DAVID B. MALAMENT

ON RELATIVE ORBITAL ROTATIONIN RELATIVITY THEORY*

I want to consider this question within the framework of relativity theory: given twopoint particles X and Y, if Y is rotat ing relative to X , does it follow that X is rotatingrelative to Y? To keep the discussion as simple as possible, I'll allow X and Y to be testparticles .

As it stands, the question is ambiguous. Roughly speaking, one wants to say that"Y is rotating relative to (or around) X;" at least in the sense I have in mind, if "thedirection of Y relative to X" is "changing over time ." What must be explained is howto understand the quoted expressions. There is a perfectly straightforward way to doso within Newtonian particle mechanics (section 1), where there is an invariantnotion of "time," and "space" is assumed to have Euclidean stru cture. At all times,there is a well-defined vector that points from X to Y, and one can use it to define theangular velocity ofY relative to X.

But the situation is more delicate in relativity theory. Here no such simple inter­pretation of "relative rotation" is available, and some work is required to make senseof the notion at all. (It seems to me unfortunate that this is often overlooked by partieson both sides when it is debated whether relativity theory supports a "relativist" con­ception ofrotation.) In section 2, I'll consider one way ofdefining the "angular veloc­ity of Y relative to X" (Rosquist 1980) that does not presuppose the presence ofspecial background spacetime structure (e.g., flatness , asymptotic flatness, stationar­ity, rotational symmetry), and can be explained in terms of simple (idealized) experi­mental procedures. I'll also derive an expression for the angular velocity of Y relativeto X in the special case where the worldlines ofX and Yare (the images of) integralcurves of a common background Killing field . Finally, in section 3, I'll tum to theoriginal question.

SECTION 1

For purposes of motivation, let us first consider relative rotation within the frame­work of Newtonian particle mechanics. Here we can associate with the particles, atevery time t , a relative position vector 1x y(t) that gives the position of Y relativeto X. (We can think of the vector as having its tail coincident with X and its headcoincident with Y.) The inverted vector, 1 yx(t) = - 1x Y( t ) gives the position of X

175

A. Ashtekar et al. (eds.), Revisiting the Foundations ofRelativistic Physics, 175-190.© 2003 Kluwer Academic Publishers.

176 DAVID B. MALAMENT

relative to Y at time t. Let us take for granted that the particles never collide (so that~r xy(t ) is non-zero at all times), and consider the normalized vector:

ItXy(t )~r xy(t)

If xy(d '

We can think of it as giving the direction ofY relative to X at time t. The (instanta­neous) angular velocity ofY relative to X at time t is given by the vector cross prod­uct:

-7 -7 d-7QXy(t) = n Xy( t ) x d/n xy(t )).

Notice that it is not here presupposed that X is in a state ofuniform recti linear motion.X (and Ytoo) can w~gle so long as ItXy( t ) has a well-defined derivative. Notice alsothat if Ityx(t ) and Q yx(t) are defined in the obvious way, by interchanging the roles

-7 -7 -7 -7of X and Y, then n yx(t ) = - n Xy(t ) and Q yx(t) = QXy(t) . We will be interested

in two assertions.(i) Y is not rotating relative to X

~ d -7QXy(t) = 0 (or, equivalently, d/n x y(t )) = 0) for all t .

(ii) Y is rotating relative to X with constant angular velocity (i.e., in a fixed planewith constant angular speed):

d ~d/QXy(t )) = 0 for all t .

What is important for present purposes is that both assertions are manifestly sym­metric in X and y'l It is the purp ose of the present modest note to show that the situa­tion changes, and changes radically, when one passes to the context of genera lrelativity. We show with an example in section 3 that there it is possible for Y to benon-rotating relative to X; and yet for X to be rotating relative to Y with constant(non-zero) angular velocity. Moreover, the X and Y in question can be chosen so thatthe distance between them is constant (according to any reasonable standard of dis­tance) . And the distance can be arbitrari ly small. (Of course, it remains to explain theinterpretation of relative orbital rotation in genera l relativi ty on which these claimsrest .)!

ON RELATIVE ORBITAL ROTATION IN RELATIVITY THEORY 177

SECTION 2

Let us now tum to the relativity theory. In what follows, let (M , gab) be a relativisticspacetime structure, i.e., a pair consisting of a smooth, connected 4- manifold M , anda smooth semi-Riemannian metric gab on M of Lorentz signature (+ I , - I, -I , -1 ).3Let yx and y y be smooth, non-intersecting timelike curves in M representing,respectively, the world lines of X and Y. (We will not always bother to distinguishbetween the curves and their images.) We will follow Rosquist (1980) , and define ateach point on yx a vector Q a that may be interpreted as the "instantaneous (appar­ent) angular velocity ofY relative to X."4

Imagine that an observer sitting on particle X observes particle Y through a tubulartelescope. We can take the orientation of his telescope at a given moment to deter­mine the "(apparent) direction of Y relative to X' at that moment; and we can repre­sent the latter as a unit vector, orthogonal to yx . In this way, we pass from the curvesyx and y y to a (normalized, orthogonal) direction field v a on yx' Once we have thefield v a in hand , we are almost done . We can~hen define Q a in terms of v" in closeanalogy to the way we previously defined QXY in terms of Ii xy . We need onlyreplace the "time derivative" of Ii X y with the Fermi derivative of v" along yx . Theconstruction is shown in fig. I.

wo rld line of X

Figure 1.

world line of Y

/ca (tangent to null geodesic)

Let Sa be the four-velocity of X, i.e., a future-directed. ' timelike vector field onyx' normalized so that SaSa = I . We assume that given any point p on yx' there is(up to reparametrization) a unique future-dire cted null geodesic that starts at some

178 D AVID B. MALAMENT

point (or other) on y y and ends at p . This amounts to assum ing that X can always seey, and never sees multiple images of y'6 Let Aa be the (future directed, null ) tangentfield to this geo desic (given some choice of parametr ization). We arr ive at the direc­tion vector v " (ofY relative to A) at p by starting with -Aa at tha t poin t, then project­ing it orthogonal to Sa and finally normalizing the resultant vector:

(Equiva lently, v" is the unique vector at p in the two-plane spanned by Sa and Aasuch that vaSa = 0, y aya = - I , and yaAa> 0. ) The Fermi derivative of v" in thedirection Sa,

( a )'a)' )'''V 11/g 11/ - '" "'11/)'" "y ,

is just the component of the directional derivative s"V"ya orthogona l to Sa, i.e., thespatial component of the derivative as detennined relative to X . We arrive at theangular velocity n a

of Y with respect to X at each point on yx ' in effect, by takingthe cross product there of ya with (ga11/ - SaS11/) S"V"y11/ in the three-plane orthogo­nal to Sa:

In ana logy to the conditions form ulated in section I , we say

(i ') Y is not rotating relative to X if

n a = 0 (or, equivalentl y,' (ga11/ - SaS11/)S"V"y11/ = 0) at all points on yx ;

(in Y is rotating relative to X with constant angular velocity if

These conditions have a natural physical interpretation . Consider again ourobserver sitt ing on particle X and observing Y through his tubul ar tele scope. Condi­tion (i ') holds iff the orientation of his telescope is constant as determined relative tothe "compass of inertia." So, for example, we might position three gyroscopes at Xso that their axes are mutually orthogonal." The orientation of the telescope tube atany moment can then be fully speci fied by the angl es form ed between each of thethree axes and the tube. Condition (i ") cap tures the requirement that the thre e anglesremai n constant. Condition (ii) captures the requirement that the three gyroscopescan be positioned so that the telescope tube is at all times orthogonal to one of thethree, and its angles relative to the other two assume the characteristic, sinuso idal pat­tern ofun ifonn circular motion (with respec t to elapsed proper time).

ON RELATIVE ORBITAL ROTATION IN RELATIVITY THEORY 179

We now consider the special case where yx and y y are integral curves of a back­ground future-directed, timelike Killing field 1:a . In this case , there is a strong sensein which the particles X and Y remain a constant distance apart .? To match our nota­tion above !/ 2we express r " in the form 1:a = 1:~a , with ~a~a = 1 and1: = (1:'\,) . Associated with ~a is a vorticity (or twist) vector field

a 1 abcdr- n r-(0 = 2£ '-Jb v c'-Jd ·

We want to derive an expression for Q a in terms of (Oa. To do so, we direct attentionto the one-parameter group of local isometries {rs } associated with r ", i.e., the"flow maps" of which 1:

ais the "infinitesimal generator." Given anyone null geode­

sic segment running from y y to Yx ' it 's image under each map F, is another nullgeodesic segment running from y y to yx . (This follows immediately. Since y y andyx are integral curves of t ", each is mapped onto itself by r s . Since T, is an isom­etry, it preserves all structures that can be characterized in terms of the metric gab'

and that includes the class of null geodesics.) The collection of maps {rs } in itsentirety, acting on the null geodesic segment, sweeps out a two-dimensional subman­ifold S, bounded by y y and yx' through every point of which there passes a(unique) integral curve of 1:

aand a (unique) null geode sic segment running from 't r

and yx (see figure 2).

Yx

timeli ke Killing field , a - ---- ---

null geod esic field "a ------------

Figure 2.

Yy

180 DAVID B. MALAMENT

Thus, we have on S two fields tangent to S: the timelike Killing field r" , and afuture-directed null geodesic field "AaC'A"VI"Aa= 0 and "A""A" = 0) that is preservedby each map r s ' or, equivalently, that is Lie derived by the Killing field 'C

a, i.e.,

With this equation in hand , it is a matter of routine computation to derive an expres­sion for Qa in terms of (j)a .

Proposition: 10 Let S, t", and "Aa be as in the preceding paragraph (and let v", "Aa,

and (j)a be the corresponding fields on S, as defined earlier in this section). Then, atall points on S,

(We have formulated the proposition in terms of the relative velocity of Y withrespect to X. But, of course, a corresponding statement holds if the roles of X and Yare reversed. One just has to remember that the reversal brings with it a different two­dimensional submanifold S and a different null field "Aa.)

Proof Since "Aa, gab (and r") are Lie derived by the Killing field t" , so are all fields

definable in terms of them . In particular, V b is Lie derived by 'Ca . Thus ,

O = lin m _ lin m = (r- II)n m _ lin ( r-m)'C v IIV V v II'C 'C.., v IIV V V II 'C..,

= (r- II)n m _ lin r-m _ r-m lin'C.., v IIV 'CV v fl'" .., V v II'C'

So

r-"n m = lin r-Ill+ ( - I )r-m lin.., v flV V v fl'" 'C.., V v II'C

and, hence,

The final equality follows from the fact that eabcd;b;d = 0 (since eabcd

is anti-sym­metric in the indices ' b' and ' d ' ). To proceed further, we use the following expres-

ON R ELATIVE ORBITAL ROTATION IN R ELATIVITY THEORY 181

sion for 'VI/S d that holds for any unit timelike field S a proportional to a Killing field!' :

Direct substitution yields:

Qa _ abcdr- 1/ ( r-P r + r- r-Ill n r- _ abcdr- 1/ r-P r- - c ":J bYcY c l/dp r":J ffi ":J I/":J Y m":Jd) - - c ":J bYcY cl/dpr ":J ffi

dabc r- I/ r-P r o;: a o;: b o;: C r- I/r- P r= - e c dllpr":JbY cY":J ffi = 6u [IIU pU r j ":JbY cY ":J co

= 6SbY cyl asbffici = SbY c( y aS bffic . . . _ y CSbffia + ...)

= (Ycffic ) ya + ffia .

(The second equality follows from the fact that V"S" = 0; the fourth from the factthat c dabcCdllPr = _60

a[lI() bp()c r ] : For the latter, see (Wald 1984,432).) .

We claimed above that relativity theory allows for the possibility that there be twopoint particles X and Y , a constant distance apart, such that Y is non-rotating rela­tive to X , but X is rotat ing relative to Y with constant (non-zero) angular velocity.Our strategy for producing an example in section 3 is this. We exhibit a spacetimewith a future-directed, timelike Killin g field 'ta = 'tSa

, and two integra l curves of thefield , yx and yy, such that the following conditions hold .

(a) ffia = 0 on Yx

(b) a 0 b (a r-ar- )r-I/ n III 0ffi:t on y y, ut g m - ":J ":Jm ":J Y IIffi = on Yr-(c) Whether working from y y to Yx, or from Yx to yy, the associated future­directed null geodesic field 'Aa that is Lie derived by 't a (as in the constructionabove) is everywhere orthogonal to ffi a .

Thi s will suffice. Consider the condition in (c). If the connecting null field 'Aais

orthogonal to ffi a , then the direction field v" induced by 'Aais also orthogonal to

ffi a :

So, by the proposition, Q a( y wr t X) = ffia on 'Ix - and Q a(X wr t Y) = ffia on 'l r ­So by (a) and (b) ,

Q a( y wrt Xl = 0 on 'tx - whil e

Q a 0 b (a r-a r- ) r- ll n QIIl 0(X wr t Y) :t on y y, ut g III - ":J ":Jm ":J Y II (X wrt Y) = on y y,

as desired.

182 DAVID B. MALAMENT

SECTION 3

The example we present in this section is a bit artificial. But it does have the virtue ofsimplicity. It will be relatively easy to identify the necessary elements of structure­the time like Killing field r", and the integral curves yx and y y - and verify that theysatisfy conditions (a)-(c). Given how very stringent the conditions are , it is of someinterest, perhaps, to have any simple example at all.

In constructing the example, we start with Godel spacetime (M, gab) in itsentirety and then, at a certain point, shift attention to a restricted model of form(0, gabIO) , where 0 is an open subset of M . The restricted spacetime is, in somerespects, much better behaved than the original. In particular, it does not admit closedtimelike curves. Indeed, it satisfies the stable causality condition. (But , unlike theoriginal, of course, it is extendible.)

In what follows , we take Godel spacetime'? to be the pair (M, gab)' where M isthe manifold R 4

and gab is characterized by the condition that given any point p inM , there is a global (adapted) cylindrical coordinate system t, r, <p, y on M suchthat t(p) = rep) = y(p) = a and

(We use "s h r' and "ch r' to stand for 'sinh r ' and ' cosh r ' .)Here -oo <t<oo ,O~r<oo, -oo<y<oo , and O~<p<oo with <p = a identified with <p = 21t; Il isan arbitrary positive constant. (We will assume a point p has been chosen, once andfor all, and work with the corresponding coordinate system.) The metric gab is asolution to Einstein's equation

for a perfect fluid source

1 2 - 1with four-velocity 11 a = (21lf (d /dt)a, mass density p (l61tGIl) , and iso-

2 - Itropic pressure p = (161tGIl) .

The field (d /dt/ is everywhere timelike, and defines a temporal orientation on(M, gab). The integral curves of (d /dt)a will be called "matter lines" (since thefour-velocity 11a of the fluid source is everywhere proportional to (d /dt)a) .

In the appendix, we give an explicit expression for a volume element eabcdon

(M, gab) in terms ofcoordinates t, r, <p, y . It defines an orientation on (M, gab)·In Godel spacetime, (d /dt/, (d /dy)a , and (d /d<p)a are all Killing fields and so,

therefore, are all linear combinations of these fields . We will be interested, specifically,

O N R ELATIVE O RBITAL ROTATION IN R ELATIVITY T HEORY 183

in the field

Since

° 2 4 2 2't 't o = 411 [I + 2(sh r - sh 1')+ 4sh 1' ]

2 2 2411 [I + 2(sh r )(ch r )J,

it follows immediately tha t

(I ) 'to is everywhere timelike,

It is also clear that

(2) the coordinate functi ons I' and y are constant on all integral curves of 't o . 13

If the cons tant value of I' is 0, the integral curve is a matter line (since (d/dcp)o = °where I' = 0) , characterized by its y value. We call it an "axis curve." If the con­stant value of I' is stric tly positive, we can picture it as a helix that wrap s around anaxis curve (the one with the same y value)."

If, as above, we express 't o in the form 't o = 't~o , with ~o~o = I , the vorticityfield asso ciated with ~o comes out to be:

r; 2 2(3) 0)0 = 22.; 2(sh r);ch 1')2 (d /dyt

(4 11 ) [ 1+ 2(sh r)( ch 1' )]

(The computation requires just a bit of work. We present it in the appendix.) It fol­lows immediately that

(4) 0)0 = °¢:::> I' = 0. 15

It also follows that

(5) ( ° ~ o~ )~I/ n 111 ° hg 111 - '" "'111 '" V I/O) = everyw ere.

In fact, the stronger condition ~I/VI/O)° = °holds everywhere. This follows because~I/ VI/r = 0 , by (2) above, and (d /dY)o is covariantly constant, i.e., V,ld/dy t . 16

So

r;:., 2 22.;2 (sh r)(ch 1') ~I/ V (d /d )02 2 2 '" 1/ Y(4 11 )[ 1 + 2(sh r )(ch 1')]

0.

We are now well on our way. If we take yx to be any integral curve of 't o withI' = 0 , and 't r to be any one with r :;t: 0 , conditions (a) and (b) listed at the end ofsec tion 2 will be automatically satisfied. So it only rem ains to consider condition (c).

To satisfy the orthogonali ty constraint in (c), we need to further restrict the choiceof yx and y y so that the y coordinate function has the same (constant) value on bothcurves. Let 1'0 be any positive real number and let Yo be any real whatsoever. Let yxbe an integral curve of 'to with constant values I' = 0, Y = Yo' and let y y be onewith constant values I' = 1'0' Y = Yo' The following conditional claim about null

184 D AVID B. M ALA MENT

geodesics follows easily.(6) ({there exists a null geodesic that intersects both 't v and Yx, and if ,,0is thetangent field to the curve, then " a roo = 0 at all points on the curve.

For assume there is such a curve o with tangent field ,,0. Since ,,0 is a geodesicfield , we have ""V,l,,o Ko) = 0 for all Killing fields KO. 17 In particular, taking K O tobe (o loyt, "o(oloy)o is constant on cr. But " o(oloy)o = - 4 /-1.2(" oV aY). So" oV aY is constant on cr. If the constant value of this function were not 0, the value ofthe coordinate y would have to increase or decrease along o - contradicting the factthat the initial and final points share the value Yo. So it must be the case that"o(oloy)o is 0 at all points on cr. But , by (3), roO is everywhere prop ortional to(o loy)o. So "o roo = 0 at all points on cr.

Now it only remains for us to consider the existence and uniqu eness of null geo­desics running betwe en yy and yx . But here, for the first time , things get sticky. Wewant to be able to assert that an observer on one of the particl es will see the other atall times , but not see it in more than one position on the celestial sphere. It is a curi­ous fact about null geodesics in Godel spacetime that this will simply not be the case,in general. It turns out that if s h ro> 1 (i.e., if r0> In( 1 + J2 )), the observer willnot see the other part icle at all. And if s h r0 ~ 1, he will, in genera l, see multipleimages of the other. Roughly speaking, this results from the fact that photons act likeboomerangs in Godel space time . Any future or past directed null geodesic that start sat a point on yx moves outward (with monotonically increasing r value) until itreaches the criti cal radiu s rc = ln(l + J2 ), and then moves inward (with monotoni­cally decreasing r value) until it hits yx again; and then the process starts all over,"So, it can happen, for example, that two past-d irected null geodesics start out in dif­ferent directions from a point on yx ' and both intersect y y, though at differentpoints. One hits y y on the way out. The other hits it on the return trip in.

To avoid this complication, we now impose the requi rement that ro < rc ' restrictattention to the open subse t

0 = {qE M : r (q ) <rc },

and consider (0, goblO) as a space time model in its own right (with the temporal ori­entation and orientation inherited from the original).19 Then we can make the desiredexistence and uniqueness claim concerning null geode sics.

(7) Given any point qx on yx' there is a unique point q y on y y such that thereexists a future directed null geodesic running from q y to qx ; and symmetrically,with the roles of X and Y reversed.

That this is true follow s alone from the qualitative description of past and futuredirected null geode sics just given (the boomerang effect). We sketch the proof in theappendix.

This puts all the needed pieces of the example together. We now revert to the dis­cuss ion at the end of sec tion 2.

ON RELATIVE ORBITAL ROTATION IN RELATIVITY THEORY 185

APPENDIX: NEEDED FACTS ABOUT GODEL SPACETIME

(A) Derivation offormula (3) in section 3

Let k be any real number and let 'Ca be the Killing field (d ldt)a + k(d ld<p)a. Ifwerestrict attention to the (open) region where it is timelike, we can express 'C

ain the

form 'Ca = 'C~a , with ~a~a = 1. We claim that the vorticity associated with ~a (inthis region) is given by

r; 2 2 r; 4£.,a = ",2 +k(2sh r -l) +k ",2sh r (:l/:I)aUI 2 r;:; 2 2 4 2 0 oy .

(41l )[ 1 + k2 '"2s h r + k (s h r - sh r)]

If k = j2 , this reduces to (3).In the derivation, we use the following basic relations:

2 t: 2(d ldt)a = 41l «dt)a + ",2sh r(d<P)a)

2 t: 2 4 2(dld<P)a = 41l (",2sh r(dt)a+ (sh r -sh r)(d<P)a)

2 a . «: r;:;2 4 2'Ca = 41l «l +k",2sh r)(dt)a +(",2sh r +k(sh r- sh r))(d<P)a)

2 a 2 r;:; 2 242r = r 'ta = 41l (1 + k2",2sh r + k (sh r - sh r)),

and we work with the volume element defined" by

eabed = f(dldt)[a(d ldr)b(dld<p/(dldy)dl where f = 4 -4!(l61l )(sh r)(ch r)

° abedSmce e 'Cb'Cd = 0 , we have

So we start by deriving an expression for 'C[b V e'Cdlo First, note that since

V[e(dt)dl = 0 = V[e(d<P)dj '

2 r;:; 2 r;:; 2 42V[e'Cd1 = 41l ((dt)l dVd1 + k",2sh r) + (d<P)[dVd",2sh r + k(sh r -sh r)))

= (41l 2(k2 j2 (sh r)(ch r)(dt)[d(dr)eJ)+

2(sh r)( ch r)(j2 + k(2sh2r

- 1))d<pld(dr) el)'

186

Hence,

where

DAVID B. MALAMENT

t: r;;. 2 4 2K = - k2",2(s h r)(ch r)(",2sh r + k( sh r - s h z)

+ (I + kJ2sh2r

)2(sh r)(ch r)(J2 + k(2sh2r

- I»r; 2 2 t: 4= 2(sh r)(ch r)[",2 + k(2sh r - I) + k ",2 sh r].

So,

a I ab ed n 1 4 abe d d d d(0 = -2£ '!Ib v C'!dl = - 2(16/1 )K£ (t)lb( r) c( ep)dl

2'! 2'!

= ~(l6/14)K f (4 1r1(- (d/ dyt )

2'!t: 2 2 r; 4

",2 +k(2shr -l) +k",2shr ( .., / ..,)a2 r;;. 2 2 4 2 0 oy .

(4/1 )[1 + k 2",2sh r +k (s h r - sh r)]

(B) Proofsketch ofclaim (7) in sec tion 3

Let qx be any point on Yx and let ~a be any past-directed (non-zero) null vector atqx such that 'A"V"y = 0, Let (J be the (unique) inextendible, past-directed nullgeode sic starting at qx whose tangent at that point is ~a. Let its tangent field be 'Aa.

The r coordinate on (J starts at 0 and increases (monotonically) through all valuesless than rC' SO there is exactly one point q on (J whose r value is ro' 21 Let thecoordinates of q be (t , ro, ep' Yo ) ' (We know, from the discussion after (5), that'A/V,,y is constant on (J . Since it is 0 at qx , it must be 0 at all points. So the value ofthe Y coordinate must be Yo at all points on (J.) The point q need not fall on Yy.

We have so far considered just one inextendible, past-directed null geodesic start­ing at qx along which Y has the constant value Yo. But the entire class of these isgenerated by taking the image of (J under "rotations" of form

(t, r, rp, y ) -7 (t, r, ep + epo'y),

i.e., under isometries generated by the Killing field (d/dep t . One of these isometricimages of (J does intersect 'tv (since there is som e point q y on 't r and some eposuch that q y has coordinates ( r, r o' ep + ep 0 ' Yo»' The time reversed, i.e. , future­directed, version of this curve qualifie s as a null geodesic running from a point q y ony y to qx ' So we have established the existence claim in (7). And uniqueness follows

*

O N R ELATIVE ORBITAL ROTATION IN RELATIVITY THEORY 187

easily as well. Suppose (J J and (J2 are both past-directed null geodesics starting atqx that intersect y r- Then since both arise as images of (J under rotations of the sortjust described, and since these maps preserve the value of the coordinate t , the inter­section points share a common value of t . But there can be only one point on y yhaving any particular value of t. (This follows because y y is a future directed time­like curve, and (see note 19) the coordinate function t is strictly increasing on all suchcurves.)

The argument for the symmetr ic claim (with the roles ofX and Y interchanged) isvery much the same. But now, in addition to considering "ro tations" (as above), wealso consider " timelike translations" of form

(t, r , rp,y) ~ ( t + to' r , cp, y),

i.e., isometries generated by the Killing field (% t)o . Let q y be any point on Y v­Essentially the same argument as we have just considered shows that given any pointon yx' there is a unique point on y y such that there exists a f uture-directed null geo­desic running from the first point to the second. By moving to the image of this curveunder, first, a timelike translation and, then, a rotation, we arrive at a future-directednull geodesic (J that starts at a point qx on yx and ends at q y. This gives us exist­ence. For uniqueness, suppose there were a second point q' x on yx and a null geo­desic (J' running from q' x to qy. By first sliding (J' up or down so that q' x ismapped to qx' and then rotating it, we could generate a future-directed null geodesicthat starts at qx , but ends at a point on yy distinct from q y- and this we know isimpossible.

University ofCalifornia

NOTES

It is a pleasure to dedicate this paper to John Stachel, and thank him here for the encouragement andsupport he has given me over the years. (It is also a pleasure to thank Robert Geroch, Howard Stein,and Robert Wald for helpful comments on an earlier draft.)

I. It should be emphasized that this does not imply that all claims about "orbital rotation" are symmetricwithin the framework of Newtonian physics. For example, let X be a particle sitting at the center ofmass of the solar system. The earth and the sun both rotate relative to X (and relative to each other) inour sense; and X rotates relative to both the earth and the sun in that sense. But there is this asymm e­try between the motion of X on the one hand, and that of the earth and the sun on the other: X is non­acce lerating, while both the eart h and the sun have non-zero acceleration vectors that point toward X .This captures ail e sense in which one might say that the earth and sun are rotating around X , but notconversely.

2. The discussion to this point has been cast in terms of textbook Newtonian parti cle mechanics. It mightbe asked what, if anything, changes when one passes to the Cartan formulation of Newtonian theory inwhich gravity is treated as a manifestation of spacetime curvature. (Rather than thinking of point par­ticles as being deflected from their natural straight trajectories by the presence of a gravitational poten­tial, one thinks of them as traversing the geodesics ofa non-flat affine connection.) The short answer isthat our notion of relative angular velocity carries over in a natura l way, and condi tions (i) and (ii)remain symmetric. (More problematic is the notion oforbital rotation considered in the preceding note

188 D AVlD B . M ALAMENT

since it makes reference to the "acceleration" of particles in a gravitational field. But it can be refor­mulated (in terms of the presence of background spacet ime symmetries) and remains asymmetric) .)It would take us too far afield to sort this all out here.

3. Definitions ofthe technical terms used here and in what follows can be found, for example, in (Wald1984). (Strictly speaking, a few minor transpositions will be necessary since Wald works with the sig­nature (- I, + 1, + 1, + 1) rather than ours.)

4. We might have written ' nXy a' , but that notat ion is potentia lly mislea ding . The identification indices

' X Y' should not be confused with tensor or spinor indices . In what follows, it will usually be clear

from context whether we are talking about the angular velocity of Y relative to X, or of X relative to

Y. But when there is danger of confusion , we will write ' n~y wn X)' or ' n~x WI1 y)"

5. In what follows, we assume that (M, g ab) is temporally orientable and a particular temporal orienta­

tion has been se lected. We also assume that it is orientable and a volume element Eabcd has been

selected . (A smooth field Eabcd on M qualifies as a volume element if it is completely anti-symmetr ic

( Eabcd = E[abcdJ) and normalized so that Eabcd

Eabcd = - 4 !) Neither the assumption of temporal ori­

entability nor orientabi lity is really necessary. We can, alternat ively, restrict attention to appropriate

local neighborhoods of M . But the assumptions are convenient and, in fact, the spacetime we will use

for our example in section 3 (Godel spacet ime) is temporally orientable and orientable.6. This is a substantive assumption, and will play a role in the presentation of our example in section 3.7. The equivalence here corresponds perfectly to that in (i) in section I, and the proof is essentially the

same . na= 0 iff the three vectors ~a, v", and «gam -~a~m)~nVnvm are linearly dependent. But

since ~a and v" are non- zero, and both va and «gam - ~a~m)~nvnv m are orthogonal to ~a , this

condition holds iff (gam - ~a~m)~nvnvm is proportional to v" , But (l m - ~a~m)~nvnvm is orthog­

onalto v" (since va~a = 0 and vm~nVnvm = ( 1 /2 ) ~n Vn(vmvm ) = ( 1/2)~nVn(-I) = 0) . So

na = 0 i ff (gam - ~a~m)~nVnvm = 0 , as claimed.

8. If they are positioned so as to be orthogo nal at some initial moment, they will remain so.9. For example, the distance between them is constant as determined by the time it takes a light signal to

complete a round trip passage from one particle to the other and back- as measured by clocks sittingon the respec tive particles. Indeed, the distance between them is constant according to any notion ofdistance that can be formul ated in terms of the spacetime metric g ab and the curves YX and 't v- sincethey are all preserved under the flow maps associated with 't

a.

10. The proposition is slightly more general than the one proved in (Rosquist (980). He worked with a unittimelike vector field ~a that is Born rigid (i.e., has vanishing scalar expans ion and shear) and geode­sic . These two conditions imply that ~a is proport ional to a Killing field, but not conversely. Rosquistalso limited attention to the case where, in our notation, v"00" = O.

II . Every unit timelike field ~a whatsoever satisfies

V"~d = e "d + W nd + ~,,~mVm~d '

where

rAnd every such field satisfies, Endpr~Pw

r = w" d (as one can verify by direct substitution for 0) in

the left side express ion). But if 't~a is a Killing field for some scalar field r , e"d = O. (This follows

ON RELATIVE ORBITAL ROTATION IN R ELATIVITY THEORY 189

from Killing's equation, Vr('~s) + Vk'~r) = 0.)

12. For an indication of what Godel spacetime "looks like," see the diagrams in (Hawking and Ellis 1973;Malament 1984).

13. This is equivalent to the claim that , "V"r = 0 and , nVnY = O. The first equation holds since

(d ldt)n(d ldr)n = 0 = (dl dq»n(dld r )n and, hence ,

n " " ,,2 2 no = , (u lur)n = , (-4ft (d r) n) = ~4ft ' V nr .

The argument for the second equation is similar.14. This picture, while helpful, is potentially misleadin g in one respect. As we sha ll see, a partic le whose

worldline is one of these helic es can qualify as non-rotating relative to a particle whose worldline is anaxis curve.

15. This fact explains the choice of the coefficie nt J2 in our expression for r". We wan t (fJ" to be 0 at

points where r = O. As we show in the appendix, the vort icity associated with the general field

(d ldt)a + k«()ldq»a (where it is timelike) comes out to be

r::. 2 2 r::. 4.J2 + k(2sh r -I) +k .J2s h r ( " I,,)a

2 r::. 2 2 4 2 o uY(4ft )[1 + k 2 .J2s h r +k (sh r -sh r)]

This reduces to

2 - I t: a(4ft) ( .J2 - k)(dldy)

at r = O.

16. V (a(d ldY)b) = 0 , since «()/()y)a is a Killing field; and V1a«()ldY) bl = 0 , since

2V[a(dldY)bl = - 4 ft V [aV bV' = O.

17. We have'AnVnO...aKa) = 'Aa'AnV nKa + Ka'AnV n'Aa

= O.The first term in the sum vanishes because Ka is a Killing field (and , so, V (nKa) = 0). The seconddoes so because Aa is a geodesic field.

18. See (Lathrop and Teglas 1978) for an analytic characterization of geodesics passing through pointswhere r = 0 , and see (Hawking and Ellis 1973) for a picture.

19. The coordinate map t : 0 ~ R qualifies as a global time function on (0, g abIO)' i.e., it increases

along all future-directed timelike curves . (Hence ther e cannot be any closed timelike curv es in

(0, gabl0) ') The assertion is equivalent to the claim that the vector field (Vat) is timelike and future-

directed (i .e., (Va tHVat) > 0 and «()/()t)a(Vat) > 0) on O. But this is clear since

2(Vat)(Vat) = ; - s h r

2on 0 ,

4ft (I +sh r)

and (d ldt)a(Vat) = everywhere. (The expression for (Va tHVat) follows from the fact that the

inverse metric is:

ab 1 2 4"" a " " b 2 4"" a" " bg 2 2 4 [(sh r -s sh rHulut) (u lut) - (sh r + sh rHulur) (u lur)4ft (sh r + sh r)

_ (sh 2r + sh4 rH() /()y)a(d ldy)b _ «()/() q> )a«()ldq» b + 2J2sh 2r«()/()t) (a(dldq» b)] .)

20. Let R be the closed set of poi nts where r = O. Since the fields (d l()t)a, «()Idr)a, (d l()q» a, and

«()Idy)a are linearly independent on M - R , there must exist some function{ defined on M - R for

190 D AVID B. M ALAMENT

which the equation holds. We can determine f , up to sign, with the following calculation:

- (4 !) = €abcd€abcd = .r(dl d t) la(dl dr )b(dl d q> / (dldy / J

• (dl dt)la(dldr) b(d ld q» c(dl dY )d)

= .r(dl dt )la(d l dd (dl d q>/( dl dy/ J

2 4 4 , 4• (4~ ) « sh r - sh- r ) - 2sh r)(dt) [a(dr)b(dq» c(dY)dJ

r2 , 4 4 2 4'= - J (4~-) (s h r + sh r) - '-2'

(4!)

The volume elements on M - R defined by the two solutions for / have well defined limits at points

in R. Once those limit values are included, we have a (smooth) volume element on all of M .

21. It is precisely here that the present argumentwould break down if we had not restricted attention to O .

REFERENCES

Hawking, S. W.and G. F. R. Ellis. 1973. The Large Scale Structure ofSpace-Time. Cambridge: CambridgeUniversity Press.

Lathrop, J. and R. Teglas. 1978. "Dynamics in the G6del Universe." II Nuovo Cimento 43 B: 162-1 71.Malament, D. 1986. "Time Travel in the Godel Universe." PSA 1984. Vol. 2 (Proceedings of the Philoso­

phy of Science Association Meetings, 1984): 91- 100.Rosquist, K. 1980. "Global Rotation." General Relativity and Gravitation 12: 649-664.Wald, R. 1984. General Relativity. Chicago: University of Chicago Press.