On the Concept of Rigidity in the Theory of Relativity T. Y. THOMAS

Communicated by R. A. TOUPXN

Contents P~e w t. Rigid motions . . . . . . . . . . . . . . . . . . . . . . . . . . . 30t w 2. Conditions on the velocity vector for a motion to be rigid . . . . . . . . . 303 w 3. Newtonian approximation of the rigidity conditions . . . . . . . . . . . 305 w 4. Application to the motion of a system of material bodies . . . . : �9 �9 �9 307 w 5. Remark on irrotational motions . . . . . . . . . . . . . . . . . . . 308

w 1..Rigid motions Denote by

(~.t) ds~ = h,~n(x) , l ~ , Ix ~

the square of the differential element of distance ds in the four-dimensional continuum or Einstein-Rieman.n space E, where the indices A and B are summed over the range 0, 1, 2, 3. We assume that the signature of the form (t.t) is such that it becomes the usual expression for-the element of distance in the special theory of relativity at an arbitrary point P of the continuum as the result of a suitable coordinate transformation, i.e. that

(t.2) ds~ = c~ d t ~ - ~,id z' dzi,

where the constant c is the velocity of light in vacuo, the ~ j are the Kronecker deltas, and there is a summation on the indices i, j over the values t, 2, 3. As indicated in the above equations (t.t) and 0.2), we shall use capital Roman indices for indices having the range 0, t, 2, 3, while lower case Roman indices will be restricted to the range t, 2, 3- I t will suffice for the analytical require- ments of this note for the coefficients has in (t.t). i.e. the components of the metric tensor h of E, to be continuous and to have continuous first partial derivatives with respect to the coordinates x a of the space.

The differential ds has a dual interpretation. I t is considered, as stated above, to be the element of distance in the space E, and it is also assigned the meaning of an invaria~ time differential. O n the basis of this latter viewpoint the velocity vector W of points of E has components "W ~ given by

dx,4 (t.3) WA = -d~--

I t follows, as an immediate consequence of (tA), that

0 .4 ) hart W ~ W n = t ,

302 T . Y . TI~OMAS :

i.e., the velocity vector W is a unit time-like vector. We assume that the com- ponents W A can be represented by continuous and differentiable functions of the coordinates x ~ x 1, x 2, x 3 in the region R ( E under consideration.

Now consider two nearby events or points P~ and P2 of R; i.e., the coordinate differences d x a of the points P1 and P2 are arbitrarily small, having the distance d s = 0 ; such points P~ and P, may be called equivalent or contemporaneous in the sense that they can be regarded as the limits of points whose proper (eigen) time interval ds approaches zero. Thus the coordinate differences dx a of the points P1 and P2 satisfy the condition

(1.5) hAB d x A d x B = O,

in which the coefficients hA B c a n be taken to be the components of the metric tensor at the point P1. Also

~haB(P~) dx c, ha B (P~) = ha B (P~) + 0 . c

to within terms of the first order in the d x "a, where the has (P1) and hA B (P~) are the values of the components h A B at P1 and P, respectively. Hence, to within terms of the second order in the quantities d x a, we can write

h a , (P2) d x a d x" = 0,

corresponding to the equation (t. 5). This use of differentials or "small" quantities enables us to express, to the required degree of approximation, the local con- ditions with which we shall be concerned in what appears to be a more intuitive and natural manner than would otherwise be possible.

Denote by L t and L 2 the world lines of the points P1 and P,, and let us refer these lines to the arc length s as parameter. Then the point P1 will be displaced along L 1 in the small time interval 8s into a point PI' with coordinates

(t.6) x A + W A ~s,

provided P~ has the coordinates x a, and provided W a are the velocity components at Px. Similarly the point P, will have the coordinates x a + d x A and the velocity

components W A + WA, B d x s,

where the comma denotes partial differentiation; also, in the same time interval 8s, the point P, will be displaced into the point P~' on Lz with coordinates

(t.7) (x a + dx a) + (W a + W a B d x B) ~s.

Subtracting the expressions (1.6) and (t.7), we obtain the differences in the coordinates of the points Px' and P~, namely

dxa + W a B d x S ~s.

In accordance with (tA) the square of the distance between the points P~ and P~ is therefore given by

(t.8) h 'C l) ( d xC + WC a d xa ,~ s ) ( d x~ + W , d ~ ,~ s ) ,

wl~ere hcD are the components of the metric tensor at the point Px'.

Rigidity in relativity 303

We shall say thai the motion defined by the vdocity vector W in R is locally rigid, or simply rigid /or brevity, i/ the distance between every two nearby con- temporaneous points Pt and P2 in R remains unchanged as the result o/ a small displacement ~s o[ the points along their world lines. In conformity with our approximation procedure, involving the use of differentials, this statement implies that the motion in R will be rigid if the square of the distance between the above points Px' and P,' into which the points Px and P~ are displaced in the small time interval Os, will be the same as the square of the distance between the points Pt and P,, i.e., will be zero, to within terms of the third order in the quantities dx a and Os. The term rigid body or rigid medium will be used for convenience in referring to a region R which is assumed to be in a state of rigid motion.

w 2. Conditions on the velocity vector for a motion to be rigid

Suppose, for simplicity, that coordinates are chosen such that h a B,c vanishes at the point P1- Then the quantities ]~D in (t.8) have the values hcD of the components of the metric tensor at P~ if we neglect terms of the second and higher order in r hence (1.8) becomes

~(2.t) h A s d ~ d x S + (Wa,B + WB, A dx a dx 8 0s,

to within third-order terms in the quantities dx a and 6s, where the W A and W n which appear in this expression are the covariant components of the velocity vector at P~. But the first set of terms in (2.t) vanishes by (t.5), and the second set of terms gives the required expression .for the change in the square of the distance between the two points Px and P~ in the time interval ~s; equating this expression to zero and taking cognizance of the fact that there is no difference between partial and covariant differentiation at Px due to the vanishing of the derivatives ha n,c at this point, we obtain

(2.2) (WA;~ + rvB;a) d . ~ dx e = 0,

where the semicolon is used to denote covariant differentiation. Thus the motion determined by a velocity [idd W will be rigid q (2.2) holds at arbitcary points P o/the/ield/or all variations dx a such that (t.5) is sa~isl~ied.

Tlie symmetric tensor e having the components

ca8 = ~ fcG;~ + WB;A

will be called the rate o/ strain tensor in, conformity with the terminology in ordinary continuum mechanics. By use of coordinates such t h a t (tA) has the form (t.2) at a n arbitrary point P of.the velocity field, the required conditions (2.2) for rigid motion can be written

(2.3) eoodtl+ 2~oldx~ dt + eiidxi d xi = 0

for variations dr, dx i such that

(2.4) ct d t~ = (d X x)" + (d x t ) ~ + (d x a) t.

But we must also have

(2.5) eoodtZ" 2e~ d t + eiidxi dx i= 9'

304 T .Y. THOMAS:

since we can obviously replace dr, dx ~ b y dr,, - - d x ~ in (2.3). Subtracting corre- sponding members of (2.3) and (2.5), we obtain

(2.6) ~o~ dx~ dt -~ O.

Hence eo~ must vanish at the point P, relative to the special coordinate system employed, since the variations dx ~ can be chosen arbitrarily subject to the con- dition dt=~ O. The conditions (2.3) therefore reduce to

8O0 (2.7) eoodta-~ - ~ i j d x ' d ~ = (--~- ~,j-~- ~ij)dx 'dx~=O,

on account of (2.4). But (2.7) implies tha t

800 ~i" (2"8) ~ ' J = - - c '

The above results can be expressed by writing

s hAB J (2.9) eAB---- c'

when the form 0.2) for the element of distance is applicable. In fact if A, B = i , j the relations (2.9) become (2.8) and if A, B = 0 , j these relations give eo j=0; finally, if A, B----0, 0 it follows tha t (2.9) is satisfied identically. I t is now clear tha t we must have

(2.t0) eAB=~hA~

in an arbi t rary coordinate system, where the quant i ty ff is a scalar. This follows from the invariant character of the relations (2.t0) and the fact that they have been shown to hold for ~----eoe/C z relative to the above special coordinate system.

By covariant differentiation of (t.4) we see that

(2.tt) W'~ W~;B=O.

Hence if we multiply (2.t0) by W a and WB, sum on the repeated indices, and then make use of (t .4) and (2.tt), we find tha t the scalar ~0 must be equal to zero. The following result has now been proved. The comtitiw,r for a vdocity/ie2d W to produce a rigid motion is the, vanishing o/ i ts rate o/strain tensor L

x The derivation ot.this condition for a velocity field W to determine a rigid mot ion is 'not unrelated to the usual derivation of Killing's equations (3.t2). See, for example, SCHOUTES~ J. ~., Rioci-Calculus, 2rid Ed.i Springer, Berlin, p. 548 (t954); also, .TRUESDELL, C., & TOUPIN, R., The Classical Field Theories, Encyclopedia of Physics, Vol. I I I / t , Principles of Classical Mechanics and Field Theory, Springer, Berlin, p. 350, (t960). In this article, however, we have limited our attention to the motion of contemporaneous events or points, i.e., points subject to the invariant Condition ds=O, which, in a sense, is the analogue of the time condition dt=O that is naturally assumed in treating the corresponding problem in classical mechanics. This has resulted in the derivation of the equations (3.t2) for rigid motion in the space-time continuum ,ruder weaker conditions than those imposed in the above references. I t is obviouS, of course, that the procedure depends on the indefinite metric form of the space-time continuum ~nd is not applicable in the case of the strict Riemann space for which the roetric form is positive definite.

Rigidity in relativity 305

w 3. Newtordan approximation of the rigidity conditions Let us now assume that the Einstein-Riemann space E reduces to the space

of the special theory of relativity with the form for ds s given by (t.2) in which t has the ordinary meaning of time and the x i are the coordinates of a rectangular system. If the velocity components v ~ are defined by dx~/dt, as customary, the relation between these components and the above components W a and W A can readily be deduced. Thus we have

d t I 1 (3:1) We = ~ - =

d s / d t ca~e~i-Z~_w z '

(3.2) W i = d # d t __ v g W e _ v ~ dt ds c,~i-&~--~,'

in which we have used w to denote the magnitude of the velocity v in the rec- tangular system. Also the components W a are given by

(~.9) Wo = koB W n = hoo W e - - c"

(3.4) W~ = h~n W n = - ~ i W~ - -v~ 1 6 ' - w ' "

There is, of course, no distinction between the contravariant components v ~ in (3.2) and the covariant components vi in (3.4) since rectangular coordinates are employed. Substituting the values of the components W~ given by (3.3) and (3.4) in the equations expressing the .vanishing of the rate of strain tensor ~, we obtain the following conditions for rigid motion, namely,

t t (5.5) v ~ d + t'i'~-~ 2 t - - w ' / c ' [V~(to'/Cl)'i+ V'(to2/C~)'~] = O,

Ov~ ~ ~ t v~ O(wS/ct) -- O, (3.6) a t - ~O-w'/c*) + 2 s - w , / c , o t

(3.7) ew , _ o, Ot

in which it is assumed that tot< c I. If tos and its derivatives can be neglected in comparison with c ~, the equations (3-5) and (3.6) can be replaced by

(3.8) v~,~+ v~,~ = o,

0vi I (3-9) o t - 2 to"~"

The system consisting of (3-7)~ (3-8) and (3.9) may be referred to as the Newtonian approximation to the conditions for rigid motion.

I t is observed immediately that the equations (3.8) give tim usual conditions for rigid motion in a system o f rectangular coordinates. To see the meaning

o f the remaining eonclitions (3-7) and (%9) let us first multiply (3.9) by v i and sum on the repeaied index i; this gives

Ovi I (3.t0) v~ a t = -~ (v~'i+ v,,~) v~v i,

Arch. Rational bleeh. Anal., Vol. 9 21

306 T.Y. T~OMAS:

as follows readily when we replace w" by its value in terms of the velocity components. But the fight member of (3.t0) vanishes on account of (3.8) and hence (3AO) is seen to yield the condition "(3.7). Intltoducing the components dvddt of the acceleration, we find that the equations (3.9) can be given the form

dvi (3.it) dt -- (vi'i + vi'i) vi = O,

in consequence of which the acceleration must vanish. Hence the Newtonian approximation to the conditions for rigid ~notion in the Einstein-Riemann space E yields the usual kinematical conditions (3.8) for rigid motion in Euclidean space and the Newtonian equations (3.11)/or the motion of a material medium not subiect to the action of forces.

The" above consideration involving the evaluation of the conditions for rigid motion in the continuum of the special theory of relativity eliminates, of course, the effect of gravitational forces on the material in the flow region; but also the effect of possible electromagnetic forces will be eliminated if one assumes that the conditions for rigid motion, i.e., the conditions

(3.t2) wa;B + WS;A = O,

are associated with a unified theory of relativity in which the existence o f both gravitational and electromagnetic forces depends essentially on the curvature of the space-time continuum. Moreover one must expect that internal forces, e.g. the forces determined by the stress tensor in the Newtonian theory of the mechanics of continuous media, will be lacking, since their occurrence would produce deformations in the medium in contradiction with the underlying as- sumption of rigidity. In the absence of such forces it would appear that an acceleration free motion of the medium is approximately valid for small velocities as represented in fact by the above equations (3.tt). We are thus led to conclude that the equations (3A2) furnish suitable conditions /or rigid flow in a purely gravitaional theory, or in a combined theory of gravitation and dectromagnetism, such that all forces which can act on the medium result from the structure of the space-time continuum and are annulled, as they' should be, on passage to the flat space of the special theory of relativity. This point of view is confirmed .by tile results in the following section.

To avoid possible misunderstanding,, we wish to.emphasize that the conditions (3.12) constitute, in no sense, a solution of the general kinematical problem of the corfstruction of invariant conditions for rigid flow in the four-dimensional space-time continuum, which has been treated by various authors z. In fact it is easily seen from the special character of the conditions (3.t2) that the world lines of particles in the flow region are geodesics, as shown in the next section; while this appears as a restriction on the generality of the rigidity conditions, it is not an objection in itself but rather a valuable adjunct in the solution of certain, problems.

t See J . L. S'r : Relat ivi ty: The General Theory, North-Holland Publishing CO., Amsterdam; and .Interscience Publishers, Inc., New York, 1960, p. 173, where several other references may be found. Also, R. A. TouvxN: World invariant kinematics. Archive for Rational Mech. and Anal. 1, 181--21t (1958).

Rigidity in relativity 307

w 4. Application to the motion of a system of material bodies As a particular application of the above results let us consider a system of

material bodies which are free to move under their mutual gravitational attrac- tions. We assume these bodies to be rigid as defined in Section t ; it may further- more be assumed that the bodies move in ]ree space, i.e., that the m o m e n t u m - energy tensor vanishes in the region outside the bodies, although this restriction is not necessary in the following discussion. Within the material bodies the conditions (3.t2) will now be satisfied. But multiplying (3A2) by W e, summing on the repeated index B, and making use of (2.it), we obtain

(4.t) W4. B W e_= DWA = O, ' Ds

where the symbol D denotes absolute differentiation along the world lines of the material particles within th~ bodies when these lines are referred to the arc length s as a parameter. Expanding (4.t) and introducing the contravariant components W a of the velocity vector W, we have

d2 xA dab dxC -- O, (4.2) ds' + A~c ds ds

in which the quantities A a r e t h e components of the affine connection; the equations (4.2) are the well known equat ions for the geodesics of an Einstein- Riemann space. The following result can now be stated. The world lines o] the material particles in a system o/rigid bodies moving under their mutual gravi-

rational attractions are geodesics in the Einstein-Rieraann space E.

The locus of the particles comprising the surface of any one of the above material bodies is a three-dimensional surface 27 in the four-dimensional space E. Now from a consideration of the discontinuities 3 in the components of the metric tensor h and the momentum-energy tensor T it would appear that the components of k may be assumed to be continuous and to have continuous f i .~ partial derivatives with respect to the coordinates x a over the surface 27 but that dis- continuities in the second derivatives of these components will occur on 27. On the basis of this assumption the quantities A in (4.2) will be continuous across X, and hence we can state the following result. The wor/d lines o] the particles on the sur[ace o[ any body o] the above system o[ material bodies can be regarded as geodesics in the region o] the space E exterior to the bodies. These world fines, of course, will form a congruence of curves on the three-dimensional surface 27.

I t is clear that the equations (4.1) and (4.2) are valid along the world lines of particles in any rigid medium (see italicized statement at end of Section 3), and the above restriction to purely gravitational phenomena was imposed largely because of the historical interest in this problem. Deviations of the world lines of particles fr6m geodesics will be possible only if t he conditions of rigidity (3.t2) are not satisfied, i.e., when deformations occur within the medium.

s EDELEN, D. G. B., & T. Y. THOMAS: Discontinuities in the Einstein-field fo r general momentum-energy tensors. Archive for Rational Mech. and Anal. 9, 153-- t 71 (t962); also Differential compatibi l i ty condi*dons on the momentum-energy tensor and necessary conditions for the existence of solutions to the Einstein field equations. Archive for Rational Mech. and Anal. 9, 245--254 (t962).

21"

30~. T .Y. THOMAS: Rigidity in relativity

w 5. Remark on irrotational motions

Defining the skew-symmetric tensor tp with components ~0~ B by the equations

(5.t~ vaB = �89 (w~;~ - ws;A),

let us say tha t a region R of the Einstein-Riemann space E is in irrotational motion if the tensor ~0 vanishes in R; more precisely we assume that the vanishing of the tensor ~o gives the conditions for irrotational motion subject to the limi- tat ions mentioned in the italicized statement at the end of Section 3 for rigid flow. If we mult iply (5.t) by W s and sum on the index B, we again obtain the equations (4.1) and (4.2), from which it follows that the world lines of the particles of a region in irrotational motion will be geodesics. Although such irrotational

.motion may cause the deformation of a material medium, the restrictions imposed by the vanishing of the tensor ~o are, nevertheless, sufficient to insure the geodesic character of the world lines.

Prepared under Office of Naval Research Contract Nonr-908(09), Indiana Uni- versity NR 04t 037.

Graduate Institute for Mathematics and Mechanics Indiana University, Bloomington, Indiana

(Received October 17, 1961)