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No Time Machines from Lightlike Sources

arXiv:gr-qc/9210012v1 21 Oct 1992No Time Machines from Lightlike Sourcesin 2+1 GravityS. Deser ∗Alan R. Steif ∗†AbstractWe extend the argument that spacetimes generated by two timelike particlesin D=3 gravity (or equivalently by parallel-moving cosmic strings in D=4)permit closed timelike curves (CTC) only at the price of Misner identificationsthat correspond to unphysical boundary conditions at spatial infinity and toa tachyonic center of mass.

Here we analyze geometries one or both of whosesources are lightlike. We make manifest both the presence of CTC at spatialinfinity if they are present at all, and the tachyonic character of the system:As the total energy surpasses its tachyonic bound, CTC first begin to format spatial infinity, then spread to the interior as the energy increases further.We then show that, in contrast, CTC are entirely forbidden in topologicallymassive gravity for geometries generated by lightlike sources.Among the many fundamental contributions by Charlie Misner to general relativityis his study of pathologies of Einstein geometries, particularly NUT spaces, which inhis words are “counterexamples to almost everything”; in particular they can possessclosed timelike curves (CTC).

As with other farsighted results of his which were onlyappreciated later, this 25-year old one finds a resonance in very recent studies ofconditions under which CTC can appear in apparently physical settings, but in factrequire unphysical boundary conditions engendered by identifications very similar tothose he discovered. In this paper, dedicated to him on his 60th birthday, we reviewand extend some of this current work.

We hope it brings back pleasant memories.To be published in a Festschrift for C.W. Misner, Cambridge University Press.BRX TH–336∗The Martin Fisher School of Physics, Brandeis University, Waltham, MA 02254, USA.†Present address:DAMTP, Cambridge University, Silver St., Cambridge, CB3 9EW, U.K.E-MAIL: deser@brandeis; ars1001@amtp.cam.ac.uk.1

1.IntroductionOriginally constructed by G¨odel [1], but foreshadowed much earlier [2], spacetimespossessing CTC in general relativity came as a surprise to relativists. The shock wassoftened by the fact that these solutions required unphysical stress tensor sources,and in this sense should not have been so unexpected: it is after all a tautologythat any spacetime is the solution of the Einstein equations with some stress tensor,as often emphasized by Synge.

Indeed, Einstein himself, while elaborating generalrelativity, apparently worried about geometries with loss of causality and CTC, andhoped that they would be excluded by physically acceptable sources.1 Almost twodecades after G¨odel’s work, Misner in his pioneering studies [3] of NUT space showedhow CTC could be generated as global effects, by taking local expressions for a metricand making appropriate identifications among the points.Very recently, the subject of CTC was revived in two quite different contexts. Thefirst, which we shall not discuss, involves tunneling through wormholes in D=4 grav-ity.

The second, which will be our subject here, concerns solutions to D=3 Einsteintheory with point sources or equivalently, D=4 gravity with infinite parallel cosmicstrings (since the latter system is cylindrically symmetric). We will therefore operateentirely in the reduced dimensionality.

The dramatic simplification at D=3 is thatthe Einstein and Riemann tensors are equivalent, so spacetime is locally flat wher-ever sources are absent; consequently there is neither gravitational radiation nor anyNewtonian force between particles. For these reasons, the sign of the Einstein con-stant is not physically determined, unlike in D=4.

We will adopt the usual sign here,but mention the opposite sign, “ghost” Einstein theory, at the end. Local flatnessin D=3 means that all properties are encoded in the global structure, i.e., in theway the locally flat patches are sewn together.

Indeed, we will use this geometricapproach to analyze the CTC problem. However, for orientation we will begin witha brief discussion in terms of the analytic form of the metric.Consider the general solution outside a localized physical source as given by the“Kerr” metric [4]ds2 = −d(t + Jθ)2 + dr2 + α2r2dθ2 .

(1.1)Our units are κ2 = c = 1, and α ≡1 −M/2π. The constants of motion are theenergy M and angular momentum J (space translations not being well defined [5].This interval is manifestly locally flat in terms of the redefined coordinates Θ = αθ,T = t + Jθ, but the global content lies in the different range, 0 ≤Θ ≤2πα, ofΘ corresponding to the usual conical identification, and in the time-helical structure1We thank John Stachel for telling us this.2

resulting from the fact that the two times T and T +2πJ are to be identified whenevera closed spatial circuit is completed. The interval (1.1) can clearly support CTC; forexample, the interval traced by a circle at constant r and t,∆s2 = (2πα)2(r2 −J2/α2) < 0 ,(1.2)is timelike for r < |J|/α.

However, the relevant physical question is whether theconstituent particles are ever confined within this radius; otherwise the CTC crite-rion (1.2) ceases to apply. To be sure, if we simply insert the metric (1.1) into theEinstein equations, it is valid down to r = 0; the “source” is a spinning particle withT 00 ∼mδ2(r), T i0 ∼J ǫij∂j δ2(r).

But we do not accept classical spinning particlesas physical, precisely because of their singular stress tensors, any more than we doG¨odel’s sources. Instead, one must check whether a system of moving spinless par-ticles with orbital angular momentum can support CTC.

This was the question thatwas initially raised in [4] and answered in the negative, on the simple physical groundsthat the point particles, being essentially free, will — both initially and finally — bedispersed so that the constant J would have been exceeded at t = ±∞by the radiusat which the exterior metric (1.1) is valid. Thus, CTC, if present at all, would haveto appear and then disappear spontaneously in time in an otherwise normal Cauchyevolution, and it seemed unlikely that this violation of Cauchy causality would occurin a finite time region for an otherwise non-pathological system.It was therefore quite surprising when, a year ago, Gott [6] gave an explicit con-struction of a geometry generated by an apparently acceptable source consisting oftwo massive particles passing by each other at subluminal velocities, in which CTCappear only during a limited time interval [7].

However, it was then shown both that,in these spaces CTC will also be present at spatial infinity, which constitutes an un-physical boundary condition, and that the spacetimes have an imaginary (tachyonic)total mass [8]. This is in contrast to the globally flat space of special relativity, wherea collection of subluminal particles cannot of course be tachyonic.

Indeed, the factthat in D=3 everything lies in the global properties raises a cautionary, and as weshall see, decisive, note. Let us illustrate this with one simple object lesson for thecase of static sources.

It is clear from (1.1) that a single particle cannot have a massgreater than 2π (in our units); indeed, m = 2π corresponds to a cylindrical ratherthan conical 2-space.One might suppose that, since there are no interactions in2 + 1 dimensions, two stationary particles should give rise to a perfectly well-definedmetric as long as each one separately satisfies the above inequality. This in fact isnot so; the sum of the two masses must also not exceed 2π.

If it does, the total massmust then jump to the value 4π and at least one further particle is required to bepresent, the total system now having an S2 — rather than an open — topology: G003

is essentially the Euler density of the 2-space [4]. This example reflects the presenceof effective global constraints in 2 + 1 dimensions, even though the theory is locallytrivial, so that a source distribution consisting of several individually acceptable par-ticles is not thereby guaranteed to be itself physically acceptable.

The moral appliesto the Gott pair and, as we shall see, also to its lightlike extension. We emphasizethat the pathology here is not merely that there is a total spacelike momentum,but more importantly, that the latter implies a “boost-identified” exterior geometry,namely one in which CTC will always be present at spatial infinity.

But if one allowspathology in the boundary conditions of any system, then it is no surprise that it willbe present in the interior as well! Indeed, this is just the sort of behavior that theMisner identifications [3] gave, and can be seen in the metric form of the interval aswell.

For, to say that the effective source is a tachyon, really means that the exteriorgeometry is that generated by an effective pointlike stress-tensor which replaces theT 00 ∼δ(x)δ(y) of a particle with a T yy ∼δ(x)δ(t), etc. Consequently, in (the carte-sian coordinate form of) the Kerr line element (1.1), the (x, y) space is replaced by(x, t) with a jump in t replaced by one in y [8].

The resulting metric shows that CTCdo not really appear and disappear spontaneously in some finite region where theparticles pass each other, but rather that they are always present at spatial infinity;thence they close in (very rapidly!) on the finite interaction region.Attempts to remedy these difficulties by adding more particles [9] to the system fail;it has been shown that the total momentum of any system containing the Gott timemachine is necessarily tachyonic [10]; thus, two particles constituting the Gott systemcannot arise from the decay of a pair of static particles (of allowed mass less than 2π)since the latter’s momentum is timelike [11].

If the mass of the initial static particlesexceeds 2π, the universe closes; but as was shown in [12], a closed universe will endin a big crunch just before the CTC appear. Since there is no spatial infinity, thepathology there has been transmuted into a singularity!In this paper, we extend the Gott construction to systems involving lightlike particles(“photons”).

Here too, CTC will appear just as the the system becomes tachyonic.In Section 2, we review the geometries due to a two-photon source and to the “mixed”system consisting of one photon and one massive particle [13]. These systems will beour testing ground for the existence of CTC.

In Section 3, we calculate the mass forthese two-particle systems, thereby obtaining the condition for their total momentumto be non-tachyonic. In Section 4, the condition for CTC to arise is derived and isshown to coincide exactly with the condition that the system be tachyonic.

Fur-thermore, it will be manifest that (since they first occur there) CTC exist at spatialinfinity if they are present at all. In Section 5, the analysis is extended to a moregeneral model, topologically massive gravity.

Its two-photon solution is constructed4

and is shown to exclude CTC for all positive values of the photons’ energies. This istrue for the ghost Einstein theory as well.2.Spacetimes Generated by Lightlike SourcesIn this section, we review two systems, involving lightlike sources, from which we willattempt to build a time machine.

The first consists of two non-colliding photons,the second of one photon and one massive particle. Each can be obtained by pastingtogether the appropriate one-particle solutions, which we first describe.In D=3, a vacuum spacetime is specified by the way in which locally flat patches aresewn together.

Different patches are identified using Poincar´e transformations, sincethese define the symmetries of flat space. This method of constructing solutions, inwhich the particle parameters (mass, velocity, and location) determine the transfor-mation generators, was presented2 in [4].

This procedure is, of course, completelyequivalent to the standard analytic approach of obtaining the metric from the fieldequations.The simplest example is the conical spacetime describing a particle of mass m at restat the origin of the x−y plane. This solution is obtained by excising a wedge of anglem with vertex at the origin and identifying the two edges according to x′ = Ωmx,where Ωm is a rotation by mΩm =1000cos msin m0−sin mcos m,(2.1)whose rows and columns are labelled by (t, x, y).This description is completelyequivalent to the metric form (1.1) with J=0.The geometric description of the one-photon solution can be found from the analyticsolution, or by an infinite boost of the conical static metric [13].

Consider a singlephoton moving along the x-axis with energy E and energy-momentum tensor Tµν =Eδ(u)δ(y)lµlν, lµ = ∂µu where u = t−x, v = t+x are the usual lightcone coordinates.The Einstein equations Gµν = Tµν can be solved with a plane-wave ansatzds2 = ds20 + F(u, y)du2 ,(2.2)where ds20 = −dudv+dy2 is the flat metric. This ansatz simplifies the Einstein tensorto Gµν = −12∂2F∂y2 lµlν, and reduces the Einstein equations to the ordinary differential2Such procedures are described more formally in [14].5

equation ∂2F∂y2 = −2Eδ(u)δ(y). Solving for F yields the general one-photon solution:ds2 = ds20 −2Eyθ(y)δ(u)du2(2.3)up to a homogeneous solution, of the form F = B(u)y + C(u), that can be absorbedby a coordinate transformation.If we now apply the coordinate transformation v →v −2Eyθ(y)θ(u), the metricbecomesds2 = θ(u){−dud(v −2Eyθ(y)) + dy2} + θ(−u){−dudv + dy2} .

(2.4)In this form, the geometric description of the one-photon solution becomes clear. Itcorresponds to making a cut along the u = 0, y > 0 halfplane extending from thephoton’s worldline to infinity and then identifying v on the u = 0−side with v −2Eyon the u = 0+ side.

It is easily checked that the points being identified are in factrelated by the Lorentz transformationNE =1 + 12E2−12E2E12E21 −12E2EE−E1. (2.5)This matrix corresponds to the β →1, m →0 fixed energy E =m√1−β2, limit ofthe boost-conjugated rotation matrix ΛβΩmΛ−1β .

Here Λβ is a Lorentz boost in thex-direction,Λβ =1√1 −β21β0β10001. (2.6)[This geometric construction of the one-photon solution is analogous to that of theAichelburg–Sexl one-photon geometry [15] in D=4, our null boost being the analogof their null shift.] The above formulation is not unique however; an equivalent one,more analogous to the conical solution, is obtained by boosting the cone along itsbisector rather than perpendicular to it [16].

The physics is of course independent ofsuch choices.The solution for two non-colliding photons can now be constructed by pasting to-gether the individual one-photon solutions. [It is of course not possible to constructthe two-photon solution in this way in D=4, since spacetime is not flat betweensources.] We consider two non-parallel3 photons in their center-of-momentum frame,3The solution for parallel photons can also be constructed, but it does not admit CTC for thesame reason as that given below for the one-photon solution.6

where the photons are taken to be moving with energy E respectively in the positivex-direction along y = a, (a > 0) and in the negative x-direction along y = −a.The spacetime associated with the first photon is obtained by making a cut alongthe u = t −x = 0, y > a halfplane and then identifying the point (x, x, y) on theu = 0−side with the point (x −E(y −a), x −E(y −a), y) on the u = 0+ side. Forthe second photon, one makes a cut along v = t + x = 0, y < −a and identifies(−x, x, y) on v = 0−with (−x + E(y + a), (x −E(y + a), y) on v = 0+.

The completetwo-photon geometry then consists of these two one-photon solutions simply pastedtogether along the y = 0 plane. In contrast to the massive Gott pair, no relativeboost between the two particles’ halfspaces is necessary, since the photons’ motion isalready encoded in the identification made on their respective halfplanes.The second system, consisting of a photon and a massive particle, can also be obtainedby pasting together the respective one-particle solutions.

We use a frame in whichthe massive particle is at rest at the origin in the x−y plane, and the photon ismoving in the positive x−direction along y = a > 0. For the static massive particlewe excise from the y < a/2 halfspace a wedge of angle m whose vertex is at theorigin and which is oriented in the negative y-direction, then identify the edges asusual.

For the photon, we simply translate the one-photon solution given above fromy = 0 to y = a. If m < π, then the orientation of the wedge ensures that there is nointersection with the photon’s halfplane, so that the two-particle solution is obtainedby pasting together these two halfspaces along y = a/2, again with no relative boost.If m > π, the solution is no longer obtainable by simple gluing, as the tails of thetwo sources would now overlap.3.Total Energy and Tachyon ConditionsIn this section, we calculate the total mass of the two previously described sys-tems.

In the following section, we will see that CTC arise precisely when the totalmass becomes imaginary, i.e., the system becomes tachyonic and non-physical Misneridentifications emerge. The mass can be found by composing the one-particle iden-tifications and writing the result for the complete system in the form of the generalspacetime identificationx′ = a + L(x −a) + b .

(3.1)The spatial vector a = (0, a) describes the location of the center-of-mass; the directionand magnitude of the timelike vector b define, respectively, the time axis and the time-shift along it. For a system of total mass M and velocity β in the x direction, L willbe a Lorentz transformation of the form L = ΛβΩMΛ−1β , implying in particular thatcos M = 12(Tr L −1) .

(3.2)7

The system is non-tachyonic provided M is real, implying that the right-hand sidelies in the range [−1, 1]. We now proceed to calculate L, and from it M, in terms ofthe constituent parameters of the systems constructed in the previous section.For the two-photon system, the one-particle identifications are given byx′1=a + NE(x1 −a)x′2=−a + ΩπNEΩ−π(x2 + a) .

(3.3)The conjugation of NE by a π-rotation in the second equation reflects the fact thatthe second photon is moving in the negative x-direction. Composing the two identi-fications in (3.3), we find L = NEΩπNEΩ−π, and Tr L = 3 −4E2 + E4.

Comparingwith (3.2), we obtain the conditionE > Emax = 2(3.4)for the system to be tachyonic. This condition can also be formally obtained as thelimit of the original Gott condition [6] for two masses m moving subluminally: there,the tachyonic threshold is given bysin 12m√1 −β2 > 1 .

(3.5)Clearly, the limit m →0, β →1 in this equation (with the energy fixed) yields (3.4).For the mixed system, the one-particle identifications are given byx′1=a + NE(x1 −a) ,x′2=Ωmx2 . (3.6)Composing these yields L = NEΩm; comparison of its trace with (3.2) impliescos M = cos m −(sin m)E + 14(1 −cos m)E2 .

(3.7)The criterion for tachyonic M can be expressed as a condition on E for fixed m,E > Emax = 2sin m +q2(1 −cos m)1 −cos m.(3.8)8

4.Closed Timelike CurvesWe now find the conditions for CTC to be present in the two-photon and mixedsystems. We first show that the one-photon spacetime (like the conical spacetime fora particle of non-zero mass) does not admit CTC.

This may not be obvious, since theidentification involves a timeshift, which as in the case of the Kerr solution (1.1), couldpotentially lead to CTC. Recall that the one-photon solution is characterized by theshift (x, x, y) →(x′, x′, y′) = (x−Ey, x−Ey, y) upon crossing the u = t−x = 0, y > 0null halfplane from u = 0−to u = 0+.

A CTC, γ, would have to cross this halfplane totake advantage of the timeshift (there being no CTC within flat spacetime patches).Irrespective of where γ enters the halfplane, the shifted point from which γ emergesis separated from the entry point by a lightlike interval. Therefore only particlestravelling faster than the speed of light can complete the loop in time, showing thatthe one-photon solution does not admit CTC.Now consider the two-photon solution described by the identifications (3.3).

Since, asshown above, the one-photon solution does not permit CTC, we can restrict ourselvesto curves γ that enclose both photons’ worldlines and therefore intersect both oftheir halfplanes. In order that it not become spacelike, γ must be directed oppositeto the photon whose plane it is about to cross; this sense will automatically yielda gain in time upon crossing the halfplanes.

Label the point at which γ intersectsthe u = 0, y > a halfplane by xµ1 = (x1, x1, y1) on the u = 0−side and hence byxµ′1 = (x1 −E(y1 −a), x1 −E(y1 −a), y1) on the u = 0+ side, and the point at which γintersects the other halfplane by xµ2 = (−x2, x2, y2) on the v = 0−side and hence byxµ′2 = (−x2 + E(y2 + a), x2 −E(y2 + a), y2) on the v = 0+ side. For γ to be timelike,the total traversed distance,d=|x2 −x′1| + |x1 −x′2|(4.1)=q(x1−x2−E(y1−a))2+(y1−y2)2 +q(x1−x2+E(y2+a))2+(y1−y2)2,must be less than the total elapsed time,T = (t2 −t′1) + (t1 −t′2) = E(y1 −y2 −2a) .

(4.2)For a given T, we can find the minimum value of d as a function of its arguments.The extremization occurs at x1 −x2 = T/2 and y1 + y2 = 0, with the result thatdmin = 4y1, with T = 2E(y1 −a). Therefore CTC will be present ify1/a > E/(E −2) .

(4.3)9

Recalling that y1 > a > 0, we see that the lowest allowed value is E = 2, precisely thetachyon threshold Emax of (3.4); there, y1 (and therefore also −y2) becomes infinite,i.e., CTC first arise at spatial infinity. As the energy increases, the CTC spreadinto the interior as well, but they are always present at spatial infinity, if presentat all.

[The requirement that γ be everywhere future-directed imposes no relevantconditions: The individual time segments (t2−t′1) and (t1−t′2) must each be positive,implying the inequalities E(y1 −a) > (x1 + x2) > E(y2 + a). At the minimum, theyread |x1 + x2| < E(y1 −a), and are easily satisfied, since (x1 + x2) is otherwiseunconstrained.

]Let us now find the condition for which the “mixed” system admits CTC. Sinceneither individual one-particle solution alone admits CTC, a potential CTC, γ, mustagain enclose both particles, thereby intersecting both the photon’s halfplane and thestatic particle’s wedge.

Let the point of intersection with the halfplane be labelled byxµ1 = (x1, x1, y1) on the u = 0−side and by xµ′1 = (x1 −E(y1 −a), x1 −E(y1 −a), y1)on the u = 0+ side, and that with the wedge by xµ2 = (t2, y2tan m2 , y2) on one edgeand by the rotated values xµ′2 = (t2, −y2tan m2 , y2) on the other edge. The curve γ istimelike provided the distance traversed,d =q(x1−E(y1−a)−y2 tan m2 )2+(y1−y2)2 +q(x1+y2 tan m2 )2+(y1−y2)2 , (4.4)is less than the elapsed time T = E(y1 −a).

Here, we minimize d with respect to x1and y2 for fixed T (or y1); the extremum occurs at x1 = T/2, y2 = y1cosm2 −12E(y1 −a)sinm2 and is given by dmin = 2asinm2 + (y1 −a)(Ecos m2 + 2sin m2 ). Therefore,existence of CTC requiresy1/a > E/ E −2sin m21 −cos m2!.

(4.5)Since y1 > 0, E must equal or exceed the threshold tachyon value Emax of (3.8), asis easily seen using half-angle formulas. Again, y1 is infinite at E = Emax, and as theenergy increases, CTC begin to move into the finite region.

[Here the requirementsthat the travel segments be future-directed reduce to E(y1−a) > (x1−t2) > 0, whichcan always be fulfilled by adjusting t2. ]5.Topologically Massive GravityTopologically massive gravity (TMG) [17] is of interest because, in contrast to pureD=3 gravity, it is a dynamical theory.

In [13] exact solutions for lightlike sources,including (for certain orientations) two-photon solutions, were found. Here we show10

that they do not admit CTC for any values of the photons’ energies. [We cannotconstruct the analog of the “mixed” system since the exact solution for a massivesource is not known in TMG.] The field equations for TMG are the ghost (i.e.,with the opposite sign of κ2) Einstein equations, to which is added the conformallyinvariant, conserved, symmetric Cotton tensor Cµν ≡ǫµαβDα(Rβ ν −14δνβR):Eµν ≡Gµν + 1µCµν = −κ2Tµν .

(5.1)Here µ is a parameter (whose sign is arbitrary) with dimensions of mass or inverselength. These equations can be solved exactly for a photon source [13], using theplane-wave ansatz (2.2).For a photon with energy E moving along the positivex−axis, the resulting spacetime metric is given byds2 = −dudv + dy2 + 2κ2Ef(y)δ(u)du2,f ≡(y + 1µ(e−µy −1))θ(y) .

(5.2)Observe that as µ →∞, f(y) →yθ(y) and (5.2) reduces to the ghost gravitationalone-photon solution (i.e., (2.3) with the opposite sign of κ2). Like its Einstein counter-part, the metric (5.2) can also be obtained by a cut-and-paste procedure, albeit not byusing Poincar´e transformations, since the spacetime is not flat along the u = 0, y > 0null halfplane.

After applying the coordinate transformation v →v + 2κ2Ef(y)θ(u)to remove the δ(u) factor in (5.2), it takes the formds2=−dudv + dy2 −2κ2Eθ(u)f ′(y)dudy=θ(u){−dud(v + 2κ2Ef(y)) + dy2} + θ(−u){−dudv + dy2} . (5.3)Clearly, this corresponds to identifying v on u = 0−with v + 2κ2Ef(y) on u = 0+.We can construct the two-photon spacetime, in the convenient frame where one pho-ton moves in the positive x-direction along y = a > 0 and the other in the negativex-direction along y = −a, by pasting together the one-photon solutions along they = 0 hyperplane.

This pasting is possible since, as in Einstein gravity, each of theone-particle solutions is both flat and has zero extrinsic curvature on this hyper-plane. The resulting spacetime consists in making cuts along the u = 0, y > a andv = 0, y < −a halfplanes and then identifying the point (x1, x1, y1) on the u = 0−side with (x1 +κ2Ef(y1 −a), x1 +κ2Ef(y1 −a), y1) on the u = 0+ side, and the point(−x2, x2, y2) on the v = 0−side with (−x2 −κ2Ef(−y2 −a), x2 + κ2Ef(−y2 −a), y2)on the v = 0+ side.

[We note that if the directions of both photons were reversed(corresponding to the parity operation x →−x), then this simple pasting prescrip-tion is not possible, since the individual one-photon solutions are no longer flat onthe y = 0 hyperplane. This reflects the parity violation implicit in the dependence11

of the field equations (5.1) on ǫµνρ.] The absence of CTC for all positive values ofE can be seen as follows.

Since f(y) ≥0, the time shift upon crossing a halfplanehas opposite sign relative to that of the pure gravity case, (2.3). Hence, to gain atimeshift one would have to cross the (null) halfplane from the u > 0 side, ratherthan from the u < 0 side, which is impossible for any timelike (or lightlike) curve.These conclusions obviously also apply to ghost Einstein gravity4 as well, since thelatter is just the µ →∞limit of TMG.6.ConclusionWe have examined, in the case where one or both of their sources are lightlike, thephysical difficulties associated with geometries that permit CTC in 2 + 1 Einsteingravity.

We first obtained the total energy in terms of the constituent parameters,using the geometric approach in which flat patches are identified through null boosts,and found the conditions for the systems to be tachyonic. We then showed that, asthe energy of a system first surpasses its tachyonic bound, CTC initially emerge atspatial infinity, then spread into the interior, but always remain present at infinity.This is, of course, the manifestation of the unphysical spatial boundary conditionsthat are the price paid for CTC, and are analogous to the Misner identificationsthat give rise to CTC in NUT space.

We also demonstrated, using the known two-photon solution in TMG, that this dynamical model, and therefore also its limit,ghost Einstein theory, never admits CTC.All sources of the 2+1 Einstein equations considered to date thus share the propertythat if they are physical—nontachyonic—they do not engender acausal geometries.These results add evidence in favor of both Einstein’s original hope and its recentavatar [18], that this is a universal property of general relativity.7.AcknowledgementsWe thank G. ‘t Hooft for useful conversations.This work was supported by theNational Science Foundation under grant #PHY88–04561.References[1] K. G¨odel, Rev. Mod.

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