A Lemaitre-Tolman-Bondi cosmological wormhole
aa r X i v : . [ g r- q c ] J u l A Lemaˆıtre–Tolman–Bondi cosmological wormhole
I. Bochicchio ∗ Dipartimento di Matematica e Informatica, Universit´a degli Studi di Salerno,Via Ponte Don Melillo, 84084 Fisciano (SA), Italy
Valerio Faraoni † Physics Department, Bishop’s UniversitySherbrooke, Qu´ebec, Canada J1M 1Z7
We present a new analytical solution of the Einstein field equations describing a wormhole shell ofzero thickness joining two Lemaˆıtre–Tolman–Bondi universes, with no radial accretion. The materialon the shell satisfies the energy conditions and, at late times, the shell becomes comoving with thedust-dominated cosmic substratum.
PACS numbers: 04.20.-q, 04.20.Jb, 98.80.-k
INTRODUCTION
Static and asymptotically flat wormhole solutions ofthe Einstein equations have been known for a long time[1]. The study of wormholes developed with the seminalpaper by Morris and Thorne [2], after which many solu-tions were discovered (see [3] for an extensive discussion).The possibility that inflation in the early universe mayenlarge a Planck size wormhole to a macroscopic size ob-ject was contemplated in Ref. [4]. Dynamical wormholeswere discovered and studied in Refs. [5, 6] and worm-holes in cosmological settings were contemplated in var-ious works ([7, 8] and references therein), with particu-lar attention being paid to wormholes with cosmologicalconstant Λ, which are asymptotically de Sitter or anti–deSitter according to the sign of Λ [9].After the 1998 discovery of the present acceleration ofthe universe [10] and the introduction of dark energy incosmological theories to account for this cosmic accel-eration, there were claims that phantom energy, an ex-tremely exotic form of dark energy with
P < − ρ (where P and ρ are the pressure and the energy density, respec-tively) could cause the universe to end with a Big Rip sin-gularity at a finite time in the future [11]. There was thena claim in the literature [12] that, if a wormhole accretesphantom energy, it grows to enormous size faster thanthe background universe, swallowing the entire cosmoswhich would then tunnel through the wormhole throatand re-appear in a different portion of the multiverse be-fore reaching the Big Rip singularity. This claim wasbased on qualitative arguments and was later disprovedby two classes of exact solutions of the Einstein equa-tions representing wormholes embedded in a cosmologicalbackground dominated by phantom energy [13]. Thesewormholes accrete phantom energy but, even if their ex-pansion rate differs from that of the cosmic substratuminitially, they become comoving with it as the scale factorof the Friedmann–Lemaˆıtre–Robertson–Walker (FLRW)universe in which they are embedded grows.The first class of solutions consists of a zero-thickness shell which carries exotic energy and does not perturbthe two copies of the FLRW universe which it joins. Thesecond, more realistic, class is described by a general-ized McVittie metric [14] with an imperfect fluid and aradial energy flow, with the mass of the wormhole shelldistorting the surrounding FLRW metric [13]. Another,less general, solution of the Einstein equations describinga cosmological wormhole comoving with the backgroundwas presented in Ref. [15].Cosmological wormholes are truly dynamical and inter-est in this kind of solution has developed in parallel withthe increasing attention paid to cosmological black holes[16]. Additionally, gravitational lensing by wormholeswas studied in [17] and numerical solutions interpretedas wormholes in accelerating FLRW universes were pre-sented in Refs. [18]. Recently, Maeda, Harada, and Carrhave given precise definitions for general cosmologicalwormholes and have found two new exact solutions ofthis kind [19]. An important result of this work, whichechoes a previous result of [7], is that the null energy con-dition needs not be violated in this dynamical situation,although it must be violated for static wormholes to exist[19]. It seems that the study of cosmological wormholesis developing into a promising new area of research.In this paper we propose a new analytical solutionof the Einstein field equations describing a cosmologi-cal wormhole shell joining two Lemaˆıtre–Tolman–Bondi(LTB) universes. We are led to this solution by the fol-lowing considerations: the second class of solutions inRef. [13] is inspired by the McVittie metric, which de-scribes a central inhomogeneity in a FLRW background.However, the McVittie metric needed to be general-ized by removing the McVittie “no accretion” condition G = 0 (in spherical coordinates) which forbids radialenergy flow. The goal of Ref. [13] was to describe theeffect of the accretion of phantom energy onto the worm-hole. Here, we begin by noting that inhomogeneities em-bedded in a FLRW background are usually described byusing an LTB metric [20–22], not a McVittie one. Theclassical LTB metric describes a spherically symmetricinhomogeneity in a dust–dominated FLRW background.The Bondi condition G = 0 parallels the McVittie no-accretion condition and forbids the (radial) flow of cosmicdust onto the inhomogeneity. It would be interesting toobtain a solution describing a wormhole shell joining twoidentical LTB universes and perturbing its surroundingsin the way described by the LTB metric. This is what wedo here. We obtain a wormhole shell composed of exoticmatter which expands more slowly than the cosmic sub-stratum (which becomes a spatially flat FLRW universeat late times), but eventually becomes comoving with it.The next section details how to construct the wormholeshell and satisfy the Israel–Darmois–Lichnerowicz junc-tion conditions [23] on this shell. The Einstein equationson the shell provide expressions for the energy densityand pressure of the material on the shell. Sec. III usesthe covariant conservation equation to relate the rate ofchange of the mass of shell material, the shell area, andthe flux of cosmic fluid onto the shell due to the relativevelocity between the shell and the cosmic substratum.The metric signature is − + ++, we use units in whichthe speed of light and Newton’s constant are unity, andwe follow the notations of Ref. [25]. Greek indices runfrom 0 to 3 and Latin indices assume the values 0 ,
1, and2 corresponding to the coordinates ( t, θ, ϕ ) of the spher-ical hypersurface Σ defined below.
THE LTB WORMHOLE SOLUTION
The spherically symmetric LTB line element for thecritically open universe in polar coordinates ( t, r, ϑ, ϕ ) is ds = − dt + [ R ′ ( t, r )] dr + R ( t, r ) d Ω , (1)where R ( t, r ) = (cid:18) r / + 32 p m e ( r ) t (cid:19) / (2)is an areal radius, r is a comoving radius, m e ( r ) = 4 π r Z dx x ρ ( x ) , (3) ρ ( r ) is the energy density on an initial hypersurface,a prime denotes differentiation with respect to r , and d Ω ≡ dϑ + sin ϑ dϕ . The line element (1) describesa spherical inhomogeneity in a dust–dominated universe([20–22]; for a recent review see [24]).Consider now a wormhole shell Σ at r = r Σ ( t ) whichjoins two identical copies of an LTB spacetime (this shelldescribes a wormhole created with the universe and notformed as the result of a dynamical process after theBig Bang). The wormhole shell is dynamical and moves,possibly also relative to the cosmic substratum, and its motion is described by the form of the function r Σ ( t ). Itis convenient to write the equation of this shell as [27] f ( t, r ) ≡ R − R Σ ( t, r Σ ( t ) ) = 0 . (4)To find the unit normal to Σ we first compute N µ ≡ ∇ µ f = ∇ µ ( R − R Σ ) = (cid:16) R t − ˙ R Σ , R ′ , , (cid:17) , (5)and N µ = (cid:18) ˙ R Σ − R t , R ′ , , (cid:19) , (6)and then normalize according to n µ = α N µ . Here R t ≡ ∂R/∂t and an overdot denotes a total derivative withrespect to t , i.e. , ˙ R ≡ dR/dt .The normalization n µ n µ = 1 yields α = 1 r − (cid:16) R t − ˙ R Σ (cid:17) . (7)It is convenient to introduce the radial velocity of thewormhole shell relative to the cosmic substratum v ≡ ˙ R Σ − R t | Σ , (8)where R t | Σ ≡ R t ( t, r Σ ( t )). Then α = 1 √ − v = γ ( v ) (9)is an (instantaneous [28]) Lorentz factor for the relativemotion shell–background. The unit normal is then n µ = ( − γ v, γ R ′ , , , (10) n µ = (cid:16) γ v, γR ′ , , (cid:17) . (11)The restriction of the metric to Σ is given by ds | Σ = − dt + R ′ Σ dr | Σ + R d Ω (12)or, using the fact that ˙ R Σ = R t | Σ + R ′ Σ ˙ r Σ on the shell, ds | Σ = − (1 − v ) dt + R d Ω , (13)which expresses the fact that the proper time τ of theshell is given by dτ = p − v dt , (14) i.e. , it is Lorentz-dilated with respect to the comovingtime t of the background.Using the triad n e α ( t ) , e α ( ϑ ) , e α ( ϕ ) o = np − v δ αt , δ αϑ , δ αϕ o , (15)the extrinsic curvature of the shell is given by ( a, b, c = t, ϑ , or ϕ ) K αβ = e ( a ) α e ( b ) β ∇ a n b = e ( a ) α e ( b ) β ( ∂ a n b − Γ cab n c ) , (16)where Γ cab are the Christoffel symbols of the 3–dimensional metric g ab | Σ . Eq. (16) yields K tt = γ ( ∂ t n t − Γ ttt n t ) = − v t γ − γv t v ,K ϑϑ = − Γ tϑϑ n t = γ v R Σ R t | Σ ,K ϕϕ = K ϑϑ sin ϑ . (17)The mixed components of the extrinsic curvature are K tt = γ v t (cid:0) γ v + 1 (cid:1) , (18) K ϑϑ = γ v R t | Σ R Σ = K ϕϕ , (19)while the trace is K = K tt + K ϑϑ + K ϕϕ = 2 γ v (cid:18) R t | Σ R Σ + v t v (cid:19) + γv t . (20)Since there are two identical LTB universes joining at theshell with unit normal n µ pointing outward, the jumpsof these quantities on Σ are[ K a b ] = 2 K a b , [ K ] = 2 K . (21)The Einstein equations at the shell Σ are [26][ K ab − δ ab K ] = − π S ab , (22)where S ab is the energy–momentum tensor of the mate-rial on the shell. We assume that this matter is a perfectfluid, described by S ab = ( σ + P Σ ) u (Σ) a u (Σ) b + P Σ g ab | Σ , (23)where σ and P Σ are the 2–dimensional surface densityand pressure, respectively, and u µ (Σ) is the 4–velocity ofthe shell given by u α (Σ) = d x α (Σ) dτ = ∂x α (Σ) ∂x µ dx µ dτ = γ dx µ dt ∂x α (Σ) ∂x µ . (24)The coordinates on Σ are x µ Σ = ( t, r Σ ( t ) , ϑ, ϕ ), whichyield u α (Σ) = (cid:18) γ , γ vR ′ Σ , , (cid:19) , u (Σ) α = ( − γ , γ v R ′ Σ , , . (25)As such, it is easy to see that u µ (Σ) u (Σ) µ = 1 , u µ (Σ) n µ = 0 . (26) The ( t, t ) component of the Einstein equations (22) atthe shell is σ = − γ v π R t | Σ R Σ , (27)while the ( ϑ, ϑ ) or the ( ϕ, ϕ ) component yields P Σ = γ π (cid:18) v t + 2 γ v v t + γ v R t | Σ R Σ (cid:19) = − σ γ v t π v − v . (28)Using eq. (2), one obtains R t R = p m e ( r ) r / + 3 p m e ( r ) t/ t , the Hubble parameterof the dust–dominated cosmological background, as t → + ∞ . It is also σ = − γ v π R t | Σ R Σ = − γ v π p m e ( r Σ ) r / + 3 p m e ( r Σ ) t/ σ > v <
0. A wormhole shellwith positive surface energy density must necessarily ex-pand slower than the cosmic substratum, a fact that isinterpreted as the influence of the inhomogeneity slow-ing down the expansion locally. Since the expansionrate of the background t tends to zero at late times,the shell expansion rate must also tend to zero and theshell becomes comoving. In fact, since v <
0, it is˙ R Σ = R t | Σ + R ′ Σ ˙ r Σ < R t | Σ and, since R ′ Σ >
0, it is˙ r Σ <
0. This inequality is consistent, of course, with therelation ˙ r Σ = vR ′ Σ which is easy to derive.Now, r Σ > t → + ∞ , either r Σ tends to a horizontal asymp-tote r ∞ >
0, or lim t → + ∞ r Σ ( t ) = 0 + . If r Σ → + , then R Σ = (cid:18) r / + 32 p m e ( r Σ ) t (cid:19) / −→ m e (0) = 0 and the wormhole disappears asymp-totically, which doesn’t make sense physically, and thispossibility is discarded. Hence, R Σ −→ (cid:18) r / ∞ + 32 p m e ( r ∞ ) t (cid:19) / (32)and the wormhole shell becomes comoving at late times.We conclude this section with a comment on the en-ergy conditions. The strong energy condition for a 2–dimensional perfect fluid is σ + P Σ ≥ σ + 2 P Σ ≥ σ and hence v <
0, it is σ + P Σ = σ γv t π v − v > σ + 2 P Σ = γv t π v − v > v < v → − as t → ∞ , hence v t > σ ≥ σ + P Σ ≥
0, while the null energy conditionis equivalent to σ + P Σ ≥
0. Therefore, the materialon the shell satisfies the weak, strong, and null energyconditions.
THE COVARIANT CONSERVATION EQUATION
We can now solve the covariant conservation equationprojected along the 4–velocity of the shell u a (Σ) [26], u a (Σ) ∇ b S ba = − h u α (Σ) T βα n β i . (35)It is convenient to note that u a (Σ) ∇ b S ba = u a (Σ) ∇ b h ( σ + P Σ ) u (Σ) a u b (Σ) i + u a (Σ) ∇ a P Σ = −∇ b (cid:16) σ u b (Σ) (cid:17) − P Σ ∇ b u b (Σ) (36)using the normalization u a (Σ) u (Σ) a = − u a (Σ) ∇ b u (Σ) a = 0. We now compute ∇ b (cid:16) σ u b (Σ) (cid:17) = 1 q(cid:12)(cid:12) g | Σ (cid:12)(cid:12) ∂ b (cid:18)q(cid:12)(cid:12) g | Σ (cid:12)(cid:12) σ u b (Σ) (cid:19) , (37)where g | Σ = γ − R sin ϑ is the determinant of the 3–dimensional metric g ab | Σ , obtaining ∇ b (cid:16) σ u b (Σ) (cid:17) = γR ∂ t (cid:0) R σ (cid:1) = γ ˙ MA Σ . (38)Here A Σ ≡ πR is the area of the shell and M ≡ σA Σ isthe mass of the material located on the shell. Similarly,one obtains ∇ b u b (Σ) = γ ˙ A Σ A Σ (39)and h u α (Σ) T βα n β i = 2 u α (Σ) T βα n β = − γ ρv . (40)Putting everything together, we obtain the covariant con-servation equation in the form˙ M + P Σ ˙ A Σ = − γρvA Σ . (41) This formula is interpreted physically as follows: thequantity ρ v is the flux density of cosmic fluid onto theshell caused by the relative motion of the shell with re-spect to the background. The quantity ρvA Σ is the fluxof this material; the factor 2 appears because there aretwo LTB spacetimes joining at the shell. The Lorentzfactor γ is due to the Lorentz contraction caused by theradial motion of the shell.Eq. (41) expresses the first law of thermodynamics re-lating changes over a time interval dtdM + dW = dQ Σ , (42)where dM = ˙ M dt is the variation of internal energy dur-ing dt , dW = P Σ ˙ A Σ dt is a work term due to the variationof the area of the shell, and dQ Σ is the energy input dueto the influx of cosmic fluid onto the shell. DISCUSSION AND CONCLUSIONS
We have obtained, and interpreted physically, an ex-act solution of the Einstein field equations representinga wormhole shell joining two identical LTB spacetimeswhich are dust–dominated. This solution is similar tothe wormhole solution of Ref. [13] obtained by general-izing the McVittie metric, but there are important dif-ferences. First, we adopted the no–accretion conditionof Bondi [22] which forbids radial flow of energy into thewormhole while Ref. [13], being interested in the effectof accretion, allows for radial flow with the consequencethat an imperfect fluid is needed in order to obtain so-lutions in [13]. Here, instead, we can consider a perfectfluid, the dust of classical LTB solutions [20–22]. While in[13] the conservation equation analogous to our eq. (41)has a right hand side consisting of two terms, one dueto the relative motion between shell and cosmic substra-tum, and another due to accretion, only the first termappears in our case in which there is no radial flux.An important result of [19] is that, contrary to staticwormholes, the null energy condition needs not be vi-olated for their cosmological and dynamical wormholesto stay open; here we propose a different cosmologicalwormhole solution made with material which satisfies theweak, null, and strong energy conditions on the shell. Inother words, the “stuff” necessary to keep this worm-hole throat open does not need to be very exotic. Thisfeature motivates further studies of dynamical wormholesolutions of the Einstein equations.I.B. thanks the Fonds Qu´eb´ecois de la Recherche surla Nature et les Technologies (FQRNT) for financial sup-port and Bishop’s University for the hospitality. V.F. issupported by the Natural Sciences and Engineering Re-search Council of Canada (NSERC). ∗ [email protected] † [email protected][1] H.G. Ellis, J. Math. Phys. , 104 (1973); Gen. Relat.Gravit. , 105 (1979); K.A. Bronnikov, Acta Phys. Pol.B , 251 (1973); T. Kodama, Phys. Rev. D , 3529(1978); G. Cl`ement, Gen. Relat. Gravit. , 763 (1981);D.H. Coule and K. Maeda, Class. Quantum Grav. ,955 (1990); D.H. Coule, Class. Quantum Grav. , 2353(1992); C. Barcelo and M. Visser, Phys. Lett. B , 127(1999); C. Armendariz-Picon,
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Phys. Rev. D , 1129 (1991).[27] Since r is a comoving radius, if r Σ were constant, theshell would be comoving. But r Σ depends on time, hencea relative motion between the shell and the cosmic back-ground is allowed. However, we will find below that theshell becomes comoving at late times. Moreover, onecould define N µ as ∇ µ ( r − r Σ ( t )) instead but this, ofcourse, leads to the same unit normal n µ .[28] It is not assumed that v is a constant.[29] We assume that the function r Σ ( tt