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PHYSICAL REVIEW D 98, 124008 (2018)

New for rotating dust in (2 + 1)-dimensional

Grant N. Remmen* Center for Theoretical and Department of Physics, University of California, Berkeley, California 94720, USA, and Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

(Received 31 October 2018; published 11 December 2018)

Multiparameter solutions to the Einstein equations in 2 þ 1 are presented, with stress-energy given by a rotating dust with negative cosmological constant. The matter density is uniform in the corotating frame, and the ratio of the density to the vacuum energy may be freely chosen. The rotation profile of the dust is controlled by two parameters, and the circumference of a circle of a given radius is controlled by two additional parameters. Though locally related to known metrics, the global properties of this class of spacetimes are nontrivial and allow for new and interesting structure, including apparent horizons and closed timelike , which can be censored by a certain parameter choice. General members of this class of metrics have two Killing vectors, but parameters can be chosen to enhance the symmetry to four Killing vectors. The of these geometries, interesting limits, and relationship to the Gödel metric are discussed. An additional solution, with nonuniform dust density in a Gaussian profile and zero cosmological constant, is also presented, and its relation to the uniform-density solutions in a certain limit is discussed.

DOI: 10.1103/PhysRevD.98.124008

I. INTRODUCTION aspects of the AdS3=CFT2 case (see Ref. [8] and references therein for a review, as well as Ref. [9]). In the past century of Einstein’s theory of gravity, In this paper, a class of axially symmetric progress in general relativity has been shaped in large geometries in 2 þ 1 dimensions is presented. The cosmo- part by the discovery of new exact solutions to the field ’ logical constant is negative, and the energy-momentum tensor equations. From Schwarzschild s work to the present day, is described by pressureless, rotating dust, as in the Gödel exact solutions provide useful laboratories for investigating solution. However, this class of geometries is much more the properties of relativistic gravitating systems. general, with multiple free parameters describing the dust ’ Two exact solutions of particular interest are Gödel s density, (spatially dependent) rotation rate, and circumference spacetime [1] describing a rotating dust and Bañados, profile of the spacetime. Depending on the parameters, these Teitelboim, and Zanelli’s (BTZ) [2]. Both of spacetimes can exhibit many interesting properties, including these solutions describe geometries in 2 þ 1 dimensions, CTCs, apparent horizons, spacetime boundaries, enhanced with a negative cosmological constant. Gödel’s solution symmetry algebras, and geodesic completeness. demonstrated properties of closed timelike curves (CTCs) This paper is organized as follows. In Sec. II, the class of in general relativity, while the BTZ geometry showed that solutions is stated and the corotating timelike congruence black holes can exist in three-dimensional gravity. General is examined. Next, in Sec. III the causal structure of the relativity in three dimensions provides an interesting arena spacetime is investigated, including null congruences, in which to explore the geometric properties of spacetime, Cauchy horizons, geodesic completeness, and censorship without the complications endemic to a nonzero Weyl of the CTCs. Particular limiting geometries are discussed tensor [3]. Moreover, (2 þ 1)-dimensional spacetimes with in Sec. IV, and the symmetries of the spacetime are found a negative cosmological constant are of current interest as a in Sec. V. We examine various special cases of interest in result of the AdS=CFT correspondence [4–7], in particular, Sec. VI and conclude with the discussion in Sec. VII.In Appendix, another new spacetime geometry is exhibited, with a spinning dust with Gaussian density profile, which * in a certain limit reduces to a member of the family of [email protected] geometries we present in Sec. II. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. II. A CLASS OF SPINNING DUST SOLUTIONS Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, Consider the following stationary metric in 2 þ 1 and DOI. Funded by SCOAP3. dimensions:

2470-0010=2018=98(12)=124008(11) 124008-1 Published by the American Physical Society GRANT N. REMMEN PHYS. REV. D 98, 124008 (2018)

2 dr spacetime is stationary. However, it is not static for j1 ≠ 0, ds2 ¼ −dt2 − wðrÞdtdϕ þ DðrÞdϕ2 þ ; ð1Þ NðrÞ since there do not exist hypersurfaces everywhere orthogo- nal to orbits of the timelike Killing vectors, or equivalently ’ ∇ ≠ 0 where by Frobenius s theorem [10], u½a buc . For this timelike congruence, we can define a spatial L metric hab ¼gab þuaub; the extrinsic is Bab ¼ wðrÞ¼2L j1 þ j2 ; ∇ r bua. Then we find that the expansion, measuring the 2 logarithmic derivative of the area element along the con- 2 2 L 2 L 2 DðrÞ¼−j ML þ 2c1L þ c2L ; gruence, vanishes, 1 r r 2 3 r j1j2 þ c1 r θ ¼ habB ¼ 0: ð6Þ NðrÞ¼4ð1 − MÞ þ 8 ab L j2 L 1 2 þ 4 Similarly, the shear vanishes, implying that the dust is rigidly þ 4 j2 c2 r ð Þ 3 2 : 2 rotating, j1 L 1 ς ¼ Bð Þ − θh ¼ 0: ð7Þ Here, j1, j2, c1, c2, and M are unitless constants and ab ab 2 ab −∞

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t

r r0 r0 rC φ

t rC

r0 φ rC r φ r

FIG. 1. Illustration of the null congruences orthogonal to circles of fixed r (gray arcs) on some slice of constant t (green surface). The outgoing and ingoing orthogonal null congruences have tangents given in Eq. (9) by ka and la and are illustrated by red and blue arrows, respectively. For this illustration, j1j2 was chosen to be negative, so the frame dragging vanishes on the surface at radius r0 (dotted line) where wðr0Þ¼0 and goes in the þϕ or −ϕ direction inside and outside, respectively. As the Cauchy horizon rC (dashed line) is approached, the orthogonal null congruences are dragged around the angular direction. For r

ϕ ϕ Despite the fact that k , l ≠ 0, one can verify that which is a consequence of the fact that the congruences ϕa ¼ l ϕa ¼ 0 ϕa ¼ ∂ ¼ð0 0 1Þ ka a , for ϕ ; ; giving the unit are hypersurface orthogonal. The twist one-form gauge tangent to σ. We note that k2 ¼ l2 ¼ 0, and we have field (i.e., the Hájíček one-form) is

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1 ω ¼ L lb ¼ − blc∇ a 2 qab k qa bkc vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u ð Þ ð Þ 0ð Þ − ð Þ 0ð Þ ¼ t N r w r D r D r w r 1 þ w2ðrÞ 8DðrÞ2 4DðrÞ × ðwðrÞ; 0; −2DðrÞÞ; ð13Þ where Lk denotes the Lie derivative along k. Were we to choose parameters for which we can have DðrÞ > 0 when NðrÞ¼0, the metric in Eq. (1) would have apparent horizons, where θk or θl vanish, at the zeros of NðrÞ located at r ¼ r, 2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2ð 2 þ Þ ¼ j1j2 c1 4−1 1 − ð1 − Þ j1 j2 c2 5 r L 2 M 2 : j2 þ c2 ðj1j2 þ c1Þ

ð14Þ

Whether these zeros exist (i.e., whether r is real and positive) in the spacetime depends on the relative signs and magnitudes of j1, j2, c1, c2, and M. FIG. 2. Cases in Eq. (17) for geometry with metric given in To guarantee that the angular direction is not timelike at Eqs. (1) and (2). The singularities in the rr component of the ≥ 0 0 large r, we must take c2 . If we choose c1 > , then metric [the zeros of NðrÞ] are r, while the Cauchy horizon rC is j2 ϕϕ ð Þ DðrÞ would reach a maximum at r ¼ r ¼ 1 ML. In this determined by the zero of the component of the metric D r . m c1 In Cases I and II, 0 0, while in Cases II and IV, an apparent horizon at r , where the null expansions can þ 0 ≥ 0 ≤ 0 ≥ m j1j2 c1 < .Ifc2 and c1 , then rC r. vanish. For the rest of this paper, we will consider c1 ≤ 0, so that D0ðrÞ > 0; this ensures that circles of constant r have circumferences that grow with r asymptotically, as we 0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi would intuitively expect. In that case, σ is a normal surface, 2 ¼ − c1 @1 þ 1 þ j1Mc2A ð Þ i.e., a surface for which the outward future-pointing null rC L 2 : 16 c2 c congruence has positive expansion and the inward future- 1 pointing null congruence has negative expansion. We then have the following conditions for the existence of ð Þ B. Cauchy horizon and boundary the zeros of N r in Eq. (14): This spacetime can exhibit Cauchy horizons, defining Case I: 00⇒no zero; the boundary of the nonchronological region of the spacetime where DðrÞ < 0. In this region, circles in ϕ at Case II: 00⇒one zero at rþ; by the zeros of DðrÞ, 0 1 ≥1 þ 0⇒ ð Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Case IV: M ;j1j2 c1 < one zero at r−: 17 2 c1 j Mc2 C ¼ − @1 1 þ 1 A ð Þ r L 2 : 15 c2 c1 See Fig. 2 for an illustration. The nonchronological region can extend outside the apparent horizon; i.e., it is possible ≤ 0 By the null energy condition, we will take M ≥ 0, so that to have rC >r. Indeed, with our assumptions c1 and ≥ 0 the dust has non-negative density. Further, taking c2 ≥ 0 c2 , one can show that the Cauchy horizon always ≥ and c1 ≤ 0 as noted previously, the nonchronological satisfies rC r, which implies that the congruence in C Sec. III A cannot be extended down to r [since the region is given by r 0 by Eq. (4)]. Hence, if c1 ≤ 0 and c2 ≥ 0, the zeros of NðrÞ are not truly apparent horizons where θ or θl vanish, since 6Note that the appearance of CTCs does not run afoul of the k theorem in Ref. [20], since the CTCs extend arbitrarily close to the congruence does not exist there. r ¼ 0, and thus constitute “boundary CTCs” in the sense of In the region where NðrÞ < 0, we can consider a surface Ref. [20]. μ at constant r. Let us attempt to construct the orthogonal

124008-4 NEW SPACETIMES FOR ROTATING DUST IN (2 þ 1)- … PHYS. REV. D 98, 124008 (2018) null congruences from μ. Writing one of the tangent vectors is of non-Lorentzian causal structure. We therefore exclude ¯a ¯ to such a congruence as k , we must have kϕ ¼ 0 for the this region from the spacetime and take a surface at r ¼ r congruence to be orthogonal to μ. Thus, we have to be a boundary of the geometry.

ðk¯ Þ2 k¯2 ¼ðk¯ Þ2NðrÞ − t : ð18Þ C. Geodesics and completeness r 1 þ w2ðrÞ 4DðrÞ Let us consider the evolution of a timelike geodesic in this spacetime. The geodesic equation, Since we have NðrÞ < 0 on μ and since Eq. (2) requires ≠ 0 ð Þ 0 μ j1 , Eq. (4) implies that we must have D r < on . 2 2 d xa dxb dxc By Eq. (4) again, we then have 1 þ w ðrÞ=4DðrÞ > 0 on μ, þ Γa ¼ 0 ð Þ 2 bc ; 19 which by Eq. (18) makes the requirement k¯2 ¼ 0 impos- dτ dτ dτ ¯ ¯ sible to satisfy for any choice of kt and kr. Hence, the null congruences from μ do not exist; i.e., the region NðrÞ < 0 implies

j2L6NðrÞ d2t dr dϕ dr dϕ dt 1 − 2wðrÞD0ðrÞ þ w0ðrÞ 2DðrÞ þ wðrÞ ¼ 0; r4 dτ2 dτ dτ dτ dτ dτ d2r N0ðrÞ dr 2 NðrÞ dϕ dt dϕ − þ w0ðrÞ − D0ðrÞ ¼ 0; dτ2 2NðrÞ dτ 2 dτ dτ dτ j2L6NðrÞ d2ϕ dr dϕ dt 1 þ ð4D0ðrÞþwðrÞw0ðrÞÞ − 2w0ðrÞ ¼ 0; ð20Þ r4 dτ2 dτ dτ dτ

2 where we used Eq. (4) to write 4DðrÞþw ðrÞ in terms of and r>rþ in Case III; and the two regions rr− in Case IV) can be essentially regarded as separate in this limit, we must have spacetimes. Indeed, if we replace the right-hand side of the geodesic equation with the proper acceleration, we see that 2 0 2 d r N ðrÞ dr any trajectory with nonzero dr=dτ at r ¼ r will neces- − ¼ 0; ð21Þ dτ2 2NðrÞ dτ sarily undergo infinite proper acceleration; the barrier at r ¼ r is insurmountable. 0 where we can approximate NðrÞ by N ðrÞðr − rÞ. Thus, This example suggests that the spacetime in the region ð Þ near r, we have the solution where N r is positive, bounded by r, is, in fact, geodesi- cally complete. Indeed, we can verify this with the help of v2τ2 the theorem proven in Ref. [21]. In the classification of ðτÞ − ¼ − þ τ þ i ð Þ r r ri r vi 4ð − Þ ; 22 ri r where ri and vi are constants. Note that ri can be either larger or smaller than r, so this solution applies to timelike geodesics that originate on either side of r. We find from Eq. (22) that the geodesic can never cross the surface at ¼ ðτÞ τ ¼ −2ð − Þ r r: r has a turning point when ri r =vi, at rðτÞ¼r. That is, the timelike geodesic can just touch the surface and bounce off of it, but cannot pass through; see Fig. 3. Replacing τ by an arbitrary parameter λ respecting the affine connection, the analogous conclusion applies for both null and spacelike geodesics, which also bounce off of this boundary. (Given ri and vi, whether the geodesic is timelike, null, or spacelike can be specified by choosing the initial data in the t and ϕ components of the FIG. 3. Geodesics bouncing off of a boundary at r ¼ r. geodesic equation.) This justifies our considering r to be Depicted is a family of geodesics with different initial data ri boundaries of the spacetime. That is, the connected regions and vi chosen to all intersect the boundary at some fixed of the fixed sign of NðrÞ (i.e., the three regions rr− in Case II; the two regions r

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Ref. [21], the family of metrics given in Eq. (1) fits IV. LIMITING GEOMETRY the definition of a Gödel-type spacetime provided In this section, we will keep the choice in Eq. (23) that w2ðrÞþ4DðrÞ>0 7 . By Eq. (4), this is true whenever we made in Sec. III D, allowing us to exclude the CTCs j1 ≠ 0 NðrÞ > 0 and . Hence, our geometry is within the from the geometry; as we saw in Sec. III C, we can drop the class of Gödel-type metrics (a class that includes the Gödel region r

4 2 2 2 2 d3 2 D. Censoring CTCs ds ¼ −dt − 2d1dtdϕ þ d dϕ þ dr ; ð25Þ 2 r4 ≤ 0 ≥ 0 To avoid rC >r− (while keeping c1 and c2 ), we 4 2 must take either Case II or Case IV above and further 2 2 4 L j1 where d1 ¼ Lj2, d2 ¼ c2L , and d3 ¼ 2 . Defining 4ðj2þc2Þ require the tuning of c1 and c2 such that 0 2 0 2 2 2 r ¼ d3=r, t ¼ t þ d1ϕ, and d4 ¼ d1 þ d2,wehave 2 c1 þ ¼ c2 ð Þ j2M 23 2 0 2 2 2 0 2 j1 j2 ds ¼ −ðdt Þ þ d4dϕ þðdr Þ ; ð26Þ and also j1j2 < 0. With the choice (23), the radius of the ð Þ¼0 which is just the geometry of a flat Lorentzian cylinder, outer of the N r surfaces, which we will call rH,is 1;1 1 R ⊗ S , of radius d4. In particular, at constant ϕ, this is located at simply two-dimensional , and the asymp-

totic causal structure includes null infinity. However, the ¼ ¼ ¼ ¼ − j1 ¼ j1 ð Þ 0 rH r− rC r0 L L: 24 coordinate t has a “jump” or “winding” property due j2 j2 to its dependence on ϕ, as discussed in Ref. [25].

In this case, the radius r0 where the frame-dragging reverses direction, the NðrÞ¼0 surface at r−, and the B. Near-boundary geometry ≤ 0 Cauchy horizon at rC all coincide. While requiring c1 Let us define a near-boundary coordinate x, where ≥ 0 2 − and c2 , it is not possible for the Cauchy horizon x ¼ r rH. For small x, the metric becomes to be located a finite distance inside the NðrÞ¼0 surface, rH i.e., r ≮r. 3 C 2 2 2 2L 2 2 The choice of coefficients in Eq. (23), necessary to ds ¼ −dt − 2j2Lx dtdϕ − ðc1 þ j1j2MÞx dϕ rH censor the CTCs, immediately leads us to an observation 2 2 ð Þ¼0 j2Lr dx about the boundary. Since Eq. (23) implies that D rH , − H θ 2ðc1 þ j1j2MÞ one finds from Eq. (10) that limr→rH k is not zero but is ω → 2 2 2 2 2 2 2 instead infinite. However, the twist a vanishes as r rH ¼ −dt − 2l3x dtdϕ þ l1x dϕ þ l2dx ; ð27Þ with the choice in Eq. (23). Hence, for generic c1 and M, the choice in Eq. (23) is incompatible with a smooth, where we eliminated c2 using Eq. (23) and defined 3 2 marginally trapped horizon. We stress that this is not 2 2L 2 j2LrH l1 ¼ − ðc1 þ j1j2MÞ, l2 ¼−2ð þ Þ, and l3 ¼ j2L. a curvature singularity (neither s.p. nor p.p. type in the rH c1 j1j2M Recalling that c1 ≤ 0, j1j2 < 0, and M ≥ 0,wehave sense of Ref. [23]), since all curvature scalars for this metric 2 2 c1 þ j1j2M ≤ 0, and hence the dϕ and dx terms are are regular and, moreover, explicit computation shows that ϕ the full Riemann tensor is everywhere finite in a local spacelike. The dtd term still induces frame dragging, 8 but let us consider spatial slices at constant t. The spatial Lorentz frame. Instead, rH is a boundary in the Lorentzian structure of the , a singularity in the causal metric is just structure in the sense of Ref. [24]. l2 1 ð 0Þ2 ϕ2 þð 0Þ2 ð Þ 2 x d dx ; 28 l2 7 Note that the conclusion of Ref. [22], which exhibited a class 0 of metrics equivalent to the Gödel universe, does not apply to our where we have defined x ¼ l2x. This is simply the metric metric (1), since we will see in Sec. VI A that our class of metrics of a two-dimensional plane with a conical defect. Defining is strictly larger than the Gödel solution. 0 l1 0 2 0 2 8 μ ν ρ σ μ ϕ ¼ l ϕ so that the spatial metric is simply ðx Þ ðdϕ Þ þ That is, e ae be ce dRμνρσ is finite, where the dreibein ea is 2 μν μ ν ab 0 2 0 0 defined as g ¼ e ae bη for flat metric η. ðdx Þ , ϕ takes the range 0 ≤ ϕ < 2πδ, where

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l 1 c1 effective mass is negative. Indeed, m is unbounded from δ ¼ ¼ 2 þ j2M : ð29Þ l2 j1 below; as we increase M, δ can grow without limit, and m We could make this conical defect vanish by enforcing can become arbitrarily large and negative. l1 ¼ l2, i.e., δ ¼ 1, in which case we could simply take the ¼ entire surface at r rH to be a single point, with a flat spatial V. SYMMETRIES metric, albeit with frame dragging present. Without this ¼ Let us now consider the symmetries of our metric in condition, we have a conical singularity if we take r rH ð Þ 0 to be identified as a single point. Interpreting the conical Eq. (1). For a region where N r > (e.g., r>max r), defect as a mass m in a flat background [25,26],wehave the family of geometries described by Eq. (1) has a Lie m ¼ð1 − δÞ=4G. Since δ is manifestly positive, we have algebra of Killing vectors v each satisfying the Killing ∇ ¼ 0 m<1=4G, so the universe is not overclosed. When δ>1, this equation ðavbÞ . The basis of this algebra is given by

a u ¼ð1; 0; 0Þ¼∂t; a ϕ ¼ð0; 0; 1Þ¼∂ϕ; 3 ð Þ pffiffiffiffiffiffiffiffiffiffi 2 ð Þ d ð D r Þ d 2 w r dr 2ð Þ ½4DðrÞþw ðrÞ χa ¼ NðrÞ − w r sin αϕ; −2α cos αϕ; dr sin αϕ ; 4DðrÞþw2ðrÞ 4DðrÞþw2ðrÞ 3 ð Þ pffiffiffiffiffiffiffiffiffiffi 2 ð Þ d ð D r Þ d 2 w r dr 2ð Þ ½4DðrÞþw ðrÞ ψ a ¼ NðrÞ w r cos αϕ; −2α sin αϕ; − dr cos αϕ ; ð30Þ 4DðrÞþw2ðrÞ 4DðrÞþw2ðrÞ where VI. SPECIAL CASES qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Let us now compute some special cases and limits of 2 2 2 2 Eq. (1) that are of physical interest, including the choice of α ¼ − 2j1j2c1 þ c1 þ j1½ðM − 1Þc2 þ j2M: ð31Þ j1 parameters that leads to the Gödel metric and the behavior of the geometry in the Λ→0, nonspinning, and M→0 limits. The vector ua is the global timelike Killing field we encountered previously, while the vector ϕa is the angular A. Relationship to Gödel universe Killing field associated with the axial symmetry of the Choosing spacetime. The Killing vectors χ and ψ exist only if α is an integer, so that χ and ψ are single-valued. That is, for 1 M ¼ ; general values of the parameters ðM; j1j2;c1;c2Þ, the 2 symmetry algebra is two-dimensional. j1 ¼ −2; If ψ and χ are well-defined periodic Killing vectors, so that the Lie algebra of symmetries is four-dimensional, j2 ¼ 0; then α ¼ n, i.e., c1 ¼ 1;

2 2 c2 ¼ 0; ð33Þ 2 þ 2 þ 2½ð − 1Þ þ 2 ¼n j1 ð Þ j1j2c1 c1 j1 M c2 j2M 4 32 and definingpffiffiffiffiffiffiffiffi new unitless radialpffiffiffi and time coordinates r0 ¼ arcsinh L=r and t0 ¼ t= 2L, our metric in Eq. (1) for some integer n. In that case, we can reach any spacetime reduces to the metric of the Gödel universe in 2 þ 1 point in the connected region where NðrÞ > 0 using these dimensions [1,23]: Killing vectors, by moving in t using ua,inϕ using ϕa, and a a a a in r using χ or ψ in combination with u and ϕ . Hence, 2 2 02 02 4 0 2 0 2 ds ¼ 2L ½−dt þ dr − ðsinh r − sinh r Þdϕ as in the Gödel metric, the isometry group acts transitively pffiffiffi and the spacetime region for NðrÞ > 0 is homogeneous. þ 2 2sinh2r0dϕdt0: ð34Þ Thus, if Eq. (32) is satisfied but not Eq. (23), so that CTCs extend to the region where NðrÞ is positive, the spacetime Hence, our metric constitutes a generalization of the is everywhere vicious outside the boundary, with CTCs Gödel metric. Unlike the Gödel universe, in Eq. (1) the through every point (so the Cauchy horizon disappears). general class of metrics (i) corresponds to a matter density We will discuss the relationship between Eq. (1) and the ∝ M that is a free parameter, rather than being pinned to Gödel metric further in Sec. VI A. the cosmological constant, (ii) has two distinct angular

124008-7 GRANT N. REMMEN PHYS. REV. D 98, 124008 (2018) momentum parameters j1 and j2, allowing the spin of the vectors as expected, with Eq. (32) satisfied with n ¼ 1. dust to be nonuniform and even reverse direction at different Note that the converse is not necessarily true, however: radii, and (iii) can have apparent horizons for certain choices satisfying α ¼ 1 does not imply that the metric is diffeo- of the parameters. morphic to the Gödel universe, since Eq. (32) does not One can alternatively view the Gödel solution not as pin the dust density to the cosmological constant (i.e., simply describing a dust with negative cosmological set M to 1=2) as is the case in the Gödel metric. constant, but instead as describing arbitrary matter content for which the in a local Lorentz frame B. Λ → 0 limit satisfies Gab ∝ diagð1; 1; 1Þ. For example, describing a ðM; j1;j2;c1;c2Þ Λ → 0 L → ∞ perfect fluid with arbitrary pressure, plus a cosmological For fixed , the (i.e., ) limit of the metric described in Eqs. (1) and (2) is singular. constant, in order to correspond to the Gödel solution one However, let us define the rescaled parameters must satisfy (in 8πG ¼ 1 units) ρ − p ¼ −2Λ. That is, in Λ ¼ 0 3 the case, the Gödel universe corresponds to a stiff ι1 ¼ L j1; ρ ¼ fluid with p [27]. Similarly, one can view the energy- ι ¼ momentum source for our metric in Eq. (1) as describing an 2 Lj2; 3 arbitrary perfect fluid plus cosmological constant for which κ1 ¼ L c1; 2 the Einstein tensor is L Gab ¼ diagð4M − 1; 1; 1Þ in a local 2 κ2 ¼ L c2; ρ þ Λ ¼ 4M−1 − Λ ¼ 1 Lorentz frame, that is, L2 and p L2. μ ¼ M ð Þ Taking the cosmological constant to vanish, the equa- 2 ; 35 ρ ¼ 1 L tion-of-state parameter p= 4M−1 corresponds to a stiff 1 and then take the L → ∞ limit, holding ι1, ι2, κ1, κ2, and μ fluid when M ¼ 2 in accordance with Eq. (33), a cosmo- logical constant when M ¼ 0, phantom energy when constant. In this case, the metric becomes 0 1 ¼ 3 → ∞ r . Moreover, if we impose the tuning (32), ds2 ¼ −ðdt¯Þ2 þ fðρÞdφ2 þ ; ð39Þ C fðρÞ so that the metric for r>max r is homogeneous, we can sequester the CTCs behind the boundary if we simulta- where neously impose Eq. (23). In that case, homogeneity guar- antees that there are no CTCs outside the boundary. The 4κ1 fðρÞ¼1 − 4μρ2 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ: ð Þ requirements of Eqs. (32) and (23) can simultaneously be 2 40 jι1j κ2 þ ι2 satisfied by first imposing Eq. (23) and then requiring 2jc1 þ j1j2Mj¼−j1n. We note, remarkably, that if κ1 ¼ 0, then the spatial sector With the choice of parameters (33), the parameter α in of Eq. (39) is simply a Euclideanized two-dimensional Eq. (31) is simply 1, so the Gödel metric has four Killing de Sitter (dS) space in static slicing; to make the analogy

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2 precise, one would require κ2 þ ι2 ¼ 1=4μ, so that the We can turn Eq. (44) into a metric that locally corresponds φ ϕ¯ ¼ ϕˆ − t periodicity in (the Wick-rotated time coordinate of the to AdS3 in global coordinates by defining L,in dS2 space) matches the dS length. terms of which we have ˆ2 ˆ2 −1 C. Nonspinning limit 2 ¼ − 1 þ r 2 þ 1 þ r ˆ2 þ ˆ2 ϕ¯ 2 ð Þ ds 2 dt 2 dr r d : 45 The geometry described in Eqs. (1) and (2) must always L L be spinning; there is no way to smoothly take the joint j1, If t is allowed to take values in all of R, then ϕ¯ spans the j2 → 0 limit while keeping the metric nonsingular. Specifically, the form of NðrÞ and DðrÞ dictate that we real numbers and the metric in Eq. (45) describes a covering space of AdS3 with decompactified time and angular cannot take j1 → 0. coordinate. To correspond to AdS3, we should make t However, we can set j2 to zero. In this case, the dtdϕ term periodic, 0 ≤ t<2πL. For consistency, we can then take in the metric asymptotes to zero for large r. The vorticity ˆ 2c1=j1 ¼ 1 so that ϕ ∈ ½0; 2πÞ, and we are left with the tensor (8) for the timelike congruence along ∂t then satisfies global AdS3 geometry. We note that if 2c1=j1 is any integer limr→∞Ωab ¼ 0, but the twist one-form for the null geo- ω ¼ ϕˆ ∈ ½0 2π Þ desics describedpffiffiffiffiffi in Sec. III A satisfies limr→∞ a n, we have ; n , which (for appropriate periodicity ð0; 0; ∓ c2Þ, where the þ case occurs if and only if j1 in t) corresponds simply to an n-fold cover of AdS3, which Z is negative (and vice versa). Thus, even when we send j2 to we may quotient by n to recover the single copy of the zero, information about the spin of the dust is imprinted on geometry. For any n, Eq. (23) and our M ¼ 0 choice the twist of null geodesics at arbitrarily large r, even though together then imply that Eq. (32) is satisfied, enhancing the the frame-dragging wðrÞ asymptotically vanishes. symmetry algebra (which is expected, since AdS3, in fact, possesses six Killing vectors). D. Vacuum limit Hence, we find that in the CTC-free case where we impose Eq. (23), our metric (1) obeys the extension of Let us consider the vacuum limit of the geometry Birkhoff’s theorem to 2 þ 1 dimensions [29], which states ∝ described in Eq. (1). Taking the gas density ( M) to zero, that any solution of the (2 þ 1)-dimensional vacuum we find that the Cauchy horizon is located at Einstein equations with a negative cosmological constant that is free of CTCs must correspond to AdS3, a BTZ ¼ −2 c1 ð Þ geometry [2], or a Coussaert-Henneaux [30] solution. rC L: 41 c2 VII. DISCUSSION For r>rC, let us define The metric in Eq. (1) contains six free parameters: a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi length L and five unitless constants ðM; j1;j2;c1;c2Þ. 2c1L r˜ ¼ L þ c2; ð42Þ Choosing a value of L effectively sets the length scale r of the geometry (relative to the three-dimensional Newton’s constant). The remaining five parameters specify a five- ð Þ ϕ2 → ˜2 ϕ2 so D r d r d . If we continue to impose the con- dimensional space of geometries that, for general choices of ¼ 2 dition (23) (which has simply become c2=j2 c1=j1 for the constants, remain distinct under diffeomorphisms. The ¼ 0 M ) in order to sequester the CTCs behind a boundary, parameter M can be measured in the geometry through the then the metric becomes value of Rtt, while c2 can be measured by the proper circumference of the spacetime at large r. Moreover, the ˜2 4 2 ˜2 −1 ð Þ ð Þ ð Þ 2 ¼ − 2 − j1r ϕ þ ˜2 ϕ2 þ c1 þ r ˜2 existence and location of the zeros of N r , D r , and w r ds dt dtd r d 2 2 dr : c1L j1 L (i.e., r, rC, and r0, respectively) allow various combina- tions of the remaining three parameters to be measured in ð43Þ the spacetime. These locations are all physically mean- ingful: the rr component of the metric flips sign at r, ˜ ¼ 0 The boundary is located at r , so the CTC region behind r r0 2 CTCs appear at C, and the frame dragging vanishes at . the boundary corresponds to the analytic continuation of r˜ The relative order and proper distances among these to negative values; cf. Ref. [24]. locations and the axis of rotation allow us to conclude Next, we can rescale the radial coordinate again, define that ðM; j1;j2;c1;c2Þ indeed specifies a five-dimensional 2 4π rˆ ¼jj1 jr˜, and also define ϕˆ ¼ c1 ϕ, 0 ≤ jϕˆ j < j c1 j,so 2c1 j1 j1 family of globally inequivalent geometries. This paper leaves multiple interesting avenues for further 2ˆ2 ˆ2 −1 research. We leave for future work the subjects of the 2 ¼ − 2 − r ϕˆ þ ˆ2 ϕˆ 2 þ 1 þ r ˆ2 ð Þ ds dt dtd r d 2 dr : 44 dynamical stability of these geometries and their formation L L from the collapse of a cloud of dust. The question of what

124008-9 GRANT N. REMMEN PHYS. REV. D 98, 124008 (2018) other three-dimensional geometries exhibit an analogue of 1 2 2 mðrÞ¼ e−r =a ; ðA2Þ the bouncing boundary discussed in Sec. III C is also a a2 compelling one. Further, it would be interesting to study the causal structure of the c1 > 0 case mentioned in Sec. III A, for some length parameter a. The metric for the solution is in which an apparent horizon appears at radius r and DðrÞ m 2 2 2 2 2 2 2 2 asymptotically decreases. The appearance of the analyti- ds ¼ −dt − 2adtdϕ þðr − a Þdϕ þ er =a dr : ðA3Þ cally continued de Sitter metric in Sec. VI B also merits investigation; the double Wick rotation is reminiscent of The solution has a two-dimensional isometry algebra a a a b the “bubble of nothing” solution of Ref. [31]. Finally, an generated by Killing vectors u and ϕ . Since u ∇au ¼0, examination of the spatial geodesics of this class of metrics, ua generates a timelike congruence as before, for which we how they can be embedded within a surrounding vacuum to find that the expansion θ, shear ς, and vorticity Ω all vanish. produce an asymptotically anti–de Sitter geometry, and a The solution is clearly distinct from Gödel’s universe [1] subsequent computation of their Ryu-Takayanagi surfaces (since the dust density is not uniform) and van Stockum’s [32] are well-motivated future directions from a holo- solution [15] (since the dust density is finite in the closure of graphic perspective. the geometry and goes to zero at the boundary). Such a rotating dust solution, with higher density in the center and ACKNOWLEDGMENTS exponentially falling density at large distances, may be of astrophysical use if extended to four dimensions. In a sense, It is a pleasure to thank Raphael Bousso, David Chow, this solution may be thought of as corresponding to a Metin Gürses, Illan Halpern, and Yasunori Nomura for (2 þ 1)-dimensional galaxy. However, the solution pos- useful discussions and comments. This work is supported sesses CTCs for r

APPENDIX: GAUSSIAN DUST SOLUTION 2 dr ds2 ≃−dtˆ2 þ r2dϕ2 þ ; ðA4Þ In this Appendix, we discuss an additional, apparently r2 1 − 2 new solution of the Einstein equations in 2 þ 1 dimensions. a Although not globally a member of the family of metrics where we define tˆ ¼ t þ aϕ. The metric (A4) is a member presented in Eqs. (1) and (2), we will find that it is related in of the family of Λ ¼ 0 solutions discussed in Sec. VI B; a certain limit. Consider a dust solution to the Einstein if we set the parameters in Eq. (35) to μ ¼ 1=4a2, equations with zero cosmological constant, 2 ˆ κ1 ¼ ι2 ¼ 0, κ2 ¼ a , writepffiffiffiffiffiffiffiffiffiffiffiffiffiffit in Eq. (36) as t, and send 2 2 1 r in Eq. (36) to jι1j=2a a − r , we recover Eq. (A4). − ¼ ð Þ ð Þ Rab 2 Rgab m r uaub; A1 Thus, as we would expect, the small-r (and thus nearly constant density) limit of the metric (A3) corresponds to a a with vector u ¼ ∂t as before. Here, let us take the density member of the family of uniform-density rotating dust mðrÞ to have a Gaussian profile, solutions in Eq. (1).

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