Vacuum Fluctuations and the Small Scale Structure of Spacetime

S. Carlip*

Department of Physics, University of California, Davis, California 95616, USA

R.A. Mosna† and J. P.M. Pitelli‡

Instituto de Matematica, Estatıstica e Computacao Cientıfica, Universidade Estadual de Campinas, 13083-859, Campinas, SP, Brazil(Received 8 April 2011; published 6 July 2011)

We show that vacuum fluctuations of the stress-energy tensor in two-dimensional dilaton gravity lead

to a sharp focusing of light cones near the Planck scale, effectively breaking space up into a large number

of causally disconnected regions. This phenomenon, called ‘‘asymptotic silence’’ when it occurs in

cosmology, might help explain several puzzling features of quantum gravity, including evidence of

spontaneous dimensional reduction at short distances. While our analysis focuses on a simplified two-

dimensional model, we argue that the qualitative features should still be present in four dimensions.

DOI: 10.1103/PhysRevLett.107.021303 PACS numbers: 04.60.Kz, 04.20.Gz, 04.62.+v

Introduction.—A fundamental goal of quantum gravityis to understand the causal structure of spacetime—thebehavior of light cones—at very small scales. One recentproposal [1,2], based on the strong coupling approximationto the Wheeler-DeWitt equation [3,4], is that quantumeffects may lead to strong focusing near the Planck scale,essentially collapsing light cones and breaking space upinto a large number of very small, causally disconnectedregions. Such a phenomenon occurs in classical cosmologynear a spacelike singularity, where it has been called‘‘asymptotic silence’’ [5]; it can be viewed as a kind ofanti-Newtonian limit, in which the effective speed oflight drops to zero. Near the Planck scale, asymptoticsilence could explain several puzzling features of quantumgravity, most notably the apparent dimensional reductionto two dimensions [1,2,6] that occurs in lattice models [7],in some renormalization group analyses [8], and else-where [9,10].

In cosmology, the strong focusing of null geodesicscomes from the presence of a singularity. For quantumgravity, a different source is required. Null geodesics aregoverned by the Raychaudhuri equation [11,12]. For acongruence of null geodesics—a pencil of light—withaffinely parametrized null normals ‘a, this equation tellsus that

‘ara�¼�1

2�2��ab�

abþ!ab!ab�8�Tab‘

a‘b; (1)

where the expansion

� ¼ 1

A‘araA (2)

is the fractional rate of change of area of a cross section ofthe pencil, �ab is its shear, and !ab is its vorticity. Thestress-energy tensor Tab appears by virtue of the Einsteinfield equations, which relate it to the Ricci tensor. (We usenatural units, @ ¼ c ¼ GN ¼ 1.) In particular, vacuum

fluctuations with positive energy focus null geodesics,decreasing the expansion, while fluctuations with negativeenergy defocus them.It has recently been shown that for many forms of

matter, most vacuum fluctuations have negative energy[13]. These fluctuations have a strict lower bound, how-ever, while the rarer positive fluctuations are unbounded. Akey question is then which of these dominate the behaviorof light cones near the Planck scale.In this Letter, we answer this question in the simplified

context of two-dimensional dilaton gravity, a model ob-tained by dimensionally reducing general relativity to onespace and one time dimension. We show that the positiveenergy fluctuations win, and cause a collapse of light conesin a time on the order of 15 Planck times. The reduction totwo dimensions is, for the moment, required for exactcalculations; it is only in this setting that the spectrum ofvacuum fluctuations is fully understood. But we show thatthe qualitative features leading to strong focusing are alsopresent in the full four-dimensional theory, strongly sug-gesting that our results should apply to full generalrelativity.Dilaton gravity and the Raychaudhuri equation.—Our

starting point is two-dimensional dilaton gravity, a theorythat can be described by the action [14,15]

I ¼Z

d2xffiffiffiffiffiffijgj

q �1

16�’Rþ V½’� þ ’Lm

�; (3)

where ’ is a scalar field, the dilaton, with a potential V½’�whose details will be unimportant. This action can beobtained from standard general relativity by dimensionalreduction, assuming either spherical or planar symmetry.A conformal redefinition of fields is needed to bring theaction into this form; the dilaton coupling to the matterLagrangian Lm is fixed provided the two-dimensional mat-ter action is conformally invariant. From now on, weassume such an invariance.

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In two dimensions, the Raychaudhuri equation (1) doesnot quite make sense, since there are no transverse direc-tions in which to define the area or the expansion. Itsgeneralization to dilaton gravity is simple, however. Forany model of dilaton gravity obtained by dimensionalreduction, the dilaton has a direct physical interpretationas the transverse area in the ‘‘missing’’ dimensions. We cantherefore define a generalized expansion

�� ¼ 1

’‘ara’ ¼ d

d�ln’; (4)

where � is the affine parameter. It is then an easy conse-quence of the dilaton gravity field equations that

d ��

d�¼ � ��2 � 8�Tab‘

a‘b ¼ � ��2 � 16�TL; (5)

where TL is the left-moving component of the stress-energy tensor.

We next need the vacuum fluctuations of the stress-energy tensor. For conformally invariant matter in twodimensions, Fewster, Ford, and Roman have found theseexactly [13]. A conformally invariant field is characterizedby a central charge c; a massless scalar, for instance, has

c ¼ 1. To obtain a finite value for the stress-energy tensor,one must smear it over a small interval; Fewster et al. use aGaussian smearing function with width �. The probabilitydistribution for the quantity TL in the Minkowski vacuumis then given by a shifted Gamma distribution

PrðTL ¼!Þ¼#ð!þ!0Þ ð��2Þ�ð!þ!0Þ��1

�ð�Þ e���2ð!þ!0Þ;

(6)

where

!0 ¼ c

24��2; � ¼ c

24; (7)

and # is the Heaviside step function. As noted earlier, thisdistribution is peaked at negative values of the energy; forc ¼ 1, the probability of a positive fluctuation is only 0.16.There is, however, a long positive tail, as there must be inorder that the average hTLi be zero.Before proceeding with the calculation, it is useful to

look at the behavior of the Raychaudhuri equation (5) witha constant source TL ¼ !. It is easy to see that

��ð�Þ ¼

8>>>><>>>>:� ffiffiffiffiffiffiffiffiffiffiffiffiffi

16�!p

tanffiffiffiffiffiffiffiffiffiffiffiffiffi16�!

p ð�� �0Þ if !> 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij16�!jptanh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij16�!jp ð�� �0Þ if !< 0 and j ��ð0Þj< ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij16�!jp� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij16�!jp

if !< 0 and ��ð0Þ ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij16�!jpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij16�!jpcoth

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij16�!jp ð�� �0Þ if ! � 0 and j ��ð0Þj> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij16�!jp;

(8)

where the integration constant �0 is determined from theinitial value of ��.

As noted earlier, positive energy fluctuations can quicklydrive the expansion to �1. If �� is initially negative, evensmall negative energy fluctuations—those on the cothbranch of (8)—cannot overcome the nonlinearities thatalso drive the expansion to �1. Larger negative energyfluctuations tend to defocus null geodesics, increasing theexpansion, but even these have a limited effect. The lowerbound �!0 for energy in the distribution (6) determines amaximum asymptotic value

��þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi16�!0

p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2c=3�2

q; (9)

and if the expansion starts below this value, it will asymp-tote to at most ��þ.

Moreover, the same bound on negative energy fluctua-tions means that the positive contribution to the right-handside of (5) cannot be larger than 16�!0 ¼ ��2þ. The expan-sion thus has a critical negative value. Any fluctuation thatbrings �� to a value lower than

�� crit ¼ �ffiffiffiffiffiffiffiffi2c

3�2

s¼ � ��þ (10)

is irreversible: once the expansion becomes this negative, itwill necessarily continue to decrease. This gives a qualita-tive answer to our central question—in the long run, thepositive energy fluctuations will always win, and the ex-pansion will diverge to �1. Note the crucial role of thenonlinear term in (5); a similar problem was considered in[16], but only in the approximation that �� was smallenough that this term could be neglected.To estimate the time to this ‘‘collapse’’ of the light

cones, let us assume that the initial negative energy fluc-tuations push �� to near its peak value of ��þ. We can thenask for the probability of a positive energy fluctuation largeenough to drive �� to ��crit within a characteristic time �.This can be determined from (6); for a massless scalar field(c ¼ 1), it is � � 0:065. If we now treat these fluctuationsas a Poisson process—i.e., independent events occurringrandomly with probability ��t=� in any interval �t—andignore all smaller fluctuations, the time to collapse will be

given by an exponential distribution ð�=�Þe��t=�, with amean of approximately 15:4�.This is, of course, an oversimplified picture of the com-

bined effect of many vacuum fluctuations. To obtain amore precise result, we next turn to a numerical analysisof the Raychaudhuri equation.

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Quantitative results.—Our analysis so far has been semi-classical: we consider quantum fluctuations of matter,but ignore purely quantum gravitational effects. We can,in principle, incorporate weak quantum fluctuations of themetric into the stress-energy tensor in (5), but there is noreason to trust our methods at scales at or below the Planckscale. We therefore choose the width � of the Gaussiansmearing function to be the Planck length. The overallsystem (5) and (6) is invariant under a simultaneous rescal-ing of � and the affine parameter �, so this choice is notcritical; given any cutoff �, our results can be interpreted asgiving the focusing time in units of �. For our matter field,we choose a massless scalar, that is, a conformal field withcentral charge c ¼ 1.

We proceed as follows.(1) We choose an initial condition ��0 ¼ 0, and select a

random value of the vacuum fluctuation !, with a proba-bility given by the distribution (6).

(2) We evolve forward one Planck time using (8), todetermine a new value ��1.

(3) Using ��1 as an initial condition and choosing a newvalue of!, we evolve another Planck time to determine ��2.We repeat the process, keeping track of the number ofiterations, until we land on a negative branch of (8) withthe expansion diverging to �1 during the step.

Using MATHEMATICA [17], we have performed 107 runsof this simulation. Figure 1 shows the probability densityof light cone collapse as a function of time in Planck units.The mean time to collapse is 14:73tP, where tP is thePlanck time; the standard deviation is 14:53tP. These areperhaps large enough to justify our neglect of strong quan-tum gravitational effects, but small enough to probe thecausal structure of spacetime in a physically very interest-ing region.

We see from Fig. 1 that the probability for collapse at thefirst step is approximately 0.051. This provides a usefulcheck, since this probability can be obtained directly

from the distribution (6): it is just the probability that theargument of the tangent in the collapsing branch of (8) is atleast �=2 when � ¼ 1, �0 ¼ 0. The exact result matchesour simulation.As shown in Fig. 1, after the first few steps the distribu-

tion can be fit very accurately to an exponential distribu-tion:

Prðtf ¼ nÞ ¼ �e��n with � � 0:0686: (11)

This is surprisingly close to our approximation at theend of the preceding section. Moreover, while the detailsof individual runs vary widely, almost all show �� risingquickly to near its maximum value of ��þ, confirming thestarting assumption of our approximation.Two-dimensional dilaton gravity thus appears to exhibit

short distance asymptotic silence, with vacuum fluctua-tions causing a rapid convergence of null cones.Generalizations and implications.—Our computations

have been restricted to two-dimensional dilaton gravity,primarily because this is the only setting in which theprobability distribution (6) is known. We have also re-stricted ourselves to conformally invariant matter, for thesame reason. But the qualitative features of our results arelargely independent of such details, and we expect them tocarry over to the full four-dimensional theory. In particular,(1) The vacuum fluctuations of the stress-energy tensor

in four dimensions are also expected to have a strict lowerbound and an infinitely long positive tail [13]. If anything,the positive tail seems to fall off more slowly than in twodimensions. Moreover, while our detailed results used aGaussian test function to smear the stress-energy tensor,these features hold for any sufficiently compact testfunction [18].(2) If the stress-energy tensor is bounded below, the

expansion in four dimensions has bounds exactly analo-gous to those of (8), even taking the same functionalform [19].(3) Adding real matter would change the quantitative

details of our results. But as long as that matter satisfiesthe null energy condition, it can only lead to furtherfocusing.We thus expect something akin to short distance asymp-

totic silence in four-dimensional general relativity. Weshould, however, add three caveats. First, we have treatedour sequence of vacuum fluctuations as if they were sta-tistically independent. This is not quite right: the results of[13] imply that the (Gaussian smeared) stress-energy ten-sor TL at coordinate u is weakly anticorrelated with thesame object at uþ �. The correlation drops off sharplywith distance, and should not affect our qualitative results.It may be possible, though, to take this effect into accountto produce a more accurate quantitative picture. Work onthis question is in progress.Second, the vacuum fluctuations found in [13] are fluc-

tuations of the Minkowski vacuum (or, by conformal

0 20 40 60 80 1000.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

t f

Rel

ativ

eFr

eque

ncy

FIG. 1. Probability of the expansion diverging to �1 as afunction of Planck time steps. The solid line is the exponentialdistribution (11).

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invariance, of any conformally equivalent vacuum). Whileany curved spacetime is approximately flat at short enoughdistances, this is not enough to determine a unique vacuum,and we do not know how sensitive our results are to thischoice.

Third, we have by necessity neglected purely quantumgravitational effects, which could also compete with thevacuum fluctuations of matter. This is not independent ofthe problem of choosing a vacuum; for instance, it isknown that if one chooses the Unruh vacuum near a blackhole horizon, the renormalized value of the shear term�ab�

ab in (1) is negative [20].If a proper handling of these caveats does not drasti-

cally change our conclusions, though, we have learnedsomething very interesting about the small scale structureof spacetime. The strong focusing of null cones meansthat ‘‘nearby’’ neighborhoods of space are no longer incausal contact. This sort of breakup of the causal structurehas been studied in cosmology [5], where it leads toBelinskii-Khalatnikov-Lifshitz behavior [21]: each smallneighborhood spends most of its time as an anisotropi-cally expanding Kasner space with essentially randomexpansion axes and speeds, but periodically undergoes achaotic ‘‘bounce’’ to a new Kasner space with differentaxes and speeds. It was argued in [1,2] that such a localKasner behavior could explain the apparent spontaneousdimensional reduction of spacetime near the Planck scale,while preserving Lorentz invariance at large scales.Whether or not this proves to be the case, the shortdistance collapse of light cones suggests both a newpicture of spacetime and a new set of approximationsfor short distances.

S. C. was supported in part by U.S. Department ofEnergy Grant No. DE-FG02-91ER40674. J. P.M. P. wouldlike to thank Fapesp (Grant No. 2008/01310-5) for finan-cial support and the UC Davis Physics Department forhospitality during this work. R. A.M. acknowledges sup-port from CNPq and Fapesp.

*carlip@physics.ucdavis.edu†mosna@ime.unicamp.br‡pitelli@ime.unicamp.br

[1] S. Carlip, in ‘‘Foundations of Space and Time,’’ edited byJ. Murugan, A. Weltman, and G. F. R. Ellis (CambridgeUniversity Press, Cambridge, England, to be published).

[2] S. Carlip, AIP Conf. Proc. 1196, 72 (2009).[3] C. J. Isham, Proc. R. Soc. A 351, 209 (1976).[4] M. Henneaux, Bull. Math. Soc. Belg. 31, 47 (1979).[5] J.M. Heinzle, C. Uggla, and N. Rohr, Adv. Theor. Math.

Phys. 13, 293 (2009).[6] L. Anchordoqui, D. C. Dai, M. Fairbairn, G. Landsberg,

and D. Stojkovic, arXiv:1003.5914.[7] J. Ambjørn, J. Jurkiewicz, and R. Loll, Phys. Rev. D 72,

064014 (2005).[8] M. Reuter and F. Saueressig, Phys. Rev. D 65, 065016

(2002).[9] L. Modesto, Classical Quantum Gravity 26, 242002

(2009).[10] J. J. Atick and E. Witten, Nucl. Phys. B310, 291 (1988).[11] A. Raychaudhuri, Phys. Rev., 98 1123 (1955).[12] S.W. Hawking and G. F. R. Ellis, The Large Scale

Structure of Space-Time (Cambridge University Press,Cambridge, England, 1973).

[13] C. J. Fewster, L. H. Ford, and T.A. Roman, Phys. Rev. D81, 121901 (2010).

[14] D. Grumiller, W. Kummer, and D.V. Vassilevich, Phys.Rep. 369, 327 (2002).

[15] D. Louis-Martinez, J. Gegenberg, and G. Kunstatter, Phys.Lett. B 321, 193 (1994).

[16] J. Borgman and L.H. Ford, Phys. Rev. D 70, 064032(2004).

[17] MATHEMATICA 8, Wolfram Research, Inc., Champaign, IL,2010.

[18] C. J. Fewster and S. P. Eveson, Phys. Rev. D 58, 084010(1998).

[19] J. D. E. Grant, Ann. Henri Poincare 12, 965 (2011).[20] P. Candelas and D.W. Sciama, Phys. Rev. Lett. 38, 1372

(1977).[21] V. A. Belinskii, I.M. Khalatnikov, and E.M. Lifshitz, Adv.

Phys. 19, 525 (1970); 31, 639 (1982).

PRL 107, 021303 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending8 JULY 2011

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