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^iQ j TECHNICAL NOTE

>— GEOMETRICAL AND PHYSICAL INTERPRETATION OF CO THE WEYL CONFOEMAL CURVATURE TENSOR

cri .ST- =r ^^ F.A.E. Pirani, King's College, London CD f^ \ > o ü » ""^f —rtw* and

A. Schild, University of Texas, Austin, Texas and Institute for Advanced Studies, Dublin.

1 May 1961 -:■•■•.■. -'■«•,

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The research reported in this document has been sponsored in part by the Wright AIT Development Division of the AIR RESEARCH AND DEVELOPMENT COMMAND, UNITED STATES .JR FORGE, through its European Office.

ABSTRACT

The Weyl (conformal curvature) tensor of space-time is

interpreted geometrically in terms of the behaviour of

congruences of null geodesies. The corresponding physical

interpretation provides, in principle, a means for the

measurement of physical components of the Weyl tensor with

light rays alone, without the use of clocks or rigid rods.

GEOMETRICAL AND PHYSICAL INTERPRETATION OF

THE WEYL CONFOKMAL CURVATURE TENSOR

by

F.A.E. Pirani and A, Schild

1. Introduction; Conformal-invariant methods

This paper outlines, without proofs, a geometrical and

physical interpretation of the Weyl tensor in the space-time of

general relativity. Proofs and further details will be published

elsewhere. The interpretation embodies a method whereby the

physical components of the Weyl tensor could, in principle, be

measured by observations of light-rays alone, without the use

of clocks or rigid rods. The possibility of such measurements

may be inferred from the fact that null geodesies, which

represent light rays, arc invariant under conformal transformations

of Riemannian space-time, while proper time along time-like lines

is not invariant (nor are time-like geodesies invariant). In

vacuum, the same measurements will yield corresponding components

of the Riemann curvature tensor, since the Riemann and V/eyl

tensors coincide wherever Einstein's vacuum field equations hold.

In a previous paper (Pirani 1956), a physical interpretation

was given to the Riemann tensor, and a method of measuring its

components by observations of test particles was described. The

2.

whole discussion there was explicitly metrical, and required the

use of clocks (or rigid rods) as well as light rays. It was

understood that a particular Riemannian space-time was given in

advance, and the arguments depended on the measurement of proper

times.

In ehe present paper, on the other hand, the arguments are

entirely cenfurma 1 -invar 1 ant °, both geometrical and physical

interpretations refer not to a particular Riemannian space-time,

hut to a whole class of space-times which may be obtained from

one another hy conformal transformations of the metric. Such a

class of Riemannian space-time is called a conformal space-time;

thus a conformal space-time C/.\ is a (sufficiently) differentiahle

manifold endowed at each of its points P with a real infinitesimal

null cone

(l) g dx dx =0.

Physically, the nail cone is the history of a wave front of light

collapsing to and emitted from the event P. The quadratic form (l)

must have hyperbolic normal signature in order that the proper

distinctions between past and future may be preserved.

The null cone (l) determines the metric e g in any of

the Riemannian space-times of the conformal class up to a grange 0 o

factor, e which is an arbitrary function of position. A

1. Latin indices a,fc,c,,.. range and sum over 1,2,5,4.

5.

particular Riemannian space-time may be selected by assigning the

gauge. It is often easier to carry out proofs of theorems in a

particular gauge, exhibiting their conformal-invoriance afterwards,

than to devise a strictly confonnal-invariant proof.

It is evident that the ratio of the magnitudes of two vectors

or of tv.ro simple bivectors at the same point, and the angle between

two directions at the same point (especially, the orthogonality of

two directions at the same point) are well-defined in a conformal

space-time. It is in fact easj to show how these quantities may

be determined by experiments with light signals alone.

In § 2 we give conformal-invariant definitions of null

geodesies and of preferred parameters on them. In § 3 we give a

conformal-invariant definition of infinitesimal shea^, and state

our main result, which connects the second parameter-derivative of

the shear with the conformal curvature tensor. The infinitesimal

shear was introduced by Sachs (196] a,b) in his analysis of null

geodesic congruences, and many of the ideas employed here were

developed originally, in metrical form, by him.

2. Null geodesies and preferred parameters

A null hypersurface is defined, conformal-invariantly, as

a hypersurface which is tangent at each of its points to the

infinitesimal null cone at that point. Equivalently, a null hyper-

surface contains at each of its points exactly one null direction.

A null geodesic is a null curve which lies entirely in a null

hypersurface (this manifestly conformal-invariant definition

reduces to the usual cne as soon as a gauge is assigned).

Physically, a null geodesic is the world-line of a light ray, that

is, the history of a light pulse.

We show now how to define a conformal-invariant preferred

parameter u along any null geodesic of a given congruence. Let t

"be the selected geodesic and Zix its infinitesimal tangent vector

at any point P (Fig. l). Let (". , i , and t be three neighbouring

null geodesies, t, and -t being chosen so that the connecting

vectors S-.x , 6 x from P to ^ -C are orthogonal to Ax , and

C , being chosen so that the connecting vector 6 ,x from u to 5 3

-j^, , is not orthogonal to Ax . Then a parameter u may be defined

along -C by the condition

area of parallelogram spanned by Ax /Au and 0 x (2) ^— = constant along €

area of parallelogram spanned byö-jx" and 6 „x

5.

It can be shown that this definition determines the parameter u

up to a linear transformation with coefficients constant along -C ,

irrespective of the choice of the neighbouring geodesies 'f-, , -lip,

^£ , and of the choice of connecting vectors from P.

If the gauge is chosen so that the denominator of (2) is

constant along i- (which corresponds to a zero magnification rate

along € in the terms of Sachs's analysis), then the parameter u

may be identified with the usual preferred parameter along a null

geodesic in the corresponding Riemannian space (cf Synge and

Schild 1949, p.46).

$. Infinitesimal shear and its propagation

We now define in a conformal-invariant way the infinitesimal

shear for a congruence of null geodesies (ef Sachs 196la, p. ).

Let Sp and S be infinitesimal 2-eIements orthogonal to a null

geodesic ^ at neighbouring points P and Q of t- , and let C be an

infinitesimal circle with centre P, lying in Sp (Pigt 2). Those

null geodesies of the congruence which meet 3^ in the circle C B - p

will meet Sn in an ellipse E. The infinitesimal shear of the

congruence from P to Q, d£, is defined by the equation

/,\ length of major axis of E n „,_ (5) a ' = 1 + 2d£. length of minor axis of E

It may be shown that both d£ and the major axis e of the

ellipse E are determined uniquely by the congruence of null geodesies

and the points P arid Q,, and that de is independent of the choice

of the orthogonal 2-elements Sp and S^.

Ellipse £

Circle «T

Fig. 2

The physical interpretation of this construction has been

given, for the metrical case by Sachs (1961 a) 5 Suppose that light

passes normally through a small fl<t transparent circular disc and

falls normally on a screen nearby. At a certair instant, (P), the

disc becomes momentarily opaque, and throws a shadow on the s-creen

(:i). Refraction of the light by the gravitational field makes

the shadow elliptical. The null geodesies represent the light

rays; the circle C represents the periphery of the disc and the

ellipse E the periphery of the shadow. Sachs has shown that the

shape, size and orientation of the shadow depend only on the choice

of P and Q and not on the velocities of the disc or the screen.

It is evident from the above construction that de is conformal-

invariant. It can be shown that the orientation of the shadow

also is conformal-invariant; clearly, the size is not.

We can now state our main result, which relates the

propagation of the shear along a null geodesic to the conformal

tensor. It is that

A2 „ ^ A c . (A) ä. £ a a^_ dx_ b W du2

0 bed du du a

Here C , , is Weyl's conformal curvature tensor: in any Riemannian

space of the conformal class,

C bed = R bed + ^[dR c] + Vd6 c] - /3 gb[ä0 c]R'

a R . -.is the Riemann curvature tensor, defined for example by the

bed '

commutation rule for covariant differentiation, V, , - V, , = bjed b;dc

R 1 ,V , for any vector V ,R, = R , ia the Ricci tensor, and bed a' ^ a' DC bca

be dx b R = g R, . Also -r— is tangent vector to the geodesic, and P

DO du a a

is the projection operator in the direction of the major axis e

of shears P e = e , P f =0 for every vector fc orthogonal a a . a to e ,

Equation (4) determines a physical component of the tensor

C , ,. Since C , -is irreducible under transformations of the bed bed.

local orthogonal frame at any point (i.e. under Lorentz

\ a transformations), all the components of C , , may be determined

8.

"by examining enough congruences of null geodesies, that is, "by-

carrying out appropriate experiments with light rays.

Acknowledgements

We are grateful to J. Ehlers, J. N. Goldberg, R. K. Sachs and

L. Witten for many stimulating discussions.

One of us (A.S.) is indebted to the Research Institute of

the University of Texas for a grant during the tenure of which part

of this work was carried out.

References

Pirani, F.A.E. Acta Physica Polonica 15, 389-405 (1956).

Sachs, R.K. a. Mimeographed lecture notes. King's College, London, (1961).

b. Phil. Trans. Roy. 3oc. in press.

Synge, J.L. and Schild, A. Tensor Calculus, Toronto, (1949).

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