aa r X i v : . [ h e p - t h ] M a y The nature of the gravitational vacuum Samir D. MathurDepartment of PhysicsThe Ohio State UniversityColumbus, OH 43210, [email protected]
Abstract
The vacuum must contain virtual fluctuations of black hole microstates for eachmass M . We observe that the expected suppression for M ≫ m p is counteractedby the large number Exp [ S bek ] of such states. From string theory we learn thatthese microstates are extended objects that are resistant to compression. We ar-gue that recognizing this ‘virtual extended compression-resistant’ component of thegravitational vacuum is crucial for understanding gravitational physics. Remark-ably, such virtual excitations have no significant effect for observable systems likestars, but they resolve two important problems: (a) gravitational collapse is haltedoutside the horizon radius, removing the information paradox; (b) spacetime ac-quires a ‘stiffness’ against the curving effects of vacuum energy; this amelioratesthe cosmological constant problem posed by the existence of a planck scale Λ. Essay awarded an honorable mention in the Gravity Research Foundation 2019 Awards for Essayson Gravitation. lassical general relativity is expected to fail if we encounter curvature singularities.Two singularities of special interest are the central singularity of a black hole and thebig bang singularity of cosmology. Both the associated spacetimes exhibit horizons, andthe evolution equations are similar as well: the interior of a uniform collapsing dust ballmaps, under time reversal, to a region of the dust cosmology.But the questions we face are very different in the two cases. With black holes, we askhow information can escape from the horizon in Hawking radiation. With cosmology, wewonder why the cosmological constant is not order Λ ∼ l − p , which would curl spacetimeinto a planck sized ball.In this essay we will argue that these two very different sounding questions are resolvedby a common hypothesis: the vacuum of quantum gravity contains virtual black holemicrostates that are extended compression resistant objects . The lesson from black holes
Consider a collapsing shell of mass M . In semiclassical general relativity, this shell willpass unimpeded through its horizon radius r h = 2 GM . The light cones turn ‘inwards’ for r < r h (fig.1), so if we assume that causality holds in our theory then no effect emanatingfrom the singularity at r = 0 can alter the vacuum nature of the region 0 < r < r h . Witha vacuum around the horizon, we are trapped by the information paradox [1, 2].String theory has provided remarkable progress on this problem. In this theory wecan try to explicitly construct all objects with mass M . For the cases studied so far,we do not find the above structure of the semiclassical hole. Instead, we find fuzzballs :horizon sized quantum objects with no horizon or singularity [3]. Fuzzballs radiate fromtheir surface like a normal body, so there is no information puzzle.So what alters the semiclassical collapse of the shell? In [4] it was argued that thecollapsing shell has a probability P ∼ e − π ( Mmp ) (1)to tunnel into a typical fuzzball microstate. While P is tiny, as expected for tunnelingbetween macroscopic objects, we must multiply by the number of fuzzball states N thatwe can tunnel to [5]: N ∼ e S bek ∼ e A G ∼ e π ( Mmp ) (2)We then find that the overall probability for transitioning to fuzzballs is P total ∼ P × N ∼ Figure 1: (a) A shell of mass M is collapsing towards its horizon. (b) If the shell passes throughits horizon, then the information it carries is trapped inside the horizon due to the structure oflight cones. Virtual black hole microstates
If fuzzballs exist as real objects of mass M , then the vacuum must contain virtualfluctuations of these objects. We expect the amplitude of such a fluctuation to be tinyfor M ≫ m p . But we argue that the large number N (eq.(2)) of possible fuzzballsoverwhelms this suppression, and makes the virtual fuzzballs an important componentof the gravitational vacuum.Two important properties of these virtual black hole microstates we obtain from theirstring theory constructions. First, they are extended objects, with a radius R ( M ) closeto the horizon radius 2 GM . Second, they are very compression-resistant. This can beunderstood from the fact that if we have energy M , then no more than S bek ( M ) fuzzballsshould fit in a region of radius 2 GM . Compressing the fuzzballs to a smaller volume willhave to raise the energy; a simple computation gives the equation of state p = ρ , thestiffest possible [8].Let us collect the above observations to get a picture of the vacuum of quantumgravity. This vacuum contains virtual extended compression resistant objects , which wecall vecros for short. These vecros are nothing but virtual black hole microstates, butnow endowed with the properties that we have observed from explicit constructions instring theory. The largeness of their degeneracy (2) will make them a crucial player forboth the information paradox and the cosmological constant problem. Resolving the information paradox
String theory respects causality: signals do not propagate outside the light cone. Thisplaces a stringent constraint on where a collapsing shell must transition to fuzzballs. Ifthe shell passes unimpeded through its horizon radius r h , then no effects in string theory The role of virtual black holes has also been considered in other approaches; e.g. [6, 7]. C' A cosmology is quite different from a black hole. Instead of a compact object we havea homogeneous infinite solution. We do not expect anything special at the cosmologicalhorizon; rather the question is: what prevents the spacetime from curling up everywherein response to a vacuum energy density Λ ∼ l − p ? We shall argue that the vecros givespacetime a stiffness which counteracts this vacuum energy.To see this, consider 2-dimensional Euclidean space as a toy model. Let the vacuumcontain disc shaped objects (vecros) centered around each point ~r with all possible radii0 < R < ∞ . Note that when this space is flat, a disc with radius R has a circumference C ( R ) = 2 πR . 3ow suppose we bend this space to create a hemispherical depression centered at r = 0 with curvature radius R c (fig.2). A disc centered at r = 0 with radius R now has,for the same radius R , a smaller circumference C ′ ( R ). The ratio α = C ′ /C is almostunity for vecros with small R , bur when R ∼ R c , then C ′ /C starts becoming significantlyless than unity; i.e., there is significant compression of such vecros.We model the compression resistance by taking a number α in the range 0 < α < α . From fig.2 we see thatthis puts no restriction of the curvature radius R c itself, but requires that this curvaturepersist only for a radial distance L that is L . R c . For α close to unity, we find thecondition L < L max with L max ≈ R c (6(1 − α )) (4)In particular, the only space with constant curvature everywhere will be flat space.This is the crucial point: for the actual gravitational theory, since there is no newconstraint on R c , there will be no change to the local gravity Lagrangian L = R + aR + . . . .But any curvature radius R c can only be maintained over length scales L . R c ; else wewill encounter a stiff resistance from squeezing the virtual extended objects of radius R & R c in the vacuum.A star of radius R star creates curvature with curvature radius R c ≫ R star . Thiscurvature lasts only over a radial distance L ∼ R star ≪ R c , so the vecros will feel nosignificant compression, and no observable effects will arise for stellar structure. Thesame holds for objects like galaxies or clusters where the gravitational field is weak.On the other hand there are two situations where we do get L ∼ R c : (i) an objectcompact enough to make a black hole (i.e., R ∼ GM ) and (ii) a region of a homogeneouscosmology with radius R ∼ H − where H − is the cosmological horizon. Thus vecros willnot affect the usual tests of general relativity, while they will affect both the formationof black hole horizons and dynamics at the scale of the cosmological horizon.More generally, we can consider a distribution function D ( R ) giving the density ofvecros with radius R . If D ( R ) vanishes for R > R max , then we can maintain a curvatureradius R max for arbitrarily large regions, since there are no vecros with R & R max tocompress. This will allow spacetimes with a nonvanishing effective cosmological constant.But we see that the value of Λ is set by D ( R ) rather than the vacuum energy density ρ . Summary: In retrospect it is not surprising that we should have to worry about virtual blackhole microstates. Black holes are universal objects in all theories of gravity, and theirdegeneracy S bek is large. String theory has told us that the microstates have radius R ≈ GM and are compression resistant; this sets the stage for the role of these ‘vecros’in gravitational dynamics.We have seen that a shell trying to cross its horizon r h = 2 GM tries to crush vecrosof radius R ≈ GM ; this stops the collapse and generates a string star. Most solutions ofthe cosmological constant problem involve fine-tuning Λ. But we have argued that any4niform curvature radius R c will tend to crush vecros with R & R c ; this forces flatnessdespite a nonvanishing vacuum energy. Thus understanding the vecro component of thegravitational vacuum may resolve many conundrums of nature. 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