arXiv:gr-qc/9810023v1 7 Oct 1998 hc h togeeg odto SC utb iltd On violated. be must (SEC) condition energy strong the which fteohreeg conditions energy other curvature, surpris- the spatial More positive of is with condition.) universes (SEC–violation sufficient for not ingly, [7–9]. too but violated Not clas- necessary be of a must occur. (SEC) gravity condition to sical energy minimum bounce strong absolute a the the such surprisingly, what for ask are the by and requirements turnaround of “bounce” reassessment the under- a a model universe going we for (FRW) Friedman–Robertson–Walker Letter, ripe this a but is In time [3,4], the theorem situation. that singularity feel cosmological we first development the the after of os- declined in largely Interest ambigu- universes quite cillating barrier?). uni- turnaround angular-momentum the oscillating (cusp? of of ous nature discussions Unfortu- the older leave a verses the universe. from of present evolution many our cyclical nately, of a sin- with incarnation big-bang it previous the avoid replace They and gularity [3–6]. cosmology bang big ieetcnet pligteeeg odtost the [21–23]. to formation conditions galaxy energy of a the utility epoch in applying demonstrated The been context: recently [10–15]. different has universe approach for an the state such of of of content equation making matter the without the to conditions information commitment much energy particular as the any extract from to possible ex- is as idea or model-independent key singularity a the In crunch analysis, singularity). big big-bang a a hibiting undergoing (as to expansion subsequent opposed and instant compression, collapse, maximum a of undergoes that universe FRW a simply “natural”. is which for big-bounce ex- model known a the inflation-inspired with an discuss compatible exhibit are we and physics, violations Finally SEC exten- which evaded. modern to be tent their can and cosmolog- [16–20] various [3,4] sions the theorems how singularity examples ical also specific and with [10–15], show the wormholes to applied traversable analysis Morris–Thorne model-independent bounce the the to of similar analysis model-independent a present shall aifid npriua,w xii niflto-nprdm inflation-inspired an exhibit we particular, In satisfied. h Morris–Thorne the buc”fo rvosclas,rte hna explosion an than a rather as collapse, to previous a from “bounce” Abstract: iia odtosfrteceto faFriedman–Robertso a of creation the for conditions Minimal siltn nvre 12 r lentvst standard to alternatives are [1,2] universes Oscillating h R omlg sdsrbdb h erc[3–6] metric the by described is cosmology FRW The is wormhole, Tolman or universe, baby bouncing A omnwormhole Tolman nti etrw netgt the investigate we Letter this In + rvral wormholes traversable hoeia iiin o lmsNtoa aoaoy Los Laboratory, National Alamos Los Division, Theoretical ++ hsc eatet ahntnUiest,SitLus M Louis, Saint University, Washington Department, Physics esbettebuc oagnrlmdlidpnetanaly model-independent general a to bounce the subject We . need 1 etme 98 L 1998; September (10 amnMolina–Par´ısCarmen n hwta hr sawy noe eprlrgo surrou region temporal open an always is there that show and , evoae.We violated. be minimal odtosudrwihteceto foruies ih ar might universe our of creation the which under conditions none ”bounce” dli hc i onei “natural”. is bounce big a which in odel A T rmabgbn iglrt.Sc onei oeie refe sometimes is bounce a Such singularity. big-bang a from E 1 -dSpebr1,2018) 11, September X-ed + h te hand, other the vlae ttebounce the at evaluated oeqatte fitrs are, interest of quantities Some h ieo h nvrei iiu.W aeti to this take We subscript a minimum. use a and is zero, universe time the be of size the respectively. sections, spatial bolic with ihteeaanbigsrc inequalities. strict being again these with eprrsls ic ewn iezr ob true a be to zero time want we Since minimum results. deeper hsi h nlg o onigbb nvre of universe, baby bouncing a for analog, Morris–Thorne the the is This en titieult.A plcto ftefunda- the implies: of now application calculus differential An of theorem inequality. mental strict a being ¨ on inequality weak This pnrgossronigtebuc o which for bounce the surrounding regions open omoe 1] Seas 1,euto 1.2,pg 104] page (11.12), [12–15].) equation [11, and also (See [10]. wormholes n atVisser Matt and ds ohv one hr utb oetm twhich at time some be must there bounce, a have To o R nvreteEnti qain eueto reduce equations Einstein the universe FRW a For 2 k = 1 ,or 0, +1, = ∃ − > t ˜ ems have must we , d dt 2 2 [ln( p dt : 0 lms e eio87545 Mexico New Alamos, + = all 2 uai mutandis Mutatis a a ρ ∀ − ++ )] a ( ˙ t = t h te nrycniin a aiybe easily can conditions energy other the ∗ 8 sor 63130-4899 issouri ) ∈ > − πG 2 “flare-out” ;¨ 0; = 1 8 o yeshrcl a,o hyper- or flat, hyperspherical, for 1  ( –akruies rma from universe n–Walker ;and 0; πG 3 − i ln h ie fta ple to applied that of lines the along sis 1  t, ˜ − a dr 2 t a > a  ∗ 0) a a ¨ 0: = kr 2 a a ˙ snteog o rvn our proving for enough not is + 2 2 ∪ 2 a + ∗ (0 a a + ∗ ˙ odto o traversable for condition 2 2 ≥ d , a for k r hr ilb similar be will there , t ˜ 2 + 2 dt 2 ∗ ¨ ) 0 (  ( a . dn h oneover bounce the nding 2 dθ t odnt quantities denote to a 3 , k 2 ,wt hsnow this with 0, 6= ) 2  > > a sin + . 0 , 0 2 . s u oa to due ise θdφ 2 )  rred (5) (6) (1) (4) (3) (2) , 1 a¨ a˙ 2 k ρ + p = − + + (In particular, any of these three conditions automat- 4πG  a a2 a2  ically implies violation of the WEC, SEC, and DEC.) 1 d2 k Thus a bounce in a hyperbolic (k = −1) universe must = − ln a + , (7) violate all the pointwise energy conditions, a bounce in 4πG  dt2 a2  a spatially flat (k = 0) universe is on the verge of vio- lating all the energy conditions, and a sufficiently gentle 1 a¨ a˙ 2 k bounce in a hyperspherical (k = +1) universe exhibits ρ − p = +2 +2 4πG a a2 a2  a “window of opportunity” that requires more detailed analysis. 1 1 d2(a3) k = +2 , (8) Secondly, by working in suitable open regions surround- 4πG 3a3 dt2 a2  ing the bounce [and using the strict inequalities (3)–(4) derived above] we obtain the stronger results and 3 a¨ ∃ Bounce + [k =6 +1] ⇒ NEC violated, (20) ρ +3p = − . (9) 4πG a ∃ Bounce ⇒ SEC violated. (21) By the strict inequalities discussed above [equations (3)– (4)], there will be open temporal regions surrounding the Thus the energy condition violations are minimized by bounce for which taking the universe to be hyperspherical (k = +1) and −1 1 k by making the bounce sufficiently gentle:a ¨∗ ≤ a∗ . In ρ + p< . (10) this case it is easy to check that NEC, WEC, and DEC 4πG a2  are satisfied, and only SEC need be violated. Indeed we only need SEC to be violated in some open temporal 1 k region surrounding the bounce, and it is quite possible to ρ − p> . (11) all 4πG a2  satisfy the point-wise energy conditions at sufficiently early and late times:

−1 ρ +3p< 0. (12) ∃ Bounce + [k =+1;a ¨∗ ≤ a∗ ] ⇒ NEC, WEC, DEC satisfied; SEC violated. (22) For comparison, the standard point-wise energy con- ditions are the null energy condition (NEC), weak en- This is to be contrasted to the situation for Morris– ergy condition (WEC), strong energy condition (SEC), Thorne traversable wormholes, wherein (for spherically and dominant energy condition (DEC). Their specializa- symmetric wormholes) there is an open spatial region sur- tions to a FRW universe have previously been discussed rounding the throat over which the NEC (and therefore in [21–23]. Basic definitions are given in [3,11]. also the WEC, SEC, and DEC) must be violated [10–13]. Generalization of all these results to spacetimes more NEC ⇐⇒ (ρ + p ≥ 0). (13) general than the FRW universes, along the lines of [14,15] is certainly possible, and we intend to address this issue more fully in a subsequent paper [24]. WEC ⇐⇒ (ρ ≥ 0) and (ρ + p ≥ 0). (14) A simple specific example of a geometry that satisfies NEC, WEC, and DEC but violates SEC is

SEC ⇐⇒ (ρ +3p ≥ 0) and (ρ + p ≥ 0). (15) 2 2 2 2 2 ds = −dt + a∗ + β t dr2
DEC ⇐⇒ (ρ ≥ 0) and (ρ ± p ≥ 0). (16) × + r2(dθ2 + sin2 θ dφ2) , (23) 1 − r2  As applied to a bouncing baby universe we have: provided we take β < 1. Note that this is the temporal First, by working at the bounce itself [where we only have analog of the toy model traversable wormhole considered the weak inequality (2)] on page 398 of the Morris–Thorne article [10]. Explicit ∃ Bounce + [k = −1] ⇒ NEC violated, (17) calculation of the stress-energy components yields 2 2 2 2 3 a∗ + β [1 + β ]t ρ = > 0. (24) 2 2 2 2 ∃ Bounce + [k =0;a ¨∗ > 0] ⇒ NEC violated, (18) 8πG (a∗ + β t )

−1 2 2 2 2 2 ∃ Bounce + [k =+1;a ¨∗ >a∗ ] ⇒ NEC violated. −1 a∗[1 + 2β ]+ β [1 + β ]t p = < 0. (25) 2 2 2 2 (19) 8πG (a∗ + β t )

2 So that rapidly realized that the original black hole singularity 2 2 2 2 2 theorem (which also uses the SEC [3,4]) could be modi- 2 a∗[1 − β ]+ β [1 + β ]t ρ + p = > 0. (26) fied to produce more powerful theorems that used weaker 2 2 2 2 8πG (a∗ + β t ) energy conditions (e.g, the NEC [3,4]). It was commonly believed (in at least some circles) that the cosmological 2 2 2 2 2 singularity theorems could be similarly strengthened, and 2 a∗[2 + β ]+2β [1 + β ]t ρ − p = > 0. (27) there are in fact a number of newer cosmological singular- 2 2 2 2 8πG (a∗ + β t ) ity theorems that use the NEC (but at the cost of adding other rather strong conditions) [16–20]. How does the 2 2 present discussion evade the consequences of these theo- −6 a∗β ρ +3p = < 0. (28) rems? 2 2 2 2 8πG (a∗ + β t ) The theorems of [16,17] are stated using the WEC but really only need the NEC. However the key assumption The condition β < 1 is used to keep ρ+p positive definite made there is that the universe is open in the mathemat- and prevent violations of the NEC. The pressure is not ical sense (which in a FRW universe implies the universe positive in this toy model, (nor does it need to be positive is either hyperbolic, k = −1, or flat, k = 0). Thus these to satisfy the energy conditions). singularity theorems are compatible with the results of A second specific example of a geometry that also sat- this Letter since we have explicitly shown that a bounce isfies NEC, WEC, and DEC but violates SEC is in hyperbolic or flat FRW universes requires NEC viola- 2 2 2 2 tion. ds = −dt + a∗ cosh (Ht) The closed universe singularity theorem of [18] uses 2 dr 2 2 2 2 a very strong technical requirement (compact localized × + r (dθ + sin θ dφ ) , (29) 1 − r2  past light cones) explicitly violated by our models. The more recent results in [19,20] are also compatible with −1 provided we take H

2 2 ficult to violate the NEC, WEC, and DEC; violations 1 3(1 − a∗H ) ρ H2 > . of these energy conditions typically being due to (small) = 3 + 2 2 0 (30) 8πG  a∗ cosh (Ht) quantum effects. On the other hand, it is rather easy to violate the SEC, leading some researchers to refer to the SEC as “the unphysical energy condition”. Violations of −1 (1 − a2H2) p H2 ∗ < . the SEC are generic to classical scalar fields [3,11], to in- = 3 + 2 2 0 (31) 8πG  a∗ cosh (Ht)  flationary spacetimes [16–20], to spacetimes with (cosmo- logically large) positive cosmological constants [21–23], So that to certain mean-field quantum field theories [25,26] and 2 2 other quantum mechanical situations [8], and signifi- 1 2(1 − a∗H ) ρ + p = ≥ 0. (32) cantly, to classical relativistic fluids with two body in- 8πG 2 2 a∗ cosh (Ht) teractions [9,27]. A particularly wide class of Tolman wormholes can also be constructed by Wick rotating Eu- 2 2 clidean wormholes back to Lorentzian signature [28]; the 1 4(1 − a∗H ) ρ − p = 6H2 + > 0. (33) Wick rotated Euclidean wormholes typically satisfying 8πG  2 2  a∗ cosh (Ht) all energy conditions except the SEC. These observations are important in that they elevate the discussion of this −6 Letter from a mere mathematical curiosity to an issue ρ +3p = H2 < 0. (34) 8πG that merits serious attention. An inflation-inspired model: The generic feature com- −1 The condition H

3 and the inflaton field satisfies the equation of motion to thank LAEFF (Laboratorio de Astrof´ısica Espacial y F´ısica Fundamental; Madrid, Spain), and Victoria Uni- a˙ ∂V (φ) φ¨ +3 φ˙ + =0. (37) versity (Te Whare Wananga o te Upoko o te Ika a Maui; a ∂φ Wellington, New Zealand) for hospitality during final stages of this work. The key feature relevant for the present discussion is that

2 ρφ +3pφ =2φ˙ − 2V (φ). (38)

If the inflaton field bounces at the same time as the ge- ometry (i.e., φ˙∗ = 0), then the inflaton field can be used as the natural candidate for providing the SEC violations Electronic mail: [email protected] required to support the bounce: Electronic mail: [email protected] 235 ˙2 [1] A. Einstein, Berl. Ber. , (1931). (ρ +3p)total = (ρ +3p)normal +2φ − 2V (φ). (39) [2] R.C. Tolman, Phys. Rev. 38, 1758 (1931). [3] S.W. Hawking and G.F.R. Ellis, The large scale structure ˙2 of space-time, (Cambridge, England, 1973). (ρ + p)total = (ρ + p)normal + φ . (40) [4] R.M. Wald, General Relativity, (Chicago University Press, 1984). [5] C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravita- (ρ − p)total = (ρ − p)normal +2V (φ). (41) tion, (Freeman, San Francisco, 1973). Thus adding a spatially-homogeneous inflaton field to [6] P.J.E. Peebles, Principles of Physical Cosmology, normal (energy condition satisfying) matter preserves the (Princeton University Press, 1993). [7] J.D. Bekenstein. Phys. Rev. D 11, 2072–2075 (1975). NEC, WEC, and DEC, but can easily lead to violations [8] L. Parker and S.A. Fulling, Phys. Rev. D 7, 2357–2374 of the SEC. In this sense inflation (either old or new in- (1973). flation; but not chaotic or eternal inflation) is naturally [9] L. Parker and Y. Wang, Phys. Rev. D 42, 1877–1883 compatible with the bounce scenario. With typical es- (1990). timates for the inflaton VEV being of order the GUT [10] M.S. Morris and K.S. Thorne, Am. J. Phys. 56, 395 energy scale, we would similarly estimate the bounce to (1988). occur when the radius of the universe is about one GUT [11] M. Visser, Lorentzian wormholes — from Einstein to distance scale (about 1 000 Planck lengths). This is cer- Hawking, (AIP Press, New York, 1995). tainly a small distance, even by particle physics stan- [12] D. Hochberg and M. Visser, Phys. Rev. D 56, 4745–4755 dards, but because it is so much larger than the Planck (1997). scale, we may still reasonably hope for the applicability [13] D. Hochberg and M. Visser, in The Internal Structure of of semiclassical quantum gravity — thus we now hold Black Holes and Spacetime Singularities, edited by L.M. out the reasonable hope for a big bounce that not only Burko and A. Ori, (Institute of Physics Press, Bristol, evades the classical singularity theorems but also evades 1997). the necessity for dealing with the full theory of quantum [14] D. Hochberg and M. Visser, Phys. Rev. Lett. 81, 746 gravity. (1998). In summary, (1) replacing the big bang with a big [15] D. Hochberg and M. Visser, Phys. Rev. D57, 044021 bounce violates the SEC but does not necessarily violate (1998). 72 any of the other energy conditions, and (2) violating the [16] A. Borde and A. Vilenkin, Phys. Rev. Lett. , 3305– SEC is relatively easy and can be achieved at the classi- 3308 (1994). cal level, without needing to appeal to quantum effects. [17] A. Borde and A. Vilenkin, “The impossibility of steady- Proceedings of the Eighth Yukawa Of course, even more exotic variations can be contem- state inflation”, in Symposium on Relativistic Cosmology, Japan, 1994; gr- plated. You can consider the effects of violating all the qc/9403004. energy conditions [29], including the NEC, or even more [18] A. Borde, Phys. Rev. D 50, 3692–3702 (1994). boldly you can consider having the universe bootstrap it- [19] A. Borde and A. Vilenkin, Int. J. Mod. Phys. D5, 813– self into existence via a chronology violating region [30]. 824 (1996). A key aspect of this Letter is that extreme steps of this [20] A. Borde and A. Vilenkin, Phys. Rev. D 56, 717–723 type are not necessary: the big bang singularity can be (1997). tamed with relatively mild modifications of the standard [21] M. Visser, Science 276 (1997) 88-90. cosmological model. Indeed, there are simple extensions [22] M. Visser, Phys. Rev. D 56 7578 (1997). of either old or new inflation for which such a bounce is [23] M. Visser, “Energy conditions and Galaxy formation”, “natural”. To appear in the proceedings of the Eighth Marcel Gross- mann Meeting on General Relativity; Jerusalem, June Acknowledgments: This research was supported by the 1997; gr-qc/9710010. U.S. Department of Energy. Matt Visser also wishes [24] D. Hochberg, C. Molina-Par´ıs and M. Visser, in prepara-

4 tion. [25] B. Rose, Class. Quant. Grav. 3, 795–995 (1986). [26] B. Rose, Class. Quant. Grav. 4, 1019–1030 (1987). [27] H. Kandrup, Phys. Rev. D 46, 5360–5366 (1992). [28] D.H. Coule, Class. Quant. Grav. 10, L25–L28 (1993). [29] J. Levin, Phys. Rev. D57, 7611 (1998). [30] J.R. Gott III and Li–Xiu Li, Phys. Rev. D58, 023501 (1998).

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