In a very interesting article entitled 'Einstein and the Rigidly Rotating Disk', JohnStachel (1980) has traced the genesis of the general theory of relativity based on theconcept of a curved spacetime. As he remarks, the rigidly rotating disk 'seems toprovide a "missing link" in the chain ofreasoning that led him (Einstein) to the crucialidea that a nonflat metric was needed for a relativistic treatment of the gravitationalfield' . The chain of reasoning begins essentially with space and time being combinedinto spacetime as a consequence of the special theory of relativity. After that, equivalence principle connecting gravitation to accelerated frames was the first step towardsthe formulation of the general theory of relativity. Now comes the rigidly rotating diskas the "missing link" . The inertial force, namely the centrifugal force, experienced byobservers fixed on the rotating disk is identified as equivalent to a gravitational field.Einstein showed that these observers find the ratio of the circumference to the radiusof a circle around the axis of rotation to be greater than 27T. This immediately leads tothe conclusion that the geometry associated with the rotating disk is non-Euclidean.In other words , the presence of gravitation leads, in analogy with the Gaussian theoryof surfaces, to a metrical theory of four dimensional curved spacetime. The general theory of relativity is boru and the inertial forces lie dormant under the cover ofcurved spacetime. Although all gravitational phenomena are considered using generalspacetime metrics that are solutions to the Einstein field equations, the basic conceptsarising in the case of the rigidly rotating disk are still important. Some of these concepts can be formulated precisely and fruitfully in the case of stationary, axisymmetricspacetimes . Such a spacetime can be considered to be a broad generalization of that ofthe rigidly rotating disk. For instance, in contrast to the observers fixed on the rotatingdisk, there are the rest observers outside the disk. The idea and the existence of globalrest observers in an arbitrary stationary axisymmetric spacetime become fundamentally important for the study of physical phenomena. It is often claimed that one ofthe major consequences of general relativity is the abolition of the notion of forces.On the other hand, analogues of inertial forces in general relativity have been recentlyformulated . They can be used in analysing the effects ofrotation inherent to the spacetime as well as in the case of orbiting particles. There are two other phenomena thatare of considerable interest in studying rotation , namely gyroscopic precession and
gravito-electromagnetism. All these three can be elegantly inter-connected utilizingspacetime symmetries.
Whereas the considerations above may be viewed as arising from the generalization of the rotating disk, the rigidl y rotating frame associated with the disk has recently emerged in the most practical and unexpected form. Thi s is in connection withthe Global Positioning System. One wonders what Einstein might have thought of amachinery, making use of his good old rotating disk, that is not merel y utilitarian butalso carries with it the ominous potential for destructive usage .
2. RIGIDLY ROTATING DISK IN FLAT SPACETIME
Within the framework of general relativity, one can study the rigidly rotating disk.For this purpose, let us start with the flat spacetime. The line element in cylindricalcoordinates is given by
(I)
with c = G = 1. Static observers in the inertial frames ofreference at each space pointfollow the Killing vector field ~a = c5g This field is orthogonal to the hypersurfacest = constant and they can synchronise their clocks with respect to time t since ~a =(1, 0, 0, 0) = t ,«. In order to go over to the disk rotating at the uniform angular speeedw, we make the coordinate transformation,
t = t' , r = r' , </> = </>' + wt' , z = z' ,
Then the line element for the spacetime geometry of the rotating disk is given by,
ds2 = (1 - w2r'2) dt ,2 - 2wr,2d</>'dt' + dcr'2 ,
(2)
(3)
where dcr ,2 = dr '? + r,2d</>,2+ dz'2.We note two new features in the line element (3) in comparison with the original
one (1). We have 900 = (1 + 2<Pc), where the centrifugal potential <Pc = _ ~w2r2 .
There is also the term 903 = -wr,2which shows the rotation inherent to the spacetimeof the disk. Stationary observers fixed at spatial points on the disk follow the Killingvector field adapted to them, r = c5f,. Then ~a = (900, 0, 0, 903) and therefore isnot hypersurface orthogonal. If it were, it could have been expressible in the form~a = a (3,a , where a and (3 are scalar functions. On the other hand , cons ider thevector field Xa = r - 9S&rja , where rja = 15;1, is the Killing vector field giving rise to
9 3 3 '¥
rotational symmetry about the z' - axis. It is easy to verify that Xa = (1,0,0, - w)and Xa = (1, 0, 0, 0) = t ,a' Hence Xa is orthogonal to t = constant hypersurfaces.In fact Xa is nothing but the original Killing vector field ~a of the inertial observersexpressed in the rotating coordinates.
To sum up, stationary observers fixed on the rigidly rotating disk follow worldlinesalong t' which is no longer global synchronous time for these observers. However theycan define a hypersurface orthogonal timelike vector field which turn s out to be thet - lines of the inertial observers in the flat spacetime. Also , the metric component900 contains the Newtonian centrifugal potential <Pc from which one can construct the
308 c. V. VISHVESHWARA
centrifugal force acting on the observers fixed to the disk as measured by the inertialobservers.
In the case ofa rigidly rotating frame , one can trivially recover the original nonrotating flat spacetime by the global transformation inverse to the one given in equation(2) . In a genuine stationary spacetime with rotation, this is impossible although locally one can obtain a flat Minkowskian metri c. Nevertheless, the ideas pertaining tothe rigidly rotating disk can be generalized to an arbitrary stationary spacetime withaxial symmetry as we shall see in the next sections.
3. THE GLOBAL REST FRAME
As has been pointed out , an axisymmetric stationary spacetime is the broad generalization of the rotating disk. Two classic examples of such a spacetime is the Schwarzschild and the Kerr. The Schwarzschild spacetime is static, or equivalently, possessesno inherent rotation since the source, egothe black hole , is non-rotating. On the otherhand, the Kerr spacetime is stationary and has rotation. In both cases it is importantto identify the global rest frame akin to the frames of observers at rest outside therotating disk.
The Schwarzschild metric is written as
where m =~; M = Mass ; c = G = 1.It admits the Killing vector fields ~a = og and T]a = og correponding respectivelyto time symmetry and axial or rotational symmetry with respect to z - axis. Theycommute with each other and their scalar product ~aT]a = 903 = 0 since it is a staticspacetime. As the metric is spherically symmetric there exist two more rotationalKilling vectors which would be absent in a spacetime which is only axially symmetric.The timelike Killing vector field , in its covariant form, can be written as:
~a = 900t ,a · (5)
This shows that ~a forms a hypersurface orthogonal congruence ie., orthogonal to surfaces t = constant . It is therefore irrotational. In otherwords, observers followingthe timelike Killing field trajectories with (r , B,¢) = cons tant have no mutual rotation and, further more , share a common synchronous time t. At any given moment,t = constant is the entire three dimensional space as seen by these observers. Theseare the rest observers and their spatial frame of reference is the rest frame .
Let us now consider the more complicated case of the stationary spacetime withaxial symmetry as exemplified by the Kerr line element :
(2m ) 2mar 2
1 - E dt 2 +2~ sin Bdtd¢
2ma 2r 2 . 2 2 L: 2 2(a2 + r2-L:- sin B)sin Bd¢ - ~ dr - L:dB (6)
R IGID LY ROTATING DISK REVISITED 309
Where m = Mass, J = ma = Angular Momentum, I: = r 2 + a2cos20,~ = m 2
- 2mr +a2. It admits the two Killing fields ~a = 88 and TJ a = 8~ giving rise
as before to the time and rotational or axial symmetries respectively. They commutewith each other, but their scalar product ~aTJa = g 03 -f=. 0 since the spacetime hasinherent rotation. This shows up in the fact that the vorticity of the ~a congruence isnon-zero:
a 1 abed c c -J. 0 (7)WE; = A e <"b<"e;d r
where ea bed is the completely antisymmetric Levi-Civita symbol. Consequently ~a
field is not globally hypersurface orthogonal and the stationary observers are not restobservers. On the other hand, consider the vector field
(8)
which is the projection of ~a orthogonal to TJ a . The vorticity of the Xa congruenceis zero. The frames adapted to Xa were called locally non-rotating reference frames(LNRF) by Bardeen (1970). Actually Xa forms a hypersurface orthogonal congruence.
(9)
Observers following Xa are in fact the global rest observers and frames adapted tothem form the global rest frames. Properties of Xa congruence and the global restframes were studied in detail by Greene, Schucking and Vishveshwara (1975). The
rest observers revolve in circular orbits with the angular speed Wo = -~ = - ~,TI Tic 9 3 3
whereas ~a_ observers are stationary fixed at spatial points! The former share a com-mon synchronous time 1. If two circular light rays start at the same moment froma point in opposite directions, they will return to that point at different moments oftime with respect to the stationary observer. On the other hand, the rest observer willreceive them at the same moment, since in a given interval of time one ray travelslonger than the other and the rest observers will have moved exactly by this distanceby then. This motion is a manifestation of the so-called inertial frame dragging . Whenthe spacetime has orthogonal transitivity, ie., there are no cross terms between (t, ¢)and (r, 0), gtr = gt () = g()</> = gr </> = 0, one can show that Xa becomes null on a surface which is itself null. This surface is identified as the stationary event horizon. Thishappens for the Killing field ~a in the case of static spacetimes like the Schwarzschild.One can also show that the surfaces given by t = constant are maximal.
Physical phenomena are studied within the global rest frames especially since extended objects can be defined on t = con stant hypersurfaces. As we shall see, thegeneral relativistic analogues of inertial forces can be covariantly defined with respectto the global rest frames.
310 c. V. VISHVESHWARA
4. ROTATIONAL EFFECTS
There are three important aspects of rotational effects that appear in a stationaryaxisymmetric spacetime.
4.1. Gravito-Electromagnetism
There is a striking similarity between electromagnetism and gravitation. This showsup through the metric component 9 00 which behaves like the electrostatic potential ,while 900:(0: = 1 - 3) acts as the magnetic potential. Or, the timelike Killing vectorfield ~a plays the role of the electromagnetic four potential. This has been studiedextensively in the weak field approximation. I
4.2. Gyroscopic Precession
As is well known, gyroscopic precession is an important phenomenon which bringsout rotational effects. Not only does it find applications in various contexts, but alsohas been put forward as a candidate for testing the general theory of relativity itself. In flat spacetime Thomas precession is improtant in atomic physics. Spacetime curvature, as in the case of the Schwarzschild spacetime, is incorporated into thegeodesic precession, e.g. Fokker-De Sitter precession. The source angular momentumgives rise to additional effects in the case of stationary axisymmetric spacetimes. Inthe weak field limit this is related to the Lense-Thirring effect. A consequence of thisis that a gyroscope carried by a static observer in a static spacetime does not precesswhereas a gyroscope carried by a stationary obsever in a stationary spacetime doesundergo precession. Gyroscopic precession has also been studied by several authorsin the past. ?
4.3. Inertial Forces
Recently there has been a great deal of interest in the general relativistic analogues ofinertial forces. We shall be considering them in some detail in the next section.
All the three aspects of rotation could be studied on a rigidly rotating disk. As hasbeen already mentioned , we study them in the more general framework ofa stationaryaxisymmetric spacetime. Utilizing the two inherent symmetries, we shall indicate howthese three aspects can be inter-related in an elegant and covariant manner.
5. INERTIAL FORCES IN GENERAL RELATIVITY
First of all, one may ask why inertial forces be considered in the context of the general theory of relativity at all. After all, one of the major steps taken by the theorywas to banish the notion of forces. Nevertheless, there are advantages in constructing the analogues of inertial forces in general relativity. To begin with, the question'why?' may very well be countered by the response 'w hy not?' . After all we aremore familiar with forces, both conceptually and physically than, for instance, spacetime curvature and geodesics. But more pertinently, these forces serve as an aid in
RIGIDLY ROTATING DISK REVISITED 311
analysing rotational effects, especially in identifying those engendered by the stationary spacetime with rotation in contrast to the static case. Furthermore, as has beenmentioned earlier, inertial forces can be inter-related to gyroscopic precession andgravito electro-magnetism bringing out the essential unity among these three differentapproaches to the study of rotation in general relativity.
The original motivation for defining the inertial forces may perhaps be traced to theobservation of Abromowicz and Lasota (1986) , see also Abromowicz and Prasanna(1990) regarding the centrifugal force reversal in the Schwarzschild spacetime. In thisspacetime there exists a circular null geodesic, or a photon orbit , at r = 3m. It waspointed out in (Abramowicz and Lasota 1986), that rockets moving at different speedsin circular timelike orbits at r = 3m have no relative acceleration. They need thesame engine thrust to move. This was attributed to the fact that they need to overcomeonly the gravitational pull, the centrifugal force being zero for all of them. This wasconsidered to be a natural consequence of a light ray being a straight line, in thepresent case, the circular path at r = 3m being a 'straight line' in a curved spacetime.This centrifugal force reversal at r = 3m led to the definition and study of the generalrelativistic analogues of inertial forces.
Next, Abromowicz, Carter and Lasota (1988) formulated what they termed as theoptical reference frame. Essentially, 9 00 is factored out in the line element of a stationary axisymmetric spacetime and particle dynamics studied in the conformal spacetimewhich is identified with the optical reference frame. They showed that the spatial trajectories of the null geodesics of the original four dimensional spacetime happen to begeodesics in the optical reference frame. Further, the four acceleration of particles following geodesics in the original spacetime, when projected onto the conformal spacetime, splits into terms that may be identified as inertial forces. This formalism wasfollowed by the covariant definition of inertial forces by Abromowicz, Nurowski andWex (1993) . This may be summarized as follows in the case of stationary spacetimeswith axial symmetry.
The globally hypersurface orthogonal field defined in equation (5) is written as
(10)
That is, n a is the four velocity of the rest observers and e<P = (Xb Xb )~ is the normalization factor. Any arbitrary four velocity u a may be split into
(11)
where T a is a unit vector orthogonal to n a and v is the spatial velocity of the particlealong this direction. Denote by a tilde over any tensor its projection orthogonal ton ' , Then the four acceleration a i projected thus is made up of the four analogues ofinertial forces as follows.
(12)
312
where m = particle mass.We have then
c. V. VISHVESHWARA
Gravitational force G k = ¢, k
Centrifugal force Zk = - (rV) 2Ti'\7(Tk
Euler force E k = - 11Tk
Coriolis-Lense-Thirring force Ck = "'/V Xk
where
v= (v ecP 'Y),i u i
X k = ni
(7k;i - 7i ;d
¢ ,k = -nink,i
Tk = ecP 7k
(13)
(14)
In the case ofparticles in circular orbits with uniform motion the Euler forces E i = O.In the case of static spacetimes the Coriolis- Lense-Thirring force Ci = O.
Let us return to the phenomenon that motivated the above formulation of the inertial forces, namely the centrifugal reversal at the circular photon orbit in the Schwarzschild spacetime. Since this orbit is viewed as a straight line, one would expect thata gyroscope carried along this orbit would not precess. This happens to be true. Thisraises the question whether inertial forces and gyroscopic precession are related toeach other in stationary axisymmetric spacetimes. As we shall see, this is in fact thecase. First we shall briefly outline the covariant description of gyroscopic precessionfor this purpose.
6. GYROSCOPIC PRECESSION
A comprehensive treatment of gyroscopic precession in a stationary, axisymmetricspacetime has been given by Iyer and Vishveshwara (1993). This work makes use ofthe Frenet-Serret formalism which provides a geometric, invariant description ofa onedimensional curve. In a four dimensional spacetime, the Frenet-Serret tetrad e( a) istransported along a timelike curve which is characterized by the three scalars n , 71 and72 which are termed the curvature , the first and the second torsions respectively. Thecurvature", is the magnitude of the four acceleration of the particle following the timelike trajectory. The torsions 7 1 and 72 are directly related to gyroscopic precession.Consider a tetrad f (a) Fermi-Walker transported along the particle trajectory. Identifye(O) and f (o ) with the four velocity u a of the particle. The triad f (a ) is physicallyrealized by three mutually orthogonal gyroscopes. In general f (a ) or equivalently thegyroscopes precess with respect to e( a) . The rate of gyrosocopic precession is givenby
w~ = - h eel) + 7le(3))' (15)
Determination ofthe tetrad e(a) and the parameters 71 and 72 along a particle trajectoryprovides a complete description ofgyroscopic precession with respect to the comovingFrenet-Serret frame.
RIGIDLY ROTATING DISK REVISITED
7. APPLICATION TO AXIALLY SYMMETRIC STATIONARYSPACETIMES AND COVARIANT CONNECTIONS
Let us consider a particle following the Killing vector field,
313
(16)
where ~ is the timelike Killing vectors , 1]i the spatial Killing vector that the spacetimemay admit and w i are constants . Although our formalism is applicable to such ageneral case, we shall specialize to a circular orbit with 1]i == 1] the axial Killingvector, so that the particle describes a circular orbit with angular speed w. The fourvelocity of the particle is then
(17)
where ecP is the normalization factor. The Frenet-Serret triad e(l)' e(2 )' e(3) will bealigned respectively along the radial direction, tangential to the orbit and normal tothe orbital plane . The covariant derivative of the tetrad components with respect to theproper time (denoted by an overhead dot) satisfies a Lorentz-like equation,
(18)
Killing vector fields ~a and 1]a obey the equations.
V a;b + Vb ;a = 0 and V a;b;c = R abcdVd
.
Consequently we have
(19)
(20)
Therefore F ab is like a homogeneous electromagnetic field tensor. In fact the motionalong Killing trajectories is remarkably similar to the motion of charged particles ina homogeneous electromagnetic field as was demonstrated by Honig, Schucking andVishveshwara (1974) . Acceleration
. i i ja' = K:e(l) = Fj e(O)
is analogous to the electric field. Similarly one can show that
(21)
(22)
where * the star denotes the dual. This is like the magnetic field which producesspin precess ion in an electro-magnetic field. Hence there is a striking resemblance toelectro-magnetic phenomena which is the essence of gravito electro-magnetism.
It is straight forward to compute K:, 71, 72 and e(a) in terms of powers of Fab ande( O)' Therefore, gyroscopic precession can be described in a covariant manner forKilling trajectories and can be related to gravito-electromagnetism. This has beendone in detail in reference (Iyer and Vishveshwara 1993).
314 C. V. VISHVESHWARA
(23)
The formalism developed for inertial forces can also be applied to the Killing orbits , once again , in a straight forward manner. This has been done in detail by RajeshNayak and Vishveshwara (1998). Furthermore, in this paper covariant connectionshave been established among the inertial forces, gyroscopic precession and gravito electromagnetism. As we have seen, gyroscopic precession is represented by 71 and7 2, while the inertial forces are given by the gravitational force G i , the centrifugalforce Zi and the Coriolis force C«. In the case ofKilling orbits it can be shown (ibid.)that
71 rv a . (Z + a C )
72 rv a x (Z + aC) ,
where a is the acceeleration and a is a scalar function involving the Killing vectorfields. Thus inertial forces and gyroscopic precession are inter-related.
Now, acceleration ai = FJu j and Zi + aCi rv * Flu j , so that a and the combination (Z + aC) are equivalent to gravito-electric and gravito-magnetic fields E and Brespectively. Therefore, we have the relations
71 rv E · B
72 rv E x B
This is exactly similar to the case ofcharged particles moving in a homogeneous electromagnetic field (Honig, Schiicking, and Vishveshwara 1974). Thus the interelationsamong the three formalisms can be established in a covariant and elegant manner.The formals are considerably simplified in the case of static spacetimes for which theCoriolis force C i = o.
The results of reference (Rajesh Nayak and Vishveshwara 1998) shed further lighton some of the observations that had been made earlier. Equation (23) shows that ,in static spacetimes, (C i = 0), gyroscopic precession and centrifugal force undergosimultaneous reversal when they become zero. It can also be shown (ibid.) that thishappens at a photon orbit. This phenomenon had been studied earlier in the case of theSchwarzschild spacetime (Abramowicz and Lasota 1986; Abramowicz and Prasanna1990) and also by Rajesh Nayak and Vishveshwara (1997) in the case of Emst spacetime. There is no such correlation in the case of stationary spacetimes when C i =/= O.This had been noticed by Sai Iyer and Prasanna (1993) for the Kerr spacetime and byRajesh Nayak and Vishveshwara (1996) in the case fo Kerr-Newman spacetime.
8. BACK TO THE ROTATING DISK
In recent years, the rigidly rotating disk has made a dramatic comeback. This is in thecontext of the Global Positioning System. For a lucid account ofGPS and the relativ istic effects involved we refer the reader to the article by Neil Ashby (1998) . The usersofGPS are fixed on the rotating earth . One of the relevant processes to be consideredis the Einstein synchronization of clocks carried by these observers to set up a network of synchronized clocks . This is achieved by means of light propogation as seenby these earth bound observers . The simplest model used for carrying out calculationsin this regard makes use of the rigidly rotating frame and the resulting line element
RIGIDLY ROTATING DISK REVISITED 315
(Langevin metric) given in the equation (3) with W = WE = 7.292115 x 1O- 5r adj s,the rotation rate of the earth . A light pulse travelling between two rigidly rotating observers takes an extra amount of time to catch up with the moving receiver. This pathdependent contribution, known as the Sagnac Effect, depends on the metric component g 03 . For a light pulse traversing the equatorial circumference this additional timedelay amounts to about 207 nanoseconds. This is a significant amount consideringthe accuracy of the GPS system. Probably, no one had dreamt of a situation in whichgeneral relativity became significant on a terrestrial scale that too in such a utilitarianmanner!
Within the full machinery of the general theory of relativity, the rigidly rotatingdisk has invaded the area of exact solutions as well. Neugebauer and Meinel (Neugebauer 1996) have derived an exact solution for a rigidly rotating disk of dust. Thisis of significance on two counts . Firstly, no exact global solution of Einstein's fieldequations has been found for a rotating distribution such as a rotating star. Secondly,the solution may serve as a model for astrophysical objects such as galaxies and accretion disks. In any event, the spacetime of the disk exhibits novel features that makeit interesting in its own right. For instance , as the angular speed of rotation is increased , the exterior of the disk undergoes a phase transition and becomes the extremeKerr solution. Another unusual feature is the formation ofergospheres as the physicalparameters are varied . These and other aspects of the spacetime associated with thedisk makes it worthy of further studies .
In this article, we have tried to trace briefly the evolution of the rigidly rotatingdisk. As we have seen, it started out as the 'missing link' in the chain of reasoningthat led Einstein from the special theory ofrelativity to the general theory building upa spacetime picture of gravitation. Aspects inherent to the spacetime of the disk findtheir natural generalization in the phenomena characteristic of axisymmetric, stationary spacetimes. The disk metric has recently found amazing application in the GPSsystem . And, finally, one has now an exact solution for the disk made up of dust exhibiting interesting features . For all we know, there may be surprises still in store forus . A ride on the rigidly rotating disk cannot but tum you on!
ACKNOWLEDGEMENT
I thank Ms. G. K. Rajeshwari and Mr. Rajesh Nayak for their valuable help in thepreparation of the manuscript.
Indian Institute ofAstrophysics
NOTES
* I have known John for some three decades now as a very close friend and an esteemed colleague. It is agreat pleasure to dedi cate this article to him with affection and my wannest best wishes.
I. See (Bini , Carin i, and Jantzen 1997a, 1997b; Embacher 1984; Thome, Price, and Macdonald 1986;Jantzen, Carini , and Bini 1992; Ciufolini and Wheeler 1995).
316 c. V. VISHVESHWARA
2. See (Abramowicz, Nurowski, and Wex 1995; Barrabes, Boisseau , and Israel 1995; de Felice 1991,1994 ; Rindler and Perlick 1990; Semerak 1995, 1996, 1997).
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