along the curve. Here we choose to retain a = a( s) as a positive definite quantity as defined by by eq. (A.51) instead of redefining it to allow negative values, as was done with the curvature in E 3 . The choice for the retention of the positive definite definition stems for the confusion that could result concerning the sign of the magnitude of a covariant vector in Minkowski space, a situation that does not arise in Euclidean space.

At some initial time s0 where a =/= 0, let nil be parallel to all so that the positive sign is chosen in eq. (A.53). Then for subsequent segments between points of inflection, the positive or negative sign is chosen depending on whether nil is parallel or antiparallel to all along that particular segment. At the initial time so, the direction of nil can


be chosen so that uf..L and nf..L form a right-handed coordinate basis in the 1 + 1 dimensional subspace of M 4 .

It follows that nf..L is a spacelike unit vector either parallel or antiparallel to af..L, continuous through points of inflection, and satisfies the correct orthonormality relations,

nf..Lnf-L = -1, nf..Luf..L = 0.

A.2.3 The Unit Binormal Four-Vector

(A.54)

In E 3 , the third unit vector h characterizing the differential geometry of a curve was found from the first two ( u, n) by the use of the threedimensional cross product. In M 4 , however, the first two unit vectors do not specify a unique direction spacetime, but rather an orthogonal plane. It follows that the derivation of the third unit vector in M 4

must follow an alternative route, namely through the Gramm-Schmidt orthonormalization procedure below. Once the third unit vector is in hand, the fourth one will follow from the covariant vector product, as will be shown in the next section.

Because the world line representation x J.L = x J.L ( s) is of class C 4 , one may define the covariant third derivative vector as

(A. 55)

From eq. (A.45), the components of daf..Ljds in a given coordinate basis are

daf..L

ds [~~12 +2a1~; + ~: (v·a) 1 4 +v (~:·a) 1 4

+ v ( v · ~:) 1 4 + 4v (v ·a) 13 ~~)

which by eq. (A.38) and eq. (A.44) is

(A.56)


da11 { a2f'5 +(v·b)f'5 +4(v·a)2 1'7

IS

ds = bf'3 +3a (v·a)1'5 +v (a2f' 5 + (v·b) 1'5 +4 (v·a)2 1'7)

(A.57) The Lorentz invariant magnitude of the third derivative four-vector

~d:;; = (a21'5+(v·b) 1'4+4(v·a)2 l'~r

- (b1'3 +3a (v·a)1'5+va21'5+v (v· b) 1'5 +4v (v·a) 2 l''r · (A.58)

yielding

~ d:;; = -b2f'6 + a4/'8- (v. b )2 1'8- 6 (v. a) (a. b) 1'8 (A.59)

-7a2 (v · a) 2 1'10 - 6 (v ·a) (v · b )2 1'10 - 8 (v · a) 41'12

which can take on both positive and negative values, indicating that da 11 / ds can be either spacelike, timelike, or lightlike depending on the kinematical state of the particle. Define the function w = w(s) by

2 da 11 da 11 w =---.

ds ds Then w = w(s) is the Lorentz invariant scalar given by

w 2 = b2 f' 6 - a 4 f' 8 + (v · b) 2 1'8 + 6 (v ·a) (a· b) 1'8

+ 7 a2 (v · a) 2 1'10 + 6 (v ·a) (v · b )2 1'10 + 8 (v · a) 4 1'12.

(A.60)

(A.61)

The scalar product of third derivative da11 / ds and the curvature vector a 11 is

dall 11 -a ds

(A.62)


which gives

~aiL= [(a215+ (v·b) 14+4(v·a)2 17) ((v·a) 14)]

- [(b13 +3a (v·a)15+va21 5+v (v·b) 1 5+4v(v·a) 2 1 7)

(A.63)

daM , ) 5 ( ) ( ) 7 2 ( ) 7 ( )3 g ds a~ = - (a · b 1 - v · a v · b 1 - 3 a v · a 1 - 3 v · a 1 ,

(A.64) which is negative. Define the function ( = ( ( s) by the equation

( = (- d:: aiL r/2, (A.65)

then ( = ( ( s) is the positive definite Lorentz invariant scalar

( = ((a. b) 1 5 + (v. a) (v. b) 1 7 + 3 a2 (v. a)r7 + 3 (v. a) 3 1 9f 12 . (A.66)

The scalar product of the third derivative da~tfds and the velocity

ll ~' IS

bl3 +3a(v·a)l5 · ( v-y-y) , (A.67)

+v ( a 2 1 5 + (v ·b) 1 5 + 4 (v · a) 2 1 7)

(a2 1 5+(v·b)l4+4(v·a) 2 1 7) 1

-(b13 +3a(v·a)l5 (A.68)

+va2 l+v (v· b) 1 5+4v (v·a) 2 17) VI,

which can be shown to yield


i.e., da __.!!_ui-L = -al-L a~-', ds

which can be proven as follows:

:s (al-'u~-') dds (al-'u~-')

0

dal-' 1-' -u ds

(A.69)

(A.70)

(A.71)

From eqs. (A.64) and (A.69), the covariant third derivative dal-'jds has components along both ul-' and nw By subtracting out these components along ul-' and nl-', one arrives at a spacelike vector that is perpendicular to both ul-' and nl-',

dal-' - (dal-' . u~-') u + (dal-' . ni-L) n = daJ-L - azu - (2 a ' (A.72) ds ds 1-' ds 1-' ds ~-' a 2 ~-'

where the positive sign is used before the third term on the left-hand side because both dal-'j ds and nl-' are spacelike vectors. The positive magnitude of this vector is the Lorentz invariant scalar denoted by a =

a(s),

(A.73)

(A.74)

which reduces to


(A.75)

where the quantity inside the radical can be shown below to be positive definite. Using eqs. (A.61), (A.66), and (A.51), a is explicitly given by

1 a=

a [b2a2110- (a. b)211o + b2 (v. a)2112

+a2 (v · b) 21 12 - 2 (v ·a) (v ·b) (a· b) 1 12] 112 (A.76)

Let h11 = h11 (s) be the spacelike vector defined in eq. (A.72) divided by its positive magnitude a in eq. (A.76),

h11 = ~ [ d::- a 2 u11 - ~:a11]. (A.77)

Then h11 is a spacelike unit four-vector perpendicular to both u 11 and n11 at any point x 11 (s) along the world line, i.e.,

(A.78)

h11 n 11 = 0.

The definition of h11 in eq. (A.77), however, suffers from the same difficulty in the original definition of n 11 in that at any point where a vanishes, the vector is undefined. Let a point at which a = 0 be known as a point of torsional inflection, so that a point of torsional inflection occurs at any point at which the third derivative da 11 / ds has no component perpendicular to the plane containing u 11 and aw

To restore the continuity of h 11 through points of torsional inflection, one employs the same "inverted definition" technique as was used for nw Let the unit binormal h11 = h11 (s) be a spacelike unit vector such that at any point x 11 ( s) along the curve, either

(A.79)

where h11 is initially set parallel to ~ - a 2 u 11 - ;~ a11 at some arbitrary initial s0 for which a f:. 0. The positive or negative sign is chosen for


subsequent segments between points of torsional inflection depending on whether h 11 is parallel or antiparallel to the vector d;; - o:2 u 11 -

-,Sa11 within that segment. In general, the point s0 can be chosen to be identical to the point at which the initial direction of n 11 is chosen so that ( u11 , n 11 , h11 ) form a right-handed basis in the 2 + 1 dimensional subspace of M 4 •

To derive the components of h11 in a given coordinate basis, first note that the {0} component of the first two terms inside the bracket in eq. (A.79) is

lS

{ dda!L - a2uiL} = (v. b) /5 + 3 (v. a)2 17· s (0)

(A.80)

Multiplying through by a 2 gives

(A.81)

The { i} components of the same first two terms is

Multiplying through by a 2 gives

+a (v·a)3r 11 +v (a2(v·b)r9)

+3va2 (v·a) 2 r 11 +v (v·b)(v·a)2r 11

+ 3v ( (v · a) 4 1 13) . (A.83)

The {0} component of the numerator of the last term of eq. (A.79)

(v ·a) (a· b) 19 + (v · a) 2 (v · b)/' 11

+ 3a2 (v. a)2 Ill+ 3 (v. a)4113, (A.84)

A.2. SPACETIME CURVES

while the { i} component of the numerator of last term is

{C2 aM}(i) = 3aa2 (v·a)r5 +a(a·b)r7 +a(v·a)(v·b)~/

+ 3a (v · a) 31 11 + v (v ·a) (a· b) 1 9

+ v (v · a) 2 (v ·b) ~/ 11

303

+ 3va2 (v · a) 2 1 11 + 3v (v. a)4 ~~ 13 . (A.85)

The {0} components in eqs. (A.81) and (A.84) and the {i} components in eqs. (A.83) and (A.85), when divided by aa2 , yield the components of the the unit binormal hl.t = hl.t(s) in a given coordinate basis,

a 2 (v ·b) 1 9 - (v ·a) (a· b) 1 9

hl.t = _1_ aa2 b (a 2r 7 + (v ·a) 2 r 9)- a((a · b)r7 - (v ·a) (v · b)r9)

+v (a 2 (v ·b) 19 - (v ·a) (a· b) 19)

(A.86) To explicitly verify that hl.t is a unit spacelike vector, one may use

eqs. (A.51) and (A.76) to find aa2 ,

aa2 = [ (b2a21 1o _(a. b)2 1 12 + b2 (v. a)2 1 12 + a2 (v. b)2 1 12

-2 (v. a) (v. b) (a. b)r12). (a 2 [ 4 + (v. a) 2r 6)r12 '

(A.87)

which when factored and squared yields

a2a4 = 114 [b2a4 - az (a. b)2 + a4 (v. b)212

+ 2b2a2 (v · a) 21 2 - (v · a) 2 (a· b) 212 (A.88) - 2a2 (v ·a) (v ·b) (a· b) 12 + b2 (v · a) 4 r 4

+ a2 (v · a) 2 (v · b )2 14 -2 (v · a)3 (v · b) (a · b) 14] .

From the components of hl.t in eq. (A.86), the magnitude squared of h>, is


- 2a2 (v ·a) (v ·b) (a· b) 1 18

_ b2a4{14 _ az (a. b) 1 14

- bz (v. a)4118- az (v. a)2 (v. b)2118

- v2a4 (v. b)2 118- v2 (v. a)2 (a. b)2 118 . 2 2 + 2a2 (a. b) 114- 2b2a2 (v. a) 116

+ 2a2 (v ·a) (v ·b) (a· b) 1 16

- 2a4 (v · b) 21 16 + 2a2 (v ·a) (v ·b) (a· b) 1 16

+ 2 (v · a) 2 (a· b) 2 1 16 - 2a2 (v ·a) (v ·b) (a· b) 1 16

+ 2a2 (v ·a) (v ·b) (a· b) 1 16 - 2 (v ·a) 2 (a· b )2 1 16

+ 2 (v · a) 3 (v ·b) (a· b)r18 - 2a2 (v · a) 2 (v · b) 21 18

+ 2 (v · a) 3 (v ·b) (a· b) 1 18

+ 2a2 (v · a) 2 (v · b) 2 1 18 - 2 (v · a) 3 (v ·b) (a· b) 1 18

+2v2a 2 (v ·a) (v ·b) (a· b) 1 18], (A.89)

which after combining terms reduces to

hllhJl = (a~2 r . [ ( 1- v2) a4 (v. b)2 118

+ (1- v2) (v · a) 2 (v · b) 21 18

-2 (1- v 2 ) a2 (v ·a) (v ·b) (a· b) 1 18

- b2a4{14 + a2 (a. b )2 114 (A.90)

- 2a4 (v. b)2116- 2b2a2 (v. a)2116

+4a2 (v ·a) (v ·b) (a· b)116

- b2 (v. a)4118- a2 (v. a)2 (v. b)2118

+2(v·a)3 (v·b)(a·b)r18],

which after use of (1 - v2 ) = ,-z becomes

hilh = (-1-)2 . [-b2a4114+a2(a·b)2114 Jl aa2

- a4 (v. b)2 116 + (v. a)2 (a. b)2 116 (A.91)

+ 2a2 (v ·a) (v ·b) (a· b) 1 16

- 2b2a2 (v. a)2 116

- b2 (v. a)4 118- a2 (v. a)2 (v. b )2 118


+2(v·a)3 (v·b)(a·b)l18].

Comparison with eq. (A.88) immediately confirms that

hJ.Lh11 = -1. (A.92)

Likewise, use of the components of u 11 in eq. (A.29) explicitly gives

h 11 u,, = C1~2 ) · [a2 (v ·b) 1 10 - (v ·a) (a. b) 1 10

- a2 (v. b)18 - (v ·a) (v · b)18 (A.93)

- (v · a) 2 (v ·b) 1 10 + (v · a) 2 (v ·b) 1 10

-v2a2 (v. b) 110 + v2 (v ·a) (a· b) 1 10],

which using (1 - v2 ) = ,-2 immediately confirms that h 11 u11 = 0. Likewise, use of the components of n 11 from eq. (A.53) gives

- a2 (v ·a) (v ·b) 1 11 + (v. a) 2 (a· b) 111

- (v . a)3 (v . b) 1 13 + (v . a) 3 (v. b) 1 13

- v 2a2 (v. a) (v ·b) 1 13 + v 2 (v · a) 2 (a· b) 1 13

- a 2 (a· b) 19 + a2 (a· b) 19 - (v · a) 2 (a. b) 1 11

+ a 2 (v. a) (v ·b) 1 11 (A.94)

-a2 (v ·a) (v ·b) 1 13 + 2 (v · a) 2 (a· b) 1 11],

which reduces to

h''n = (-1 ) 11 aa3

[(1-v2)a2 (v·a)(v·b)113 (A.95)

- (1-v2) (v·a) 2 (a·b)l13

-a2 (v ·a) (v. b) 1 11 + (v. a) 2 (a. b) 1 11] ,

which using (1 - v2 ) = ,-2 immediately gives h11 n 11 = 0, and therefore the components of the unit binormal h11 in eq. (A.86) explicitly obey the orthonormality conditions eqs. (A. 78).

Finally, explicit use of the Lorentz transformations for v, a, b and 1 = 1(v) given in eqs. (A.20), (A.21), (A.22), and (A.32) shows h11 to be a covariant vector.


A.2.4 The Unit Trinormal and Orthonormal Tetrad

Given the orthogonal set of unit four-vectors uJ.L, nJ.L, and hJ.L at some point x J.L ( s) along a world line, it is possible to define a fourth orthogonal unit four-vector by using the totally antisymmetric Levi-Civita tensor Ew,.>..· Let the unit trinormal vector sl-' = sJ.L(s) of the particle at the point x J.L ( s) along its world line be defined by

(A.96)

Then from the definition of Ew,.>.. and from eqs. (A.29), (A.53), and (A.86), the {0} component of sJ.L in a given coordinate basis is

s0 = uinjbk- uinkbj- ujnibk- uknjbi + ujnkbi + uknibj, (A.97)

the first term of which is given by

. . k 1 u'n1 b =

aa3 { (Vif') ( ar·? + Vj (v ·a) /',4) · [bk ( a2'"'/ + (v · a) 2 1 9)

+ ak ((a· b)-/+ (v ·a) (v ·b) 19)

(A.98)

+vk (a2 (v · h)19 - (v ·a) (a· h)19)]}.

In the above expression, indices { i = 1, j = 2, k = 3} on the right-hand side refer to four-vector components whereas on the left-hand side they refer to ordinary vector components.

After cycling the indices of eq. (A.98) and inserting the results into eq. (A.97), the only nonvanishing terms are

[a21 10 (viajbk + cyc.perm.)

+ (v · a) 21 12 ( Viajbk + cyc.perm.)] , (A.99)

where the cyclic permutations include all antisymmetric combinations of the three indices. The above expression reduces to

so= a~3[(a2110+(v·a)2112) v·(axb)],

1 6 -v·(axh)l. (J(}'

(A.100)


Likewise, the { i = 1} component of sf.l is

si = n°n.ibk- n°nkbj + nkn°b.i- nJn°bk + 1tjnkb0 + nknjb0 . (A.lOl)

The first term of this expression is

{ ('y) ( aj'Y2 + Vj (v ·a)--/)

· [bk ( a2--/ + (v · a) 2 ·l) +ak((a·b)·/+(v·a)(v·b)19 ) (A.l02)

+vk (a2 (v·b)l9 - (v·a)(a·b)19)]},

where, as in eq. (A.97), the indices on the left-hand side refer to fourvectors while the indices on the right-hand side refer to ordinary vectors.

The second term is obtained simply by switching the indices i and j in the above expression. The third term of eq. (A.lOl) is

{ (vn) ( (v ·a) 1 4 )

. [bj ( a217 + (v. a)2 19) + a.i ((a· b) 1 7 + (v ·a) (v ·b) 1 9) (A.l03)

+vj ( a2 (v ·b) 19 - (v ·a) (a· b) ·l)]}. The fifth term of eq. (A.lOl) is

[(vn) ( an2 + Vk (v ·a) 1 4 )

(a2 (v·b)l9 -(v·a)(a·b)19)]. (A.104)

After inserting eqs. (A.l02), (A.l03), and (A.l04) into eq. (A.lOl) for the {i} component of s1, and interchanging indices for the other components, the remaining nonvanishing terms are

s' = 0'~3 [ a21 10 ( ajbk - akbj) + (v · a) 2 1 12 ( ajbk - akbj) J ,

s' 0'~3 [ ( a21 10 + (v · a) 2 1 12 ) (ax b);],

s' 1 6

-(axb);1. a a

(A.l05)


From eqs. (A.lOO) and (A.105), the components of the unit trinormal

sf-! in a given coordinate basis are

sf-!=_!:_ { (v ·(a X b))·l

era 6 (axb)'f (A.106)

The magnitude of sf-! is

(A.107)

The derivation of the result will depend on the ordinary vector iden

tities,

(p X q) · (r x s) (p · r) (q · s)- (p · s) (q · r), (A.l08)

(px(qxr)) = (p·r)q-(p·q)r. (A.109)

Using the vector identity eq. (A.l08),

(v x (a X b))· (v X (a X b))= v2 1a X bl 2 - (v ·(a X b)) 2 . (A.llO)

From the vector identity eq. (A.l09),

(vx(axb)) (v ·b) a- (v ·a) b, (A.lll)

lv X (a X b)l 2 b2 (v. a) 2 - a2 (v · b) 2 + 2 (v ·a) (v ·b) (a· b).

(A.112)

Inserting eqs. (A.llO) and (A.112) into eq. (A.107) gives

s~-'sJ-t (cr1a) 2 [(v2 -l)iaxbl2 -b2 (v·a) 2 (A.113)

-a2 (v · b) 2 + 2 (v ·a) (v ·b) (a· b)] 1 12 .

Likewise, application of the vector identity eq. (A.l09) gives

(A.114)


which subsequently gives

s 11 s 11 (:a) 2 [ ( -b2a2 +(a· b) 2 ) 1-2 - b2 (v · a) 2

-a2 (v · b) 2 - 2 (v ·a) (v ·b) (a· b)] 1 12 .

(A.l15)

Multiplying the top and bottom of the right-hand side by a 2 gives

11 -( 1) s sr, - a2a4 [ -b2a4l14 + a2 (a. b)2 1 14 _ 2b2a2 (v. a)2 1 16

- a4 (v · b )2 1 16 + 2a2 (v ·a) (v ·b) (a· b) 1 16

+ (v. a)2 (a. b )2 116- b2 (v. a)4 118

- a2 (v. a)2 (v. b)2 118

+2 (v · a) 3 (v ·b) (a· b) 1 18]. (A.l16)

Comparison with eq. (A.88) immediately confirms that s1Ls 11 = -1,

i.e., that s 11 has a spacelike unit magnitude. Use of the component expressions for u 11 , n 11 , and h 11 easily completes the verification of the the relations.

s 11 s11 = -1,

u 11 s 11 = 0,

n 11 s11 = 0, h 11 s 11 = 0.

Together with eqs.(A.36), (A.54), and (A.78), these relations complete the required orthonormality conditions for the vectors u 11 , n 11 , h 11 ,

and sw It can be verified that s 11 transforms as a covariant vector under Lorentz transformations.

With the expression eq. (A.l06) for the components of s 11 in hand, use of the expression

(A.117)

yields the more compact expression for the components of the unit binormal h 11

1 { [(v x a) · (b x a)Jr9

hf' = -- . aa2

[(v · (b x a)) (v X a)+ a X (b X a)] 1 9 (A.118)


Since the four-vectors u11 = u11 (8), n11 = n11 (8), h11 = h11 (8) and 811 = 811 (8) given respectively in eqs. (A.29), (A.45 and A.53), (A.l18), and (A.106) form an orthogonal basis set over spacetime at any point x 11 ( 8) along the world line of the particle, they may be said to com prise a co-moving tetrad 12 of unit four-vectors for the moving particle, i.e., at any given proper time 8, the set

(A.l19)

spans the coordinate basis, with the last three vectors spanning the normal subspace orthogonal to the unit tangent vector uw All four of the covariant vectors in the set eq. (A.l19) evolve in 8, but they remain mutually orthogonal and of constant unit magnitude for all s along the world line.

A.3 The Covariant Serret-Frenet Equations

The covariant vectors of the orthonormal tetrad { ul1 nl1 h11 311} comprise the M 4 generalization of the spatial vectors { u n h} found in the Serret-Frenet Equations (A.9) for the three-dimensional Euclidean space. Since the orthonormal tetrad { u 11 n 11 h11 8 11 } spans spacetime at any point along the world line, the derivatives of the vectors with respect to arc length,

{ du 11

ds dn 11

d8 dh 11

ds ds11 }

d8 ' (A.120)

must therefore be linear combinations of u11 , n 11 , h11 and 8 11 at any point x1,(8), i.e.,

du 11 B0ou11 + Bo1 n11 + Bozh11 + B03 s11 ,

d8

dn 11 Btou11 + Bu n11 + B12h11 + B13s11 , --

d8

dh11 B2oU11 + B21 n11 + B22h11 + B23811 , (A.l21)

d8

d8 11 B3oU 11 + B31 n11 + B32h11 + B33811 , (A.l22) - =

d8 12 J. L. Synge, Relativity: The Special Theory (North-Holland, Amsterdam, 2nd ed.

1965).

A.3. THE COVARIANT SERRET-FRENET EQUATIONS 311

where the coefficients Baf3 = Ba(3(s) are functions to be determined. The system of linear differential equations above can be written in matrix form

(A.l23)

The notation ()af3, a, {3 = 0 - 3 for the components is purely formalistic, and it is not meant to suggest that the 4 X 4 matrix in eq. (A.l23) is a covariant tensor of rank 2. Like the Levi-Civita tensor c: ~"vd, this matrix is an invariant matrix, in that all of its components are separately invariant under Lorentz transformations (i.e., under the passive coordinate basis transformations).

Before deriving the explicit form of the components of this matrix, one may ascertain the general form of the matrix through use of the orthonormality relations. Since all four vectors are unit vectors, they must be perpendicular to their derivatives,

(A.l24)

which means that the diagonal components of the matrix are all vanishing,

Boo = Bu = B22 = ()33 = 0. (A.l25)

From the orthogonality relations of u~",

0, (A.l26)

which gives

(A.l27)


dull dnll ds nJL - ds Uw

Likewise the remaining orthogonality relations

d~ (nllhJL) = 0,

d -(nils ) 0, (A.l28) ds JL

d -(hils ) 0, ds JL

give the conditions

dnJL dhJL ds hJL - ds nJL,

dnll dsll (A.129) ds sJL - ds nJL,

dhJL dsJL ds sJL - ds hw

Eqs. (A.l27) and (A.129) would seem at first glance to suggest that the matrix in eq. (A.123) is totally antisymmetric. Inserting the eqs. (A.121) into the first set of conditions eq. (A.l27), however, gives

(A.l30)

Using the orthonormality conditions, these become

801 Ow,

(A.l31)


In contrast, inserting the eqs. (A.l21) into the second set of conditions eq. (A.l29) gives

(A.132)

which after use of the orthonormality relations reduce to

(A.l33)

From the results in eqs. (A.131) and (A.l33), the matrix is symmetric in the time-space components (i.e., Bo; = B;o) but is antisymmetric in the space-space components (i.e., B;j = -Bj;), and therefore the linear system eq. (A.l23) can be written

(A.l34)

where the components are continuous functions defined by

~1(s) dui-' 1-' --n ds '

6(s) _ dn~-'h~-' ds '

6(s) dhi-' 1-' --s ds '

(A.135)

~4(s) dni-' 1-'

+ ds 8 '


~5(s) _ du 11 h11

ds '

In deriving the explicit form of these matrix components, the sign changes in the definitions of n 11 and h11 will be temporarily suppressed. Once all the components have been derived, the possibility of a sign change at points of inflection and points of torsional inflection will be restored and the meaning of such a reversal will be discussed.

Use of the equation

immediately gives

a,

~6 = 0.

(A.136)

(A.137)

(A.138)

To find the functional form of the coefficients of dn 11 / ds, consider a point along the world line for which a =/= 0. Taking the derivative with respect to s of the components of n11 in eq. (A.53) gives

dn11

ds 1 da 11 1 da 11

----a+--. a 2 ds a ds

(A.139)

From the definition of a in eq. (A.51), its derivative with respect to S IS

da ds

da ds

d -(-a af-L)l/2 ds 11 '

which using the definition of (from eq. (A.66) is

da ( 2

ds a

(A.140)

(A.141)

(A.142)

A.3. THE COVARIANT SERRET-FRENET EQUATIONS 31.5

Inverting the definition of hi-' in eq. (A.79) for the covariant third derivative gives

da~-' ( 2 - = o:2 u~-' + -n~-' + ah~-'. ds o:

Inserting eqs. (A.l42) and (A.143) into eq. (A.139) gives

dn~-'

ds

(A.l43)

dn~-' a = o:u~-' + -h~-'. (A.l44)

ds o: From this result, one infers that the vector dn~-' / ds has components

along only ul-' and hi-', with the component along ul-' equal to o:, as required by the symmetry conditions eqs. (A.l31). In particular,

6 = a

' 01

~4 = 0.

01 =F 0,

(A.l45)

The last remaining nonzero matrix component to be determined is 6. Consider a point along the curve at which o: =/: 0 and a =/: 0. From the definition of hi-' in eq. (A.79), its derivative with respect to proper time is

ds

where the coefficients are defined as

1 da ( 2

(2 da 2 d ((2) ---01 -- -ao:2 ds ds o:2 '

o:2 da do: ---- 2o:-a ds ds'

(A.l47)

(A.l48)


and where d2ap,jds2 = d4 x!ljds4 is the covariant fourth derivative of the particle position x !l = x !l ( s) with respect to proper time. Differentiation of the components of da!ljds in eq.(A.57) with respect to s gives the components of this vector

(v ·d) -y6 + 3 (a· b) -y6 + 13a2 (v ·a) -y8

+ 13 (v ·a) (v ·b) -y8 + 28 (v. a) 3 -y 10

d-y4 + 6b (v ·a) -y6 +a [ 4a2-y6 + 4 (v ·b) -y6 + 19 (v · a) 2 -y8 J +v [(v ·d) -y6 + 3 (a· b) -y6 + 13a2 (v ·a) -y8

+ 13 (v ·a) (v · b) -y8 + 28 (v. a)3 -ylO J (A.149)

where the vector d = d(t) was defined previously as the fourth derivative of x = x(t) with respect to frame timet. The covariant fourth derivative can be shown to have the Lorentz invariant magnitude

-dz-ys- (v. d)2 ~~10 + 9 (a. b )2 '"YIO- 8a2 (a. d) '"Ylo

- 8 (v ·b) (a· d) -y 10 - 12 (v ·a) (b ·d) -y 10

- 8a2 (v ·a) (v ·d) -y12 - 8 (v ·a) (v ·b) (v ·d) -y 12

- 30 (v ·a) (v ·b) (a· b) -y 12 - 6a2 (v. a) (a. b) -y 12

- 16a6 -y 12 - 16a2 (v · b) 2 -y 12 - 32a4 (v ·b) -y 12

- 36b2 (v · a) 2 -y 12 - 38 (v · a) 2 (a· d) -y 12

- 87a4 (v · a) 2 -y 14 - 91 (v · a) 2 (v · b) 2 -y 14

- 98 (v · a) 3 (a· b) -y 14 - 178 (v · a) 2 (v ·b) -y 14

- 351a2 (v · a) 4 -y 16 - 326 (v · a) 4 (v ·b) -y 16

- 280 (v · a)6 -y 18. (A.150)

Using eq.(A.l49) for the covariant fourth derivative and eq. (A.147) for dh~"/ ds verifies that

dh~"

ds Up, 0,

dh~" aja, ds nil (A.151)


dh~-'

ds h~-' = 0.

as required by previous conditions (that dhl-'/ds · n~-' is equal to positive afo: means that the component of dhl-'jds along nl-' is -ajo:, as required by the antisymmetry conditions).

Taking the scalar product of dh~-'/ds in eq. (A.147) with s~-' from eq. (A.106) and using the orthogonality conditions gives

dh~-' 1 d2 a~-' -s ----s ds ~-'-a ds ~-'"

(A.152)

Using eqs. (A.106) and (A.149), the scalar product of d2a11 jds2 with s 11 immediately yields the simple expression

1 d2a11 1 10 -;;Tss~-' = a2o: d · (b x a) 'Y .

Defining the Lorentz invariant scalar function TJ = TJ( s)

TJ = -d · (b X a) 7 10,

the vector dh 11 / ds can then be written

dh 11 a TJ -d = --nJ.t + -2-sM,

s o: ao:

(A.153)

(A.154)

(A.155)

and the last remaining matrix component is given at any point where o: f. 0 and a f. 0 by

TJ 6 = -2-. ao:

(A.156)

Finally, the components of s~-' in eq. (A.106) explicitly verify the required conditions

ds11

ds u 11 0,

ds~-'

ds n 11 0,

(A.157) ds11 TJ ds h 11 a2o: ' ds11

ds 811 0.


The linear system eq.(A.l23) can at last be written as

_!}__ [~:] = [~ ds hJ.L 0 sJ.L 0

6 0

-6 0

0 0 l [ uJ.L l ~2 0 nJ.L 0 6 hJ.L '

-6 0 sJ.L

(A.l.S8)

where the matrix components at any point where o:-::/= 0 and a -::/= 0 are

6 6(s)=o:,

(J

6 6 (s) = -, (A.l59) 0:

6 'TJ = 6 (s) = -2-, a o:

where o: = o:(s), a = a(s), and 'TJ = ry(s) are defined respectively in eqs. (A.51), (A.75), and (A.154).

The linear system eq. (A.l58) are the covariant versions of the SerretFrenet equations for a one-dimensional world line in flat spacetime M 4 .

Likewise the three intrinsic curvature coordinates for a world line in A14 are the matrix elements ~1, 6, and 6, which are Lorentz invariant scalar functions that vary in value with proper time s along the world line and completely determine the geometric structure of the world line up to background transformations of the coordinate basis. The function 6 ( s) = o:( s) has already been labeled as the curvature. In analogy with three-dimensional Euclidean space, let the functions 6 = 6 ( s) and 6 = 6(s) be known respectively as the first torsion and second torsion.

Because of the directional changes built into the definitions of nJ.L and hJ.L, the matrix components eqs. (A.l59) must reflect the sign behavior of o: and a. Specifically, at a point of inflection, all three independent components may switch signs together. At a point of a torsional inflection, the component 6 may switch signs. Without loss of generality, the sign ambiguity will continue to be suppressed in the sequel for purposes of functional clarity. Moreover, as intrinsic curvature coordinates, it is only the absolute values of 6, 6, and 6 that determine the local

geometry. 13


A.4. THE ACTIVE LORENTZ TRANSFORMATION 319

Furthermore, just as the torsion k2(s) in E 3 is defined by eq. (A.lO) and not by eq. (A.ll) so that it is a continuous function along the curve, the intrinsic coordinates 6 and 6 are continuous functions defined by eqs. (A.l35) and not by eqs. (A.l59), which are not valid at points of inflection and points of torsional inflection.

This completes the derivation of the covariant Serret-Frenet equations and the Lorentz invariant intrinsic curvature coordinates as functions of arc length for a point particle in spacetime.

A.4 The Active Lorentz Transformation

An issue that arises in the discussion of relativistic particle motion is the correct definition and use of the instantaneous rest frame that follows an accelerated observer.14 Although such a frame is not an inertial frame, and thus it serves no role in the postulates of relativity, it nevertheless remains an interesting issue, especially from the standpoint of measurements by this frame.

The use of this type of coordinate system was suggested by Cartan,15 and is known as the repere mobile (Fr. "moving reference"). The use of such a frame to record measurements introduces several types of interesting distortions of order of magnitude of the acceleration. 16

At some arbitrary initial proper time s0 , consider a Lorentz frame with i and x axes that happen to be parallel respectively to the particle's four-velocity ul-' and four-acceleration al-L, which is assumed for now to be constant. The directions of the y and z axes of the Lorentz frame are arbitrary up to a rotation around the i- x plane. At this initial time, the repere mobile of the moving particle is defined to be instantaneously identical to this background Lorentz frame, which is initially arbitrary in two of its spatial axes but nevertheless uniquely defines the repere mobile for all future values of proper time. From measured acceleration lal, the direction of the axes at some later proper time swill be given

14 See Section 2.8. 15 E. Cartan, Exposes de Geometrie, V, (Hermann, Paris, 1935), reprinted in CEuvres

Completes, Partie III: Geometrie dif]erentielle Divers (Editions du Centre National de la Recherche Scientifique, Paris, 1984).

16 See C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation(Freeman, N.Y., 1973).


in terms of the basis vectors at time s0 by the matrix equation

[i(s)l [cosha(s~s0 ) sinh a (s- so) 0 0 l [i(so) l x ( s) _ sinh a ( s - so) cosh a (s- so) 0 0 x (so) y (s) - 0 0 1 0 y (so) z (s) 0 0 0 1 z (so)

(A.160)

A.4.1 The Fermi-Walker Operator

The repere mobile is likewise used in the Fermi-Walker transport model17

to evolve a four-vector along the world line of an accelerated observer undergoing constant acceleration aw A four-vector q~" that is traveling with the accelerated observer (such as a spin vector) will undergo proper time evolution according to the differential equation

(A.161)

where Qp,v is the operator

(A.162)

The Fermi-Walker operator Qp,v generates a "pseudo-rotation" of the vector qt-L through the timelike plane containing the vectors 1Lp, and a~"

but leaves untouched the components orthogonal to this plane. The main drawback of the repere mobile and the Fermi-Walker trans

port model is that although it can be generalized easily to accelerations that are variable in magnitude, it is strictly limited to the case of acceleration that is constant in direction. In particular, the evolution of the basis vectors in eq. (A.160) allows for changes in the direction of the four velocity u~" but not of the acceleration aw This limitation is the result of the "non rotating" requirement, and it precludes the application of the repere mobile to such simple examples as a particle traveling at constant relativistic speed on a circular path. At the present time, to the knowledge of the authors, there is no known way to uniquely specify the instantaneous rest frame for a general observer undergoing arbitrary time-dependent acceleration. 18

17 E. Fermi, Atti. R. Accad. Rend. Cl. Sc. Fis. Mat. Nat. 31, 21 (1922); S. Weinberg, Gravitation and Cosmology (Wiley, N. Y., 1972).

18 C. W. Misner, K. S. Thorne, and J. A. Wheeler, op. cit., footnote 16.

A.4. THE ACTIVE LORENTZ TRANSFORMATION 321

It will be shown below that the co-moving tetrad in the covariant Serret-Frenet equations furnishes the most natural choice for the comoving reference frame for an accelerated observer with arbitrary timedependent acceleration. It will be shown that the evolution of this basis set in general includes a spatial rotation, and that it reduces to the Cartan repere mobile when the acceleration is constant.

A.4.2 The General Co-Moving Frame

Because the covariant orthonormal tetrad { uJ.L nJ.L hJ.L sJ.L} spans spacetime at any value of s along the world line, one may consider them to instantaneously form a coordinate basis at any value s, i.e., at some time s the set { uJ.L nJ.L hJ.L sJ.L} can be considered as a set of unit vectors

{i x y i} which define a basis (t, x, y, z). It follows that the evolution

of the tetrad { uJ.L nJ.L hJ.L sJ.L} at time s to the tetrad { u~ n~ b~ s~} at s + ds can expressed as an equivalent active infinitesimal Lorentz transformation.19 The basis formed by the orthonormal tetrad must not be confused with the observation frame coordinate basis, which is fixed in s for a given observer. Transformations of the co-moving tetrad are active and depend explicitly on s, whereas the Lorentz frame transformations between observers are passive and occur independently of s (i.e., "simultaneously" for all s). Under the active evolution, the frame-dependent observables such as v(t) do not transform by eqs. (A.20), (A.21), (A.22), and (A.23), but rather as

v = v' + a''Y(v)dt,

a a' + b' 'Y( v')dt,

(A.l63)

b b' + d''Y( v')dt,

( dd)' d = d' + dt 'Y(v')dt.

Likewise the basis vectors uJ.L, nJ.L, hJ.L, and sJ.L do not separately transform as covariant vectors under the active transformations but as the

19 J. L. Synge, op. cit., footnote 12 on p. 310.


four components of a single covariant vector (or more appropriate, as the four components of a single spacetime coordinate basis.

Since the differential geometric properties represented in the arclength evolution of the co--moving tetrad do not explicitly depend on position, it follows that the corresponding active coordinate transformation ought to be a member of the infinitesimal homogeneous Lorentz group, which includes the Lorentz boosts and rotations, but excludes spacetime displacements of the origin.

At some value of s, let x represent the co-moving tetrad basis vectors { u~-' n~-' h~-' s~-'} and let x' represent the co-moving tetrad basis vectors

{ u~ n~ b~ s~} at time s+ds. Let A(s) be the infinitesimal homogeneous

Lorentz transformation that gives the new basis tetrad x' in terms of the old one x, i.e.,

x' = A(s)x. (A.l64)

Since the active evolution is infinitesimally close to the identity operator I, the transformation A = A( s) can be represented to first order in ds as

A(s) =I+ L(s), (A.l65)

where L ( s) represents the infinitesimal part of the transformation away from the identity. In particular, the operator L = L(s) is a linear combination of infinitesimal generators of the homogeneous Lorentz group,20

(A.l66)

where [(i is the boost generator along the l axis, Ji is the rotation generator around the z axis, and the qi are infinitesimal parameters.

In particular, the generators I<x, Jx, and Jz have the representation in M 4 as

[ ~i -z 0

~] , 0 0 0 0 0 0

(A.l67)

[~ 0 0

~l 0 0 0 0 -z 0 +i 0

(A.168)

20 L. H. Ryder, Quantum Field Theory (Cambridge Univ. Press, Cambridge, 1985).

A.4. THE ACTIVE LORENTZ TRANSFORMATION

[~~ 6~.. 0 0] . T ~

323

(A.l69)

Let L generate an infinitesimal transformation away from unity of the form

(A.l70)

where o<jy is an infinitesimal boost along the x axis, and Of) and 01/J are infinitesimal Euler rotations around the x and z axes respectively.

Then using the matrix representations of the generators Kx, lx, and Jz, the operator L can be represented as

1 = o<P o [ o o<P

0 -01/J 0 0

0

01/J 0

-oB

(A.l71)

The covariant Serret-Frenet equations (A.l58) can be written as

dx ~( ) _ ds =.:: s x, (A.l72)

where :=: = :=:(s) is the 4 X 4 matrix of which the components are the intrinsic curvature coordinates (6(s) ,6(s) ,6(s)).

To first order in s, the evolution from the tetrad x at time s to x' at time s + ds can be written as

_, _ dx x = x + ds ds, (A.l73)

which using eq. (A.l72) is

x' = (I+ :=:(s) ds) x. (A.l74)

From eq. (A.l65), the active Lorentz transformation of the tetrad can be written

x' = (I+ L(s)) x. (A.l75)


Comparison with eq. (A.174) yields the correspondence between operators

L(s) = 3(s)ds. (A.l76)

Comparison of the components of 3(s) with eq. (A.171) reveals that the active infinitesimal Lorentz transformation corresponding to evolution of the tetrad in time consists of a boost along the principal unit normal nil- (i.e., along the x axis of magnitude o¢), an Euler rotation around the principal unit normal nil- (i.e., around the x axis of magnitude MJ), and another Euler rotation around the unit trinormal s,, (i.e., around the z axis of magnitude o'lj;). The infinitesimal magnitudes of the boost and the rotations are explicitly

a¢ 6ds,

o'lj; 6ds, (A.l77)

ae 6ds,

which can be written as differential equations for the active Lorentz parameters as functions of s,

a¢ ds

a,

o'lj; a (A.l78)

ds ' a

ae , ds a 2a

Consider the case where the acceleration all- of the particle is constant in time. Then dall-jds vanishes, and subsequently all points along the world line are points of torsional inflection. From eqs. (A.135), 6 = 6 = 0 and the equivalent active Lorentz transformation A(s- s0 ) for finite time separation s- s0 reduces to the form

[

cosh a (s- so)

A(~s) = sinh<> ~s- s0 )

sinh a (s- so) cosh a (s- so)

0 0

0 0 1 0

0 0 0 1

(A.179)

which is identical to the repere mobile operator given by eq. (A.160).

A.5. CONCLUSIONS 325

A.5 Conclusions

It has been shown that the differential geometry of an arbitrary world line in flat Minkowski spacetime is given by a covariant set of SerretFrenet equations, which are differential equations (in proper time) of a set of four covariant unit vectors which form an orthonormal set at any point along the curve. The three independent matrix elements of the 4 x 4 matrix of the linear system are the three Lorentz invariant intrinsic curvature coordinates for the world line that uniquely determine the curve (up to background basis transformations) as a function of any arbitrary parameterization. Unlike the situation for nonrelativistic motion in E 3 , these intrinsic coordinates furnish a complete description of the special relativistic particle kinematics independent of the choice of parameterization of the world line.

Moreover, it was shown that the four orthogonal unit vectors comprise a covariant tetrad which acts as a co-moving local rest frame for an arbitrary accelerated observer. It was shown that the active evolution of these vectors as a coordinate basis in general includes both a Lorentz boost and an Euler rotation. For constant particle acceleration, this co-moving basis reduces to the Cartan repere mobile which is used in the Fermi-Walker transport model. The co-moving tetrad in the covariant Serret-Frenet equations therefore furnishes the most natural candidate for the local rest frame of an observer undergoing arbitrary time-dependent acceleration.

Appendix B

The Solutions Derived by Cook

As mentioned in the Introduction and in Chapter 7, one of the most frequently cited articles regarding this dynamical theory is the 1972 work by Cook1 on the classical two-body problem, as well as the follow-up paper on the quantum version. 2 The articles are important for the discussion here because they include a solution to the V = -k/ p two-body potential using an invariant parameter T as the dynamical evolution time. Although in the article the parameter is called the proper time, it does appear that this assumption is actually made,3 and therefore the results hold for an arbitrary correlated representation as well. 4 It is useful, therefore, to examine the significance of these two articles in regard to the results derived in Chapter 7 of this work.

The classical dynamical principle assumed by Cook is, in formal terms at least, identical to the principle assumed in this work. Thus Section II of Cook's classical paper contains a discussion of the relativistic Hamiltonian theory that is formally equivalent to the basic equations presented in Chapter 5 here. That is, the classical world lines are assumed to be parameterized by a common time T, with the event separation assumed to be spacelike in general.

1 .J. L. Cook, Aust . .J. Phys. 25, 117 (1972). 2 .J. L. Cook, ibid., p. 141. 3 Notwithstanding the equation following eq. 18 on p. 121 of Cook's work, there is

no explicit assumption of the mass shell constraint. 4 See Section 4.1.2.

327

328 APPENDIX B. THE SOLUTIONS DERIVED BY COOK

In Section III, however, Cook derives the equations of motion for two

charged particles of finite mass with mutual interaction using a method

that differs completely from the one presented here. In particular, in. eq. (26) he assumes the familiar action-at-a-distance along the retarded

light cone, which cannot describe the many-body system. 5 The inter

action at a common T is then estimated from the light cone interaction using the familiar assumption of low particle acceleration, e. g., as dis

cussed in Synge.6 Thus in this particular section of Cook's work, there

is no direct assumption of a covariant potential in the form V = V(p),

and the results so obtained are low-order approximations that in no way

correspond to the results in Chapter 7. This conclusion is notwithstand

ing the fact that in order to perform this analysis, Cook introduces the identical pseudospherical coordinates (7.2) used in this work.

In Section IV, Cook discusses the weak-field approximation to general relativity. In particular, he introduces a five-dimensional fiat space

time metric tensor with a signature ( -1, 1, 1, 1, -1), in which the fifth

component represents the world time T. As in conventional general

relativity/ the weak gravitational field is introduced as a linearized per

turbation upon the fiat spacetime metric. Although the results in this

section of his work have broad implications for the extension of the T

dynamics to field theory, the most important result from the standpoint

here is that in the suitable limits, he obtains (in eq. 52) the potential

V = -kjp. 8

In Section VI, Cook discusses the two-body interaction in the form

V = V(p) (Cook uses the symbol S for the two-body spacelike separation). In Section VI (a), he obtains the reduced solutions for the 1+1-dimensional harmonic oscillator potential discussed in Section 5.6 above.

Although Section VI (b) bears the title "Coulomb Law of Force," it can be seen from eq. (81) in his work that the interaction is given by an ap

proximated two-body field theoretical Hamiltonian which yields a com

plicated expansion in powers of p for the center-of-mass term in eq. (83).

5 See Section 1.2. 6 Cf. fig. 1 of Cook with the figures in J. L. Synge, Proc. Roy. Soc. A 177,

118 (1940). Synge uses a method of successive approximations to find an expression

for the two-particle interaction. 7 C. Moller, Theory of Relativity, (Oxford Univ. Press, 1952). 8 This potential is actually obtained in the limit of low acceleration only, a fact

that demonstrates that the difficulties inherent in field theory appear in a T formalism

as well. By contrast, it is asserted here that the expression V = -k/ pis valid to all

orders for two-body special relativistic systems in the limit of Newtonian strength.

329

It can be easily seen that the Hamiltonian does not arise from the covariant many-body theory studied here for the potential V = -k/ p.

Consequently, the complicated radial orbit solution that Cook obtains in eq. (85) does not correspond to the solutions presented here. It is not

surprising, therefore, that the complicated semi-classical hydrogen spec

trum Cook obtains from this Hamiltonian in eq. (88) contains incorrect

terms.9

Section VI (c) of his work provides the reduced solution to the two

body system with the inverse cube potential, V = -k/ p2 . As mentioned in Section 5.4.9 above, Cook showed that this system does not possess

bound orbits. The fact that the corresponding nonrelativistic version

of this potential possesses bound orbits that are, however, open may

allow a consistent interpretation of the result in Chapter 5 regarding

the fact that all bound orbits of the covariant two-body problem are

closed. It is suggested here that it may be possible to prove that, in

fact, the only two-body covariant potentials V = V(p) that produce bound orbits, i.e., orbits that reach turning points ~ = 0 on both ends,

are the inverse square potential and the harmonic oscillator. Although

the latter case was not studied here in detail, this result may be expected

on the grounds that in nonrelativistic mechanics, these are the only two

potentials V = V(r) that produce closed orbits in general, according to Bertrand's theorem. 10 The limitation of the two-body system to these

two potentials, however, need not be regarded as a disadvantage to the

theory as a whole. This is because the use of other such potentials in

nonrelativistic classical mechanics does not arise from consideration of fundamental interactions, which are those modeled in the many-body

dynamics here by event-to-event correspondence.

It is in Section VI (d), under the title "Planetary Motion under

Gravity,'' that Cook discusses the system studied in Chapter 7 of this work, i.e., the two-body reduced motion for the inverse square potential,

F = -k/ p. The radial orbit solution found at the bottom of p. 135 of his

article is, in fact, identical to the Type I reduced solution in eq. (7.19)

above. Likewise, eq. (90c) of his work is identical to the expression here

0 In Section VI of his quantum mechanical paper (see footnote 2), Cook uses this

same complicated (and approximate) Hamiltonian to get an incorrect quantum me

chanical expression for the hydrogenic spectrum (specifically, on the top of p. 159).

Cook does not discuss the quantum version of the potential V = -k/ p. 10 See Section 3-6 of H. Goldstein, Classical Mechanics (Addison-Wesley, Reading,

Mass., 2nd ed. 1980).

330 APPENDIX B. THE SOLUTIONS DERIVED BY COOK

for</>= ¢((3) in eq. (5.136). Cook does not, however, obtain the Type II

solution (7.20), nor does he derive the semi-classical hydrogenic for this

potential. The most interesting result from the standpoint here is probably

eq. (91), which gives the plane-polar orbit r = r(¢) of the reduced motion

as observed in the center-of-mass rest frame.U By substitution of the

azimuthal relation (3 = (3(¢) into the radial orbit solution p = p((3), he

obtains

~= ~; [J1-Elcos2 </>-E2cos¢], (B.1)

where E1 and E2 are constants that depend on the initia.l conditions.

From this last result, it can be seen that in general, the nonvanishing of

c 1 in eq. (B.1) means that the reduced plane-polar trajectory 1' = r·(¢)

observed in the center-of-mass rest frame differs from the nonrelativistic

Kepler solution. In the limit E1 ---+ 0, the trajectory reduces to the

Newtonian expression. The lack of perihelion precession in the bound orbits discovered in

Chapter 7 is implicit in Cook's solution as well, and moreover Cook

shows that the overall shape of the reduced orbit, and thus the parti

cle trajectories, differs from the nonrelativistic Newtonian trajectories.

The deviations from the nonrelativistic orbits are obtained for the case

of the Solar System and plotted in table 1 on page 137 of his article. In particular it is found that eq. (B.1) predicts that the radial distance of the planets from the sun are smaller than the values calculated by

Newtonian mechanics, with the deviation being most pronounced at the

aphelion and the perihelion of a planet's orbit. Moreover, the planet either lags behind, or moves in advance of, its angular position as pre

dicted by nonrelativistic theory. Cook concluded, however, that in the case of planetary motion the magnitude of the deviations are insignifi

cant and cannot be measured. 12

The values given by Cook in table 1 on page 137 of his article can

not, however, be taken to be the actual radial and temporal separations

predicted by the covariant many-body theory. In the case of the radial

coordinate, the reason is that the orbital equation r = r( </>) derived by

11 Cf. the solution r' = r'(<P') by Piron and Reuse, which is discussed in Section 7.6. 12 The predicted effect is most pronounced for the inner planets, since they move

the swiftest. The deviations for the radial distance are, however, on the order of one part in 108 . The time discrepancies for the angular position, which in principle should

be more easily measured, are on the order of 10-2 seconds.

331

Cook in eq. (91) provides the radial separation in the center-of-mass rest frame at a common value ofT. In general, this separation is not directly observable by the center-of-mass observer, since the two events at a common value ofT occur at different values of the time t of the frame. Conversely, at a common value of frame time t, the observer records the spatial positions of the particles at two different values of the dynamical time T. Thus in order to find the correct radial separation at a common value of the time t of this frame, it is necessary to evaluate the orbital positions at two different dynamical times.

This is shown in fig. 5.2 on p. 139, in which it can be seen that the radial separation r = r ( T) is not measured directly by the centerof-mass observer, since the events that define this separation occur at two different frame times in general. On the other hand, the radial separation in the center-of-mass rest frame, r = r ( t), is defined between two different dynamical times. 13

Moreover, in regard to the time discrepancy of the orbital position quoted by Cook, it must be remembered that the overall temporal motion of the center of mass must be added to the reduced solutions, as depicted above in fig. 5.1 on p. 129 of this work. This introduces a temporal scaling between T and the center-of-mass frame time which in general will alter the values given under the heading of "Time variation" in the fourth column of Cook's table.

Nevertheless the predictions made by Cook for this system remain interesting, and his numerical results may be taken as order-of-magnitude estimates of the effect. The possible application of these results to establish a critical experiment for the classical theory are discussed in further detail in Chapter 8.

12'This is not true in the Piron-Reuse frame, of course, since .6.t(T) = 0 in this frame, as discussed in Section 5.4.16. The observations, however, are assumed to be recorded in the center-of-mass rest frame.

Appendix C

The No Interaction Theorem

The theorem first presented in 1963 by Currie, Jordan, and Sudarshan1

has come to be known as the no interaction theorem, but a more appropriate name for the result might be the no acceleration theorem, since it makes no statement about the form of the force itself, but rather about the motion of the particles in the configuration space of the system. Although the proof in Currie2 is given in terms of the n-body system, for purposes here it is necessary to consider only the single particle moving in an inertial frame.

Consider therefore a single point particle observed by an inertial observer using the coordinates (xk, t), where here the subscripts k, £, and m shall be assumed to take on the values

k,f,m= 1,2,3. (C.1)

The observer records the position of the particle as a function of the time of the frame in the vector form

(C.2)

From the three spatial coordinates of the frame, it is possible to construct

1D. G. Currie, T. F. Jordan, and E. C. G. Sudarshan, Rev. Mod. Phys. 35, 350 (1963).

2 D. G. Currie, J. Math. Phys. 4, 1470 (1963), to which we shall refer in the sequel simply as Currie.

333

334 APPENDIX C. THE NO INTERACTION THEOREM

the six-dimensional phase space of the particle motion,

(C.3)

where Pk = pk(t) are the three components of the particle momentum. As in Currie, however, it is necessary to consider only the generalized coordinates, qk = qk(t). Let the assumption be made that the world line of the particle is a continuously differentiable one-dimensional curve.

Following Dirac,3 let ten functions of the phase space coordinates be defined in the form

{Pk, Jk, I<k, H},

which are assumed to obey the relations

[h, Pel 0,

[Jk, Pel ckfm Pm,

[Jk, Jel Ckfm Jm,

[H,hl 0,

[H, Jkl 0,

[I<k, Pel 8HH,

[I<b Jel Ckfm I<m,

[I<k, Hl Pk, [Kk, Kg] -Ekem f(m,

where the classical Poisson bracket is defined in the usual way.

(C.4)

(C.5)

In other words, it is assumed that the ten functions in (C.4) are the ten generators of the Poincare group. That is, the functions generate the background coordinate transformations4 in the following manner:

Pk :::} spatial translation along the Xk axis,

Jk :::} ordinary rotation around the Xk axis,

(C.6)

I< k :::} boost along the x k direction,

H :::} time translation.

3 P. A. M. Dirac, Rev. Mod. Phys. 21, 335 (1949); see also E. P. Wigner, Ann. Math. 40, 149 (1939).

4 See Section 2.6.

335

Moreover, it is assumed that H is the Hamiltonian of the system. Conversely, from the identifications in ( C.6), the Poisson bracket re

lations (C.5) may be demanded on the grounds of relativistic invariance of the world line.

According to the theory of canonical transformations, 5 an arbitrary phase space function G generates a change qk ----+ q~ in the configuration space variables, which to lowest order is given by

(C.7)

where g here is the infinitesimal parameter of the transformation.6

For the ten functions in (C.4), it is therefore useful to define the ten respective infinitesimal parameters

(C.S)

from which one obtains the ten infinitesimal canonical transformations,

I qk qk +de [qk, Pe], (C.9) I qk qk +Be [qk, Je], (C.lO) I

% qk + ae [qk, Ke], (C.ll) I qk + ( [qk, H] . (C.l2) qk

From the statements in (C.6), it is straightforward to provide physical interpretations of the ten infinitesimal parameters appearing in the canonical transformations (C.9)-(C.l2). The essence of the proof in Currie is the subsequent derivation of the world line conditions,

I % qk +de 8ke, (C.13)

I qk + Beckfmqm, (C.l4) qk q~(O) qk(O) + ae qe(O)vk(O), (C.15)

I qk qk + ( Vk, (C.l6)

where vk(t) = dqk/dt is the frame speed of the particle. It is important to note that in eq. ( C.l5), it has been assumed that the particle is at the time t = 0 according to the observer.

5 H. Goldstein, Classical Mechanics (Addison- Wesley, Reading, Mass., 2nd ed. 1980).

6 The transformation law (C. 7) obviously applies to any function of the phase space coordinates (qk,Pk)· As mentioned, however, we are interested only in the effect of these generators upon the position coordinates qk = qk(t).


Identifying the world line conditions with the canonical transformations in eqs. (C.9)-(C.l2) gives

[qk, Pt] 8kt,

[qk, h] Ek£mqm,

[qk, Ke] qevk,

[qk, H] Vk,

from which it follows immediately that

This last result may be called Currie's equation.

(C.l7)

(C.l8)

(C.l9)

(C.20)

(C.21)

In Section VII of reference 2, Currie shows that eq. (C.21) demands that the particle moves as a free particle, i. e., without acceleration. Put another way, eq. (C.21) demands that the world line of the particle be straight.

In order to evaluate whether or not this conclusion is in fact correct, it is useful to examine the derivation in Currie of the world line conditions (C.l3)-(C.l6).

The derivation of the first condition, eq. (C.l3), follows immediately from the identification of Pk as the generator of spatial translations of the origin of the coordinate frame. The parameter dk is the infinitesimal distance along the Xk axis over which the origin of the coordinate system is translated, according to the first observer. Under such a transformation, the Xk position components of all points in the configuration space are shifted in magnitude by the same amount dk.

Likewise, the derivation of the second condition, eq. (C.l8), follows in the usual straightforward way from the identification of Jk as the generator of the rotations of the coordinate frame around the Xk axis.

In particular, under an infinitesimal rotation of the frame axes by an angle fh around the x1 axis, the coordinates of a point in the configuration space change by

q1,

q2 cos 01 - q3 sin 01 ,

q3 cos 01 + q2 sin 01,

(C.22)

which to lowest order is

q~

q~

q~

ql,

q2- fh q3,

q3 + fhq2,

from which the condition (C.l8) is obtained immediately.

337

(C.23)

An important note regarding these first two conditions is that the transformations (C.l7) and (C.l8) apply equal well to all points in the configuration space, whether or not they happen to be particle events, and in particular they apply to the entire particle world line at once. This is a consequence of the fact that the transformations of the Poincare group are background coordinate transformations.

In order to obtain the third world line condition, eq. (C.l9), Currie makes the following assumptions: (i) that the magnitude of the boost angle ak =tanh f3k is infinitesimal, i.e.,

(C.24)

and secondly, (ii) that before the transformation, the particle is located at t = 0 of the coordinate frame.

Thus two limiting assumptions have been made here. The first assumption follows from the stipulation that the transformation be infinitesimal in magnitude. This demand is completely in accordance with those made in the derivations of the first two world line conditions.

The second assumption, on the other hand, is quite different in its meaning, for here it has been assumed that the transformation is not applied to the entire length of the world line of the particle, but only to an infinitesimal region of the world line near the time t = 0.

Thus, under a Lorentz boost of arbitrary magnitude along the Xk

axis, the time coordinate of an arbitrary point qk = qk(t) on the world line transforms as

t' = t cosh ak - qk sinh ak. (C.25)

Under the assumption of an infinitesimal magnitude of the boost transformation, eq. (C.25) becomes

(C.26)

Under assumption (ii) of an infinitesimal region of transformation, this becomes

(C.27)


The corresponding boost transformation for the spatial position is

(C.28)

which, since t = 0, reduces to

(C.29)

Expanding q~ in a Taylor series in powers of o:e gives, to lowest order,

(C.30)

By the chain rule, dq~ dq~ dt' do:e dt' do:e '

(C.31)

which by eq. (C.27) is dq' dq' k - k do:e - -qe dt' · (C.32)

Using eq. (C.27) to replace the evaluations at o:e = 0 with evaluations

at t' = 0, the Taylor expansion becomes

dq' I qk(O) = q~(O)- o:e qe(O) dt~ t'=o. (C.33)

By identifying the particle velocity in the second frame as

I dq~ ve = dt'' (C.34)

the expansion is written as

(C.35)

The final step necessary to obtain the third world line condition is the

assumption that, to lowest order, the particle velocities are the same

before and after the transformation, i.e.,

(C.36)

from which one obtains the world line condition eq. (C.15).

The fourth world line condition, i.e., the relation for the time transla

tion generator, is obtained by the stipulation that during the background

339

transformation, the time coordinate of each event changes from t tot', which differs from t by the infinitesimal magnitude (, i.e.,

t' = t- (. (C.37)

Since the event is held stationary during the transformation, it follows that the position in the new frame at the new time t' must be the same as the position in the old frame at t, i.e.,

(C.38)

which, using eq. (C.37), may be written as

(C.39)

Expanding this in powers of the infinitesimal parameter ( gives, to lowest order,

dq' I qk(t) = q~(t)- dt, (=0. (.

Currie then makes the assumption that

dq' dq' _k __ k

d( - dt''

and using eq. (C.36) once again, eq. (C.40) may be written as

which is the fourth world line condition, eq. (C.l6).

(C.40)

(C.41)

(C.42)

The four world line conditions having been established in the above manner, eq. (C.21) follows, from which it is possible to derive

[[qk. H], H] = 0. (C.43)

Using eq. (C.20), Currie identifies the quantity on the left-hand side of this expression as the acceleration, which vanishes. That is, the world line of the particle is straight.


C.l Comments on the Proof

Having outlined the proof in Currie, let us examine it from the standpoint of the discussion in Chapters 2 and 3 of this work. The first question one may legitimately ask concerns the identification made in eq. (C.41) during the derivation of the fourth world line condition. In particular, is the quantity dqk/ d( the velocity of the particle according to the frame observer?

Following Section 2.6, the quantity ( measures the difference in the time coordinate recorded between two Lorentz obs~vers. Thus, under a background coordinate transformation, it is defined globally throughout spacetime, as depicted in fig. 2.5 on p. 44. Being a fixed change for all points in spacetime, it must be a constant for every point along the length of the world line.

Explicitly, by the chain rule,

dqk dqk dt d{- dt d(' (C.44)

and thus the identification in eq. (C.41) can be made only in the case dtjd( = 1. But since (is constant for all points in spacetime, it follows that d(jdt = 0, and thus dtjd( is undefined.

Put another way, ( is not the time coordinate measured by any Lorentz observer, and thus the derivative of the position of the particle with respect to this quantity is not the particle velocity, which is a time-dependent kinematical quantity. Rather, the quantity dqk/d( is identical to the extrakinematical quantity dxjd(tl.t) in eq. (2.26) in the limit fl.t ---7 0.

This being so, eq. (C.20) is actually given by

dqk [qk,H] = d{' (C.45)

where it has been assumed, for the sake of the argument here, that dqkJ d( = dqk/ d(. It follows immediately that the quantity on the lefthand side of eq. (C.43) is not the second derivative of the position with respect to the frame timet, and thus, given the assumptions made above, the theorem in Currie makes no statement regarding the particle acceleration.

Based on this argument, it would seem that since the central relation of the proof, eq. (C.21), depends explicitly eq. (C.41), which itself is not

C.l. COMMENTS ON THE PROOF 341

true, it must be concluded that in general the no interaction theorem makes no statement at all regarding the motion of the particle.

In order to restore any hope of using eq. (C.21) to prove the result, it is necessary to make a completely different interpretation regarding the generators above. In particular, it may be assumed that the Hamiltonian generator H is not, in fact, identical to the generator of the time translations of the origin of the background frame. That is, the actual set of generators in (C.4) comprises the eleven functions

(C.46)

where the functions before the semicolon are the generators of the background coordinate transformations of the Poincare group, and where H is the Hamiltonian generator. That is, the function T in the above set is the generator of time translations of the origin of the frame, and it therefore satisfies the Poisson bracket relations for H in eqs. (C.5). From this standpoint, it is not necessary to make any assumption regarding the Poisson bracket relations of H with the generators of the Poincare group.

Based on this assumption, eq. (C.l2) becomes

(C.47)

where, as before, (is the infinitesimal magnitude of the time translation of all points in spacetime. It follows that eq. (C.45) in the discussion in the previous section regarding the fourth world line condition should actually be written as

(C.48)

On the other hand, as a. canonical generator, the Hamiltonian H provides the infinitesimal transformation

(C.49)

where dt is the differential change in the frame time according to a given observer. It is not necessary here to make any assumption regarding the relationship between ( and dt.

To lowest order during a. time dt, the position qk of the particle changes according to the relation

I dqk qk = qk + dt dt. (C.50)


Comparison of this to eq. (C.45) immediately yields

(C.51)

Thus Currie's equation (C.21) is re-established between the Poincare group generator ]( k and the Hamiltonian H, and the left-hand side of eq. (C.43) may be interpreted as the particle acceleration.

It must be noted, however, that eq. (C.49) is a local relation, in that it applies only to a given point along the world line, and not to the entire world line at once. This is because H does not generate frame transformations, but rather transformations within a given coordinate system. 7 That is, eq. (C.51) is defined only over an infinitesimal range dt. This is the same situation that arose in the derivation of the world line condition for the boost generator Ke, although the reasons for this restriction are difference in the two cases. In the case of the boost generator, the restriction arises through a limiting process. In the case of the Hamiltonian, the restriction is an explicit requirement of the nature of the transformation itself.

It follows that Currie's equation (C.21), and subsequently the assertion of vanishing acceleration in eq. (C.43), is valid only over an infinitesimal duration of time. Likewise, the conclusion that the world line is straight is valid only over an infinitesimal length of the particle trajectory in spacetime.

This conclusion, however, is precisely what had been assumed in both eq. (C.36) and (C.50). It is simply the statement that any particle moves locally as a free particle, which is the foundation of all of kinematics. Likewise, any one-dimensional curve in a metric space is, to lowest order, straight.8 The no interaction theorem makes no statement at all about the acceleration over finite lengths of the world line.

7 See Sections 2. 7 above. 8 See Section 3 . .5 regarding the intrinsic curvature coordinates of the world line.

Appendix D

Classical Pair Annihilation

In Section 1. 7, it was mentioned that one of the principal motivations for Stiickelberg1 in formulating the covariant Hamiltonian mechanics was that it allowed a possible classical explanation of the phenomenon of pair creation and annihilation. Specifically, this is accomplished by the assumption that the system of two annihilating particles is treated, in the view of the dynamics, as a single particle with a world line that curves back against the sense of the time axis. The world time T increases monotonically along the curve, The curving of the single-particle world line is presumed to be due to some external covariant force, which is expressed in the form F'".

It is interesting here to examine the Stiickelberg proposal in light of the solutions developed in this chapter. It must be emphasized that, at the present time, this is a formal comparison only, since the phenomenon of pair creation and annihilation is, strictly speaking, a quantum process.

It is first useful to examine the phenomenon from the standpoint of a frame observer. In general, we shall consider only the annihilation process here, with the features of the corresponding model of pair creation being symmetric with respect to frame time. The pair annihilation process is depicted in fig. D.l. From the standpoint of a global inertial observer, the positron and electron move towards each other from infinite separation at t = -oo with asymptotic speeds v' that are equal in the center-of-mass rest frame, as it is conventionally defined in eq. (4 .. 5).

In the conventional classical description, the total energy of the sys-

1 E. C. G. Stiickelberg, Helv. Phys. Acta 14, 372, 588 (1941).

343

344 APPENDIX D. CLASSICAL PAIR ANNIHILATION

t

collision at t=O

Figure D.l: The conventional representation of pair annihilation is de

picted, in which the positron and electron emerge from infinite separa

tion at equal asymptotic speeds v' in the center-of-mass rest frame and

collide at t = 0.

tern is (D.l)

where m here is the mass of the electron and 'Y' = \11 - v'2 .

According to the depicted observer, the two particles collide and

annihilate near the time t = 0. The terminology "near" is used here

because it may not be assumed-in the classical sense, at least-that

the two particles actually travel along the straight-line asymptotes that

intersect at t = 0. The particles would obviously collide at this time

if there were no mutual interaction between the particles. The fact

that there is a mutual attraction between the positron and electron,

however, leads to the conclusion that there may be a deviation from the

asymptotic trajectories.

345

The covariant description of the same process is shown in fig. D.2. The annihilation process is depicted as the evolution in T of a single particle of mass m. The solution to equations of motion is therefore given by a single particle vectorfunction x'" = x'" ( T), where the subscript may be dropped, since it is a one-particle system. The doubling back of the world line against itself means that the time component of the generalized velocity, dt I dr, is positive along one leg and negative along the other. 2 As mentioned in Section 1.7, the most problematic feature of this classical model is that the "particle" corresponding to both the positron and electron achieves speeds greater than unity along a finite interval of the world line.3 In fig. D.2, this interval is the portion of the world line lying between the times t = tc (at which the slope of the line is unity) and t = t" (at which dt I dr vanishes). The time t = tc is called the critical time. The time t = t" is called the inflection time. 4 The time t = 0, at which the particles would intersect if they were to travel on straight line paths, is called the intersection time. The particular point in spacetime at which the asymptotes intersect is called the intersection point.

One may legitimately ask whether it is possible, from the equations of motion presented in this chapter, to derive single-particle solutions such as depicted in fig. D.2. From a glance at fig. 6.3, it appears that such solutions are possible from this potential under the following assumptions: (i) the reduced Lagrangian,

L _ ~m dx~" dx'" _ V(p) - 2 dr dr ' (D.2)

is now taken to be the total Lagrangian of the system. That is, there is no center-of-mass term, and the reduced mass is the mass of the single particle, i. e, the electron; and (ii) that the potential V = -kl pis taken to be a function of a timelike separation p = p(r), not between two particle events, but between the the single-particle event and the origin of a privileged coordinate system, identifiable as the center-of-mass rest

2 This feature was appropriated by Feynman for the description of the quantum process. See Chapter 1.

3 Considering the discussion in Chapter 6, it might be surprising to the reader that we shall now consider the possibility of particle trajectories that reach unit speed in a finite dynamical time. This particular feature is obviously necessary, however, for any discussion of the model presented in this section.

4 The terminology "collision time" is avoided here because, strictly speaking, from the single-particle point of view, no collision occurs.


m Eo=-

2

t = t"

, I ~I ~I

I

I I

I

't= -oo

t

intersection point

\ \

\ \ \-a \~

\ 't= +oo

Figure D.2: The Stiickelberg one-body model of pair annihilation is represented here by the Type II solution to the timelike potential V = -k/ p. The single particle achieves the speed of light at Pc, indicated by points A and B along the world line. The source of the interaction is the origin of the coordinate system. The collision is observed to occur at t" < 0 in the center-of-mass rest frame.

frame in the conventional sense. The origin of this frame is chosen to coincide with the intersection point of the asymptotes. The solution is then given by the Type II solution eq. (6.27) for the range 0 < e2 < 1, as depicted in fig. 6.3. It is useful to note that the interaction here is that of one particle in an external potential (the source of which is yet unspecified), and thus the electron-positron "particle" does not interact with itself. This follows from the fact that r is single-valued along the curve.

The fact that p is timelike here means that the 1 + 1 generalized

coordinates take the form

tanh f3 X

t '

the inverses of which are

x p sinh /3,

t = pcosh/3.

347

(D.3)

(D.4)

It is useful to distinguish the coordinates (D.3) from the two-body version of the same coordinates, in that not only are the roles of x and t switched in eq. (D.3), but the~ symbol has been removed as well. This latter feature is a consequence of the use of one-body description.

From eq. (6.43), the asymptotes lie along the hyperbolic angle j31

given by

from which it follows that

I 1 cosh f3 = -,

e2

• 1 Jl- e§ smh f3 = --2 -. e2

(D.5)

(D.6)

At any point along the world line, the particle frame speed v is given by

(D.7)

Differentiation x = p sinh f3 and t = p cosh f3 with respect to f3 gives

dx

df3 ~~sinh f3 + p cosh {3,

:~ = ~~cosh f3 + p sinh /3,

which, together with (6.27), gives the particle speed v = v(f3) as

cosh f3 - e2 v=

sinh !3

(D.8)

(D.9)

(D.lO)


In the asymptotic limit, (3 --+ (3', and from eqs. (D.5) and (D.6), the frame speed is given by

From this, it follows that I 1

'Y = -, e2

which gives the total frame energy (D.1) of the system as

(D.ll)

(D.12)

(D.13)

It is useful to notice here that unlike the two-body case, e2 is not given by a ratio of the particle masses. Rather the relativistic eccentricity of the 1+1-dimensional solution is given by the initial asymptotic speeds of the positron and electron in the conventional center-of-mass rest frame.

It is useful to find expressions for the critical time as well as the inflection time. At the critical time tc, v--+ 1, and thus from eq. (D.10),

1 = coshf3c- e2,

sinh f3c (D.14)

where f3c is the hyperbolic angle of the critical points, labeled A and B in fig. D.2. Solving for f3c gives

e2 + 1 hf3 - _2_ cos c- ' 2e2

(D.15)

or alternatively, . h (3 e~- 1

Sln c= --. 2e2

(D.16)

By inserting eq. (D.15) into eq. (6.26), the spacetime distance Pc at the critical point is

_!._ = mk (~ (1 _ e2)) • Pc A2 2 2

(D.17)

At this point, it is useful to eliminate the angular momentum A from this expression. The total invariant energy of the system Eo is given by the usual rule as

1 2 Eo= 2mc, (D.18)

349

(cf. eq. (D.13), which gives the frame energy). Inserting this last expression into eq. (6.107) gives

k c

A- )1- e~'

i. e., for one-body motion

A2 1- e~ k2 --;;r-·

Inserting this last expression into eq. (D.17) gives

2k Pc = --2.

me

(D.19)

(D.20)

(D.21)

Inserting the electromagnetic expression fork from eq. (6.9) gives

(D.22)

which from eq. (1.7) is exactly twice the classical radius of the electron. The critical time tc according to the center-of-mass observer is given by eq. (D.15) as

k (d + 1) tc = PcCoshf3c =- -- . m e2

(D.23)

It is also useful to find the time of the inflection point, i.e., the time after which the center-of-mass observer records the particles no longer to exist. At f3 = 0, p-+ t, and eq. (6.26) gives

A2 t" = - (1- e2)-1 •

mk (D.24)

Inserting eq. (D.19), as well as the appropriate power of c, into the above expression gives

II k t --- mC3' (D.25)

which using eq. (6.9) is equal in magnitude to the time for light to transverse the classical electron radius (1.7). Expressed in seconds, this lS

t" = -9.46 x 10-23 sec. (D.26)


This last result is negative because the inflection point occurs before the intersection point, which was taken to be at t = 0. In the center-of-mass rest frame, the positron and electron achieve superluminal speeds during the frame time interval between tc and t", which is

" _ 11 _ k ( -e~ + e1 - 1) ut - t - tc - --3 . me e2

(D.27)

It should not be surprising that this interval goes to infinity in the limit of e2 --+ 0, since from eq. (D.l2) this is equivalent to the assumption that the positron and electron start asymptotically at the speed of light.

It may noticed that so far we have avoided discussion of the emitted photons. A photon is, by definition, a purely quantum phenomenon, and thus no classical theory can hope to represent photonic processes in a completely self-consistent fashion. Nevertheless, for formal selfconsistency, one might propose a model of the two photons as a single classical particle of well-defined energy and momentum that, in the case of pair annihilation, travels along both branches of the forward light cone in fig. D.2. That is, it emerges, in the limit r --+ -oo, from infinite forward frame time t, travels down the light cone to the origin, and then travels back out, as T --+ +oo, to infinite forward frame time once again. This formal representation has the advantage that the energy is conserved from the standpoint of the center-of-mass frame observer.5

5 1t must be emphasized that the formal addition of the classical photonic particle is not necessary for the conservation in 7' of the total invariant energy. It is added only in order to conserve frame energy. If the photons are considered to be "emitted" at the origin, then there is a finite time (D.26) during which the frame energy is not conserved, i. e., during which neither particles nor photons exist.

Bibliography

Books

V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, N.Y., 1978).

A. 0. Barut, Electrodynamics and Classical Theory of Fields and Particles (Macmillan, N.Y., 1964) [Dover ed., N.Y., 2nd ed. 1980].

P. G. Bergmann, An Introduction to the Theory of Relativity (PrenticeHall, N.Y., 1942).

H. A. Bethe and E. E. Salpeter, The Quantum Mechanics of One- and Two-Electron Atoms (Academic Press, N.Y., 1957).

J. Binney and S. Tremaine, Galactic Dynamics (Princeton U niv. Press, Princeton, N.J., 1987).

E. Cartan, Theory of Spinors (Hermann, Paris, 1966) [Dover ed., N.Y., 1981].

E. Cartan, CEuvres Completes, P. III: Geometrie differentielle Divers (Editions du Centre National de la Recherche Scientifique, Paris, 1984).

C. Coen-Tannoudji, B. Diu, and F. Laloe, Mecanique Quantique, Tome II (Hermann, Paris, 1973).

P. A. M. Dirac, Lectures in Quantum Mechanics (Yeshiva Univ. Press, N.Y., 1964).

A. Einstein, The Meaning of Relativity, (Princeton Univ. Press, Princeton, N. J., 1922) [5th ed. 1988].

J. R. Fanchi, Parameterized Relativistic Quantum Theory (Kluwer Academic Press, Dordrecht, 1993).

351

352 BIBLIOGRAPHY

Frontiers in Numerical Relativity, eds. C. R. Evans, L. S. Finn, and D. W. Hobill (Cambridge Univ. Press, 1989).

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 2nd ed. 1980).

S. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamics Systems, and Bifurcations of Vector Fields (Springer, N.Y., 1983).

L. Infeld and J. Plebansky, Motion and Relativity (Polish Acad. of Sciences, Physical Monographs, Warsaw, 1960: English trans. publ. by Pergamon, N.Y.).

J.D. Jackson, Classical Electrodynamics (Wiley, N.Y., 2nd ed. 1975).

T. F. Jordan, Linear Operators for Quantum Mechanics (Duluth, Minn., 1969).

C. Lanczos, The Variational Principles of Mechanics (Univ. of Toronto Press, Toronto, 1949) [Dover ed., N. Y., 1986].

M. Lipschutz, Theory and Problems of Differential Geometry (McGrawHill, N. Y.,1969).

H. A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl, The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity (Metheun and Co., London, 1923) [Dover ed., N.Y., 1952].

R.A. Matzner and L.C. Shepley, Classical Mechanics (Prentice Hall, Englewood Cliffs, N.J., 1991).

C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, N. Y., 1973).

W. Pauli, The Theory of Relativity (Pergamon, N.Y., 1958) [Dover ed., N. Y., 1981].

Proceedings of the International Conference on Parameterized Relativistic Quantum Theory, Houston, Tex., 1998, ed. L. P. Horwitz (Fanchi, Houston, Tex., 1998).

Proceedings of the Workshop of Constraint's Theory and Relativistic Dynamics, Arceti, Firenze, 1986, eds. G. Longhi and L. Lussanna (World Scientific, Singapore, 1987).

BIBLIOGRAPHY 353

Relativistic Action-at-a-Distance: Classical and Quantum Aspects, Lecture Notes in Physics, Vol. 162, J. Llosa ed. (Springer, Berlin, 1982).

W. Rindler, Essential Relativity: Special, General, and Cosmological (Springer, N.Y., 2nd ed. 1977).

N. T. Roseveare, Mercury's Perihelion: From Le Verrier to Einstein (Clarendon, Oxford, 1982).

L. H. Ryder, Quantum Field Theory (Cambridge Univ. Press, 1985).

B. F. Schutz, A First Course in General Relativity (Cambridge Univ. Press, 1990).

A. Sommerfeld, Atomic Structure and Spectral Lines, Vol. I (Methuen, London, 3rd ed. revised 1934; translated from the 5th German ed. by H. L. Brose).

M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. II (Publish or Perish, Berkeley, 2nd ed. 1975).

J. L. Synge, Relativity: The Special Theory (North-Holland, Amsterdam, 2nd ed. 1965).

E. F. Taylor and J. A. Wheeler, Spacetime Physics (Freeman, San Francisco, 1966).

S. Weinberg, Gravitation and Cosmology (Wiley, N. Y., 1972).

Papers on the Covariant Many-Body Theory

R. Arshansky and L. P. Horwitz, "The Quantum Relativistic Two-Body Bound State I. The Spectrum," J. Math. Phys. 30, 66 (1989).

L. Burakovsky, "Towards Resolution of the Scalar Meson Nonet Enigma," Found. Phys. 27, 315 (1997).

L. Burakovsky, L. P. Horwitz, and W. C. Schieve, "New Relativistic High-Temperature Bose-Einstein Condensation," Phys. Rev. D 54, 4096 (1996) .

. J. L. Cook, "Solutions of the Relativistic Two-Body Problem," Aust. J. Phys. 25, 117, 141 (1972).

L. P. Horwitz, "On the Definition and Evolution of States in Relativistic Classical and Quantum Mechanics," Found. Phys. 22, 421 (1992).

354 BIBLIOGRAPHY

L. P. Horwitz, "Dynamical Group of the Relativistic Kepler Problem," J. Math. Phys. 34 645 (1993).

L. P. Horwitz, R. I. Arsha.nsky, a.nd A. C. Elitzur, "On the Two Aspects of Time: The Distinction a.nd its Implications," Found. Phys. 18, 1159 (1988).

L. P. Horwitz a.nd C. Piron, "Relativistic Dynamics," Helv. Phys. Acta. 46, 316 (1973).

L. P. Horwitz a.nd W. C. Schieve, "Horseshoes in a. Relativistic Hamiltonian System in 1+1 Dimensions," Phys. Rev. A 46, 743 (1992).

L. P. Horwitz, W. C. Schieve, a.nd C. Piron, "Gibbs Ensembles in Relativistic Classical a.nd Quantum Mechanics," Ann. Phys. (N. Y.) 137, 306 (1981).

L. P. Horwitz, S. Sha.shoua., a.nd W.C. Schieve, "A Manifestly Covariant Relativistic Boltzmann Equation for the Evolution of a. System of Events," Physica. A 161, 299 (1989).

M.C. Land, "Particles a.nd Events in Classical Off-Shell Electrodynamics," Found. Phys. 27, 19 (1997).

C. Piron a.nd F. Reuse, "The Relativistic Two Body Problem," Helv. Phys. Acta. 48. 631 (1975).

F. Reuse, "On Classical a.nd Quantum Relativistic Dynamics," Found. Phys. 9, 865 (1979).

W. C. Schieve, "For Professor Horwitz on his Sixty-Fifth Birthday," Found. Phys. 26, 1575 (1996).

W. C. Schieve a.nd L. P. Horwitz, "Chaos in the Classical Relativistic Mechanics of a. Damped Duffing-like Driven System," Phys. Lett. A 152,140 (1991).

E. C. G. Stiickelberg, "Rema.rque a propos de la. Creation de Pa.ires de Pa.rticules en Theorie de Rela.tivite" Helv. Phys. Acta. 14, 588 (1941).

E. C. G. Stiickelberg, "La. Meca.nique due Point Materiel en Theorie de Rela.tivite et en Theorie des Quanta.," Helv. Phys. Acta. 15, 23 (1942).

M. A. Trump a.nd W. C. Schieve, "The Synchronization Problem m Covariant Relativistic Dynamics," Found. Phys. 27, 1 (1997).

BIBLIOGRAPHY 355

M. A. Trump and W. C. Schieve, "Off-Shell Dynamics in Flat Spacetime," Found. Phys. 27, 389 (1997).

M.A. Trump and W. C. Schieve, "Classical Scattering in the Covariant Two-Body Coulomb Potential," Found. Phys. 28, 1211 (1998).

M. A. Trump and W. C. Schieve, "Perihelion Precession in the Special Relativistic Two-Body Problem," Found. Phys. 28, 1407 (1998).

Other Papers on Special Relativistic Mechanics

R. Bakmajian and L. H. Thomas, Phys. Rev. 92, 1300 (1953).

R. Balescu and T. Kotera, Physica 33, 558 (1967).

R. Balescu, T. Kotera, and E. Pina, Physica 33, 581 (1967).

A.A. Chernikov, X. Titel, G. Vattay, and G.M. Zaslavsky, Phys. Rev. A 40, 4072 (1989).

D. G. Currie, J. Math. Phys. 4, 1470 (1963).

D. G. Currie, T. F. Jordan, and E. C. G. Sudarshan, Rev. Mod. Phys. 35, 350 (1963).

P. A.M. Dirac, Proc. Roy. Soc. London A167, 148 (1938).

P. A. M. Dirac, Rev. Mod. Phys. 21, 335 (1949).

A. Einstein, Bull. Am. Math. Soc. April 1935, p. 223.

E. Fermi, Atti. R. Accad. Rend. Cl. Sc. Fis. Mat. Nat. 31, 21 (1922).

R. P. Feynman, Phys. Rev. 76, 749 (1949).

V. A. Fock, Phys. Z. Sowjetunion 12, 404 (1937).

J. Hakim, J. Math. Phys. 8 1315, 1379 (1967).

P. Havas, Phys. Rev. 74, 456 (1948).

P. Havas and J. Plebansky, Bull. Am. Phys. Soc. 5, 433 (1961).

P. Havas, in Statistical Mechanics of Equilibrium and Non-Equilibrium, J. Meixner, ed. (North-Holland, Amsterdam, 1965).

L. P. Horwitz and F. Rohrlich, Phys. Rev. D 24, 1528 (1981).

W. Israel and H. Kandrup, Ann. Phys. (N.Y.) 152, 30 (1984).

356 BIBLIOGRAPHY

H. E. lves and C. R. Stilwell, J. Opt. Soc. Amer. 28, 215 (1938); 31, 369 (1941).

F. Jiittner, Ann. Phys. (Leipzig) 34, 856 (1911).

G. Longhi, L. Lusanna, and J. M. Pons, J. Math. Phys. 30, 1893 (1989).

A. Mangeney, Ann. Phys. (Paris) 10, 191 (1965).

W. Pauli, Z. Physik 43, 601 (1927).

A. Peres and N. Rosen, Nuov. Cim. 18, 644 (1960).

E. Piiia and R. Balescu, Acta Phys. Austriaca 28, 309 (1968).

G. N. Plass, Rev. Mod. Phys. 33, 37 (1961).

H. P. Robertson, Rev. Mod. Phys. 21, 378 (1949).

F. Rohrlich, Ann. Phys. (N.Y.) 13, 93 (1961).

F. Rohrlich, Phys. Rev. D 23, 1305 (1981).

J. L. Synge, Proc. Roy. Soc. A 177, 118 (1940).

L. H .. Thomas, Rev. Mod. Phys. 17, 182 (1945).

L. H. Thomas, Phys. Rev. 85, 868 (1952).

H. Van Dam and E. P. Wigner, Phys. Rev. 138, B1576 (1965).

H. Van Dam and E. P. Wigner, Phys. Rev. 142, 838 (1966).

J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 17, 157 (1945).

J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 21, 425 (1949).

E. P. Wigner, Ann. Math. 40, 149 (1939).

Index

Abraham-Lorentz law 6, 10 abruptness 6, 35, 40, 66-70, 287,

297, 321 transformation 49, 288

acceleration constant 38, 77, 284, 320 covariant 66, 218, 293

components of 293 evolution of 321 magnitude of 73

frame components of 35 definition of 35, 287 magnitude of 35 transformation 49, 288

proper time 66-67 variable 35, 77, 319, 340

accelerator beam 27 4 accelerometer 41 action

invariant 9, 13, 124 action-at-a-distance 10 action variables 164, 259 aesthetics 284 allowability criterion 286 analytical mechanics 16, 121 angular momentum

azimuthal 162, 225 covariant 133

components of 144 invariant 188, 207

357

definition of 133 values of 144

arc length Euclidean 32, 281 Minkowski 7,32, 285

measurement of 42 atom 85

Rydberg 276 baryon 276 Bertrand's theorem 159, 328 Boltzmann equation 185 boost, see Lorentz boost boost angle 44, 324, 333 boost vector 140, 288

definition of 45 bound motion 153, 196, 230, 253,

261, 275, 329 canonical transformation 123, 333 causality 105

propagation of 16 center of mass

covariant 109 definition of 18, 109

motion 111, 128, 173 temporal speed 111, 128, 168,

191, 206 lower bound 215

time of 331 center-of-mass rest frame

conventional 19 covariant 109

358

chaos relativistic 184

charge of electron 275

clock 286 c.m. frame 253, 331 frame 101 local 286

Clock Hypothesis 42 closed orbit 329 collision 344 co-moving frame 286, 319

basis of 50, 72-74, 280 definition of 50

co-moving trihedron 72, 282 configuration space 108, 124, 146 constraint dynamics 15, 52, 124 contour integral 260 coordinates

canonical 52, 164 curvature 72, 91, 281 generalized 99, 122, 181 hyperbolic 137 pseudospherical 137

Coulomb potential 18, 187, 225, 260, 327

correspondence rule for potentials 140

Currie's equation 336 curvature 281

absolute 281 coordinates 76, 91, 281 frame 72 invariant 75 radius of 73, 281 vector 281

curve analytic 38

continuous 38, 207 regular 279

damping relativistic 184

Index

Darwin Lagrangian 8, 15 degrees of freedom 21, 146, 261 derivatives

arc length 65-71 covariant 65, 70, 91 frame 34

dilation frame 36, 100, 222, 275, 291,

343 evolution of 40, 48, 294 measurement of 40, 291

geometric 64, 69, 77, 284 invariant 89, 104

Dirac equation 10 displacement

differential 33, 92, 125, 286 transformation of 50, 54 virtual 107

distance Euclidean 57

Duffing oscillator 183 dynamical correspondence 16 dynamical law 99, 119, 268, 279 eccentricity

relativistic 189, 192, 217, 230, 347

electron 85, 224, 270, 343 radius of 6, 349

energy c. m. 109, 260, 350 conserved 7, 18, 99, 119, 125,

268 frame 343 infinite 5

Index

invariant 105, 119, 125 kinetic 15, 105, 107 potential 105 reduced 106, 147, 161, 188 total 147, 169, 268, 348

entropy 98 equal-mass limit 218 equivalence class 62, 280 Euler-Lagrange equations

c.m. 131 covariant 124, 131 hyperbolic 143 reduced 131

Euler rotation 324 event 2233

classification 58 components of 32 correlation 112, 324 "evolution" of 92 interaction 85 particle 32 stationary 43

event dynamics 18, 92 event neighborhood 54, 107

definition of 32 event state 105

definition of 91 experiment 268

critical 186 lves-Stilwell 41

extrakinematical quan. boost 45, 48, 140, 243, 288 definition of 43 derivative of 45

Fermi-Walker transport 77, 320 Feynman model 26 fictitious particle

meaning of 103, 110

field tensor 134 field theory 286, 328

limitations of 180 fine structure 23, 262 force

covariant definition of 125

driving 184 external 343

four vector components of 59, 290 continuous 59 definition of 53 magnitude of 55 scalar product 59 static 59 two-particle 55

359

fourth derivative 33, 40, 186, 285, 321

transformation of 288 frame, see Lorentz frame free particle 52, 88, 119 frequency 262

driving 184 Galaxy 42, 85 Gaussian model 4 generalized mechanics 123 general relativity 269 generator

boost 322 infinitesimal 322 rotation 322

geodesic 270 Gibbs ensemble

relativistic 185 globular cluster 42 gravitation 85, 226, 270, 284 Hamiltonian 108, 126, 161. 259

360

covariant 18, 108, 126 vanishing 86

Hamiltonian dynamics 2, 327 Hamilton-Jacobi equation

reduced 161, 259 Hamilton's equations 8, 18

covariant 108, 125 hyperbolic 160 two-body 160

harmonic oscillator 182 helix 265 history

n-body 33, 52, 60 particle 33

hydrogen 22, 260, 275, 329 hyperhelix 76 hyperplane 19 implicit representation 189 inertial observer, see Lorentz frame inflection

point of 74, 281, 296 initial conditions 11, 215 instruments

co-moving 41 integrable system

criterion for 146 integrals

contour 260 isolating 146

interaction advanced 9 electrostatic 190, 253 fundamental 328 gravitational 190 light cone 10 long-range 84, 344 net 88 point 84

retarded 5, 9, 15 spacelike 17

Index

interaction constant 190, 217 intersection

point of 345 irreversibility 98 lves-Stilwell exp. 41 Kepler problem

conventional 244, 329 relativistic 18, 178, 187

kinematic particle state 32 time-dependence 34

kinematic quantity 31, 283 kinematic record

complete 39 kinetic energy

invariant 15, 10.5, 107 Lagrangian

covariant 123 hyperbolic 137 one-body 7, 345 reduced 17.5 two-body 19, 126

Lagrangian dynamics 121 frame-dependent 19, 180

Laplace-Runge-Lenz vector covariant 188

Least Action principle 108, 124 libration 260 Lienard-Wiechert pot. 5, 18 light 4 light cone 10, 15, 196, 234

passage through 58, 213 striking of 157

Liouville's theorem covariant 185

local observables definition of 41

Index

locus of events 37

Lorentz boost 45, 243, 333 as "evolution" 95 graphic rep. 44 magnitude 45 "timelike" 111

Lorentz force 4 Lorentz frame

definition of 32, 319 privileged 112 world lines in 37, 319

Lorentz invariance 96 Lorentz transformations

active 321 co-moving 50, 72 definition of 43 fixed 52 graphical 44, 4 7 infinitesimal 336 succession of 95

magnetostatic limit 8 manifest covariance mass

constant 82, 119 dynamical 15 equal 218, 221 finite 224 infinite 8, 21, 180 point 85, 119 reduced 19 relativistic 270 total 19 122 variable 124

mass shell 13, 104, 271 in Wigner-Van-Dam 17

mechanical system 16 Melnikov function

covariant 184 Mercury 272 meridional plane 154, 233 meson 276 Minkowski space 53,280, 324

norm 56 momentum

c.m. 131 covariant 124

definition of 103 frame 180

definition of 83 hyperbolic 143 off-shell 13 particle 83 relative 131 total 83

motion sickness 35 moving vehicle 35 muon 42

361

n-body system 1, 4, 23, 180, 268, 279, 327, 333

history of 33 region of 42, 336 transformation of 41

neutrino 276 Newton's law 5, 35 Newton's method 160, 202 Newtonian theory 4, 82, 190, 271,

329 no interaction theorem 52, 333 off-shell dynamics 2, 125 one-body limit 8, 20, 179, 221,

241 orbit 63, 77, 330

closed 155 definition of 37 Kepler 178

362

reduced 196, 240 spacetime 150

orbit equation 188, 191, 284 azimuthal 150, 227 radial 150, 223, 330

orthonormal tetrad 310 definition of 73

orthonormality relations 75, 282, 297, 301

osculating plane 72 osculating subspace 75, 137 pair annihilation 25, 224, 343 parameter

arbitrary 13, 280 arc length 63, 280, 327 correlated 17, 78, 86, 327 dynamical 13 invariant 24, 63 proper time 13, 327 staggered 63 superfluous 219 symmetries of 62 transformation 39, 63, 250 world time 60

parameterized theory 2 particle event

definition of 32 perihelion 159, 259, 264, 329 perturbation 26, 184 phase space 14

relativistic 108, 115, 184 photon 106, 350 physical motion

continuity 59 observation of 62

piecewise solution 284 Piron-Reuse frame 174,243,258,

331

definition of 135 pitch

of helix 263 Planck's constant 22 plane

meridional 154

Index

of reduced motion 24 Poincare group 43, 322, 333 Poincare invariant point particle 85, 119, 287, 319 Poisson bracket 185 position

covariant 99 components 57 definition of 54 derivatives 70 parameterized 63

frame components 34, 288 definition of 34, 287 transformation of 4 7, 288

positron 25, 224, 343 potential

frame-dependent 19, 140, 177 infinite 222 invariant 105, 268 inverse cube 159, 328 inverse square 18, 187, 225,

260, 327 scalar 126, 140, 165, 187 s pacelike 2, 1 7 timelike 214, 343 two-body 126, 22, 233, 260 vanishing 88 vector 7, 20

precession 22, 159, 240, 259, 264, 329

projectile 284

Index

proper time 286 definition of 42 measurement of 42 of system 111

proton 23 quantum mechanics 4, 23 190,

276, 327, 343 semi-classical 22, 259, 269 wave equation 141

radiation electromagnetic 4, 15

radiative reaction Dirac model 9 Dirac equation 10 Lorentz model 6

radius electron 6, 349 of curvature 73, 281

reduced motion 18, 127, 142, 173, 187, 225, 233

Lagrangian of 17 5 plane of 25, 134, 149

relativity postulate 52, 319 repere mobile 315 representation

arbitrary 63, 280 correlated 75, 86, 327 implicit 189 natural 63, 280 regular parametric 59, 280 staggered 63

rotation generator 322

Rydberg atom 276 scalar product 36, 288 scattering theory 1 semi-classical theory 22, 259, 269,

329

separation covariant 96, 138, 381

definition 55 magnitude 58 spacelike 15

lightlike 287 timelike 343

363

separation constant 162 separation coordinates 101, 128,

187, 225, 260 Serret-Frenet eqs. 72, 282 signal velocity 16 simultaneity

dynamical 16, 82 hyperplane of 16

Solar System 273, 329 space probe 27 4 spacetime

curved 270 flat 32

specific heat 186 speed

frame 206, 222, 284 definition of 36 derivative of 40 transformation 50, 291

of light 213, 345 spin 23, 262, 270

vector 320 statistical mechanics 185, 276 stellar dynamics 85 stopwatch

calibration 43 Stiickelberg model 25, 191, 224,

343 superluminal behavior 213, 350 symmetries

universal 42

364

Synchronization Postulate 112 Taylor expansion

frame time 35 lowest order 40

third derivative, see abruptness three-body system 277 time

backwards 224 c.m. frame 12, 331 critical 345 dynamical 81, 213, 279, 327 geometric 12 of inflection 345 of intersection 345 scale of 62

time dependence 159, 200, 245 time evolution 40

generator of 341 of world line 47

torsion frame 72 invariant 318 second 76, 318

torsional inflection point of 74, 319

trajectory 240, 263 frame

definition of 37 uniqueness of 37

turning point of 284 uniqueness of 37

translation generator 341 of origin 44, 48, 333

triangulation 273 turning point 156, 196, 230, 329

of projectile 284 Twin Paradox 69

Index

two-body system 93, 112, 181, 223, 327

Lagrangian 126 orbit equation 150 reduced motion 100 separation vector 58

uniform motion principle of 42

unit binormal 72, 282, 301, 309 unit normal 72, 282, 296 unit tangent 68, 71, 73, 281 unit trinormal 306 unphysical solutions 160, 229, 264 velocity

covariant 100 components 66, 70 magnitude 66, 70 evolution of 321

Einstein 290 components 8 magnitude of 14

frame definition of 35, 40, 287 transformation of 49, 288

generalized 99 wave packet

centroid 26 Wheeler-Feynman theory 10, 17 work function 16, 106 world line 24, 285

analytic 66, 26:3 angle made by 37 definition of 37 evolution 47 invariance 43, ;~33 observed 119 parameterized 60 solutions for 169

Index

straight 336 world time 189

definition of 86

365

Fundamental Theories of Physics

44. P.Ptak and S. Pulmannova: Orthomodular Structures as Quantum Logics. Intrinsic Properties, State Space and Probabilistic Topics. I 991 ISBN 0-7923- I 207-4

45. D. Hestenes and A. Weingartshofer (eds.): The Electron. New Theory and Experiment. 1991 ISBN 0-7923- I 356-9

46. P.P.J.M. Schram: Kinetic Theory of Gases and Plasmas. 1991 ISBN 0-7923-1392-5 47. A. Micali, R. Boudet and J. Helmstetter (eds.): Clifford Algebras and their Applications in

Mathematical Physics. 1992 ISBN 0-7923-1623-1 48. E. Prugovecki: Quantum Geometry. A Framework for Quantum General Relativity. 1992

ISBN 0-7923-1640- I 49. M.H. Mac Gregor: The Enigmatic Electron. 1992 ISBN 0-7923- I 982-6 50. C.R. Smith, G.J. Erickson and P.O. Neudorfer (eds.): Maximum Entropy and Bayesian

Methods. Proceedings of the I I th International Workshop (Seattle, 1991). I 993 ISBN 0-7923-2031-X

5 I. D.J. Hoekzema: The Quantum Labyrinth. I 993 ISBN 0-7923-2066-2 52. Z. Oziewicz, B. Jancewicz and A. Borowiec (eds.): Spinors, Twistors, Clifford Algebras and

Quantum Deformations. Proceedings of the Second Max Born Symposium (Wroclaw, Poland, 1992). 1993 ISBN 0-7923-2251-7

53. A. Mohammad-Djafari and G. Demoment (eds.): Maximum Entropy and Bayesian Methods. Proceedings of the 12th International Workshop (Paris, France, 1992). 1993

ISBN 0-7923-2280-0 54. M. Riesz: Clifford Numbers and Spinors with Riesz' Private Lectures to E. Folke Bolinder and

a Historical Review by Pertti Lounesto. E.F. Bolinder and P. Lounesto (eds.). I 993 ISBN 0-7923-2299- I

55. F. Brackx, R. Delanghe and H. Serras (eds.): Clifford Algebras and their Applications in Mathematical Physics. Proceedings of the Third Conference (Deinze, 1993) I 993

ISBN 0-7923-2347-5 56. J.R. Fanchi: Parametrized Relativistic Quantum Theory. I 993 ISBN 0-7923-2376-9 57. A. Peres: Quantum Theory: Concepts and Methods. 1993 ISBN 0-7923-2549-4 58. P.L. Antonelli, R.S. Ingarden and M. Matsumoto: The Theory of Sprays and Finster Spaces

with Applications in Physics and Biology. I 993 ISBN 0-7923-2577 -X 59. R. Miron and M. Anastasiei: The Geometry of Lagrange Spaces: Theory and Applications.

I 994 ISBN 0-7923-259 I -5 60. G. Adomian: Solving Frontier Problems of Physics: The Decomposition Method. 1994

ISBN 0-7923-2644-X 61 B.S. Kerner and V.V. Osipov: Autosolitons. A New Approach to Problems of Self-Organiza-

tion and Turbulence. I 994 ISBN 0-7923-2816-7 62. G.R. Heidbreder (ed.): Maximum Entropy and Bayesian Methods. Proceedings of the 13th

International Workshop (Santa Barbara, USA, 1993) 1996 ISBN 0-7923-2851-5 63. J. Perina, Z. Hradil and B. Jurco: Quantum Optics and Fundamentals of Physics. I 994

ISBN 0-7923-3000-5 64. M. Evans and J.-P. Vigier: The Enigmatic Photon. Volume 1: The Field B(3l. 1994

ISBN 0-7923-3049-8 65. C.K. Raju: Time: Towards a Constistent Theory. 1994 ISBN 0-7923-3103-6 66. A.K.T. Assis: Weber's Electrodynamics. 1994 ISBN 0-7923-3137-0 67. Yu. L. Klimontovich: Statistical Theory of Open Systems. Volume 1: A Unified Approach to

Kinetic Description of Processes in Active Systems. 1995 ISBN 0-7923-3199-0; Pb: ISBN 0-7923-3242-3

Fundamental Theories of Physics

68. M. Evans and J.-P. Vigier: The Enigmatic Photon. Volume 2: Non-Abelian Electrodynamics. 1995 ISBN 0-7923-3288-1

69. G. Esposito: Complex General Relativity. 1995 ISBN 0-7923-3340-3 70. J. Skilling and S. Sibisi (eds.): Maximum Entropy and Bayesian Methods. Proceedings of the

Fourteenth International Workshop on Maximum Entropy and Bayesian Methods. 1996 ISBN 0-7923-3452-3

71. C. Garola and A. Rossi (eds.): The Foundations of Quantum Mechanics- Historical Analysis and Open Questions. 1995 ISBN 0-7923-3480-9

72. A. Peres: Quantum Theory: Concepts and Methods. 1995 (see for hardback edition, Vol. 57) ISBN Pb 0-7923-3632-1

73. M. Ferrero and A. van der Merwe (eds.): Fundamental Problems in Quantum Physics. 1995 ISBN 0-7923-3670-4

74. F.E. Schroeck, Jr.: Quantum Mechanics on Phase Space. 1996 ISBN 0-7923-3794-8 75. L. de Ia Peiia and A.M. Cetto: The Quantum Dice. An Introduction to Stochastic

Electrodynamics. 1996 ISBN 0-7923-3818-9 76. P.L. Antonelli and R. Miron (eds.): Lagrange and Finster Geometry. Applications to Physics

and Biology. 1996 ISBN 0-7923-3873-1 77. M.W. Evans, J.-P. Vigier, S. Roy and S. Jeffers: The Enigmatic Photon. Volume 3: Theory

and Practice of the B(3) Field. 1996 ISBN 0-7923-4044-2 78. W.G.V. Rosser: Interpretation of Classical Electromagnetism. 1996 ISBN 0-7923-4187-2 79. K.M. Hanson and R.N. Silver (eds.): Maximum Entropy and Bayesian Methods. 1996

ISBN 0-7923-4311-5 80. S. Jeffers, S. Roy, J.-P. Vigier and G. Hunter (eds.): The Present Status of the Quantum

Theory of Light. Proceedings of a Symposium in Honour of Jean-Pierre Vigier. 1997 ISBN 0-7923-4337-9

81. M. Ferrero and A. van der Merwe (eds.): New Developments on Fundamental Problems in Quantum Physics. 1997 ISBN 0-7923-4374-3

82. R. Miron: The Geometry of Higher-Order Lagrange Spaces. Applications to Mechanics and Physics. 1997 ISBN 0-7923-4393-X

83. T. Hakioglu and A.S. Shumovsky (eds.): Quantum Optics and the Spectroscopy of Solids. Concepts and Advances. 1997 ISBN 0-7923-4414-6

84. A. Sitenko and V. Tartakovskii: Theory of Nucleus. Nuclear Structure and Nuclear Interaction. 1997 ISBN 0-7923-4423-5

85. G. Esposito, A.Yu. Kamenshchik and G. Pollifrone: Euclidean Quantum Gravity on Manifolds with Boundary. 1997 ISBN 0-7923-4472-3

86. R.S. Ingarden, A. Kossakowski and M. Ohya: Information Dynamics and Open Systems. Classical and Quantum Approach. 1997 ISBN 0-7923-4473-1

87. K. Nakamura: Quantum versus Chaos. Questions Emerging from Mesoscopic Cosmos. 1997 ISBN 0-7923-4557-6

88. B.R. Iyer and C.V. Vishveshwara (eds.): Geometry, Fields and Cosmology. Techniques and Applications. 1997 ISBN 0-7923-4725-0

89. G.A. Martynov: Classical Statistical Mechanics. 1997 ISBN 0-7923-4774-9 90. M.W. Evans, J.-P. Vigier, S. Roy and G. Hunter (eds.): The Enigmatic Photon. Volume 4:

New Directions. 1998 ISBN 0-7923-4826-5 91. M. Rectei: Quantum Logic in Algebraic Approach. 1998 ISBN 0-7923-4903-2 92. S. Roy: Statistical Geometry and Applications to Microphysics and Cosmology. 1998

ISBN 0-7923-4907-5

the geometry of world - springer978-94-015-9303... · 2017-08-27 · the differential geometry of...

Documents