# A singularity theorem for Einstein–Klein–Gordon theory

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## Abstract

Hawking’s singularity theorem concerns matter obeying the strong energy condition (SEC), which means that all observers experience a nonnegative effective energy density (EED), thereby guaranteeing the timelike convergence property. However, there are models that do not satisfy the SEC and therefore lie outside the scope of Hawking’s hypotheses, an important example being the massive Klein–Gordon field. Here we derive lower bounds on local averages of the EED for solutions to the Klein–Gordon equation, allowing nonzero mass and nonminimal coupling to the scalar curvature. The averages are taken along timelike geodesics or over spacetime volumes, and our bounds are valid for a range of coupling constants including both minimal and conformal coupling. Using methods developed by Fewster and Galloway, these lower bounds are applied to prove a Hawking-type singularity theorem for solutions to the Einstein–Klein–Gordon theory, asserting that solutions with sufficient initial contraction at a compact Cauchy surface will be future timelike geodesically incomplete. These results remain true in the presence of additional matter obeying both the strong and weak energy conditions.

## Keywords

Energy conditions Singularity theorems Classical fields## 1 Introduction

*n*is the spacetime dimension, \(\sigma \) is the shear scalar and \(R_{\mu \nu }\) is the Ricci tensor. Assuming that the geometry is a solution to the Einstein equations

*effective energy density*(EED) for \(\rho \). Evidently, the sign of \(\rho \) is crucial. If \(\rho \ge 0\), that is, if the strong energy condition (SEC) holds, then the right-hand side of (3) is negative, driving \(\theta \rightarrow -\infty \) in finite proper time. This is incompatible with geodesic completeness and implies the existence of a singularity.

Senovilla [27] has described the skeleton of the singularity theorems in terms of a ‘pattern theorem’ with three ingredients. An *energy condition* establishes a focussing effect for geodesics, while a *causality condition* removes the possibility of closed timelike curves and a *boundary or initial condition* establishes the existence of some trapped region of spacetime. The goal of the singularity theorems is to show that the spacetime contains at least one incomplete causal geodesic; we will divide singularity theorems into ‘Hawking-type’ and ‘Penrose-type’, depending on whether they demonstrate timelike or null geodesic incompleteness respectively. While Hawking-type results are based on the SEC, Penrose-type results assume the null energy condition (NEC), \(T_{\mu \nu }k^\mu k^\nu \ge 0\) for all null \(k^\mu \).

For these reasons there has long been interest in establishing singularity theorems under weakened energy assumptions. Examples include [2, 3, 4, 26, 29, 30], in which various averages of the energy density or related quantities are required to be nonnegative if the average is taken over a sufficiently large portion of a (half-)complete causal geodesic, or at least is intermittently nonnegative [2]. Our approach in this paper follows [10], in which (generalising results from [17]) it was shown among other things that suitable lower bounds on local weighted averages of \(\rho \) are sufficient to derive singularity theorems of Hawking and Penrose type, even if \(\rho \) is not everywhere positive or has a negative long-term average.

The bounds adopted in [10] were inspired by the Quantum Energy Inequalities (QEIs) that have been established in various models of quantum field theory (see [9] for a recent review). However, there is a significant gap between the results of [10] and a semiclassical Hawking-type singularity theorem, because there is so far no QEI version of the SEC. The purpose of this paper is to show that the classical nonminimally coupled massive Klein–Gordon field obeys lower bounds on \(\rho \) of the type considered in [10]. The general approach is parallel to methods used in [11] to obtain averaged versions of the weak and null energy conditions for the classical nonminimally coupled scalar field. Elsewhere, we will use our results to establish QEI analogues of the SEC (cf. [12]); here, we use them to derive a new Hawking-type singularity theorem for the Einstein–Klein–Gordon system. In a completely different direction, we mention that the methods of [10], and therefore bounds of the type developed here, could be used in other problems in relativity. See, for example [21], in which a version of Hawking’s area theorem is proved under weakened hypotheses.

*f*is a real-valued smooth and compactly supported function. Here, it is not assumed that the background spacetime and field together solve the Einstein–Klein–Gordon equations. We derive lower bounds on such averages that depend only the values of \(\phi \), but not its derivatives. The bounds also depend on \(\gamma \) and

*f*together with its derivatives and are valid for all values of the coupling \(\xi \) in the interval \([0,2\xi _c]\) where \(\xi _c\) is the conformal coupling constant (\(\xi _c=1/6\) for \(n=4\)). We investigate the behaviour of these lower bounds under scaling of

*f*and also derive constraints on the time for which \(\rho \) can be more negative than some given value. Section 4 addresses similar questions for worldvolume averages of \(\rho \) obtaining bounds valid on an interval containing \([0,\xi _c]\) for dimensions \(n\ge 4\). In the special case of flat spacetime, one may prove that the average value of \(\rho \) over all spacetime is nonnegative. In Sect. 5, we return to worldline bounds, now adapted to the special case of solutions to the Einstein–Klein–Gordon system and obtaining a slightly refined bound which also holds in the presence of additional matter provided that it obeys the weak and the strong energy condition. This bound is used in our discussion of singularity theorems in Sect. 6. There, we first establish a Hawking-type singularity theorem using methods taken from [10] and then apply it to the Einstein–Klein–Gordon theory with or without additional matter using our worldline bounds. This provides an analogue to the Penrose-type singularity theorem for the nonminimally coupled scalar field discussed in [10]. Finally, we conclude in Sect. 7 with a discussion of the magnitude of the initial contraction needed to ensure timelike geodesic incompleteness according to our results.

Our sign conventions are the \([-,-,-]\) of Misner, Thorne and Wheeler [23] i.e., our metric signature is \((+,-,-,\ldots )\), the Riemann tensor is defined as \(R^{\phantom {\lambda \eta \nu }\mu }_{\lambda \eta \nu }v^\nu =(\nabla _{\lambda }\nabla _{\eta }-\nabla _{\eta }\nabla _{\lambda })u^{\nu }\), and the Einstein equation is \(G_{\mu \nu }=-8\pi T_{\mu \nu }\). We write the d’Alembertian with respect to the metric *g* as \(\Box _g = g^{\mu \nu }\nabla _\mu \nabla _\nu \) and work in *n* spacetime dimensions unless otherwise stated. Except in Sect. 7 we adopt units in which \(G=c=1\).

## 2 The non-minimally coupled field

*R*is the Ricci scalar. The constant

*m*has dimensions of inverse length, which would be the inverse Compton wavelength if one regarded (7) as the starting-point for a quantum field theory with massive particles. The Lagrangian is

*m*or \(\phi \) can be made large. We also observe a guaranteed violation whenever the field derivatives vanish, as we are left with a manifestly negative term.

## 3 Worldline strong energy inequality

*f*be a real-valued and compactly supported function \(f \in C_0^2(\mathbb {R})\). We are interested in expressions of the form

### Theorem 1

*M*,

*g*), where

*M*is a manifold with dimension \(n\ge 2\). Let \(T_{\mu \nu }\) be the stress-energy tensor of a scalar field with coupling constant \(\xi \in [0,2\xi _c]\) and

*f*a real valued function of compact support. Then “on shell”

*f*. Writing \(\phi _{\text {max}}\) for the maximum field amplitude of the field along the inertial trajectory \(\gamma \),

*f*. Let us now assume that

*f*has unit \(L^2\)-norm. Introducing the rescaled function

*I*of proper time with duration \(\tau _0\). This gives

*f*with compact support in an interval of length \(\tau _0\) (see Ref. [11, 13] for similar arguments). In the case where

*I*has infinite duration, this simplifies to \(\mathrm{sup}_{\gamma (I)}\rho \ge -(1+2\xi )m^2/(n-2)\mathrm{sup}_{\gamma (I)}|\phi |^2\) and we see that violations of the SEC beyond the level permitted in the infinite duration limit are possible only on timescales \(\tau _0\ll \xi ^{1/2}m^{-1}\), and not at all if \(\xi =0\).

## 4 Worldvolume strong energy inequality

*f*(

*x*) as a smearing function with compact support, and writing \(V^\mu = f(x)U^\mu \), the averaged EED for the nonminimally coupled scalar field is “on shell”

### Theorem 2

*M*is a manifold with metric

*g*and dimension \(n \ge 3\), \(T_{\mu \nu }\) the stress-energy tensor of a scalar field with coupling constant \(\xi \in [0,\xi _v]\) and

*f*a real valued function on

*M*with compact support, then “on shell ”

*f*. First let \(\phi _{\text {max}}\) be the maximum amplitude of the field

*f*. However, in order to keep \(\phi _{\max }\) constant for all rescaled smearings, we extend its definition to the entire manifold.) Consider a translationally invariant unit timelike vector field \(U^\mu \) and define the rescaled smearing function \(f_\lambda \) for \(\lambda >0\) to be

*f*by \(f_\lambda \), the right-hand side of Eq. (50) becomes

## 5 A worldline inequality for the Einstein–Klein–Gordon system

The inequalities proved in Sects. 3 and 4 are valid for solutions to the Klein–Gordon equation on an arbitrary fixed background spacetime. In this section we discuss how our worldline bound can be adapted to provide more specific information about solutions to the full Einstein–Klein–Gordon system.

### Theorem 3

*M*,

*g*) and

*f*a real valued function of compact support. Then

This inequality has the advantage that the left-hand side is geometric, while only non-geometric terms appear on the right-hand side. It will enable us to prove a singularity theorem for this system.

In fact the same conclusions hold if \((M,g,\phi )\) solves the Einstein–Klein–Gordon system with additional matter obeying both the strong and weak energy conditions.

### Theorem 4

Suppose \((M,g,\phi )\) is a solution to the Einstein–Klein–Gordon equation with any additional matter obeying both the strong and weak energy conditions. Subject to the other conditions of Theorem 3, the bound (67) is valid.

### Proof

## 6 A Hawking-type singularity theorem

In this section we establish a Hawking-type singularity theorem with a weakened energy condition. A similar Penrose-type singularity theorem was discussed in [10].^{1} We then use the result of Sect. 5 to obtain a Hawking-type singularity theorem for the non-minimally coupled Einstein–Klein–Gordon theory.

### Theorem 5

*M*,

*g*) be a globally hyperbolic spacetime of dimension \(n>2\), and let

*S*be a smooth compact spacelike Cauchy surface for (

*M*,

*g*). Suppose that

- a.
\(\tau _0>0\) is chosen such that the congruence of future-directed unit-speed geodesics issuing orthogonally from

*S*can be continued to the past of*S*for a proper time of at least \(\tau _0\) with a smooth velocity field \(U^\mu \) and expansion \(\theta =\nabla _\mu U^\mu \) (such a \(\tau _0\) exists by the construction of a normal neighbourhood for the compact surface*S*); - b.there are positive constants
*Q*and \(\tilde{Q}\) such that, along each future complete unit speed timelike geodesic \(\gamma :[-\tau _0,\infty ) \rightarrow M\) issuing orthogonally from*S*one has an inequalitywhere \(|| \cdot ||\) is the \(L^2\)-norm, for all smooth, real-valued$$\begin{aligned} \int R_{\mu \nu }\dot{\gamma }^\mu \dot{\gamma }^\nu f(\tau )^2\,d\tau \le Q(\Vert f'\Vert ^2+ \tilde{Q}^2\Vert f\Vert ^2), \end{aligned}$$(75)*f*compactly supported in \((-\tau _0,\infty )\); - c.for some \(K>0\), (i) the inequalityholds along every future-directed unit-speed geodesic \(\gamma (\tau )\) issuing orthogonally from$$\begin{aligned} \nabla _U \theta |_{\gamma (\tau )}+\frac{\theta (\gamma (\tau ))^2}{n-1}\ge Q(\tilde{Q}^2-K^2) \qquad \text {on }(-\tau _0,0] \end{aligned}$$(76)
*S*at \(\tau =0\), and(ii) the expansion \(\theta \) on*S*obeysAlternatively, it is sufficient if (c)(i) holds with$$\begin{aligned} \theta |_S <-\tilde{Q}\sqrt{Q(n-1)+Q^2/2}-\frac{1}{2}Q K \coth {(K\tau _0)}. \end{aligned}$$(77)in place of (76), and (c)(ii) holds either as before (for some \(K>0\)) or with (77) replaced by$$\begin{aligned} \nabla _U \theta |_{\gamma (\tau )} \ge 0 \qquad \text {on } (-\tau _0,0], \end{aligned}$$(78)$$\begin{aligned} \theta |_S < -\tilde{Q} \sqrt{Q(n-1)+Q^2/2}-\frac{Q}{2 \tau _0} . \end{aligned}$$(79)

*Remarks:* 1. Note that hypotheses (a) and (c) refer to the recent past of the Cauchy surface *S*, and therefore would in principle be amenable to observational confirmation. 2. The proof shows that the expansion of the geodesic congruence normal to *S* must actually diverge to \(-\infty \) at finite time. From this perspective it may seem strange that (76) can be satisfied if \(\dot{\theta }\) is large and positive on \((-\tau _0,0]\), thus apparently pushing the expansion in the positive direction. However this is just an expression of the averaged bound (75): large positive values of \(R_{\mu \nu }\dot{\gamma }^\mu \dot{\gamma }^\nu \) (i.e., SEC violation) in the recent past must be counterbalanced by (even larger) negative values in the future, which drive the expansion to \(-\infty \). Similar reasoning appears in explorations of ‘quantum interest’ [14, 16]. 3. The constants *Q* and \(\tilde{Q}\) can be global or be allowed to vary between geodesic congruences if that leads to a tighter bound. 4. Clearly \(\tau _0\) may be replaced in hypothesis (c) by any \(\tilde{\tau }_0\in (0,\tau _0]\), giving useful additional freedom. Reducing \(\tilde{\tau }_0\) means that (76) can perhaps be satisfied with a smaller value of *K*, although this needs to be weighed against any consequent increase in \(Q K \coth {(K\tilde{\tau }_0)}\). In any case there is an optimum value of \(\tilde{\tau }_0\) for any fixed function \(\theta \). 5. Finally, the requirement for (*M*, *g*) to be globally hyperbolic can be relaxed–see Remark (2) following Theorem 5.1 in [10].

### Proof

*S*

*-ray*\(\gamma \)—that is, \(\gamma \) is a unit-speed geodesic, issuing orthogonally from

*S*, so that the Lorentzian distance from each \(\gamma (\tau )\) to

*S*is precisely \(\tau \), for all \(\tau \in [0,\infty )\)—\(\gamma \) is complete by assumption. There is a neighbourhood of \(\gamma \) in \(J^+(S)\) in which the Lorentzian distance \(\rho _S(p)\) from

*p*to

*S*is smooth. (We choose conventions so that \(\rho _S\) is positive for timelike separation.) In this neighbourhood, the velocity field \(U^\mu =\nabla ^\mu \rho _S(p)\) is a smooth future-directed unit timelike vector field which is irrotational and orthogonal to

*S*. We now restrict to the geodesic \(\gamma \) and write \(\theta (\tau ):=\nabla _\mu U^\mu |_{\gamma (\tau )}\) for the expansion along \(\gamma \). By the above properties, \(\theta (\tau )\) is a smooth solution to Raychaudhuri’s equation

*f*. We proceed to prove that, contrary to what has just been shown, Eq. (80) can have no smooth solution on \((-\tau _0,\infty )\) under these conditions; indeed, the solution must tend to \(-\infty \) at finite positive time.

*h*supported on \([-\tau _0,\infty )\) with \(h(\tau )=e^{-c\tau /(n-1)}\) on \([0,\infty )\) and so that the second inequality in

*r*). This contradicts the existence of the solution shown above and therefore demonstrates that the spacetime is future timelike geodesically incomplete.

Now if (79) holds then Eq. (77) holds for some \(K>0\) because the former is just the \(K\rightarrow 0+\) limit of the latter. Therefore we have conditions (78) and (77) and the previous paragraph shows that (c)(i, ii) both hold. \(\square \)

We now apply this theorem to the more specific setting of Einstein–Klein–Gordon theory, using the worldline bound of Theorem 3

### Corollary 1

*M*,

*g*) is globally hyperbolic, let

*S*be a smooth spacelike Cauchy surface for (

*M*,

*g*) and suppose that \(\phi \) obeys global bounds \(|\phi |\le \phi _{\text {max}}< (8\pi \xi )^{-1/2}\) and \(|\nabla _{\dot{\gamma }}\phi |\le \phi '_{\text {max}}\) along all unit speed timelike geodesics \(\gamma \) issuing orthogonally from

*S*. Assume hypotheses (a) and (c) of Theorem 5, where

*Q*and \(\tilde{Q}\) are given by

*M*,

*g*) is future timelike geodesically incomplete.

### Proof

*f*. Next, using the inequality

*f*. Thus Eq. (67) implies that Eq. (75) holds with

*Q*and \(\tilde{Q}\) given by Eq. (87). This supplies hypothesis (b) of Theorem 5 and as we also assume hypotheses (a) and (c) the result follows. \(\square \)

As in Theorem 5, one could replace the hypotheses (76) and (77) by the alternatives Eqs. (78) and (79).

We can easily generalize Corollary 1 to include additional matter provided it obeys the weak and the strong energy condition. The proof of the following Corollary is immediate by using Theorem 4 in place of Theorem 3.

### Corollary 2

In spacetime dimension \(n>2\) and with coupling \(\xi \in [0,\xi _c]\), let \((M,g,\phi )\) be a solution to the Einstein–Klein–Gordon equations with additional matter obeying both the strong and weak energy conditions. If all the other hypotheses of Corollary 1 are satisfied then (*M*, *g*) is future timelike geodesically incomplete.

## 7 Discussion

This paper has accomplished two main goals. First, we have established worldline and worldvolume bounds on the effective energy density of the nonminimally coupled scalar field. Elsewhere, these bounds will be used as the basis for a quantum energy inequality variant of the strong energy condition for the quantized field. Second, we have shown that our bounds can be used to derive a Hawking-type singularity theorem for the Einstein–Klein–Gordon theory, by applying methods developed in [10]. This result was generalized to include additional matter given that it obeys the weak and the strong energy condition. The overall message here is that sufficient initial contraction on a compact Cauchy surface is enough to guarantee timelike geodesic incompleteness, even though the non-minimally coupled Klein–Gordon theory does not obey the SEC.

*G*and

*c*, the constants

*Q*and \(\tilde{Q}\) become

*m*has units of inverse length. Thus

*Q*is dimensionless, while \(\tilde{Q}\) has dimensions of inverse time, as required for dimensional correctness in Eq. (77) with \(\tau _0\) having dimensions of time and consequently

*K*being an inverse time. Evidently both

*Q*and \(\tilde{Q}\) become very large if \(\phi _{\text {max}}^2\) is allowed to be close to the critical value \(c^4/(8\pi G\xi )\). However, \(Q\le 4\) if \(\phi _{\text {max}}^2\) does not exceed half the critical value, for example, so it is no great restriction to take

*Q*of order 1. Turning to \(\tilde{Q}\), the second term is \(Q^2(\phi _{\text {max}}'/\phi _{\text {max}})^2\) up to numerical factors. The ratio \(\phi _{\text {max}}'/(mc\phi _{\text {max}})\) is dimensionless and it would be reasonable to assume bounds so that this quantity does not greatly exceed unity. Therefore \(\tilde{Q}\) would be reasonably expected not greatly to exceed

*mc*. The remaining ingredients in the contraction bound Eq. (77) are the timescale \(\tau _0\) and constant

*K*, which depend on the history of the solution prior to

*S*; we may assume for the purposes of discussion that the corresponding terms in Eq. (77) are not large. Accordingly, the initial contraction required to ensure geodesic incompleteness might be expected to be of the order of

*mc*, i.e., the characteristic frequency of the (minimally coupled) Klein–Gordon operator (recall that

*m*is an inverse length in this discussion).

For the purely classical Einstein–Klein–Gordon theory, this seems very reasonable. But—with a view to semiclassical quantum gravity—what if the scalar field is supposed to describe an elementary particle, with a correspondingly small mass? In this situation *m* is replaced in Eq. (91) by \(mc/\hbar \), the inverse Compton length for a particle of mass *m*, so \(\tilde{Q}\) is of the order of the inverse Compton time and our previous reasoning would suggest that a huge initial contraction would be required to guarantee geodesic incompleteness. For example, with the physical values of *G*, \(\hbar \) and *c* in \(n=4\) dimensions, and with *m* on the order of the pion mass, \(m=140\text {MeV}/c^2\), the initial contraction would be of the order \(2\times 10^{23}{\text {s}}^{-1}\) (using the value \(\hbar =6.6\times 10^{-22}\text {MeV.s}\)) by such arguments. This would call into question the utility of the singularity result for this situation.

*n*, in a thermal state of temperature \(T<T_m\), where \(T_m=mc^2/k\) sets a natural temperature scale for the model, beyond which its applicability might be doubtful. Here,

*k*is Boltzmann’s constant. In this regime, the expectation value of the Wick square is

*n*dimensions). The derivation of this estimate is given in Appendix A. If we take \(\phi _{\text {max}}^2\sim \langle {:}\phi ^2{:}\rangle _T\) then the factor

*Q*is reasonably assumed to be given by

*T*to parameterise the maximum acceptable field amplitude in this way leads to a contraction estimate

For a pion in \(n=4\), the leading factor on the right-hand side \(m/m_{\text {Pl}}=5.9 \times 10^{-20}\), while \(mc^2/ \hbar \), as we mentioned, is \(2\times 10^{23} \text {s}^{-1}\) and \(T_m=1.7 \times 10^{12}\text {K}\). However, the remaining factor decays very rapidly as \(T/T_m\) is reduced below 1. For example, if \(T=10^{-2}T_m\) then \((T/T_m)^{(n-2)/4} (K_{(n-2)/2}(T_m/T) )^{1/2}=6.8 \times 10^{-24}\), thus bringing the overall contraction needed down to the order of \(10^{-19} \text {s}^{-1}\). This is indeed very small: If a volume were subject to contraction at this constant fractional rate, it would require approximately 100 times the current age of the universe to halve its size. Our calculation has allowed for a maximum temperature scale of \(T=10^{10}\text {K}\) – the temperature of the universe approximately one second after the Big Bang. Repeating the calculation for the Higgs mass \(125\text {GeV}/c^2\), a minimum contraction of order \(10^{-14}\text {s}^{-1}\) is required, but this time allowing a temperature up to \(T=10^{13}\text {K}\), the temperature of the Universe at age \(0.0001\text {s}^{-1}\).

Summarising this discussion: a model in which the field mass is taken equal to an elementary particle mass would need very little initial contraction to guarantee that either there is geodesic incompleteness or that, at the least, the solution evolves to a situation where the natural energy scales associated with the field approach those of the early universe. At this stage, a macroscopic observer might be forgiven for believing that a singularity had occurred!

Hawking and Ellis [19] discuss the violation of an average SEC by a pion. Their heuristic analysis led them to argue that the convergence of timelike geodesics would not be influenced by SEC violation on scales greater than \(10^{-14} \text {m}\). By contrast, our analysis instead rigorously shows that even for extremely small initial contractions a singularity is inevitable.

The obvious extension of this work is the derivation of a quantum strong energy inequality for the non-minimally coupled field. To examine if such an inequality can be used to prove a Hawking-type singularity theorem, it is also necessary to find estimates on the timescales for averaging over which the curved spacetime approximates the Minkowski results.

It should be noted that to fully understand whether the dynamics are driven towards singularities at the semi-classical level requires a semiclassical analysis that takes backreaction into account in a dynamical way. Positive results include [6], which calculates geometrical fluctuations from (passive) quantum fields, and a result on ANEC with transverse smearing [15]. However, the calculation of backreaction, even in the second order in perturbation theory brings significant technical challenges to the problem. Finally, the question of whether full quantum gravity can resolve singularities via a “big bounce” or not remains open.

## Footnotes

## Notes

### Acknowledgements

We thank Roger Colbeck for comments on the text. This work is part of a project that has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 744037 “QuEST”. P.J.B. thanks the WW Smith Fund for their financial support.

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