Towards the Raychaudhuri Equation Beyond General Relativity

Towards the Raychaudhuri Equation Beyond General Relativity

Abstract

In General Relativity, gravity is universally attractive, a feature embodied by the Raychaudhuri equation which requires that the expansion of a congruence of geodesics is always non-increasing, as long as matter obeys the strong or weak energy conditions. This behavior of geodesics is an important ingredient in general proofs of singularity theorems, which show that many spacetimes are singular in the sense of being geodesically incomplete and suggest that General Relativity is itself incomplete. It is possible that alternative theories of gravity, which reduce to General Relativity in some limit, can resolve these singularities, so it is of interest to consider how the behavior of geodesics is modified in these frameworks. We compute the leading corrections to the Raychaudhuri equation for the expansion due to models in string theory, braneworld gravity, theories, and Loop Quantum Cosmology, for cosmological and black hole backgrounds, and show that while in most cases geodesic convergence is reinforced, in a few cases terms representing repulsion arise, weakening geodesic convergence and thereby the conclusions of the singularity theorems.

I Introduction

General Relativity (GR) is widely expected to be an incomplete theory of dynamical spacetime. One reason for this is that GR famously predicts its own demise through the existence of singularities, as demonstrated by the singularity theorems Penrose:1964wq (); Hawking:1969sw (). An essential physical ingredient of the singularity theorems is that gravity is attractive, so that congruences of convergent timelike and null geodesics develop singularities in finite proper (affine) time. More specifically, geodesic congruences with timelike and null tangent vector fields are characterized by their expansion , respectively, which satisfy the Raychaudhuri equations Raychaudhuri ()

(timelike),
(null) (1)

for spacetime dimension , where the additional terms are non-positive; see Appendix A for details.

For the timelike Raychaudhuri equation, if the so-called “convergence condition”

(2)

holds, then the expansion of a congruence of geodesics is non-increasing. Specifically, an initially converging congruence develops a singularity in finite proper time . A similar singularity develops for null convergences in finite affine time if the null convergence condition is satisfied. These singularities in the expansion don’t necessarily imply a pathology of spacetime themselves; such caustics appear in Minkowski spacetime, for example Friedrich:1983vi (). However, the existence of strictly non-positive contributions to the right-hand side of Eq.(1), when combined with other global conditions on the spacetime manifold, form the basis for the general existence of singularities in cosmological and black hole spacetimes Penrose:1964wq (); Hawking:1969sw ().

Within GR it is possible to convert the convergence conditions into energy conditions on the types of matter. We can use the (trace-reversed) Einstein equation

(3)

(where , being the -dimensional Newton’s constant) to substitute in Eq.(1)

(timelike),
(null) (4)

where the terms are again strictly non-positive terms. Thus, all of the terms on the right-hand side of (4) are non-positive provided that matter satisfies the respective energy conditions

Strong Energy Condition,
Null Energy Condition. (5)

For an isotropic perfect-fluid energy momentum tensor with energy density and pressure , these conditions translate into

Strong Energy Condition,
Null Energy Condition. (6)

Most known classical matter obeys both the null and strong energy conditions, while vacuum energy violates the strong energy condition but still saturates the null energy condition. It may possible to violate the null energy condition with exotic forms of matter ArkaniHamed:2003uz (); Kobayashi:2010cm (), non-minimal coupling Lee:2007dh (); Lee:2010yd (), or quantum gravity effects Ford:1993bw (), though these approaches often face challenges that we will not explore further here.

However, since we expect corrections to GR of some form, we do not expect the Einstein equations (3) to always hold. Corrections to Einstein’s equations may make it possible to violate the convergence conditions and/or without violating the energy conditions (5). In particular, many corrections to GR appear perturbatively in the form

(7)

where controls the strength of the corrections and is a tensor that contains contributions from the metric, curvature, energy-momentum tensor, or additional fields. The additional term in Eq.(7) in turn shows up as an additional term on the right-hand side of the Raychaudhuri equation

(timelike),
(null) (8)

where it may in principle contribute with any sign. In this paper, we will examine corrections to the Raychaudhuri equations from four frameworks for corrections to GR: string theory, braneworld gravity, theories, and Loop Quantum Cosmology, for cosmological (and in some cases black hole) backgrounds.

While we are motivated by the existence of singularities in these backgrounds, and the promise of these alternatives to GR for resolving the singularities, we will not attempt to prove the absence of singularities in this paper. Indeed, a significant amount of work has shown that finding realistic singularity-free spacetimes under computational control is quite challenging, and we expect that true singularity resolution will require physics beyond the perturbative approach of (7). Instead, we aim for a more modest goal, that of finding corrections to GR that lead to potentially positive terms on the right-hand side of the Raychaudhuri equation Eq.(1) for timelike and null geodesics – a necessary, but far from sufficient, condition for ultimately resolving spacetime singularities.

In Section II we outline a general set of corrections to GR from string theory in dimensions, and compute the corrections to the Raychaudhuri equation for black hole and cosmological spacetimes. In Section III we compute the corrections to the 4-dimensional induced Einstein equation for the braneworld scenario in a cosmological spacetime. In Section IV we compute the corrections to the Einstein equation for so-called theories, and determine the form of the corrections in a cosmological background. In Section V we compute the corrections to the Raychaudhuri equation in a cosmological background from Loop Quantum Gravity. In Section VI we conclude with some comments on the potential for theories beyond GR to resolve singularities.

Ii String Theory and Gauss-Bonnet Corrections

In string theory, higher-order corrections induce corrections to the action and, correspondingly, to the Einstein Equations. In particular, the corrected action at leading order in for bosonic, heterotic, type IIA/IIB takes the form Callan (); Zwiebach:1985uq (); Gross:1986mw ()

(9)

where we are ignoring the antisymmetric rank two tensor , is the dilaton, is the -dimensional Riemann tensor, and where 1

(10)

Unfortunately, (9) contains higher derivative terms in its equation of motion. However, it turns out that (9) is ambiguous up to field redefinitions of the fields to next order in . It is possible to use these field redefinitions to remove the higher order terms in the equations of motion2, giving rise to the action

(11)

where we have allowed for additional matter (including potentially a -dimensional cosmological constant) through , and is the 2nd order Gauss-Bonnet combination

(12)

Setting the dilaton to a constant, we are left with the usual Ricci curvature term and the quadratic Gauss-Bonnet term in our action

(13)

We will refer to this as the Einstein-Gauss-Bonnet (EGB) gravity action. The resulting equations of motion are

(14)

where

(15)

In order to put (14) in the appropriate form relevant for use the Raychaudhuri equation3, we will trace-reverse, giving

(16)

where now

(17)

More generally, the terms above are just the leading terms of the more generic Lanczos-Lovelock extensions of gravity Lanczos:1938sf (); Lovelock:1971yv () (see Padmanabhan:2013xyr () for a review). The Lanczos-Lovelock extensions include the set of additional terms that can be added to the gravitational action and still lead to 2nd order equations of motion. In 4 spacetime dimensions, it can be shown that the Ricci scalar and cosmological constant are the only non-topological terms that can be added. In particular, the correction term in (11) is purely topological in 4-dimensions (it is the Euler characteristic of the spacetime), not contributing to the equation of motion. For higher dimensions , there is a finite series of additional terms, increasing in powers of (terminating at some order for a given ). We will just focus on this leading order term for now, but will keep in mind that there can be additional corrections to consider.

In order to determine the form of these corrections it is necessary to calculate for specific backgrounds. For simplicity, we will restrict ourselves to corrections to black hole and cosmological backgrounds; in the subsections that follow, we will examine the corrections for these backgrounds in more detail.

ii.1 Black Holes

The Gauss-Bonnet correction terms (16) appear as part of a pertubative series of corrections, so we expect solutions for some given matter content to also be described by a pertubative series in : , where is the uncorrected General Relativity solution, is the first-order correction, and so on. It is remarkable that exact solutions of Einstein-Gauss-Bonnet gravity for black hole backgrounds are known to all orders in Boulware:1985wk (); Charmousis:2008kc (); Garraffo:2008hu (). However, since we expect the Gauss-Bonnet corrections to be only the first term in a series of corrections arising from string theory, we cannot trust these exact solutions beyond . One can therefore consider the effects of the Gauss-Bonnet correction terms on the Raychaudhuri equations up to for black hole backgrounds in two ways:

  1. Evaluate on the uncorrected Schwarzschild black hole metric.

  2. Evaluate on the known exact metric for black hole backgrounds, then expand the result to .

(with similar expressions for the null Raychaudhuri equation). While both approaches should give identical results, up to , we will consider and compare both approaches for completeness.

Perturbative Black Hole Corrections

Since we are considering actions with a constant dilaton and zero form fields, we will restrict ourselves to pure gravity solutions. We thus first begin by considering the perturbative correction terms to a -dimensional black hole solution of the vacuum Einstein equations

(18)

where is the metric of a -dimensional constant curvature manifold with unit radius (such as a sphere). The zeroth order black hole solution takes the form

(19)

where is related to the mass of the black hole by

(20)

and is the area of a unit sphere. Note that for we have .

While the Ricci tensor and scalar vanish , as is expected for a -dimensional Schwarzschild background, the Riemann tensor does not, and gives the only non-zero contribution to the Gauss-Bonnet scalar

(21)

and corrections to the Einstein equation,

(22)

We are primarily concerned with the correction terms in the and directions,

(23)
(24)

Note that these correction terms vanish identically for , which is what we expect since the Gauss-Bonnet correction is purely topological for , and only acts as a dynamical correction for .

We now consider a radial affine null tangent vector

(25)

where corresponds to radially outgoing/ingoing. The null Raychaudhuri equation takes the form

(26)

Our corrections due to the Gauss-Bonnet term appear on the right hand side, as (recall that )

(27)

Remarkably, the Gauss-Bonnet corrections vanish identically everywhere for null rays, implying that the null Raychaudhuri equation for black holes is uncorrected to leading order in .

Finally, consider a timelike geodesic described by the tangent vector

(28)

The timelike Raychaudhuri equation takes the form

(29)

where again the corrections to the Raychaudhuri equation due to the Gauss-Bonnet corrections come from the last term

(30)

There are a few things to notice about this correction term. First, it does not seem to suffer from any pathologies due to the coordinate singularity of our coordinate system at the horizon; thus, we expect it to be valid throughout the entire spacetime (although the coordinate will require careful interpretation). Second, notice that the corrections are manifestly positive, thus the corrections make the caustic/conjugate point at the location of the putative singularity at worse, not better!

We have shown that the perturbative EGB corrections to the Einstein equations do not provide divergence terms in either the null or timelike Raychaudhuri equations; in fact, convergence is strengthened in the timelike case. These results above are not particularly surprising since it is known that black hole solutions in Einstein-Gauss-Bonnet gravity still posses singularities Boulware:1985wk (); Charmousis:2008kc (); Garraffo:2008hu (), as we will now examine in more detail.

Exact Black Hole Solutions

In the previous section we considered how the Gauss-Bonnet curvature squared terms lead to perturbative corrections to the Raychaudhuri equation for pure-Einstein gravity black hole solutions. However, exact black hole solutions for Einstein-Gauss-Bonnet gravity are well-known Boulware:1985wk (); Charmousis:2008kc (); Garraffo:2008hu () (see also Callan ()), so it is also possible to calculate the right-hand-side of the Raychaudhuri equation for these fully backreacted solutions.

First, let us review the known black hole solutions of -dimensional Einstein-Gauss-Bonnet gravity Boulware:1985wk (); Charmousis:2008kc (); Garraffo:2008hu (). As before, we will work with the general spherically symmetric metric

(31)

Solutions to the Einstein-Gauss-Bonnet equations of motion (16) are Boulware:1985wk () (see also Charmousis:2008kc (); Garraffo:2008hu ())

(32)

where labels different branches of solutions, and we have defined for convenience.

Consider first the branch. To lowest order in , the metric is simply that of a Schwarzschild black hole, and is asymptotically flat as , reducing to the usual Einstein gravity solution. For this reason, this is usually called the “Einstein branch.” In contrast, the branch, often called the “Gauss-Bonnet branch,” has a non-trivial vacuum structure for

(33)

corresponding to anti-deSitter space for with a large negative cosmological constant (or deSitter space for ). For non-zero mass, the metric in this branch resembles that of Schwarzschild-anti-deSitter space with a negative mass

(34)

The analysis of the stability of this branch requires some care (see Boulware:1985wk (); Deser:2002jk ()). Note that since the curvature scale of the corresponding AdS space is non-perturbative in , it is doubtful that we can trust these solutions as solutions to perturbative string theory. Further, for there is a naked singularity at the origin Boulware:1985wk (). For these reasons, we will restrict our analysis to the Einstein branch .

Despite the curvature squared term in EGB gravity, the Einstein branch solutions (31,32) still have a curvature singularity at the origin. This singularity is surrounded by a horizon located at , given by the roots of the polynomial Boulware:1985wk ()

(35)

For , this horizon always exists; however, for , the existence of the horizon is guaranteed only for ; thus, for “microscopic” black holes, even the EGB solutions posses a naked singularity. Since the exact EGB black hole solutions still posses a curvature singularity at the origin, we are not particularly surprised by our perturbative result from the previous subsection indicating that the corrections to the null and timelike Raychaudhuri equations do not allow for terms that could prevent the formation of conjugate points.

Even though we have the exact EGB solutions, which do possess a curvature singularity, in hand, let us nevertheless compute the contribution of EGB gravity to the right-hand side of the Raychaudhuri equation to explore the way in which the EGB corrections can affect the formation of conjugate points.

Note that since the solutions (31,32) are exact solutions to the corrected Einstein equations

(36)

we can compute the curvature term in the null (timelike) Raychaudhuri equation ( respectively) directly, without needing to compute the quadratic curvature terms.

For the metric (31), we have

(37)
(38)

where a prime denotes a derivative with respect to . Note that these vanish for a Schwarzschild solution , as expected.

A radial affine null vector has the same form as in the Schwarzschild case

(39)

where now refers to the EGB corrected form (32). The curvature term in the Raychaudhuri equation is then

(40)

which again vanishes identically, as we found previously in the perturbative case. It is important to note that the vanishing result is independent of the precise functional form of , and only requires the generic structure of the metric (31).

The geodesic timelike null vector for the metric (31) takes the form

(41)

where we are assuming , which is the case for the Einstein branch of solutions. The curvature term in the Raychaudhuri equation then takes the form

(42)

The general expression for given in (32) is not particularly illuminating; however, expanding the solution in powers of , we obtain

(43)

matching our perturbative result (30), as expected.

As in the perturbative analysis, we see that the EGB corrections to the Schwarzschild black hole background give rise to either vanishing or convergent contributions to the null and timelike Raychaudhuri equations. Thus, the EGB corrections themselves, despite being quadratic in curvature, do not alleviate convergence that leads to conjugate points and singularities in these spaces.

We have restricted ourself to spherically symmetric black hole solutions for simplicity; however, since we have not found an improvement in the convergence behavior, we do not expect that deviations from spherical symmetry are likely to produce qualitatively different results.

Black hole backgrounds are not the only spacetimes of interest for the study of singularities and the Raychaudhuri equation. In the next subsection, we will explore the application of EGB gravity to cosmological spacetimes.

ii.2 Cosmology

Let’s examine the EGB corrections (17) for a -dimensional cosmological background 4

(44)

where is the metric of a -dimensional sphere. Our matter will consist of a -dimensional perfect fluid

(45)

where are the energy density and pressure of the fluid, respectively.

It is straightforward to compute the Gauss-Bonnet scalar

(46)

and the correction terms (17)

(47)
(48)

Notice that these corrections vanish identically for , as expected since for lower dimensions the EGB terms are topological and don’t contributed to the equations of motion.

From the radial affine null tangent vector

(49)

where again refers to radially outgoing/ingoing rays, we can compute the corrections to the null Raychaudhuri equation

(50)

where we wrote to simplify our result.

Clearly the first term in (50) is positive for matter that obeys the null energy condition, as usual. However, the second term can be negative if the time variation of the Hubble parameter is large enough.

In particular, let us consider perturbative solutions for the metric (44) of the form

(51)

where is the zeroth-order solution to the Einstein equation

(52)
(53)

and is the corresponding zeroth-order Hubble parameter. Inserting into (50), we obtain the expression

(54)

Assuming an equation of state , the second term is negative for equations of state . The lower bound is always greater than , so most ordinary matter will satisfy this condition. In particular, a -dimensional universe dominated by radiation can contribute a divergent term in the null Raychaudhuri equation.

Finally, consider a comoving, proper-time parameterized timelike geodesic described by the tangent vector

(55)

The corrections to the timelike Raychaudhuri equation due to the Gauss-Bonnet corrections come from the last term

(56)

where again we substituted to simplify our result. The first term is the typical term for matter, and is positive for matter that obeys the strong energy condition. The last term, proportional to , is the new contribution; we see that for (as is the case for any time-dependence), this term can be negative, giving rise to a positive divergent term in the timelike Raychaudhuri equation, potentially opposing convergence.

We have seen in both the null and timelike Raychaudhuri equations the existence of terms that can give rise to a positive (divergent) contribution to the divergence. This does not necessarily mean that cosmology with EGB gravity can evade the singularity problem, just that the usual singularity theorems do not apply in a straightforward way to these backgrounds.

In particular, consider the EGB-corrected equations of motion for the metric (44) from (16). Examining the component of (16) we have

(57)

For , solutions to (57) take the usual Einstein form; however, for , as we would expect to occur in the early universe near a big bang singularity, we have instead

(58)

It is clear that a curvature singularity still exists where , despite the presence of the higher curvature terms.

Iii Braneworld Gravity

We will consider a 5-dimensional bulk spacetime (with 5-dimensional cosmological constant ) with coordinates and metric given by Brax:2004xh ()

(59)

which is sufficiently general to capture the backreaction of the singular brane on the bulk as well as 4-dimensional homogeneous and isotropic cosmological evolution. The extra dimension has the range ; however, we will additionally impose a symmetry Horava:1995qa () , so that the covering space is reduced to , as in the RS I scenario Randall:1999ee (). We embed a (singular) 3-brane at with unit normal and induced metric ; we will use capital Latin letters for 5-dimensional coordinate indicies and lowercase Greek letters for 4-dimensional brane indicies. The 5-dimensional energy-momentum tensor has the form

(60)

where is the brane tension and is the brane matter energy momentum tensor. Note that the usual Minkowski RS I model has the solution

(61)

This can be obtained from (59) under the limits and identifications

(62)

Gravity on the 3-brane hypersurface is induced by its embedding in the extra dimensions; in particular, the induced 4-dimensional Einstein equations on the brane are Shiromizu:1999wj ()

(63)

where and , and the new tensors are defined as

(64)
(65)

where is the 5-dimensional Weyl tensor; see Appendix C for more details. The first two terms on the right-hand side of (63) are just the usual 4-dimensional cosmological constant and energy-momentum tensor sources, while the last two terms arise from the extra dimensional embedding of the brane. In order to put this into a form suitable for use with the Raychaudhuri equation we trace-reverse (63)

(66)
(67)

where we rewrote the last two terms as a correction term so that the corrections take the form (7) as outlined in the Introduction. The timelike and null 4-dimensional Raychaudhuri equations thus have the new terms arising from the braneworld

(68)
(69)

where on the right-hand sides denote non-positive terms.

Assuming the brane energy-momentum tensor is of the perfect fluid form, with energy density and pressure , we can immediately compute the tensor contribution for the metric (59)

(70)

where we evaluated the terms at , while we compute directly from the metric (59)

(71)
(72)

where a dot denotes a derivative with respect to , and a prime denotes a derivative with respect to . All of the components of the tensor are proportional to the quantity , involving a second derivative .

In order to evaluate the common quantity appearing in in the limit , we need to use the 5-dimensional Einstein equations for the metric (59) and sources (60), which take the form Brax:2004xh ()

(73)
(74)