a derivation of einsteins vacuum field equations

8/2/2019 A Derivation of Einsteins Vacuum Field Equations

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where is the gravitational potential. Thus, we leave out the question of theanalogue of the density of matter.

This limitation has three motivations

1. According to Einstein himself ([4]), ... [the vacuum equations] are theonly equations which follow without ambiguity from the fundamentalhypothesis of General Relativity.... On the analogue of the densityof matter of Poissons equation equations he says ([5]): The rightside of the [field] equations [the tensor kTik] is a formal condensa-tion of all things whose comprehension in the sense of a field theoryis still problematic. Further, he says ...this formulation was merely a

makeshift...

2. The vacuum field equations suffice to derive the Schwarzschild solutionand deduce the testable consequences of the theory that Einstein men-tioned in his work, i.e., the gravitational red shift of spectral lines, thedeflection of light by the sun and the precession of the perihelia of theorbits of the inner planets.

3. The whole derivation can be carried out from the assumption of aspace-time with a symmetric connection, rather than a metric.

Returning to Laplace equation, a little familiarity with Linear Algebrabrings to light the trivial observation that 2 is the trace of the lineartransformation given by the matrix (2/xx). This suggests toconcentrate on instead and find a relativistic analogue of this linear trans-formation.

This is what Sachs and Wu do in their approach to derive Einsteins fieldequations in [13]. To achieve this aim, they provide a geometrical interpreta-tion of in terms of acceleration of a neighbor (a vector field along a curve,to be defined below) whose analogue is readily available in the relativistictheory, namely covariant acceleration of the corresponding relativistic neigh-

bor. The map is in fact defined from Newton law of motion of freelyfalling particles (in a gravitational field). The corresponding map in Relativ-ity is likewise defined in terms of the Einsteins Law of Motion of free fallingparticles.

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Einsteins vacuum field equation, in analogy with Laplace vacuum fieldformulation of Newtonian theory can be formulated as the statement thatthe trace of the relativistic analogue of is zero.

Thus, we derive the vacuum field equations from the law of motion inboth cases.

Our approach to derive Einsteins vacuum field equations follows thatof Sachs and Wu. However, and in spite of the same starting point, ourderivation differs considerably from theirs as anyone browsing their book cansee. Among other differences, we do not require a metric but the simplernotion of parallel transport, as mentioned before.

Another difference from both Einsteins and Sachs and Wus approachesis the extensive use of infinitesimals. In fact, we place ourselves in the contextof Synthetical Differential Geometry (SDG). SDG is a mathematical theorywhose aim is to provide a rigorous foundation for infinitesimals and infinites-imal structures as used by geometers such as Sophus Lie, Elie Cartan as wellas some physicists and engineers. (See [7],[10],[11]).

In SDG, the reals R have the structure of a (non-trivial) ring, not a field,and the main axiom states one of the basic insights of the creators of calculus,namely, that in the infinitely small, every curve is a line in a unique way.To formulate this axiom, let D = {x R|x2 = 0}. An infinitesimal curve is

a function f RD

.Axiom of Kock-Lawvere (KL): The map R RRD that sends (a, b)into the curve [d a + db] is a bijection.

As an application, let f RR and x0 R. To define f(x0), the derivative

of f at the point x0, consider the infinitesimal curve : DR defined by(d) = f(x0 + d). By the axiom, (d) = a + db for a unique couple (a, b). Wedefine f(x0) = b = the slope of the line. Thus, f(x0 + d) = f(x0) + df

(x0),since clearly a = f(x0).

The usual properties of derivatives can be proved, by algebraic manipu-lations but we will not go into these matters. (See books cited above).

An important property of D that holds in practically all models is thatthe functor ()D, itself a right adjoint of ()D, has a right adjoint ()1/D.Thus () D ()D ()1/D. In particular (and this is the only propertyof D we will use) ()D preserves arbitrary unions.

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Besides D, the first order infinitesimals, there are higher order ones suchas Dn = {x R|xn+1 = 0} and D = {x R|n xn = 0}, the nilpotentreals. Infinitesimals make the main notions of Differential Geometry such asconnexion, parallel transport, covariant derivative, etc., quite intuitive and(several) proofs simple algebraic computations. As an example, the tangentbundle of a space M is simply the exponential space MD and the structuremap M : M

DM is just evaluation at 0, i.e., M() = (0). For goodspaces, the so-called microlinear spaces that play a role in SDG similar tosmooth manifolds in classical Differential Geometry, the fiber Mm =

1M (m)

is an R-module.

Before launching ourselves head on into our main concern, let us notice a

circumstance that will simplify enormously our task: the notions of velocityand acceleration of particles in Newtonian theory are defined at a singlepoint. So is the Laplacian. These notions, for their definition, require to takederivatives which according to SDG require functions to be defined only on aninfinitesimal neighborhoodof the point (rather that an ordinary neighborhoodas in classical DG). From now on everything will be infinitesimal: manifolds,geodesics, vector fields, etc.

In more details, open covers to define manifolds will be replaced by in-finitesimal covers to obtain D-manifolds.

A D-manifold of dimension n is a space having the property that each

point has a neighborhood isomorphic to Dn. Clearly, D-manifolds areclosed under product and the dimension of the product is the sum of thedimension of the factors. Furthermore, embedded (classical) manifolds insmooth toposes are D-manifolds. These manifolds are microlinear spaces.

Notice that in D-manifolds we can introduce coordinates and make cal-culations with them, just as in classical manifolds. Although our definitionsare invariant and hence independent of coordinates, these will allow to com-pare ours with the classical ones in the literature.

Trajectories will be infinitesimal curves with domain D.

Furthermore, vector fields on a D manifold will be maps Q : UUD

with U an infinitesimal neighborhood of M.

Working with these infinitesimal objects, we are able to carry out ourwhole derivation completely in the realm of the infinitesimal. This simpli-fies things considerably. Roughly speaking algebra rather than analysis is

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required to carry out our proof and in fact, the whole derivation of Einsteinsequations from the Sachs and Wu reformulation of Laplaces equation can becarried out synthetically (or axiomatically). As an example, existence anduniqueness of infinitesimal solutions of differential equations can be proved(as done e.g. in [8]). Other examples will appear below.

From a classical point of view we need to deal only with formal analysis(i.e., analysis with elements of formal power series, rather than ordinaryfunctions) to carry out our derivation.

The reader is referred to [10] and [12] for all that we need from SDG.

2 Acknowledgements

Among those who have helped and encouraged me, I must mention first andforemost Mihaly Makkai who followed a very long one person seminar atMcGill with contagious enthusiasm. He wasnt put off by the high density ofmistakes per phrase along the way and corrected errors, made criticisms, sug-gested better alternatives and kept himself and the speaker cheerful duringthe long process. I cannot thank him enough. Bill Lawvere was responsiblefor making these ideas public by obtaining an invitation for me to partici-pate in the Boston Meeting on Trends in the Mathematical Representation

of Space: Philosophical and Historical Perspectives on 2007 (November 31and December 1). His support has given me confidence to continue this work.

3 Newtons theory of Gravitation

Newtons law of motion states that the trajectory of particles of freely fallingparticles in the gravitational field obey the second-order differential equation

(t) + (((t)) = 0 for all t D.

In other words, the trajectory are solutions of the autonomous second order

differential equation G :

y + ((y)) = 0, where is the gravitational potential.

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written in vector form. Equivalently, the system

(y) + ((y1, y2, y3)) = 0 for = 1, 2, 3

written in terms of scalar components.

In invariant terms, this differential equation may be considered as a sym-metric vector field G : ED(ED)D EDD satisfying s G = G where s :EDDEDD is the map induced by the twisting map : DDDDgiven by (d1, d2) = (d2, d1). (Cf. Second order DEs and geodesics.pdf in[12]). The solutions of G are the trajectories of freely falling particles.

The fact that G is not an ordinary vector field on E (since it is definedon ED rather than E) creates problems for further developments.

Fortunately, G is reducible to a family {Gu}u of ordinary vector fieldsindexed by non zero vectors in the sense given by the following

Proposition 3.1 Let : DE be a curve with (0) = x and (0) = u,

a non zero vector in Ex. Then is a solution of G iff is an integral curveof Gu.

Proof: See section 5 (Proposition 5.2)

By defining a G-vector field to be a vector field whose integral curves aresolutions of G, we have

Corollary 3.2 Every Gu is a G-vector field.

The key notion of the Sachs and Wu approach is that of (the trajectoriesof) a swarm of infinitely close particles freely falling in the gravitational field.

Let {h}hD be the trajectories of the swarm. We may view this familyas a single map

W : DED

by defining W(t)(h) = h(t).

In fact, it is clear that we have the following one-to-one correspondencefor every v Mx =

1M (x).

Families {h : DE}hD with h(0) = v(h) for all h DMaps W : DE

D with W(t)(0) = 0(t) and W(0) = v.

Letting = 0, we have W(t)(0) = 0(t) = (t).

Then W is a vector field along in the sense that the following diagram

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ED

E

D

//

W

==||

||

||

||

||

||

||

||

|

E

is commutative.

Let : DE be a curve and x = (0), u = (0) a non-zero vector.

A neighbor of in a vector field Q is a vector field along , W : DED

such that for every h D, the curve h defined by h(t) = W(t)(h) is an

integral curve of Q.

Proposition 3.3 Let be a trajectory of a free falling particle, (0) = x,(0) = u (a non-zero vector) and letv Ex. Then there is a unique neighborWv of in Gu such that Wv(0) = v.

Proof: Define Wv(t)(h) = h(t), where h is the integral curve of Gu with

h(0) = v(h). Uniqueness is a consequence of uniqueness of solutions of DEs.(cf. [8]) 2

By an application of KL, Wv(t)(h) = 0(t) + h(t) = (t) + h(t) for aunique function .

Let u Ex be a non zero vector. By existence and uniqueness of second-order DEs there is a (unique) solution : DE of G with (0) = x and(0) = u. Thus is an integral curve of Gu. Define a map

u : ExEx

as follows. Let v Ex be arbitrary. Pick the unique neighbor Wv of in Gu

with Wv(0) = v. Define a vector field (Wv) along by

(Wv)(t)(h) = (t) + h(t)

for every t.

Finally, defineu(v) = (Wv)(0).

This vector field along is already well-known in differential geometry. Infact, leaving out the superscript v from W,

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Proposition 3.4 The following holds

W = 2W

where the connection is the Euclidean (or flat) one.

Proof: Recall that W is uniquely defined by the property hW(t) =((t + h), W(t + h))(h) W(t) for every h D. In words, we transportW(t + h) back along to (t) and subtract W(t). (Cf. covariantderiva-tion.pdf page 7 in [12].)

Since the connection is euclidean, vectors are preserved and the result isjust hW(t) = d[(t) + d(t + h)] d[(t) + d(t)] = d((t) + d

(t))

with (t) given by d(t) = (t) + d(t) (K/L).

Iterating,(2W)(t) = d((t) + d(t)).

finishing the proof of the proposition.

We now prove the main theorem of this section. Notice first that Ex iscanonically isomorphic to R3 by the isomorphism v Ex v0 R

3 wherev(d) = x + dv0 (K/L). In classical notation, v = (x, v0) with x = (x

1, x2, x3)and v0 = (v

10, v

20, v

30).

Thus, we may view u as a map u : R3R3 and (v) = W(0) =

(0).

Theorem 3.5 (cf. [13]) The map u : R3R3 is an R-linear transfor-mation. In fact (u) is the linear transformation defined by the matrix(2/xx). In particular,

u is independent of u.

Proof: Since Gu is a G-field, the trajectory h is a solution of G for eachh D, i.e.,

(h)(t) = (h(t)) for each t

where is the gravitational potential. Thus, at time t,

h(t) = [(h) ()](t) = (h(t)) ((t))

In particular, at time t = 0,

h(0) = (h(0)) ((0))

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By developing the RHS in Taylor series, we have that its th componentis

h

(2/xx)(x)v0

Then (v) = (0) = u(v) where

: R3R3

is the linear transformation defined by the matrix ((2/xx)(x)).Writing for u, we may formulate

Newtons vacuum field equation

trace() = 0In fact, this is just the Laplace equation.

NB We may view the preceding as the construction of a mapEx LinR(Ex, Ex)u u

where Ex the set of non zero tangent vectors in Ex, from the law of motionG (the Newtonian law).

Indeed, let u Ex. By existence and uniqueness of solutions of (au-tonomous) second order equations there is a unique solution : D

E

of G with (0) = x and (0) = u. Now, proceed as before starting withProposition 3.3.

The vacuum field equation is just the statement trace(u) = 0 for everyu. In this particular case, u is independent of u.

The same scheme will be used to derive the vacuum field equations inEinsteins relativistic theory of gravitation that we now discuss.

4 Einsteins Relativistic Theory of Gravita-

tionThe General Theory of Relativity or rather the Relativistic Theory of Grav-itation (as it should be properly called according to the Russian physicistV.Fock [6]) rests on the following fundamental assumptions.

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1. Space-time is a 4-dimensional curved manifold.

2. Free falling particles describe geodesics (straightest lines) in thismanifold.

3. There is a reciprocal action of matter on curvature and curvature onmatter described by Einsteins field equations. In the words of JohnWheeler: Matter tells space-time how to curve, and curved space tellsmatter how to move.

Some comments: one can say that the basic idea is to change the geometryso that will incorporate gravitation. More precisely, to abandon flat 3

dimensional Euclidean space for 4-dimensional curved space.This has the following consequence: in classical physics, the nature of

gravitation had been a problem since Newtons time. In the words of A.Koyre [9] (page 16) ...in spite of the rational plausibility and mathematicalsimplicity of the Newtonian law...there was in it something that baffled themind. Bodies attract each other, act upon each other (or at least behave asthey did). But how do they manage to perform this action, to overcome thechasm of the void that so radically separates them and isolates them fromeach other? We must confess that nobody, not even Newton, could (or can)explain, or understand this how. Newton himself, as we well know, never

admitted gravitation as a physical force.These problems were swept by the incorporation of gravitation into the

new geometry: there is no mysterious gravitational force acting on a parti-cle: the particle just follows a geodesic! It is fair to say, nevertheless, that theextent to which Einsteins theory has solved this problem has been debatedup to now.

The first two assumptions can be formulated in a straightforward way.The mathematical formulation of the third, i.e.,the field equations, is the realsubject matter of this paper. A lot of work has to be done to carry out theprogram of the introduction. We do this for the vacuum only.

We reformulate Einsteins physical postulates as follows:

Postulate 1 [Structure of space-time]: Space-time is a D-manifold Mof dimension n with a symmetric connection .

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We leave the dimension of the space-time M indeterminate (rather thann = 4), since our derivation does not depend on the value of n.

Postulate 2 [Law of motion]: Freely falling particles near an attractingbody describe geodesics. In other words, trajectories of particles under in-ertia and gravitation are solutions of the second order differential equationsdefining geodesics:

= (, )

or equivalently

= 0

(cf. affineconnections.pdf page 12 in [12])

From here one can derive the second order differential equation for geodesicsin terms of coordinates that can be found in texts of Differential Geometry(e.g. [14])

d2xk/dt2 +

ij

kij (dxi/dt)(dxj /dt) = 0

(cf. affineconnections.pdf page 14 in [12].)

Imitating what we did for the Newtonian gravitational theory, we assumethat space-time is a D-manifold, that vector fields are maps Q : UU

D

with U an infinitesimal neighborhood of M and trajectories, in particulargeodesics, curves with domain D.

Since a D-manifold M is microlinear, each tangent space 1M (m) is aR-modules (m M). (Here M : M

DM is the evaluation at 0). Amicrolinear space has dimension nif all the tangent spaces are free R-modulesof dimension n.

In the first section, Laplace field formulation of Newtons law of grav-itation was stated in terms of the linear transformation : ExEx astrace() = 0.

Trying to reproduce the computations of section 1 in our relativistic con-text, we take for G the second order DE of geodesics. Furthermore, we keepthe same notion of neighbor. For the record:

A neighbor of an integral curve of a vector field X is a vector field Walong such that the trajectories of W, namely the hs defined by h(t) =W(t)(h) are integral curves of X.

The existence of neighbors need not be postulated, but can be proved

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Proposition 4.1 (There are enough neighbors) LetX be a vector field, an integral curve of X and v Mx. Then there is a unique neighbor W of in X with W(0) = v.

Proof: cf. Section 5 (Proposition 5.1)

A G-field becomes a vector field whose integral curves are solutions of G,i.e., geodesics. We call such fields geodesic fields. Thus, a geodesic field is avector field whose integral curves are trajectories of free falling particles, i.e.,geodesics according to Postulate 2. (Recall that in the Newtonian framework,the integral curves of the gravitational field G are the trajectories of freefalling particles).

As before, we do not postulate the existence of geodesic fields either, butprove their existence:

Proposition 4.2 ( There are enough geodesic fields) For everyu Mxnot zero, there is a geodesic field Gu such that Gu(x) = u.

Proof: In fact, G is reducible to a family of vector fields Gu indexed bynon-zero vectors u Mx. Then G

u is the sought geodesic field. Cf. section 5(Proposition 5.2)

Following the analogy with Newtonian gravitation we define a map

u

: MxMx

for each non zero vector u Mx as follows: let : DM be the integralcurve of Gu with initial condition (0) = x.

Notice that (0) = Gu((0)) = Gu(x) = u.

Let now v Mx be arbitrary. Then there is a unique neighbor W of inGu with W(0) = v. Define

u(v) = 2W(0)

We would like to formulate Einsteins vacuum field equation equation as

trace(u) = 0

However, there are several problems with this formulation: first we have toprove that u is R-linear. Furthermore,u seems to depend not only on u

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but on Gu and, worst of all, it is not informative, in fact, we would like totie u to the curvature of the space.

The following takes care of all of these problems

Theorem 4.3 (Main Theorem, cf. [13])

(2W)(t) = R(t)W(t)(t)

where R is the Riemann-Christoffel tensor.

NB The definition of this tensor, as well as all the properties that we requirein our proofs can be found in Riemann-Christoffel.pdf in [12].

Corollary 4.4(v) = Ruvu

To prove the theorem we need the following

Lemma 4.5 5.4 Let : DM be an infinitesimal curve in M, p =(0) and Q a vector field defined on an infinitesimal neighborhood of p withQ(p) = 0. Then for every neighbor W of in Q there is a vector field W ofM such that [Q, W] = 0 and W = W.

NB Definitions and properties of Lie Derivatives LQ

W and Lie Brackets[X, Y] can be found in Liederivatives.pdf in [12].

Proof: cf. Section 5 (Lemma 5.4)

Proof (of the main theorem): We recall the following results (cf. [10])

1. X Y = YX+ [X, Y], provided that is symmetric

2. RXY Z = X (YZ) Y(X Z) [X,Y](Z)

The rest of the proof is now a simple computation:

2Q

W = QQW

= Q(WQ + [Q, W])= QWQ= RQ,WQ + WQQ + [Q,W]Q

= RQWQ

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Post-composing with , we obtain the desired result. In more details:

Claim 1: Q(t)() = (t + ) for every D2. Since M is microlinear, it isenough to show that (Qd1 Qd2)((t)) = (t + d1 + d2). This is clear. In fact,since is an integral curve of Q,

Qd1 Qd2((t)) = Qd1(Qd2((t))) = Qd1((t + d2)) = (t + d2 + d1)

Claim 2: (QW)((t)) = (W)(t). This is proved by starting from claim1 and the definition ofX Y (cf. [11] page 200 and [12]).

h(QW)((t)) = ([d Q(t)(d + h)], WQ(t)(h))(h) W(t)

= ([d (t + h + d)], W(t+h))(h) W(t)= ((t + h), W(t + h))(h) W(t)= h(W)(t)

The claim results from canceling the universally quantified h.

Claim 3: (RQWQ)(t) = R(t)W(t)(t). This follows from the definition of

the Riemann-Christoffel tensor, by noticing that Q((t)) = (t) M(t) and

W((t)) = W(t) M(t).

By analogy with Newtons gravitational theory, we postulate

Einsteins vacuum field equation [First Formulation]

trace(u) = 0 for all non zero u Mx

This is not the usual formulation of this equation, but it is equivalent.

First, we define the Ricci tensor as the contraction of the Riemann-Christoffel curvature tensor, i.e., in classical notation Rki =

l R

lkil. From

a more conceptual or invariant point of view, recall that the Riemann-Christoffel curvature tensor is given by a map R : MDMM

DMMDMD.

The contraction is a new map Ric : MD M MDMD defined as fol-

lows: if u, v Mx, let (u, v) : MxMx be the R-linear map defined by

(u, v)(w) = Ruwv. Then Ric(u, v) = trace((u, v)).

To connect this invariant definition of Ric with the classical expressionof the Ricci tensor we first take any basis of Mx to represent (u, v) asa matrix. By definition, Ric(u, v) is the trace of this matrix. Take, for

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instance the basis {/xi|x}i. The matrix of (u, v) relative to this ba-sis is (i((u, v)(/xj|x))ij , where {i}i is the dual basis, and its traceis Ric(u, v) =

i

i(Ru(/xi|x)v).

Next, we start with the following notations and identificationsRkij = Rijk (where i = /xi)(Rkij )l in our notation = R

lkij in Einsteins notation

Thus Rki =

l Rlkil =

l(R

kil)l =

l

l(Rilk) = trace((i, k)) =Ric(i, k)

Proposition 4.6 If is symmetric, then

Ric(u, v) = Ric(v, u)

Proof: cf. Metrics.pdf page 8 in [12].

This allows us to re-write the first formulation of Einstein vacuum fieldequation as Ric(u, u) = 0 for every non zero u Mx, since

u(v) = Ruv andhence trace(u) = Ric(u, u). But this implies that Ric = 0.

Indeed, let v, w Mx be arbitrary vectors. Write v =

viei and w =wiei. Then Ric(u, v) =

ij viwjRic(ei, ej ). But Ric(ei + ej, ei + ej) =

Ric(ei, ei)+2Ric(ei, ej)+Ric(ej, ej). By hypothesis, the LHS and the extreme

terms of the RHS are 0, so the middle term must be zero, i.e., Ric(ei, ej ) = 0.Thus, Ric = 0. In other words, we may reformulate the above equation as

Einsteins vacuum field equation [Final Formulation]

Ric = 0

which is the usual form of this equation.

5 Missing Proofs

In this final section we provide proofs for the lemma and propositions required

to prove the main theorem

Proposition 5.1 (There are enough neighbors) LetX be a vector field, an integral curve ofX andv Mx (with (0) = x). Then there is a uniqueneighbor V of in X with V(0) = v.

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Proof: Let : DM an integral curve of X and u Mx. Define V :DMD by V(t)(h) = h(t), where h is the unique integral curve of Xwith initial condition h(0) = v(h).

Thus, V(t)(0) = 0(t) = (t), since both and 0 are integral curves ofQ with the same initial condition. In other words, V is a vector field along .Furthermore, LX V = 0, by construction. Finally, V(0)(h) = h(0) = u(h),and this shows that V(0) = u.

To show uniqueness, assume that V1 and V2 are two such. Then V1h andV2h satisfy the same (first-order) differential equation, namely X, with thesame initial conditions since V1h (0) = V

1(0)(h) = v(h) = V2(0)(h) = V2h (0).

Thus, they coincide.The following generalizes a result of [13]:

Proposition 5.2 (Reduction theorem) Let Y be an autonomous secondorder equation, i.e., a symmetric vector field Y : MD(MD)D. Then thereis a family Yu of vector fields indexed by non zero vectors u Mx with the

following property: if : DR is a curve with (0) = x and (0) = u,

is a solution of Y iff is an integral curve of Yu

We say that Y is reducible to the family (Yu)u.

Call a vector field Q an Yfield iff all integral curves of Q are solutionsof Y.

Corollary 5.3 All the Yu are Y-fields.

Proof of the Proposition: By choosing appropriate coordinates we may as-sume that x = (0, . . . , 0) and u = (0, . . . , 0, (0, . . . 0, u0)) with u0 invertible.

Define a map : DnDn by the formula

(1, . . . , n1, t) = (1,...,n1,0)(t)

where (1,...,n1,0) is the solution of the equation Y starting at (1

, . . . , n1

, 0)with initial velocity u, i.e.,

(1,...,n1,0)(0) = (1, . . . , n1, 0)

(1,...,n1,0)(0) = (1, . . . , n1, 0, (0, . . . , 0, u0)).

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(For the existence and uniqueness of these solutions, see Second order DEand geodesics.pdf in [12] and [8]).

We claim that is invertible:

Write = (1, . . . , n). Spelling the initial conditions of (1,...,n1,t) interms of the components of we get

i(1, . . . , n1, 0) = i (i < n)n(1, . . . , n1, 0) = 0

for the first and

i

/t(1

, . . . , n1

, 0) = 0 (i < n)n/t(1, . . . , n1, 0) = u0

for the second.

By developing in Taylor series,

i(1, . . . , j + d , . . . , n1, 0) = i(1, . . . , j , . . . , n1, 0)+ d(i/yj)(1, . . . , j , . . . , n1, 0)

By identifying corresponding terms in this equation,

(

i

/yj

)(1

, . . . , j

, . . . , n1

, 0) = 0 (i = j)(i/yi)(1, . . . , j , . . . , n1, 0) = 1

Therefore (0, . . . , 0, 0) = (0, . . . , 0, 0) and the Jacobian of at (0, . . . , 0, 0) isu0 = 0. The claim is then a consequence of the Infinitesimal Inverse functiontheorem.pdf in [12].

We let be the inverse of .

Define the vector field Yu or simply Q by the formula

Q(x1, . . . , xn) = (1,...,n1,0)(t)

with (1, . . . , n1, t) = (x1, . . . , xn).

Claim 1: Q is a vector field and Q(x) = u.

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The fact that Q is a vector field is obvious:

Q(x1, . . . , xn)(0) = (1,...,n1,0)(t)(0)

= (1,...,n1,0)(t)= (1, . . . , n1, t)= (x1, . . . , xn)

On the other hand,

Q(x) = Q(0, . . . , 0) = (0,...,0,0)(0) = (0, . . . , 0, (0, . . . , 0, u0)) = u

by the second initial condition that satisfies.

Claim 2: The (1,...,n1,0)()s are integral curves of Q, i.e.,

(1,...,n1,0)(t) = Q((1,...,n1,0)(t)

In fact, let (1,...,n1,0)(t) = (x1, . . . , xn). We saw that, (1,...,n1,0)(t) =

(1, . . . , n1, t). On the other and, by definition of Q,

Q((1,...,n1,0)(t)) = Q(x1, . . . , xn) = (1,...,n1,0)(t

)

where (1, . . . , n1

, t) = (x1, . . . , xn).

Since (1, . . . , n1, t) = (1,...,n1,0)(t) = (x1, . . . , xn) and is a bijec-

tion, i = i

and t = t, completing the proof.

Claim 3: The integral curves of Q are precisely those of the form (t) =(x1,...,xn1,0)(xn + t), for (x1, . . . , xn) Dn.

In fact, all curves of this form are integral curves of Q : let u(t) = xn + tand (t) = (x1,...,xn1,0)(t). Then = u.

We claim that u = ( u) =

u = ( u) =

Indeed, computing the first term of the first equation at (t, d) with d D,we have

(u(t))(d) = (u(t) + d)

= ((xn

+ t) + d)= (xn + (t + d))= (u(t + d))= ( u)(t + d)= ( u)(t)(d)

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The second equation is proved by iteration.

As a consequence, we have the two equations

= ( u) = u = (Q ) u = Q ( u) = Q = u = (Y ) u = Y ( u) = Y

The first shows that curves of the stated form are integral curves of Q.

To prove the converse, let be an integral curve ofQ with initial condition(0) = (x10, . . . , x

n0). Since is bijective, there is a unique n-tuple (

1, . . . , n)such that (1, . . . , n) = (x10, . . . , x

n0 ). Define (t) = (1,...,n1,0)(

n + t)

Then, = since both are integral curves of Q with the same initialcondition: (0) = (1,...,n1,0)(n) = (1, . . . , n) = (x10, . . . , x

n0 ) = (0).

Proof of the Corollary: We have to show that if is a solution of Y, andxn D, so is , where (t) = (x

n + t). This is a consequence of the secondequation.

Lemma 5.4 Let : DM be an infinitesimal curve in M, p = (0) andQ a vector field defined on an infinitesimal neighborhood of p with Q(p) = 0.Then for every neighbor W of in Q there is a vector field W of M suchthat [Q, W] = 0 and W = W.

NB Definitions and properties of Lie Derivatives (LQW) and Lie Brackets([Q, W]) used in this proof can be found in Liederivatives.pdf in [12].

Proof: Since Q(p) = 0, by a change of coordinates we may assume that,locally around p, Q = /x1 (see Straightening out Theorem.pdf in [12]).Write W(t) =

i w

i(t)/xi|(t). Then

0 = LQW(t) =

i

wi(t)LQ(/xi|(t)) +

i

(wi)(t)/xi|(t).

Now, LQ((/xi) ) = L/x1((/x

i) ) = [/x1,/xi] . But[/xi,/xj ] = 0. This follows from the definition of Lie bracket

[/xi,/xj ]x(d1d2) = (/xj )d2 (/x

i)d1 (/xj)d2 (/x

i)d1(x)

and the definition of the vector field (/xi), namely (/xi)d(x) = x + dei.

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Since {/xi|(t)}i is a basis, (wi)(t) = 0, i.e. wi(t) = ai is a constant.

DefineWx =

i

ai/xi|x.

This vector field has the required properties. Indeed, obviously W = W.The second property is a consequence of bi-linearity of [,]

[Q, W] = [/x1, W] =

i

ai[/x1,/xi] = 0.

6 Appendix

6.1 A reformulation of Laplace equation

From what we did in the first section, we have an intuitive reformulation ofLaplace equation due to Sachs/Wu that we now present. This follows fromthe following

Proposition 6.1 Let : R 3R 3 be any linear transformation. Then

(1/4)

uS2(u) u = (1/3)Trace()

where is the volume form of S2.

Proof: Compute the integral using spherical coordinates:

x1 = cos 1x2 = sin 1 cos 2x3 = sin 1 sin 2

(0 1 , 0 2 < 2)

In these coordinates, the volume form of S2, namely = u1du2 du3 +u2du3 du1 + u3du1 du2 (see e.g. [1] page 96) becomes = sin 1d1 d2

Let (aij )ij be a matrix representation of . Then (u) u =

i,j aij uiujand a simple computation shows that

uS2

uiuj =

0 if i = j

4/3 otherwise

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Thus, uS2 (u) u = (4/3)i aii = (4/3) Trace() 2In particular, this proposition applies to the linear transformation given

by the matrix (2/xx). In this case, there is a simple intuitive inter-pretation of this equation given in [13]: Assume that you are in the middleof a cabin freely falling near the earth surface with lots of apples around you.Look at those apples situated at the same distance (which we assume small)from you and measure their acceleration towards (or away from) you. Theiraverage is 0.

In our context, we may formulate this intuitive interpretation as a precisemathematical statement by defining small as being of square 0, i.e., an

element of D.Recall from the section on Newtons theory of gravitation that the differ-

ence between the acceleration of the apple whose trajectory is h and yours(trajectory ) at time 0 is h(u).

We define the average or mean acceleration of all the apples at distanceh D from 0 to be

(1/4)

uS2h(u) u

where = x1dx2 dx3 x2dx1 x3 + x3dx1 dx2 is the volume form of S2.

Notice that since S2 = 4, the term mean is justified.Applying the previous proposition to we have the following

Corollary 6.2 The mean relative acceleration of all particles situated at dis-tance h from 0 is

(1/4)

uS2h(u) u = (h/3) trace()

where = x1dx2 dx3 x2dx1 x3 + x3dx1 dx2 is the volume form of S2.

The Sachs and Wu interpretation is the statement that the integral in thecorollary is 0 for every h D. But this is equivalent to the statement that

2 = 0, since = and trace() = 2.

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6.2 The first two postulates of Einstein

The aim of this section is to show how to derive postulates 1 and 2 fromother principles discussed in the literature.

Following Einstein, space-time is assumed to be a 4-dimensional semi-Riemannian manifold M with metric g. Recall that a semi-Riemannian man-ifoldis a manifold with a metric g having the property that around each point

p there are local coordinates with

(g)(p) =

where the s are the entries of the matrix1 0 0 00 1 0 00 0 1 00 0 0 1

In particular, ()(p) = 0 (see metrics.pdf corollary 0.7 page 6 in

[12]).

Ad Postulate 1: From g we can manufacture a symmetric connexion .(Details in metrics.pdf in [12].)

The unique symmetric connection compatible with a given metric is called,improperly according to Spivak ([14] II page 256), the Levi-Civita connectionfor the metric.

Ad Postulate 2: This axiom is introduced as such in General Relativityunder the name Equation of motion and it is deduced/motivated by theso-called equivalence principle (see Einstein [2] and Weinberg [15]).

I will not go into the questions of validity of this principle (see V.Fock[6]), or the shortcomings of Weinbergs proof, but point out that Postulate2 can be deduced from a very weak point-wise form of the equivalenceprinciple that seems un-objectionable and may be formulated as follows:

Let : DM = Rn be the trajectory of a free falling particle and pa point of . Whenever are local geodesic coordinates around p (i.e., with(p) = 0), the trajectory satisfies

(d2k/dt2)(p) = 0

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at the point p.

From this principle we deduce that is indeed a geodesic with equation

d2xk/dt2 +,

k(dx/dt)(dx/dt) = 0

An indication of the proof: since k is a function of the x, a simple applica-tion of the chain rule gives (at p)

0 =

( k/x)(d2x/dt2) +,

(2k/xx)(dx/dt)(dx/dt)

By multiplying this equation by x/k and adding with respect to , weobtain

d2xk/dt2 +

(k)x(dx/dt)(dx/dt) = 0

since and x are inverse of each other.

By applying the transformation rule for at point p, (k)x = (k)x.

Since p was arbitrary, this concludes the proof.

NB Recall that going from coordinates x to x, is transformed as follows

()x = ijk (kij )x(xi/x)(xj /x)(x/xk)+

i(2xi/xx)(x/xi)

(cf Affineconnections.pdf in [12]).

References

[1] H.Cartan, Formes differentielles, Hermann, Paris, 1967

[2] A.Einstein, The Foundations of the General Theory of Relativity, in ThePrinciple of Relativity: A collection of original papers on the Special and

General Theory of Relativity, Dover Publications, Inc. 1952

[3] A.Einstein, The Meaning of Relativity (Fifth Edition), Princeton Uni-versity Press, Princeton, New Jersey, 1970

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[4] A.Einstein, L.Infeld and B.Hoffmann, The gravitational equations andthe problem of motion, in Ann.Math. 39, No 1, 65, 1938

[5] A.Einstein, Autobiographical Notes, in Albert Einstein: Philosopher-Scientist Volume 1 Edited by Paul Arthur Schilpp, Harper&Row, Pub-lishers, New York, Evanston, and London (Copyright 1949) 1959

[6] V.Fock, The Theory of Space Time and Gravitation, Pergamon Press,New York, London, Paris, Los Angeles, 1959

[7] A.Kock, Synthetic Differential Geometry, LNM Lecture Notes Series 51,Cambridge University Press, Cambridge, London, New York 1981

[8] A.Kock and G.E.Reyes, Ordinary differential equations and their expo-nentials, Central European Journal of Mathematics 4(1) 2006 (64-81)

[9] A.Koyre, Newtonian Studies, Phoenix Books, The University of ChicagoPress 1968

[10] R.Lavendhomme, Basic Concepts of Synthetic Differential Geometry,Kluwer Academic Publishers, Dordrecht, Boston, London 1996

[11] I.Moerdijk and G.E.Reyes, Models for Smooth Infinitesimal Analysis,Springer-Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo,

Hong Kong, Barcelona 1991

[12] G.E.Reyes, Series of pamphlets on Synthetic Differential Geometry inthe authors web page reyes-reyes.com

[13] R.K.Sachs and H. Wu, General Relativity for Mathematicians, Springer-Verlag, New York, Heidelberg, Berlin 1977

[14] M.Spivak, A comprehensive Introduction to Differential Geometry, 5 vol-umes, Publish or Perish Inc., Berkeley, 1979

[15] S.Weinberg, Gravitation and Cosmology, Principles and Applications

of the General Theory of Relativity, John Wiley & Sons, New York,Chichester, Brisbane, Toronto, Singapore 1972

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a derivation of einsteins vacuum field equations

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