A new derivation of singularity theorems with weakened energy hypotheses

Let $(M,g_{\mu\nu})$ be a smooth Lorentzian spacetime, and let S be a smooth spacelike hypersurface in M. If $\gamma:[0,\tau]\to M$ is a smooth timelike curve, its length (i.e. the total proper time along $\gamma$) is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle L[\gamma] = \int_0^\tau |\dot{\gamma}(t)| \,{\rm d}t,\qquad |V|:=\sqrt{g_{\mu\nu}V^\mu V^\nu}. \nonumber \end{align} \tag{ 3 }$$

Let $q\in I^+(S)$ be fixed; then $\gamma$ is a critical point of the length functional among unit-speed timelike curves joining S to q if and only if it is an affine geodesic issuing normally from S. In more detail, $\gamma$ is a geodesic of this type if and only if ${\rm d}L[\gamma_s]/{\rm d}s|_{s=0}=0$ for every smooth 1-parameter family of curves $\gamma_s:[0,\tau]\to M$, $s\in (-\delta,\delta)$ obeying

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \gamma_0=\gamma, \qquad \gamma_s(0)\in S, \qquad \gamma_s(\tau)=q \nonumber \end{align} \tag{ 4 }$$

for all $s\in (-\delta,\delta)$. Introducing some terminology, the partial derivatives of $\gamma_s(t)$ (in an arbitrary system of coordinates) with respect to t and s determine the longitudinal and transverse vector fields $U^\mu = \partial \gamma_s(t){}^\mu/\partial t$, $V^\mu = \partial \gamma_s(t){}^\mu/\partial s$, obeying

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:UV} \nabla_U V^\mu = \nabla_V U^\mu, \nonumber \end{align} \tag{ 5 }$$

an identity which holds for any smooth 1-parameter family of curves. In particular, the restrictions of $U^\mu$ and $V^\mu$ to $\gamma$ yield the velocity vector $U^\mu|_{\gamma(t)} = \dot{\gamma}^\mu(t)$ and variation vector field $V^\mu|_{\gamma(t)} = {\rm d}\gamma_s(t){}^\mu/{\rm d}s|_{s=0}$.

The above discussion may be generalised to piecewise smooth curves, with the result that $\gamma$ is a critical point of L among unit-speed piecewise smooth curves if and only if it is an unbroken timelike geodesic in its proper time parametrisation. The variation field arising from a piecewise smooth variation of $\gamma$ is continuous and piecewise smooth, while the velocity field may have discontinuities (see [3, section 10.1] for further background on piecewise smooth variations).

Now suppose that $\gamma$ is an affine geodesic emanating normally from S. A point p  on $\gamma$ is a focal point to S along $\gamma$ if there is a nontrivial variation of $\gamma$, among affine geodesics issuing normally from S, with a variation field that vanishes at p . The corresponding variation vector field therefore obeys the equation of geodesic deviation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:Jacobi} \frac{D^2 V^\mu}{{\rm d}t^2} + R^\mu_{\phantom{\mu}\nu\alpha\beta}U^\nu U^\alpha V^\beta=0, \nonumber \end{align} \tag{ 6 }$$

i.e. $V^\mu$ is a Jacobi field. The other conditions on the variation imply that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:focalside} V^\mu|_{p} = 0, \qquad V^\mu U_\mu|_{\gamma(0)}= 0, \qquad K_{\mu\nu}V^\mu V^\nu|_{\gamma(0)}= V_\mu \nabla_UV^\mu|_{\gamma(0)}, \nonumber \end{align} \tag{ 7 }$$

where $K_{\mu\nu}=\nabla_\mu\xi_\nu$ is the extrinsic curvature tensor of S and $\xi^\mu$ is the normalised longitudinal vector field of this variation (in particular $U^\mu|_\gamma=\xi^\mu|_\gamma$). In fact, the first two conditions in equation (7), combined with equation (6), imply that $U_\mu V^\mu$ vanishes identically along $\gamma$.

The existence of focal points along $\gamma$ is closely related to the question of whether $\gamma$ is a local maximum of the length functional, among constant speed curves joining S to q (not necessarily geodesics). The analysis starts from a formula for the second derivative of the length functional for a variation $\gamma_s$ of $\gamma$ in which each $\gamma_s$ is a piecewise smooth constant speed curve² joining S to q and $\gamma_0$ has unit speed:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:indexform} \left.\frac{{\rm d}^2}{{\rm d}s^2}L[\gamma_s]\right|_{s=0} = I[V]:=\int_0^\tau\left(\frac{DV^\mu}{{\rm d}t}\frac{DV_\mu}{{\rm d}t} - R_{\mu\nu\alpha\beta} U^\mu V^\nu V^\alpha U^\beta \right)\,{\rm d}t + K_{\mu\nu}V^\mu V^\nu|_{\gamma(0)}. \nonumber \end{align} \tag{ 8 }$$

The quantity $I[V]$ is called the index form; it is usually presented in a polarised form as a bilinear form in two vector fields, but we will not need to do that here.

Owing to the conditions placed on $\gamma_s$, we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:Vside} V^\mu U_\mu|_{\gamma(0)}= 0, \qquad V^\mu|_{\gamma(\tau)}=0, \qquad U_\mu\frac{DV^\mu}{{\rm d}t}\equiv 0, \nonumber \end{align} \tag{ 9 }$$

where the first two conditions reflect the boundary conditions that $\gamma_s(0)\in S$, $\gamma_s(\tau)=q$, while the third is due to the restriction to constant speed curves. (To see this, note that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle U_\mu \nabla_UV^\mu=U_\mu \nabla_VU^\mu=\frac{1}{2}\nabla_V(U^\mu U_\mu)=\frac{1}{2\tau}\left.\frac{\rm d}{{\rm d}s} L[\gamma_s]^2\right|_{s=0}=0 \nonumber \end{align} \tag{ 10 }$$

using constant speed parametrisation and the fact that $\gamma_0$ is geodesic.) As $\gamma$ is an affine geodesic, these conditions imply that $\newcommand{\e}{{\rm e}} U_\mu V^\mu\equiv 0$. If V is smooth, integrating by parts in (8) and rearranging gives

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle I[V] = -\int_0^\tau V_\mu \left(\frac{D^2V^\mu}{{\rm d}t^2} + R^\mu_{\phantom{\mu}\nu\alpha\beta} U^\nu U^\alpha V^\beta \right)\,{\rm d}t + V_\mu\nabla_U V^\mu|_{\gamma(\tau)} + (K_{\mu\nu}V^\mu V^\nu - V_\mu\nabla_U V^\mu)|_{\gamma(0)}, \nonumber \end{align} \tag{ 11 }$$

and it is clear from equations (6) and (7) that $I[V]=0$ if $\gamma(\tau)$ is a focal point to S along $\gamma$ and $V^\mu$ is the corresponding Jacobi field (which necessarily satisfies the conditions (9)). In this situation the index form is not negative definite, so either the second derivative test is inconclusive or $\gamma$ is not a local maximum of L. Table 1 summarises a more detailed relationship between the index form and focal points presented in theorem 10.34 in [37]. From these logical relationships, it may be seen that the first two implications may be replaced by 'if and only if' statements. For if $I[V]<0$ for all $V^\mu$ then neither of the second or third conclusions holds, and therefore neither of the hypotheses on these lines can hold. Therefore there is no focal point in $(0,\tau]$ and so the first implication admits a converse. Similarly, if the second conclusion holds then neither of the first or third can and we deduce that there must be a focal point in $(0,\tau)$, so the implication on the second line also admits its converse. In particular, we have:

Table 1. Summary of theorem 10.34 in [37]. Here 'for some/all $V^\mu$' is to be interpreted as 'for some/all piecewise smooth $V^\mu$ obeying the conditions (9)'.

$\newcommand{\e}{{\rm e}} \not\exists$ focal point in $(0,\tau]$	$\Longrightarrow$	$I[V]<0$ for all $V^\mu$
$\newcommand{\e}{{\rm e}} \exists$ focal point in $(0,\tau)$	$\Longrightarrow$	$I[V]>0$ for some $V^\mu$
Only focal point in $(0,\tau]$ is $\tau$	$\Longrightarrow$	$I[V]\leqslant 0$ for all $V^\mu$, and $I[V]=0$ for some $V^\mu$

Proposition 2.1. There exists a focal point $\gamma(r)$ to S along $\gamma$ for $r\in (0,\tau]$ (resp., $r\in (0,\tau)$) if and only if $I[V]\geqslant 0$ (resp., $I[V]>0$) for some piecewise smooth $V^\mu$ obeying (9). Consequently: if $\gamma$ is length-maximising, then there is no focal point in $(0,\tau)$; if $\gamma$ is not length-maximising, then there is a focal point in $(0,\tau]$; if there is a focal point in $(0,\tau)$, then $\gamma$ is not length-maximising; if there is no focal point in $(0,\tau]$ then $\gamma$ is length-maximising.

Next, suppose $v^\mu$ is a unit spacelike vector tangent to S at $\gamma(0)$, extended along $\gamma$ by parallel transport. Then, any continuous piecewise smooth function f  obeying $f(0)=1$, $f(\tau)=0$ determines a continuous piecewise smooth variation field obeying (9) by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle V^\mu = f v^\mu. \nonumber \end{align} \tag{ 12 }$$

Noting that $D V^\mu/{\rm d}t = \dot{f}v^\mu$, which is spacelike, we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle I[V] = \int_0^\tau\left(-\dot{f}^2 - f^2 R_{\mu\nu\alpha\beta} U^\mu v^\nu v^\alpha U^\beta \right)\,{\rm d}t + K_{\mu\nu}v^\mu v^\nu|_{\gamma(0)}. \nonumber \end{align} \tag{ 13 }$$

But now consider an orthonormal basis $e_\mu$ ($\mu=0,\ldots,n-1$) for the tangent space to S at $\gamma(0)$, in which $e_0^\mu=U^\mu$, and apply the preceding argument to each element e_i ($i=1,\ldots,n-1$), keeping the same scalar function f . Summing, we find

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sum_{i=1}^{n-1} I[\,f e_i] = -\int_0^\tau\left((n-1)\dot{f}^2 + f^2 R_{\mu\nu} U^\mu U^\nu \right)\,{\rm d}t - K|_{\gamma(0)}, \nonumber \end{align} \tag{ 14 }$$

where we have used the fact that $U^\mu U^\nu K_{\mu\nu}=0$, so that the extrinsic curvature $K=g^{\mu\nu}K_{\mu\nu}=-\sum_i K(e_i,e_i)$. If there is no focal point in $(0,\tau]$ (resp., $(0,\tau)$) then each term on the left-hand side is negative (resp., nonpositive) for all f  obeying the boundary conditions $f(0)=1$, $f(\tau)=0$. Conversely, if the right-hand side is nonnegative (resp., positive) for some such f , then the same must be true of at least one of the terms on the left, and it follows that there is a focal point in $(0,\tau]$ (resp., $(0,\tau)$). In other words, we have sufficient conditions for the presence of a focal point in $(0,\tau)$ or $(0,\tau]$ (and consequently some necessary conditions for their absence).

Proposition 2.2. Let $\gamma:[0,\tau]\to M$ be a unit-speed timelike geodesic emanating normally from a smooth spacelike hypersurface S. If there exists a continuous, piecewise smooth f  on $[0,\tau]$ obeying $f(0)=1$, $f(\tau)=0$ and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:proptimelike} \int_0^\tau\left((n-1)\dot{f}^2 + f^2 R_{\mu\nu} U^\mu U^\nu \right)\,{\rm d}t\leqslant -K|_{\gamma(0)}, \nonumber \end{align} \tag{ 15 }$$

then there is a focal point to S along $\gamma$. If the inequality (15) holds with a strict inequality then the focal point lies before $\gamma(\tau)$.

This result may be used to deduce the existence of focal points, without using the Raychaudhuri equation [37, proposition 10.37]. Recall that the timelike convergence condition asserts that $R_{\mu\nu} U^\mu U^\nu\leqslant 0$ for all timelike $U^\mu$, which is equivalent to the SEC for solutions to the Einstein equations³.

Corollary 2.3. If $K|_{\gamma(0)}<0$, $\tau\geqslant (n-1)/|K|_{\gamma(0)}|$ , and the Ricci tensor obeys the timelike convergence criterion, then there is a focal point to S along $\gamma$, i.e. by proper time $\tau$ at the latest. If $\tau> (n-1)/|K|_{\gamma(0)}|$ the focal point occurs before time $\tau$.

Proof. Apply proposition 2.2 using the function $f(t)=1-t/\tau$, noting that the left-hand side of (15) is less than or equal to $(n-1)/\tau$ which is less than or equal to $-K|_{\gamma(0)}$ by assumption. This gives the first stated result; the second is a trivial modification. □

At this point it is helpful to explain the relationship between the index form method and the traditional approaches based on the Raychaudhuri equation. One way to determine whether the condition given in proposition 2.2 holds is to solve the variational problem of minimising the right-hand side of equation (15), treated as a functional J[f ], over smooth f  obeying $f(0)=1$, $f(\tau)=0$. Considering variations $\newcommand{\e}{{\rm e}} f+\epsilon g$ where g is smooth and obeys $g(0)=g(\tau)=0$, one finds easily that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle J[\,f+\epsilon g] = J[\,f]+ 2\epsilon \int_0^\tau\left(- (n-1)\ddot{f} + f \rho \right)g \,{\rm d}t + \epsilon^2 J[g], \nonumber \end{align} \tag{ 16 }$$

where we now write $\rho(t)=R_{\mu\nu} U^\mu U^\nu|_{\gamma(t)}$. There is at most one stationary point, namely the solution to

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:EulerLagrange} - (n-1)\ddot{f} + f \rho = 0, \qquad f(0)=1,~f(\tau)=0, \nonumber \end{align} \tag{ 17 }$$

if it exists (which it does unless there is a solution to the same equation with $f(0)=f(\tau)=0$). For this solution, one finds $J[\,f] = (n-1)\dot{f}(0)$ using an integration by parts⁴. Therefore a sufficient condition for (15) to hold is that the solution to (17) has $\dot{f}(0)\leqslant -K|_{\gamma(0)}/(n-1)$.

Assuming that f  is nonvanishing in $(0,\tau)$, we may now set $\theta=(n-1)\dot{f}/f$ and note that the Euler–Lagrange equation (17) may be rewritten as the Riccati equation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:Riccati} \dot{\theta} = -\frac{\theta^2}{n-1} +\rho, \qquad \theta(0) = (n-1)\dot{f}(0), \nonumber \end{align} \tag{ 18 }$$

with $\theta\to-\infty$ as $t\to \tau^-$. (If f  has an interior zero then we repeat the reasoning on the interval between t  =  0 and the first zero.) We see that a sufficient condition for the existence of a focal point along $\gamma$ is that the above Riccati equation fails to have a solution beyond $[0,\tau]$ for initial data with $\theta(0)\leqslant -K|_{\gamma(0)}$.

By contrast, now consider the (irrotational) congruence of unit speed timelike geodesics emanating normally from the S with velocity field $U^\mu$. The Raychaudhuri equation for the expansion $\theta = \nabla_\mu U^\mu$ gives

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{D\theta}{{\rm d}t} =R_{\mu \nu} U^\mu U^\nu-2\sigma^2 -\frac{\theta^2}{n-1} ,\qquad \theta|_{\gamma(0)} = K|_{\gamma(0)}, \nonumber \end{align} \tag{ 19 }$$

along any geodesic in the congruence, where $\sigma$ is the shear scalar, and so $\theta$ obeys the differential inequality

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:Ricc} \frac{D\theta}{{\rm d}t}\leqslant R_{\mu \nu} U^\mu U^\nu -\frac{\theta^2}{n-1} ,\qquad \theta|_{\gamma(0)} = K|_{\gamma(0)}. \nonumber \end{align} \tag{ 20 }$$

Therefore the Raychaudhuri equation leads to a similar analysis to that arising from the index form methods, though with the slight complication of working with a differential inequality rather than a differential equation. More significantly, in the index form approach one need not pass to differential equations at all, but instead try to satisfy (15) by judicious choice of trial functions f . This is exactly what was done in the proof of corollary 2.3 and will be our approach in the rest of this paper.

To complete this section, we now recall one of the simplest links between the presence of focal points and timelike geodesic incompleteness. The following result draws on [27] and [37, theorem 14.55A].

Proposition 2.4. Let S be a smooth spacelike Cauchy surface in M. (a) If S is compact and every future-complete timelike geodesic emanating normally from S contains a focal point then M is future timelike geodesically incomplete. (b) If every future-directed timelike geodesic emanating normally from S without focal points has length (strictly) less than $\tau_*$ then every future-directed timelike curve emanating from S has length (strictly) less than $\tau_*$; consequently M is future timelike geodesically incomplete.

Proof. 

• (a)  
  Assume that M is future timelike geodesic complete. As discussed in the proof of [15, theorem 5.1], based on results of [27], the hypotheses imply the existence of an S-ray; namely, a future inextendible unit-speed timelike geodesic $\gamma$ emanating (necessarily normally) from S and which is length-maximising between S and each of its points. Accordingly $\gamma$ contains no focal points to S, but as it is future-complete by assumption we obtain a contradiction.
• (b)  
  By [37, lemma 14.29 & theorem 14.44], every $q\in D^+(S)\setminus S$ is connected to S by a timelike length-maximising geodesic, which necessarily emanates normally from S and has no focal points before q. By hypothesis, this geodesic has length (strictly) less than $\tau_*$.

Now consider any unit-speed future-directed timelike curve $\gamma:[0,\tau]\to M$ with $\gamma(0)\in S$, assuming for a contradiction that $\tau> \tau_*$ (or $\tau\geqslant\tau_*$). But $\gamma(\tau)$ is also joined to S by a length-maximising geodesic of length $\tau'$ (strictly) less than $\tau_*$, so we deduce $\tau_*\geqslant\tau'\geqslant \tau>\tau_*$ (or $\tau_*>\tau'\geqslant \tau\geqslant\tau_*$) thus obtaining a contradiction. In particular, every inextendible future-directed timelike geodesic emanating from S has finite length and is therefore incomplete. □

Combining corollary 2.3 and proposition 2.4 yields one of Hawking's singularity theorems [28].

Corollary 2.5. If $\sup_S K<0$ on S (in particular, if S is compact and K  <  0) and the timelike convergence condition holds, then M is future-timelike geodesically incomplete.

The central idea of this paper is that results similar to corollary 2.3, proposition 2.4 and corollary 2.5 may be proved under weaker assumptions on the Ricci tensor, by replacing the linear function f  in the proof of corollary 2.3 by a suitable alternative. This will be described in sections 3 and 4 below. Before that, we describe how the above theory can be adapted to null geodesics, again following [37].

Let $\newcommand{\e}{{\rm e}} \gamma : [0,\ell] \to M$ be a piecewise smooth curve. Then the integral

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:energyint} E[\gamma]=\frac{1}{2} \int_0^\ell g(\gamma'(\lambda),\gamma'(\lambda)) {\rm d}\lambda , \nonumber \end{align} \tag{ 21 }$$

is called the energy or action integral, and is the natural quantity to consider in the variational theory of null geodesics because, unlike the situation for L, $E[\gamma_s]$ varies smoothly in s for any piecewise smooth variation of $\gamma$, regardless of the causal nature of $\gamma_s$. Let P be a smooth semi-Riemannian submanifold of M, $q \in M \setminus P$ and $\Omega(P,q)$ the manifold of all piecewise smooth curve segments from P to q. Then the critical points of E among curves in $\Omega(P,q)$ are geodesics emanating normally from P, and the second derivative of E in a smooth variation $\gamma_s$ of such a geodesic $\gamma=\gamma_0$ is [37, proposition 10.39]

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle \label{eqn:Esecondder} \frac{\partial^2 E[\gamma_s]}{\partial s^2} \bigg|_{s=0}=\int_0^\ell\left[(\nabla_U V_\mu)(\nabla_U V^\mu) -R_{\mu\nu \alpha \beta}U^\mu V^\nu V^\alpha U^\beta \right] {\rm d}\lambda+(U_\mu \nabla_V V^\mu)\bigg|^{\ell}_{0} , \nonumber \end{align} \tag{ 22 }$$

where $U^\mu=\gamma'(\lambda)$ and $V^\mu|_{\gamma(\lambda)} = {\rm d}\gamma_s(\lambda){}^\mu/{\rm d}s|_{s=0}$, and we assume that $\gamma$ is affinely parametrised. The right-hand side of (22) is, by definition, the Hessian $\mathcal{H}[V]$ of E at critical points; it may also be written

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle \mathcal{H}[V]=\int_0^\ell \left(\frac{D V^\mu}{{\rm d}\lambda} \frac{D V_\mu}{{\rm d}\lambda} -R_{\mu\nu \alpha \beta}U^\mu V^\nu V^\alpha U^\beta \right){\rm d}\lambda-U_\mu \mathrm{I\!I}^\mu (V,V) \bigg|_{\gamma(0)} , \nonumber \end{align} \tag{ 23 }$$

where $\mathrm{I\!I}$ is the shape tensor or second fundamental form, defined so that $\mathrm{I\!I}^\mu (V,W)$ is the projection of $\nabla_V W^\mu$ onto the subspace of vectors normal to P, for vector fields $V,W$ tangential to P. The following result is proved as proposition 10.41 of [37].

Proposition 2.6. Let P be a spacelike submanifold of M. If there are no focal points of P along a normal null geodesic $\gamma \in \Omega(P,q)$, then $\mathcal{H}[V]$ is positive semidefinite when restricted to piecewise smooth $V^\mu$ obeying $\newcommand{\e}{{\rm e}} U_\mu V^\mu\equiv 0$. If $\mathcal{H}[V]=0$ for some such $V^\mu$, then $V^\mu$ is tangent to $\gamma$.

Now let $\newcommand{\e}{{\rm e}} \gamma:[0,\ell]\to M$ be a null geodesic affinely parametrized by $\lambda$ and P a spacelike $(n-2)$-dimensional submanifold of M. Let e_i $(i=1, \dots ,n-2)$ be an orthonormal basis at $T_{\gamma(0)} (P)$. We parallel transport these vectors along $\gamma$ to get E_i $(i=1, \dots ,n-2)$. Let f  be a smooth function with $f(0)=1$ and $\newcommand{\e}{{\rm e}} f(\ell)=0$. Then

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle H[\,f E_i]=\int_0^{\ell} \left(-f'^2 -f^2 R_{\mu\nu \alpha \beta}U^\mu E_i^\nu E_i^\alpha U^\beta \right){\rm d}\lambda-U_\mu \mathrm{I\!I}^\mu (E_i,E_i) \bigg|_{\gamma(0)}. \nonumber \end{align} \tag{ 24 }$$

Now sum over all $i=1, \dots, n-2$, noting that $g^{\nu\alpha} = U^{(\nu} W^{\alpha)}-\sum_{i=1}^{n-2}E_i^\nu E_i^\alpha$ for a suitably chosen null vector W, to obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:prenull} \sum_{i=1}^{n-2} \mathcal{H}[\,f E_i]=-\int_0^{\ell} \left((n-2)\,f'^2 -f^2 R_{\mu\nu }U^\mu U^\nu \right){\rm d}\lambda-(n-2)U_\mu H^\mu |_{\gamma(0)} , \nonumber \end{align} \tag{ 25 }$$

where

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle H^\mu=\frac{1}{n-2}\sum_{i=1}^{n-2}\mathrm{I\!I}^\mu (E_i,E_i) \nonumber \end{align} \tag{ 26 }$$

is the mean normal curvature vector field of P.

This calculation may be used in conjunction with proposition 2.6 to give a sufficient condition for the existence of a focal point along $\gamma$, just as in the derivation of proposition 2.2. As there is no unique natural parametrisation of a null geodesic, it is convenient to state the result in an invariant form, regarding $\gamma$ as an unparametrised 1-dimensional submanifold of M. The notation ${\rm d}\gamma^\mu$ denotes the line element 1-form on $\gamma$, giving the tangent vector ${\rm d}\gamma^\mu/{\rm d}\lambda$ with respect to any coordinate $\lambda$ on $\gamma$, while ${\rm d}\gamma^\mu_+$ is the pseudo-1-form which is equal to ${\rm d}\gamma^\mu$ with respect to coordinates parametrising $\gamma$ as a future-directed curve. Expressions such as the right-hand side of (25) may be written in invariant form by regarding $f(\lambda)$ as the coordinate expression of a density f  of weight $-\frac{1}{2}$ on $\gamma$. Note that the combination $f^2 {\rm d}\gamma_+$ then defines a vector field along $\gamma$, given in coordinates by $f(\lambda){}^2 {\rm d}\gamma^\mu/{\rm d}\lambda$, provided that ${\rm d}\gamma^\mu/{\rm d}\lambda$ is future-pointing. It should be borne in mind that, while a density does not have invariant values at individual points, the sign of the density is invariantly defined; likewise, it makes sense to say that the density vanishes or is nonvanishing at a given point. In this notation, the result we have proved may be stated as follows.

Proposition 2.7. Let P be a spacelike submanifold of M of co-dimension 2 and let $\gamma$ be a null geodesic joining $p\in P$ to $q\in J^+(P)$. If there exists a smooth $(-\frac{1}{2})$-density f  on $\gamma$ which is nonvanishing at p  but vanishes at q and so that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eqn:nullgen} \int_\gamma \left((n-2)(\nabla_{{\rm d}\gamma}f)^2 +f^2 \mathrm{Ric}({\rm d}\gamma,{\rm d}\gamma) \right) \leqslant -(n-2) g(\,f^2 {\rm d}\gamma_+, H) |_{p} , \nonumber \end{align} \tag{ 27 }$$

then there is a focal point to P along $\gamma$; if the inequality holds strictly, then the focal point is located before q.

Note that $|\nabla_{{\rm d}\gamma} f|$ is a $\frac{1}{2}$-density on $\gamma$, while $\mathrm{Ric}({\rm d}\gamma,{\rm d}\gamma)$ is a 2-density. Accordingly, each term in the integrand of (27) is a density; similarly, $f^2{\rm d}\gamma^\mu_+$ is a vector field along $\gamma$. Written more explicitly, if $\gamma$ is parametrised by a coordinate $\newcommand{\e}{{\rm e}} \lambda\in [0,\ell]$, inequality (27) is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \int_0^\ell \left((n-2)\,f'(\lambda)^2 +f(\lambda)^2 R_{\mu\nu}U^\mu U^\nu \right) {\rm d}\lambda \leqslant -(n-2)\,f(0)^2 U_\mu H^\mu|_{p} , \nonumber \end{align} \tag{ 28 }$$

assuming $U^\mu={\rm d}\gamma^\mu/{\rm d}\lambda$ is future-directed.

Proposition 2.7 may be used to prove the following corollary (see [37, proposition 10.43]). Recall that P is said to be future-converging if $H^\mu$ is past-pointing timelike everywhere on P. In this situation we may write $H^\mu=H \hat{H}^\mu$ where H  <  0 and $\hat{H}^\mu$ is a future-pointing timelike unit vector. For any future-pointing timelike unit vector $V^\mu$ at p , we also write $L_V(\gamma)$ for the length of $\gamma$ with respect to an affine parameter in which $V_\mu {\rm d}\gamma^\mu/{\rm d}\lambda =1$ at p . We refer to $L_V(\gamma)$ as the $V$-length of $\gamma$.

Corollary 2.8. With P and $\gamma$ as in proposition 2.7, suppose additionally that P is future-converging and the null convergence condition $\mathrm{Ric}({\rm d}\gamma,{\rm d}\gamma) \leqslant 0$ holds everywhere along $\gamma$. If $L_{\hat{H}}(\gamma) \geqslant 1/|H|$ then there is a focal point to P along $\gamma$.

Proof. Choose an affine coordinate $\lambda$ on $\gamma$, so that $p=\gamma(0)$ and $\hat{H}_\mu {\rm d}\gamma^\mu/{\rm d}\lambda =1$. Then $\newcommand{\e}{{\rm e}} q=\gamma(\ell)$, with $\newcommand{\e}{{\rm e}} \ell=L_{\hat{H}}(\gamma)$. In these coordinates define $\newcommand{\e}{{\rm e}} f(\lambda)=1-\lambda/\ell$; then the right-hand side of (27) is $-(n-2)H$, while the left-hand side is less than or equal to $\newcommand{\e}{{\rm e}} (n-2)/\ell$, and the result follows by proposition 2.7. □

As in section 2.1, inequality (27) may be connected to a Riccati equation related to the Raychaudhuri equation for a null geodesic congruence.

Now we can connect the formation of focal points with future null geodesic incompleteness in the following way, drawing on the formulations in [15, 37].

Proposition 2.9. Suppose that: (i) M is globally hyperbolic with non-compact Cauchy hypersurfaces; (ii) P is a compact achronal smooth spacelike submanifold of M of co-dimension 2; and (iii) every future-complete null geodesic emanating normally from P contains a focal point to P. Then M is future null geodesically incomplete. If (iii) is replaced by: (iii)${}'$ P is future converging and every future-directed null geodesic emanating normally from P with $\hat{H}$-length at least $\newcommand{\e}{{\rm e}} \ell$ contains a focal point to P, then there is an inextendible null geodesic emanating normally from P with $\hat{H}$-length less than $\newcommand{\e}{{\rm e}} \ell$.

Proof. As described in the proofs of theorem 5.2 in [15] (see also the comparable part of [37, theorem 14.61]) conditions (i) and (ii) imply that $E^+(P):=J^+(P)\setminus I^+(P)$ is equal to the boundary $\partial J^+(P)$ and is noncompact and closed. These properties were used to show that there is an inextendible affinely parametrised null geodesic $\gamma:[0,a)\to M$ issuing normally from P and contained entirely in $\partial J^+(P)$ for some $a\in (0,\infty]$. Furthermore, $\gamma$ can contain no focal points to P, because the portion of $\gamma$ beyond any focal point would lie in I⁺ (P) [35, theorem 6.16(a)]⁵ and hence outside E⁺ (P). Assumption (iii) then entails that $\gamma$ is not future-complete and the result is proved. Alternatively, (iii)${}'$ implies immediately that $\newcommand{\e}{{\rm e}} L_{\hat{H}}(\gamma)<\ell$. □

The last two results combine to yield the Penrose singularity theorem [39].

Corollary 2.10. If, in addition to the assumptions (i) and (ii) of proposition 2.9, P is future-converging and the null convergence condition holds, then M is future null geodesically incomplete.

Proof. Corollary 2.8 implies assumption (iii) of proposition 2.9. □

Instead of the timelike convergence condition, we assume that any deviations from timelike convergence along geodesics satisfy estimates of the form

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\vertiii}[1]{{\left\vert\left\vert\left\vert #1 \right\vert\right\vert\right\vert}} \displaystyle \label{eq:Ricci_hyp} \int_I f(t)^2 R_{\mu\nu} U^\mu U^\nu|_{\gamma(t)} \,{\rm d}t\leqslant \vertiii{f}^2:=Q_m(\gamma) \|f^{(m)}\|^2 + Q_0(\gamma) \|f\|^2, \nonumber \end{align} \tag{ 38 }$$

for some $m\in\mathbb{N}$ and all smooth real-valued f  supported in the interior of the compact interval $I\subset\mathbb{R}$, where $\gamma:I\to M$ is a unit-speed timelike geodesic, $Q_m(\gamma)>0$ and $Q_0(\gamma)\geqslant 0$ are constants (independent of f , but allowed to depend on $\gamma$) of appropriate dimensions and $\|\cdot\|$ denotes the standard norm of L²(I). By a well known result [1, corollary 6.31], $\newcommand{\vertiii}[1]{{\left\vert\left\vert\left\vert #1 \right\vert\right\vert\right\vert}} \vertiii{\cdot}$ is actually a norm on $C_0^\infty(\mathrm{int}\, I)$ which is equivalent to the standard Sobolev norm $\|f\|_m:=(\sum_{r=0}^m \|f^{(r)}\|^2){}^{1/2}$ and therefore induces the same topology. Estimates of this type might arise in solutions to the Einstein equations with matter, due to properties of the matter field, such as QEIs or related constraints, see e.g. [6, 17, 18].

Then, assuming that $R_{\mu\nu} U^\mu U^\nu|_{\gamma(t)}$ is continuous on I, a simple approximation argument shows that the same bound (38) holds for all f  in the Sobolev space $W_0^{m}(I)$, which is the closure of $C_0^\infty(\mathrm{int}\, I)$ in the norm $\newcommand{\vertiii}[1]{{\left\vert\left\vert\left\vert #1 \right\vert\right\vert\right\vert}} \vertiii{\cdot}$. Each such function f  is m  −  1 times continuously differentiable on I, with a distributional m'th derivative that may be identified with an element of L²(I), and obeys the generalised Dirichlet conditions $f^{(k)}|_{\partial I}=0$ for $0\leqslant k\leqslant m-1$; see e.g. [1, theorem 4.12(III)]. As I is compact, $L^2(I)\subset L^1(I)$ so f ^(m−1) is the indefinite integral of an element of L¹(I), and is therefore absolutely continuous.

Let S be a compact smooth spacelike Cauchy surface in M with extrinsic curvature K. Suppose that $\gamma:[0,\tau]\to M$ is a unit speed, future-directed, timelike geodesic emanating normally from S and write $\rho(t)= R_{\mu\nu} U^\mu U^\nu|_{\gamma(t)}$. By proposition 2.2, $\gamma$ contains a focal point to S if there is a piecewise smooth f  on $[0,\tau]$ with $f(0)=1$ and $f(\tau)=0$, such that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle J[\,f]\leqslant -K|_{\gamma(0)}, \nonumber \end{align} \tag{ 39 }$$

where

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle J[\,f]=\int_0^\tau\left((n-1)\dot{f}(t)^2 + f(t)^2 \rho(t) \right)\,{\rm d}t. \nonumber \end{align} \tag{ 40 }$$

Our aim is to estimate J[f ] for specific functions f  defined below, using (38) to control the contribution of $\rho$. The problem to be faced is that no function with $f(0)=1$ can belong to $W_0^m([0,\tau])$; we address this by applying (38) to a 'rounded off' function and making further assumptions on $\rho$ near t  =  0 to estimate the error incurred. The rounding off could be performed in many ways. We now turn to the two scenarios mentioned above, beginning with the situation in which timelike convergence holds for small positive values of $\tau$.

4.1.1. Scenario 1.

Suppose that $\rho(t)= R_{\mu\nu} U^\mu U^\nu|_{\gamma(t)}$ is a smooth function on $[0,\tau]$ that is initially negative, $\rho\leqslant \rho_0\leqslant 0$ on $[0,\tau_0]$ for some $0<\tau_0<\tau$. That is, timelike convergence (equivalent to the SEC) holds initially, with strict inequality if $\rho_0<0$.

Defining a piecewise smooth function on $[0,\tau]$ by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:scen1fdef} f(t) = \left\{\begin{array}{@{}ll@{}} 1 & t\in [0,\tau_0) \nonumber \\ I(m,m;(\tau-t)/(\tau-\tau_0)) & t\in[\tau_0,\tau], \end{array}\right. \nonumber \end{align} \tag{ 41 }$$

we note that $f(0)=1$, $f(\tau)=0$. We will prove the following:

Lemma 4.1. For the function f  given by (41) and $\rho$ satisfying (38) on $[0,\tau]$ we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:scenario1} J[\,f]\leqslant \nu_*:= (1-A_m)\rho_0 \tau_0 + \frac{Q_m(\gamma) C_m}{\tau_0^{2m-1}} + Q_0(\gamma) A_m\tau + \frac{(n-1)B_m}{\tau-\tau_0} + \frac{Q_m(\gamma) C_m}{(\tau-\tau_0)^{2m-1}}. \nonumber \end{align} \tag{ 42 }$$

Consequently, if $-K|_{\gamma(0)}\geqslant \nu_*$ then $\gamma$ contains a focal point to S.

Proof. The consequences are immediate by proposition 2.2, so it is enough to establish the estimate on J[f ]. Defining a piecewise smooth function on $[0,\tau]$ by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:scen1phidef} \varphi(t) = \left\{\begin{array}{@{}ll@{}} I(m,m;t/\tau_0) & t\in [0,\tau_0) \nonumber \\ 1 & t\in[\tau_0,\tau], \end{array}\right. \nonumber \end{align} \tag{ 43 }$$

we note that $f\varphi$ is m  −  1 times continuously differentiable, with $(\varphi f){}^{(m)}$ existing and continuous everywhere except $t=\tau_0$, where it has a finite jump discontinuity. Therefore $\varphi f\in W_0^m([0,\tau])$, and writing $f^2=(\varphi f){}^2 + (1-\varphi^2)\,f^2=(\varphi f){}^2 + 1-\varphi^2$, we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \int_0^\tau f(t)^2 \rho(t) \,{\rm d}t &\leqslant \int_0^{\tau_0} (1-\varphi(t)^2) \rho(t) \,{\rm d}t + Q_m(\gamma) \|(\varphi f)^{(m)}\|^2 + Q_0(\gamma) \|\varphi f\|^2\nonumber \\ &\leqslant \rho_0\int_0^{\tau_0} (1-\varphi(t)^2)\,{\rm d}t + Q_m(\gamma) \|(\varphi f)^{(m)}\|^2 + Q_0(\gamma) \|\varphi f\|^2 \nonumber \end{align} \tag{ 44 }$$

using (38), along with the assumption $\rho\leqslant \rho_0$ on $[0,\tau_0]$. Now $\varphi f$ is equal to $\varphi$ on $[0,\tau_0]$ and f  on $[\tau_0,\tau]$, and the L²-norms of derivatives of $(\varphi f)$ decompose accordingly. Indeed, using suitably rescaled versions of (36) (see (A.6)),

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \|\varphi f\|^2 = A_m \tau_0 + A_m(\tau-\tau_0)= A_m\tau,\qquad \|(\varphi f)^{(m)}\|^2 = \frac{C_m}{\tau_0^{2m-1}} + \frac{C_m}{(\tau-\tau_0)^{2m-1}} . \nonumber \end{align} \tag{ 45 }$$

Thus, we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \int_0^\tau f(t)^2 \rho(t) \,{\rm d}t \leqslant \rho_0\tau_0 (1-A_m) + \frac{Q_m(\gamma)C_m}{\tau_0^{2m-1}} + \frac{Q_m(\gamma)C_m}{(\tau-\tau_0)^{2m-1}} + Q_0(\gamma) A_m\tau \nonumber \end{align} \tag{ 46 }$$

and since $\|f'\|^2 = B_m/(\tau-\tau_0)$, the estimate on J[f ] is complete. □

Now using propositions 2.2 and 2.4 the following theorem is immediate.

Theorem 4.2. Let $(M,g)$ be a smooth globally hyperbolic spacetime of dimension n  >  2 and let S be a smooth compact spacelike Cauchy surface in M. Suppose that, for some $\tau>0$, there is an integer $m\geqslant 1$ and constants Q_m and Q₀ so that:

• (i)  
  the Ricci tensor obeys (38) along every unit-speed future-directed timelike geodesic $\gamma$ of length $\tau$ emanating normally from S, with $Q_m(\gamma)\leqslant Q_m$, $Q_0(\gamma)\leqslant Q_0$;
• (ii)  
  there exists $\rho_0\leqslant 0$ and $\tau_0\in (0,\tau)$ so that along every such geodesic, $R_{\mu \nu} \dot{\gamma}^\mu \dot{\gamma}^\nu |_{\gamma(t)}\leqslant \rho_0$ for $t\in [0,\tau_0]$;
• (iii)  
  the initial extrinsic curvature of S satisfies

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle -K \geqslant \min\left\{\frac{(n-1)}{\tau_0} ,\nu_* \right\} \nonumber \end{align} \tag{ 47 }$$

  everywhere on S, where $\nu_*$ is given by (42), but with $Q_k(\gamma)$ replaced by Q_k. (Recall that $\nu_*$ depends on $\tau$.)

Then no future-directed timelike curve emanating from S has length greater than $\tau$ and M is future timelike geodesically incomplete.

We conclude this section with two remarks. First, since a focal point in $[0,\tau']$, for $\tau'<\tau$ certainly implies the existence of a focal point in $[0,\tau]$, we can replace $\nu_*$ by

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \nu_{**}: = (1-A_m)\rho_0 \tau_0 + \frac{Q_m C_m}{\tau_0^{2m-1}} + \min_{\tau'\in (\tau_0,\tau]}\left(Q_0 A_m\tau' + \frac{(n-1)B_m}{\tau'-\tau_0} + \frac{Q_m C_m}{(\tau'-\tau_0)^{2m-1}}\right) \nonumber \end{align*}$$

in the statement of lemma 4.1 and theorem 4.2 if we wish. Similarly, If $\gamma:[0,\infty)\to M$ is future complete and $-K|_{\gamma(0)}$ is greater than or equal to the minimum value of $\nu_*$ over $\tau\in (\tau_0,\infty)$ then $\gamma$ contains a focal point to S.

Second, if $(n-1)/\tau_0\leqslant \nu_*$ on S then our result simply reduces to the original Hawking singularity theorem, and no future-directed timelike curve emanating from S has length greater than $\tau_0$, as in proposition 2.4 and corollary 2.5). Therefore it is important to show that there are situations in which $\nu_*<(n-1)/\tau_0$, so that our result represents a nontrivial improvement.

For this purpose, it is convenient to write $\tau=(1+\mu B_m)\tau_0$ for $\mu>0$, whereupon we may discard the term containing $\rho_0$ in the definition of $\nu_*$ to give

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \nu_* \leqslant \frac{Q_m C_m}{\tau_0^{2m-1}}+ Q_0 A_m\tau_0 + \frac{n-1}{\mu\tau_0} + Q_0 A_m B_m\tau_0 \mu + \frac{Q_m C_m}{(B_m\tau_0)^{2m-1}\mu^{2m-1}}. \nonumber \end{align} \tag{ 48 }$$

Clearly $\nu_*\geqslant (n-1)/\tau_0$ if $\mu\leqslant 1$, so we restrict to the case $\mu>1$ and use $1/\mu^{2m-1}<1/\mu$, obtaining

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:nuEFG} \nu_*\tau_0 \leqslant E + F\mu + \frac{G}{\mu}, \nonumber \end{align} \tag{ 49 }$$

where

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle E = \frac{Q_m C_m}{\tau_0^{2(m-1)}}+ A_m Q_0\tau_0^2 ,\qquad F=A_m B_m Q_0 \tau_0^2 ,\qquad G = n-1 + \frac{Q_m C_m}{B_m^{2m-1}\tau_0^{2(m-1)}}. \nonumber \end{align} \tag{ 50 }$$

As the right-hand side of (49) takes its minimum at $\mu = \sqrt{G/F}$, it follows that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \nu_*\tau_0 \leqslant \left\{\begin{array}{@{}ll@{}} E+2\sqrt{FG} & F\leqslant G \nonumber \\ E+F+G & G<F \end{array}\right. \nonumber \end{align} \tag{ 51 }$$

(the second case is uninteresting because G  >  n  −  1). Accordingly, the conditions

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle F\leqslant G\quad{\rm and}\quad E + 2\sqrt{FG}<n-1 \nonumber \end{align} \tag{ 52 }$$

imply $\nu_*\tau_0<n-1$ (evidently this can only be satisfied if $F<(n-1)/4$). In particular, it holds if $Q_0\tau_0^2 \ll 1$ and $Q_m/\tau_0^{2(m-1)} \ll 1$, in which case the optimising value of $\mu$ is much larger than unity,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mu\sim \sqrt{\frac{n-1}{A_mB_m Q_0\tau_0^2}} \nonumber \end{align} \tag{ 53 }$$

leading to the prediction of a focal point within a timescale

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eqn:timescale} \tau\sim \sqrt{\frac{(n-1)B_m}{A_m Q_0}}, \nonumber \end{align} \tag{ 54 }$$

with initial extrinsic curvature obeying

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eqn:extcurv} -K|_{\gamma(0)} > \nu_*\sim \frac{n-1}{\tau_0}\sqrt{\frac{4 A_m B_m Q_0\tau_0^2}{n-1} }\ll \frac{n-1}{\tau_0} \nonumber \end{align} \tag{ 55 }$$

(assuming that E and F are of comparable order, so that $E\ll \sqrt{F}$). Importantly, this level of extrinsic curvature is not sufficient to guarantee a focal point within time $\tau_0$, the period in which we have assumed timelike convergence to hold. In this regime, we also have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \nu_*\sim \frac{2(n-1)B_m}{\tau}, \nonumber \end{align} \tag{ 56 }$$

which is of the same order of magnitude as the initial contraction that would be needed to guarantee a focal point in $(0,\tau]$ assuming timelike convergence (see corollary 2.3). Therefore our result provides a Hawking-type singularity theorem with weakened energy conditions, but without drastically increased contraction.

4.1.2. Scenario 2.

Let us now drop the assumption that timelike convergence holds for small positive times along geodesics leaving S. Starting with a single unit-speed timelike geodesic $\gamma:[0,\tau]\to M$, with $\gamma(0)\in S$, extend $\gamma$ backwards in time, still as a unit-speed timelike geodesic, to obtain $\gamma:[-\tau_0,\tau]\to M$. We assume that (38) holds on the extended geodesic, for all $f\in W_0^m([-\tau_0,\tau])$ obeying generalised Dirichlet boundary conditions at $t=-\tau_0,\tau$.

As in Scenario 1, we aim to estimate J[f ] for a piecewise smooth function f  on $[0,\tau]$ obeying $f(0)=1$, $f(\tau)=0$, namely

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:scen2fdef} f(t) = \left\{\begin{array}{@{}ll@{}} I(m,m;(\tau'-t)/\tau') & t\in [0,\tau'] \nonumber \\ 0 & t\in [\tau',\tau], \end{array}\right. \nonumber \end{align} \tag{ 57 }$$

where the free parameter $\tau'\in [0,\tau]$. Defining

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:Hhat1} \hat{L}_1(\tau')=\frac{Q_m(\gamma) C_m}{(\tau')^{2m-1}} + \frac{(n-1)B_m}{\tau'} + A_m Q_0(\gamma) \tau' \nonumber \end{align} \tag{ 58 }$$

and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:Hhat2} \hat{L}_2(\tau_0') = \frac{Q_m(\gamma) C_m}{(\tau_0')^{2m-1}} + A_m (Q_0(\gamma)-\rho_{\min}) \tau_0' , \nonumber \end{align} \tag{ 59 }$$

and also

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:H12} L_1(\tau) = \min_{\tau'\in (0,\tau]} \hat{L}_1(\tau') ,\qquad L_2(\tau_0) = \min_{\tau_0'\in (0,\tau_0]}\hat{L}_2(\tau_0'), \nonumber \end{align} \tag{ 60 }$$

we prove:

Lemma 4.3. For the function f  given by (57), and $\rho$ satisfying (38) on $[-\tau_0,\tau]$ we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle J[\,f]\leqslant \hat{L}_1(\tau') + L_2(\tau_0) . \nonumber \end{align} \tag{ 61 }$$

Consequently, if $-K|_{\gamma(0)}\geqslant L_1(\tau) + L_2(\tau_0)$ then there is a focal point to S along $\gamma$ in the proper time interval $[0,\tau]$.

Proof. We have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle J[\,f] = \frac{(n-1)B_m}{\tau'} + \int_0^{\tau'} f^2 R_{\mu\nu} U^\mu U^\nu\,{\rm d}t . \nonumber \end{align} \tag{ 62 }$$

To estimate the integral, we first extend the domain of f  to create a function $\tilde{f}\in W_0^m([-\tau_0,\tau])$ that agrees with f  on $[0,\tau]$ and is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \tilde{f}(t) = \left\{\begin{array}{@{}ll@{}} 0 & t\in [-\tau_0,-\tau_0'] \nonumber \\ I(m,m;-t/\tau_0') & t\in [-\tau_0',0), \end{array}\right. \nonumber \end{align} \tag{ 63 }$$

on $[-\tau_0,0)$, where $\tau_0'\in(0,\tau_0]$ is arbitrary. We then estimate

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \int_0^{\tau'} f^2 R_{\mu\nu} U^\mu U^\nu\,{\rm d}t = \int_{-\tau_0}^{\tau} \tilde{f}^2 R_{\mu\nu} U^\mu U^\nu\,{\rm d}t - \int_{-\tau_0'}^{0} \tilde{f}^2 R_{\mu\nu} U^\mu U^\nu\,{\rm d}t \nonumber \end{align} \tag{ 64 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \leqslant Q_m(\gamma) \|\tilde{f}^{(m)}\|^2 + Q_0(\gamma)\|\tilde{f}\|^2 -\rho_{\min} \int_{-\tau_0'}^{0} \tilde{f}^2 \,{\rm d}t \nonumber \end{align} \tag{ 65 }$$

where the norms are those of $L^2(-\tau_0,\tau)$ and the constants $Q_k(\gamma)$ refer to the extended geodesic, while $\rho_{\min} = \min_{[-\tau_0,0]} \rho$. The right-hand side may be evaluated by using our results on incomplete Beta functions, giving an overall upper bound

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle J[\,f] \leqslant \hat{L}_1(\tau') + \hat{L}_2(\tau_0') \nonumber \end{align} \tag{ 66 }$$

and the first part of result follows on using the freedom to optimise over $\tau_0'$. The second is proved using proposition 2.2, and optimising over $\tau'\in (0,\tau]$. □

The minimum used to define L₂ may be found explicitly, giving

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle L_2(\tau_0) = \frac{2mQ_m(\gamma)}{2m-1} ((Q_0(\gamma)-\rho_{\min})A_m)^{1-1/(2m)} ((2m-1)C_m)^{1/(2m)} \nonumber \end{align} \tag{ 67 }$$

for $Q_0(\gamma)-\rho_{\min} > (2m-1)Q_m(\gamma)C_m/(A_m\tau_0^{2m})$, and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle L_2(\tau_0) = \frac{Q_m(\gamma) C_m}{(\tau_0)^{2m-1}} + A_m (Q_0(\gamma)-\rho_{\min}) \tau_0 \nonumber \end{align} \tag{ 68 }$$

otherwise. A closed form expression for $L_1(\tau)$ is not possible for general m. However, one may note that $\hat{L}_1$ has a single minimum in $(0,\infty)$ and no other critical points. As the only positive contribution to $\hat{L}_1'$ arises from the last term, it follows that if $Q_0\tau^2\leqslant (n-1)B_m/A_m$ then the gradient at $\tau$ is negative and $L_1(\tau)=\hat{L}_1(\tau)$.

By analogy with theorem 4.2 we now have:

Theorem 4.4. Let $(M,g)$ be a smooth globally hyperbolic spacetime of dimension n  >  2 and let S be a smooth compact spacelike Cauchy surface in M. Suppose that, for some $\tau,\tau_0>0$ there is an integer $m\geqslant 1$ and constants Q_m and Q₀ so that

• (i)  
  every unit-speed future-directed timelike geodesic of length $\tau$ emanating normally from S can be extended as an affine geodesic to $\gamma:[-\tau_0,\tau]\to M$ with $\gamma(0)\in S$, along which the Ricci tensor obeys (38) with $Q_m(\gamma)\leqslant Q_m$, $Q_0(\gamma)\leqslant Q_0$;
• (ii)  
  there exists a finite lower bound $\rho_{\min}$ so that $R_{\mu \nu} \dot{\gamma}^\mu \dot{\gamma}^\nu |_{\gamma(t)}\geqslant \rho_{\min}$ for every such geodesic and all $t\in [-\tau_0,0]$ (we emphasise that $\rho_{\min}$ is uniform in $\gamma$);
• (iii)  
  the extrinsic curvature of S satisfies

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle -K \geqslant L_1(\tau) + L_2(\tau_0) \nonumber \end{align} \tag{ 69 }$$

  everywhere on S, where the functions L_i are given by (58)–(60), with $Q_k(\gamma)$ replaced by Q_k ($k=0,m$).

Then no future-directed timelike curve emanating from S has length greater than $\tau$ and M is future timelike geodesically incomplete.

Under the assumption of timelike convergence, the conclusions of this result hold provided the initial contraction satisfies $-K\geqslant (n-1)/\tau$ on S (corollary 2.3 and proposition 2.4). It is important to note that there are circumstances in which the contraction required by theorem 4.4 is of a similar order.

For example, suppose that $\rho_{\min}\geqslant 0$ and the following conditions hold

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle Q_0\tau^2\ll B_m/A_m, \qquad \tau_0\ll \tau\lesssim \frac{\tau_0^{2m-1}}{Q_m C_m}. \nonumber \end{align} \tag{ 70 }$$

Then certainly $Q_0\tau^2\leqslant B_m/A_m$, so $L_1(\tau)=\hat{L}_1(\tau)$ as noted above. As $Q_mC_m/\tau^{2(m-1)}\lesssim$$(\tau_0/\tau){}^{2(m-1)}\ll 1$ (for m  >  1) we find that $L_1(\tau)=\hat{L}(\tau)\sim (n-1)B_m/\tau$. In the case m  =  1 the assumptions imply $Q_1\ll 1$ and hence $Q_1C_1/\tau\ll (n-1)/\tau$, giving the same conclusion for $L_1(\tau)$ as before. Turning to L₂, the requirement that $\rho_{\min}\geqslant 0$ implies

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle L_2(\tau_0)\leqslant \hat{L}_2(\tau_0) \leqslant \frac{Q_m C_m}{(\tau_0)^{2m-1}} + A_m Q_0\tau_0. \nonumber \end{align} \tag{ 71 }$$

Under our assumptions, the second term is small relative to $B_m/\tau$, while the first is $\lesssim 1/\tau$. Overall,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:scen2approx} L_1(\tau)+L_2(\tau_0)\lesssim \frac{(n-1)B_m+1}{\tau}, \nonumber \end{align} \tag{ 72 }$$

which is of comparable order to $(n-1)/\tau$ at least for small values of m. Thus the contraction required by theorem 4.4 is comparable to $(n-1)/\tau$.

A feature of theorem 4.4 is that increasing $\rho_{\min}$—the extent to which timelike convergence fails at small negative times—decreases the required initial contraction. This is because the QEI type constraints require that a period in which timelike convergence fails is followed shortly by a period in which timelike convergence is satisfied even more strongly. This is analogous to the quantum interest effect [6, 15, 22, 25]. If $\rho_{\min} \leqslant 0$ then theorem 4.4 typically overestimates the required contraction and scenario 1 should be used instead.

Let $\gamma$ be any null geodesic in M. Using the same invariant notation as in proposition 2.7, we suppose that the Ricci tensor obeys

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\vertiii}[1]{{\left\vert\left\vert\left\vert #1 \right\vert\right\vert\right\vert}} \displaystyle \label{eq:Ricci_hypnull} \int_\gamma f^2 \mathrm{Ric}({\rm d}\gamma,{\rm d}\gamma) \leqslant \vertiii{f}^2:=Q_m(\gamma;w) \|w^{1-m} \nabla_{{\rm d}\gamma}^m f\|^2 + Q_0(\gamma;w) \|w f\|^2, \nonumber \end{align} \tag{ 73 }$$

for every smooth compactly supported $(-\frac{1}{2})$-density f  on $\gamma$, and every choice of smooth positive density w on $\gamma$, so that $Q_k(\gamma;\lambda w)=\lambda^{2(k-1)}Q_k(\gamma;w)$ for any $\lambda>0$. Here $\|\cdot\|$ is the intrinsic L²-norm on $\frac{1}{2}$-densities defined on $\gamma$, and appropriate powers of the density w are inserted to ensure that the arguments of these norms have the correct weight as densities. It is assumed that $Q_m(\gamma;w)>0$ and $Q_0(\gamma;w)\geqslant 0$.

Now let P be a future converging achronal spacelike submanifold of M of co-dimension 2 with mean normal curvature vector field $H^\mu$. Suppose that $\gamma$ is a future-directed null geodesic emanating normally from P. Extending $\hat{H}_\mu$ by parallel transport along $\gamma$, we define a density $w=\hat{H}_\mu {\rm d}\gamma^\mu_+/{\rm d}\lambda$. Next choose an affine coordinate $\lambda$ on $\gamma$, so that $p=\gamma(0)$ and $\hat{H}_\mu {\rm d}\gamma^\mu/{\rm d}\lambda =1$. Then w  =  1, $\newcommand{\e}{{\rm e}} q=\gamma(\ell)$, with $\newcommand{\e}{{\rm e}} \ell=L_{\hat{H}}(\gamma)$ and equation (73) becomes

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eqn:Riccinull} \int_0^\ell f(\lambda)^2 R_{\mu \nu} U^\mu U^\mu {\rm d}\lambda \leqslant Q_m(\gamma) \| f^{(m)} \|^2+Q_0 (\gamma) \| f\|^2 . \nonumber \end{align} \tag{ 74 }$$

Writing $\rho(\lambda)= R_{\mu\nu} U^\mu U^\nu|_{\gamma(\lambda)}$, and by proposition 2.7, there is a focal point to P along $\gamma$ if there is a piecewise smooth f  on $\newcommand{\e}{{\rm e}} [0,\ell]$ with $f(0)=1$ and $\newcommand{\e}{{\rm e}} f(\ell)=0$, such that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle J[\,f]\leqslant -(n-2)H |_{\gamma(0)}, \nonumber \end{align} \tag{ 75 }$$

where

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle J[\,f]=\int_0^\ell \left((n-2)\,f'(\lambda)^2 + f(\lambda)^2 \rho(\lambda) \right)\,{\rm d}\lambda. \nonumber \end{align} \tag{ 76 }$$

The two scenarios following are analogous to the ones in section 4.1. In the first, we will suppose that the NEC is satisfied at small positive values of $\lambda$ while in the second, we impose conditions on $\rho$ at small negative values of $\lambda$.

4.2.1. Scenario 1.

Suppose that initially the NEC is satisfied, and so let $\rho(\lambda)= R_{\mu\nu} U^\mu U^\nu|_{\gamma(\lambda)}$ be a smooth function on $\newcommand{\e}{{\rm e}} [0,\ell]$ that is initially negative, $\rho \leqslant \rho_0\leqslant 0$ on $\newcommand{\e}{{\rm e}} [0,\ell_0]$ for some $\newcommand{\e}{{\rm e}} 0<\ell_0<\ell$.

Defining $\newcommand{\e}{{\rm e}} f:[0,\ell]\to\mathbb{R}$ by (41) with $\newcommand{\e}{{\rm e}} \ell$ instead of $\tau$ and $\newcommand{\e}{{\rm e}} \ell_0$ instead of $\tau_0$, we can estimate J[f ] following the exact steps of the proof of lemma 4.1. The following lemma is immediate using proposition 2.7.

Lemma 4.5. For $\rho$ satisfying (74) on $\newcommand{\e}{{\rm e}} [0,\ell]$ we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:scenario1null} J[\,f]\leqslant \nu_*:= (1-A_m)\rho_0 \ell_0 + \frac{Q_m C_m}{\ell_0^{2m-1}} + Q_0 A_m\ell + \frac{(n-2)B_m}{\ell-\ell_0} + \frac{Q_m C_m}{(\ell-\ell_0)^{2m-1}}. \nonumber \end{align} \tag{ 77 }$$

Consequently, if $-(n-2)H|_{\gamma(0)}\geqslant \nu_*$ then $\gamma$ contains a focal point to P.

There is an obvious adaptation of this result to future-complete null geodesics by minimising over $\newcommand{\e}{{\rm e}} \ell\in (\ell_0,\infty)$. Using propositions 2.7 and 2.9 the following theorem is immediate.

Theorem 4.6. Let M be globally hyperbolic with non-compact Cauchy hypersurfaces and let P be a compact achronal future converging spacelike submanifold of M of co-dimension 2 with mean normal curvature vector $H^\mu=H \hat{H}^\mu$ where $\hat{H}^\mu$ is a future-pointing timelike unit vector. Suppose that for some $\newcommand{\e}{{\rm e}} \ell>0$ there is an integer $m\geqslant 1$ and constants Q_m and Q₀ so that:

• (i)  
  the Ricci tensor obeys (73) along every future-directed null geodesic $\gamma$ of $\hat{H}$-length $\newcommand{\e}{{\rm e}} \ell$ emanating normally from P, with $Q_m(\gamma)\leqslant Q_m$, $Q_0(\gamma)\leqslant Q_0$;
• (ii)  
  there exists $\rho_0\leqslant 0$ and $\newcommand{\e}{{\rm e}} \ell_0\in (0,\ell)$ so that along every such geodesic $\gamma$, the inequality $\mathrm{Ric}(\gamma',\gamma')\leqslant \rho_0 g({\rm d}\gamma,\hat{H}){}^2$ holds along the initial portion of $\gamma$ with $\hat{H}$-length $\newcommand{\e}{{\rm e}} \ell_0$;
• (iii)  
  the mean normal curvature of P satisfies

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle -(n-2)H \geqslant \min\left\{\frac{n-2}{\ell_0} ,\nu_* \right\} \nonumber \end{align} \tag{ 78 }$$

  everywhere on P, where $\nu_*$ is given by (77) and depends on $\newcommand{\e}{{\rm e}} \ell$.

Then there is an inextendible future-directed null geodesic emanating from P with $\hat{H}$-length less than $\newcommand{\e}{{\rm e}} \ell$. In particular, M is future null geodesically incomplete.

Similarly to the timelike version if $\newcommand{\e}{{\rm e}} (n-2)/\ell_0\leqslant \nu_*$ on P our result reduces to the original Penrose singularity theorem as in proposition 2.9. So we can examine situations in which $\newcommand{\e}{{\rm e}} (n-2)/\ell_0\geqslant \nu_*$ following the exact similar analysis as in section 4.1.1 but again with with $\lambda$ instead of t, $\newcommand{\e}{{\rm e}} \ell$ instead of $\tau$, $\newcommand{\e}{{\rm e}} \ell_0$ instead of $\tau_0$ and equation (74) instead of (38).

Then for $\newcommand{\e}{{\rm e}} Q_0\ell_0^2 \ll 1$ and $\newcommand{\e}{{\rm e}} Q_m/\ell_0^{2(m-1)} \ll 1$ there is a focal point before parameter

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eqn:ell} \ell \sim \sqrt{\frac{(n-2)B_m}{A_m Q_0}}, \nonumber \end{align} \tag{ 79 }$$

if the magnitude of the mean curvature vector of P satisfies

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eqn:meancurv} -(n-2)H > \nu_*\sim \sqrt{4 A_m B_m Q_0(n-2)} \sim \frac{2B_m(n-2)}{\ell}. \nonumber \end{align} \tag{ 80 }$$

Again, this is of the same order as the mean curvature needed to guarantee a focal point if the null convergence condition held (corollary 2.8).

4.2.2. Scenario 2.

In Scenario 2 we drop the assumption that the NEC holds. We instead extend the previously defined $\gamma$ to negative values of $\lambda$ to get $\newcommand{\e}{{\rm e}} \gamma:[-\ell_0,\ell]\to M$. Next we assume that equation (74) holds on the extended geodesic, for all $\newcommand{\e}{{\rm e}} f\in W_0^m([-\ell_0,\ell])$ obeying generalised Dirichlet boundary conditions at $\newcommand{\e}{{\rm e}} \lambda=-\ell_0,\ell$.

Similarly to the timelike case we want to estimate J[f ] and we can follow the same analysis with $\lambda$, $\newcommand{\e}{{\rm e}} \ell$ and $\newcommand{\e}{{\rm e}} \ell_0$ instead of t, $\tau$ and $\tau_0$. Set

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:Hhat1null} \hat{L}_1(\ell')=\frac{Q_m(\gamma) C_m}{(\ell')^{2m-1}} + \frac{(n-2)B_m}{\ell'} + A_m Q_0(\gamma) \ell' \nonumber \end{align} \tag{ 81 }$$

and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:Hhat2null} \hat{L}_2(\ell_0') = \frac{Q_m(\gamma) C_m}{(\ell_0')^{2m-1}} + A_m (Q_0(\gamma)-\rho_{\min}) \ell_0' , \nonumber \end{align} \tag{ 82 }$$

and also

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:H12null} L_1(\ell) = \min_{\ell'\in (0,\ell]} \hat{L}_1(\ell') ,\qquad L_2(\ell_0) = \min_{\ell_0'\in (0,\ell_0]}\hat{L}_2(\ell_0'). \nonumber \end{align} \tag{ 83 }$$

Then the proof of the following lemma is analogous to the one for lemma 4.3 where equation (74) is used instead of (38), and finally using proposition 2.7, optimising over $\newcommand{\e}{{\rm e}} \ell'\in (0,\ell]$.

Lemma 4.7. For $\rho$ satisfying (74) on $\newcommand{\e}{{\rm e}} [-\ell_0,\ell]$ we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle J[\,f]\leqslant \hat{L}_1(\ell') + L_2(\ell_0). \nonumber \end{align} \tag{ 84 }$$

Consequently, if $\newcommand{\e}{{\rm e}} -(n-2) H\geqslant L_1(\ell) + L_2(\ell_0)$ then there is a focal point to P along $\gamma$ in $\newcommand{\e}{{\rm e}} [0,\ell]$.

More explicitly, we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle L_2(\ell_0) = \frac{2mQ_m(\gamma)}{2m-1} ((Q_0(\gamma)-\rho_{\min})A_m)^{1-1/(2m)} ((2m-1)C_m)^{1/(2m)} \nonumber \end{align} \tag{ 85 }$$

for $\newcommand{\e}{{\rm e}} Q_0(\gamma)-\rho_{\min} > (2m-1)Q_m(\gamma)C_m/(A_m\ell_0^{2m})$, and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle L_2(\ell_0) = \frac{Q_m(\gamma) C_m}{(\ell_0)^{2m-1}} + A_m (Q_0(\gamma)-\rho_{\min}) \ell_0 \nonumber \end{align} \tag{ 86 }$$

otherwise. Our main result in this scenario is:

Theorem 4.8. Let M be globally hyperbolic with non-compact Cauchy hypersurfaces and let P be a compact achronal future converging spacelike submanifold of M of co-dimension 2 with mean normal curvature vector $H^\mu$. Suppose that for some $\newcommand{\e}{{\rm e}} \ell,\ell_0>0$ there is an integer $m\geqslant 1$ and constants Q_m and Q₀ so that:

• (i)  
  every future-directed null geodesic of $\hat{H}$-length $\newcommand{\e}{{\rm e}} \ell$ emanating normally from P may be extended to the past, to give a geodesic $\gamma$ with $\newcommand{\e}{{\rm e}} L_{\hat{H}}(\gamma)=\ell+\ell_0$ so that the Ricci tensor obeys (73) along $\gamma$ with $Q_m(\gamma)\leqslant Q_m$, $Q_0(\gamma)\leqslant Q_0$;
• (ii)  
  there exists a finite lower bound $\rho_{\min}$ so that $\mathrm{Ric}({\rm d}\gamma,{\rm d}\gamma) \geqslant \rho_{\min} g(\hat{H},{\rm d}\gamma){}^2$ on the portion of each such geodesic $\gamma$ to the past of P (we emphasise that $\rho_{\min}$ is uniform in $\gamma$);
• (iii)  
  the mean normal curvature of P satisfies

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle -(n-2)H \geqslant L_1(\ell) + L_2(\ell_0), \nonumber \end{align} \tag{ 87 }$$

  where the functions L_i are given by (81)–(83), with $Q_k(\gamma)$ replaced by Q_k ($k=0,m$).

Then there is an inextendible future-directed null geodesic emanating from P with $\hat{H}$-length less than $\newcommand{\e}{{\rm e}} \ell$. In particular, M is future null geodesically incomplete.

Similarly to the timelike case it is important to note that there are circumstances in which the mean normal curvature required by theorem 4.8 is of the same order as that in corollary 2.8.

Suppose that $\rho_{\min}\geqslant 0$ and the following conditions hold

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle Q_0\ell^2\ll B_m/A_m, \qquad \ell_0\ll \ell\lesssim \frac{\ell_0^{2m-1}}{Q_m C_m}. \nonumber \end{align} \tag{ 88 }$$

Then $\newcommand{\e}{{\rm e}} Q_0\ell^2\leqslant B_m/A_m$, so $\newcommand{\e}{{\rm e}} L_1(\ell)=\hat{L}_1(\ell)$. Similar to the timelike case $\newcommand{\e}{{\rm e}} L_1(\ell)=\hat{L}(\ell)\sim (n-2)B_m/\ell$.

Turning to L₂ we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle L_2(\ell_0)\leqslant \hat{L}_2(\ell_0) \leqslant \frac{Q_m C_m}{(\ell_0)^{2m-1}} + A_m Q_0\ell_0 , \nonumber \end{align} \tag{ 89 }$$

and overall

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eqn:scen2estnull} L_1(\ell)+L_2(\ell_0)\lesssim \frac{(n-2)B_m+1}{\ell}, \nonumber \end{align} \tag{ 90 }$$

which is of comparable order to $\newcommand{\e}{{\rm e}} (n-2)/\ell$ at least for small values of m.

If $\rho_{\min} \leqslant 0$ then theorem 4.8 typically overestimates the required H and scenario 1 should be used.

Consider first the timelike case. Assume that the scalar field magnitude admits global bounds on a timelike geodesic $\gamma$ parametrized by proper time $\tau$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\fmax}{\phi_{{\rm max}}} \displaystyle |\phi| \leqslant \fmax \leqslant (8\pi \xi)^{1/2}, \qquad |\nabla_{\dot{\gamma}} \phi| \leqslant \fmax' . \nonumber \end{align} \tag{ 93 }$$

Then, on $\gamma$, and following theorem 3 and corollary 1 of [6], we have that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eqn:ekgencon} \int_\gamma R_{\mu \nu} \dot{\gamma}^\mu \dot{\gamma}^\nu f(\tau)^2{\rm d}\tau \leqslant Q(\| \dot{f} \|^2+\tilde{Q}^2 \| f\|^2) , \nonumber \end{align} \tag{ 94 }$$

for any real valued function f  of compact support, with constants Q and $\tilde{Q}$ given by

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\fmax}{\phi_{{\rm max}}} \displaystyle \label{eq:QQtdef} Q=\frac{32\pi \xi \fmax^2}{1-8\pi \xi \fmax^2} , \qquad \tilde{Q}^2=\frac{(1-2\xi)\lambda^{-2}}{4\xi(n-2)}+\left(\frac{8\pi \xi \fmax \fmax'}{1-8\pi \xi \fmax^2}\right)^2 . \nonumber \end{align} \tag{ 95 }$$

Equation (94) is of the form of equation (38) with m  =  1, Q₁  =  Q and $Q_0=Q\tilde{Q}^2$. To estimate the values of these constants we first reinsert the units and restore the constants G and c

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\fmax}{\phi_{{\rm max}}} \displaystyle \label{eq:QQtdefdim} Q=\frac{32\pi \xi G\fmax^2/c^4}{1-8\pi \xi G\fmax^2/c^4} , \qquad \tilde{Q}^2=\frac{(1-2\xi)\lambda^{-2}c^2}{4\xi(n-2)}+\left(\frac{8\pi \xi G\fmax \fmax'/c^4}{1-8\pi \xi G\fmax^2/c^4}\right)^2 . \nonumber \end{align} \tag{ 96 }$$

It is interesting to estimate these values in a situation where $\lambda$ is the reduced Compton wavelength of an elementary particle, making appropriate choices for $\newcommand{\fmax}{\phi_{{\rm max}}} \fmax$ and $\newcommand{\fmax}{\phi_{{\rm max}}} \fmax'$. In [6] the value of $\newcommand{\fmax}{\phi_{{\rm max}}} \fmax$ was estimated by considering a quantized scalar field in Minkowski spacetime of dimension n, in a thermal state of temperature $T<T_\lambda$, where $T_\lambda=c \hbar/(\lambda k)$ is the reduced Compton temperature of the particle and k is Boltzmann's constant. The temperature $T_\lambda$ defines a scale beyond which the model cannot be trusted. With these considerations $\newcommand{\fmax}{\phi_{{\rm max}}} \fmax^2 \sim \langle {:}\phi^2{:}\rangle_T$ and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle Q\sim (\ell_{{\rm Pl}}/\lambda)^{n-2}(T/T_\lambda)^{(n-2)/2} K_{(n-2)/2}(T_\lambda/T) , \nonumber \end{align} \tag{ 97 }$$

as was shown in [6] (in that reference the mass was used instead of the reduced Compton length). Here $K_\nu$ is a modified Bessel function of the second kind and $\newcommand{\e}{{\rm e}} \ell_{{\rm Pl}}$ is the Planck length.

From equation (96) the second term of $\tilde{Q}^2$ is proportional to $\newcommand{\fmax}{\phi_{{\rm max}}} Q^2 (\fmax'/\fmax){}^2$. On dimensional grounds the ratio $\newcommand{\fmax}{\phi_{{\rm max}}} \fmax'/\fmax$ is proportional to $c/\lambda$. For $\lambda$ taken as the Compton length of elementary particles $Q\ll 1$, as will be seen shortly. Therefore the first term of $\tilde{Q}^2$ is expected to be much larger than the second and so $\tilde{Q} \sim c/\lambda$.

In order to give a quantitative, and partly heuristic, illustration of our results, we consider the following toy model. Suppose the universe were described by a Einstein–Klein–Gordon model, in which the characteristic length scale of the scalar is the reduced Compton wavelength of a pion. Given an expansion rate of the universe, and other conditions, drawn from actual cosmological data, would one be able to conclude that the universe is necessarily past timelike geodesically incomplete? To do this we must consider the time-reverse of the analysis presented above, so the question is whether the extrinsic curvature of surfaces of constant cosmic time—the Hubble parameter—is sufficiently positive; that is, whether the expansion rate is sufficiently large. We bear in mind that the SEC is violated in our actual universe, due to the dominant effect of dark energy, with the ratio of pressure to energy density being very close to  −1. This motivates two different calculations: (a) using parameters drawn from an era in which the SEC did hold, as an instance of Scenario 1, and (b) using parameters corresponding to the present time, and assuming that the SEC will continue to fail for some time $\tau_0$ into the future, as a (time-reversed version of) Scenario 2. We will show in each case that, using the pion as the matter model, the expansion rate of the actual universe would be sufficient to conclude past geodesic incompleteness. In fact, there is a caveat to these results, because the values of parameters Q₀ and Q₁ were derived on the basis that the temperature scale T is not exceeded. This means that what the heuristic argument actually shows is that, on timescales within the age of our actual universe, the toy model universe must display either past geodesic timelike incompleteness on a timescale of the age of our actual universe, or locations where the temperature scales exceed T. For brevity, we will describe either of these occurrences as a singularity.

First, we consider Scenario 1, in which the SEC is satisfied for time $\tau_0$. For a neutral pion in n  =  4 dimensions with mass $135~{\rm MeV}/c^2$, we have $\newcommand{\e}{{\rm e}} \ell_{{\rm Pl}}/\lambda=1.11 \times 10^{-20}$ and $T_\lambda=1.56 \times 10^{12}~{\rm K}$. (All calculations are made to higher precision but reported to 3S.F.; however it is really the orders of magnitude that are of interest.) Taking $T=10^{-2}T_\lambda$ (corresponding to the temperature of our universe about $1~{\rm s}$ after the big bang) gives an estimate of $Q \sim 5.66 \times 10^{-87}$. Then $Q_1=Q \ll 1$ and $Q_0\sim Q (c/\lambda){}^2=2.39\times 10^{-39}~{\rm s}^{-2}$. Thus if $\tau_0$ is of the order of the reduced Compton time ($4.87\times 10^{-24}$ s), we have $Q_0 \tau_0^2 \sim Q \ll 1$. In that case the initial contraction is given by equation (55) with the units restored

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:nustar} \nu_*\sim \lambda^{-1} c \sqrt{12 A_1 B_1 Q} \sim 3.09 \times 10^{-20}~{\rm s}^{-1} . \nonumber \end{align} \tag{ 98 }$$

The maximum allowed temperature for this case is $T=1.56\times 10^{10}~{\rm K}$, while the timescale on which the singularity occurs is given by equation (54) as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:piontimescale} \tau \sim \lambda c^{-1} \sqrt{\frac{3B_1}{A_1}} Q^{-1/2} \sim 1.94 \times 10^{20}~{\rm s} , \nonumber \end{align} \tag{ 99 }$$

or about $6.15 \times 10^{12}$ years.

For comparison, in our actual universe, and assuming the $\Lambda$CDM model, the SEC was most recently obeyed at time t₁ when

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Omega_\Lambda (t_1)=\frac{\Omega_b (t_1)}{2}+\Omega_r (t_1) . \nonumber \end{align} \tag{ 100 }$$

Here $\Omega_x=\rho_x/\rho_{{\rm crit}}$, $\rho_{{\rm crit}}$ is the critical energy density, $\Lambda$ corresponds to dark energy, m to matter (baryonic and cold dark matter) and r to radiation. Using the different evolution of each energy density component we find that the redshift when the SEC was last satisfied z₁ is given by the solution of

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Omega_{\Lambda_0}-\frac{\Omega_{m_0}}{2}(z_1+1)^3-\Omega_{r_0} (z_1+1)^4=0 , \nonumber \end{align} \tag{ 101 }$$

where $\Omega_{x0}$ are the respective quantities today. From the most recent results published by the PLANCK collaboration [2] we have that $\Omega_{\Lambda_0}=0.6889 \pm 0.0056$ and $\Omega_{m_0}=0.311 \pm 0.002$ which gives a redshift of z₁  =  0.642 for when the SEC was last obeyed. From the first Friedmann equation we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{H^2(t_1)}{H_0^2}=\Omega_{\Lambda_0}+\Omega_{r_0} (z_1+1) ^4+\Omega_{m_0}(z_1+1)^3 , \nonumber \end{align} \tag{ 102 }$$

where $H(t)$ is the Hubble parameter and H₀ its value today. Again from the PLANCK collaboration [2] we get that $H_0=(2.184 \pm 0.016) \times 10^{-18} ~{\rm s}^{-1}$. That gives $H(t_1)=3.14 \times 10^{-18} ~{\rm s}^{-1}$ at the time when the SEC was last obeyed. This exceeds the minimum threshold (98) by two orders of magnitude, so the toy model universe would necessarily have a past singularity, using these parameters. Partly because the threshold is exceeded by such a margin, the estimated timescale (99) for the location of the singularity is accordingly rather pessimistic. Nonetheless, it is remarkable that it is within two orders of magnitude of the age of the universe at the relevant time.

If the characteristic scale is replaced by the reduced Compton wavelength of more massive particles, the same calculations produce higher expansion thresholds beyond which a singularity is inevitable. For a proton, with mass $938~{\rm MeV}/c^2$, and a maximum temperature $T=1.09\times 10^{11}~{\rm K}$, the threshold is $1.50\times 10^{-18}~{\rm s}^{-1}$, which is still marginally exceeded by the measured value. The timescale for the singularity is now $1.27\times 10^{11}$ years. On the other hand, for a Higgs particle, with mass $125~{\rm GeV}/c^2$ and with $T=1.44 \times 10^{13}~{\rm K}$ (the temperature of the Universe at age $10^{-4}~{\rm s}$), the threshold is $\nu_*=2.68 \times 10^{-14}~{\rm s}^{-1}$. This threshold is larger than the observed Hubble parameter by 4 orders of magnitude, so our results would be inconclusive in that case. These illustrations are intended purely to show that the results we have obtained are quantitative and capable of producing plausible cosmological results in the toy models for the thresholds beyond which a singularity is inevitable, and the timescales on which they occur.

Moving to Scenario 2 where the requirement that the SEC holds is dropped, we are taking into account the behaviour of $\rho$ just after the time $\tau=0$ at which the Hubble parameter is measured (recall that we are using time-reversed versions of our earlier results). Here we assume that the SEC is violated for the time interval $[0,\tau_0]$, and $\rho_{{\rm min}} >0$ which as we mentioned is compatible with current cosmological observations.

First we observe that for the pion $Q_0^{-1/2}\sim 6.49 \times 10^{11}$ years is large in comparison to the lifetime of the universe. Then $Q_0 \tau^2 \ll B_1/A_1$ for focal points occurring within a timescale $\tau$ of the order of ten times the age of the universe. As Q₁ is so small, there is a large range of possible choices of $\tau_0$ so that $\tau_0 \ll \tau < \tau_0/(Q_1 C_1)$ and so that SEC fails for times in $[0,\tau_0]$, from $10^{-68}~{\rm s}$ up to $10^{-2}\tau$, say.

Given these assumptions the approximation of equation (72) is reasonable, and our results show that a singularity is inevitable within about 10 times the age of the universe, for Hubble parameter

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle H(0) > \frac{3B_1+1}{\tau} \sim 10^{-18} ~{\rm s}^{-1} , \nonumber \end{align} \tag{ 103 }$$

which on the order of the current values from PLANCK $(2.184 \pm 0.016) \times 10^{-18} ~{\rm s}^{-1}$ [2].

If the characteristic scale is based on the proton mass, $Q_0\tau^2\sim 0.1$ if $\tau$ is the age of the universe; however the minimum threshold on $H(0)$ is now $\sim 10^{-17} ~{\rm s}^{-1}$, so the measured value would not be sufficient to conclude the existence of a singularity. For the Higgs field, $Q_0^{-1/2} \sim 2.36 \times 10^{6}$ years and the approximation $Q_0 \tau^2 \ll 1$ is only valid for $\tau$ less than 10⁻⁴ times the age of the universe, requiring therefore approximately 10⁴ times the measured Hubble parameter to infer that a singularity is inevitable in our toy model.

These results give similar orders of magnitude for the required initial extrinsic curvature to those computed in [6] obtained with different methods (and phrasing the conditions for future timelike geodesic incompleteness). An advantage of the current method is that it can specify the timescale on which the focal points appear.

Now we turn to the null case. For any solution to the Einstein–Klein–Gordon equation in which the field magnitude obeys a global bound $\newcommand{\fmax}{\phi_{{\rm max}}} |\phi|\leqslant \fmax < (8\pi \xi){}^{1/2}$, it was shown in [15] that

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\fmax}{\phi_{{\rm max}}} \displaystyle \label{eqn:repineq} \int_\gamma f^2\mathrm{Ric}({\rm d}\gamma,{\rm d}\gamma) \leqslant 16\pi \xi \fmax^2 \int_\gamma \left(\nabla_{{\rm d}\gamma}\frac{f}{\sqrt{1-8\pi \xi \phi^2}} \right)^2 , \nonumber \end{align} \tag{ 104 }$$

for all smooth compactly supported $(-\frac{1}{2})$-densities f  on $\gamma$, where we have written the expression derived in [15] invariantly and adapted it to our sign conventions. Given any positive density w on $\gamma$ this implies

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eqn:ekgnull} \int_\gamma f^2\mathrm{Ric}({\rm d}\gamma,{\rm d}\gamma) \leqslant Q(\| \nabla_{{\rm d}\gamma}f \|^2+\tilde{Q}(\gamma; w)^2 \| w f\|^2) , \nonumber \end{align} \tag{ 105 }$$

where

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\fmax}{\phi_{{\rm max}}} \displaystyle \label{eq:QQtdefnull} Q=\frac{32\pi \xi \fmax^2}{1-8\pi \xi \fmax^2} , \qquad \tilde{Q}(\gamma;w)=\frac{8\pi \xi \fmax }{1-8\pi \xi \fmax^2} \sup_\gamma \frac{|\nabla_{{\rm d}\gamma} \phi|}{w}. \nonumber \end{align} \tag{ 106 }$$

Equation (105) is of the form of equation (73) with m  =  1, $Q_1(\gamma;w)=Q$ and $Q_0(\gamma;w)\,=$ $Q\tilde{Q}(\gamma;w){}^2$. Note that Q₁ in this case is independent of both $\gamma$ and w, while $Q_0(\gamma;w)$ is independent of any specific parametrisation of $\gamma$.

As described in section 2.2 we fix the affine parametrization and w in the following way: define P to be a future converging achronal spacelike submanifold of M of co-dimension 2 and let $\gamma$ be a future-directed null geodesic emanating normally from P. Extending $\hat{H}_\mu$ by parallel transport along $\gamma$ and choosing an affine coordinate $\lambda$, so that $p=\gamma(0)$ and $\hat{H}_\mu {\rm d}\gamma^\mu/{\rm d}\lambda =1$, we have w  =  1 and $\newcommand{\e}{{\rm e}} q=\gamma(\ell)$, with $\newcommand{\e}{{\rm e}} \ell=L_{\hat{H}}(\gamma)$.

Then equation (105) becomes

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \int_\gamma f^2(\lambda) R_{\mu \nu} U^\mu U^\nu {\rm d}\lambda \leqslant Q(\gamma) \left(\| f' \|^2+\tilde{Q}^2(\gamma) \| f\|^2 \right) , \nonumber \end{align} \tag{ 107 }$$

and

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\fmax}{\phi_{{\rm max}}} \displaystyle \tilde{Q}(\gamma)=\frac{8\pi \xi \fmax }{1-8\pi \xi \fmax^2} \sup_\gamma |\phi'(\lambda)| . \nonumber \end{align} \tag{ 108 }$$

Now we want to estimate Q and $\tilde{Q}$. We consider a massless field and, as in the timelike case, we work in a hybrid model: a quantized scalar field in a thermal state of temperature T. In the massless scalar field case the Wick square of a KMS state with temperature T is

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\nord}[1]{{:}#1{:}} \displaystyle \langle \nord{\phi^2} \rangle_T=\frac{T^{n-2}}{2^{n-2} \pi^{(n-1)/2}} \frac{\Gamma(n-2)}{\Gamma(\frac{n-2}{2})} \zeta(n-2) , \nonumber \end{align} \tag{ 109 }$$

where $\zeta$ is the Riemann zeta function. Similarly, if $U^\mu$ is any null vector with U⁰  =  1 then

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\nord}[1]{{:}#1{:}} \displaystyle \langle \nord{(U^\mu \nabla_\mu \phi) (U^\nu \nabla_\nu \phi) } \rangle_T=\frac{T^n}{3 \cdot 2^{n-4} \pi^{(n-1)/2}} \frac{\Gamma(n)}{\Gamma(\frac{n-1}{2})} \zeta(n) . \nonumber \end{align} \tag{ 110 }$$

For $\newcommand{\fmax}{\phi_{{\rm max}}} \newcommand{\nord}[1]{{:}#1{:}} \fmax^2 \sim \langle \nord{\phi^2} \rangle_T$, $\newcommand{\fmax}{\phi_{{\rm max}}} \newcommand{\nord}[1]{{:}#1{:}} \fmax'^2 \sim \langle \nord{(U^\mu \nabla_\mu \phi) (U^\nu \nabla_\nu \phi) } \rangle_T$ and restoring the units we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle Q \sim (T/T_{{\rm pl}})^{n-2} , \qquad {\rm and} \qquad \tilde{Q} \sim Q \frac{kT}{\hbar} , \nonumber \end{align} \tag{ 111 }$$

where $T_{{\rm pl}}$ is the Planck temperature.

Let us consider Scenario 1. For n  =  4, and a temperature $T \sim 10^{7}$K which is of the order of a newly formed neutron star [38] we have $Q_1=Q \sim 10^{-50} \ll 1$ and $Q_0 \sim 10^{-114} ~{\rm s}^{-2}$. We can consider $\newcommand{\e}{{\rm e}} \ell_0$ as a measurement in light seconds of distance along null rays, measured by an observer at rest on the hypersurface P. Then assuming it is much larger than 10⁻⁵⁷ light seconds, we have $\newcommand{\e}{{\rm e}} Q_0 \ell_0^2 \ll 1$.

The required magnitude of of the mean curvature vector of P to form a focal point is given by equation (80) with the units restored

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle - H \sim \frac{kT}{2 \hbar} \sqrt{8 A_1 B_1 Q^3} \sim 10^{-57} ~{\rm s}^{-1} . \nonumber \end{align} \tag{ 112 }$$

The contraction in [15] was found to be

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle -H > \tilde{Q} \sqrt{Q(Q+(n-2))}+QK \coth{(K\ell_0)} , \nonumber \end{align} \tag{ 113 }$$

where we have corrected some factors of 2. Using the previous values of Q and $\tilde{Q}$ and assuming that the second term which depends on the history of the solution does not get too large we get $-H > (kT/\hbar) Q^{3/2} \sqrt{2} \sim 10^{-57} ~{\rm s}^{-1}$ which agrees with our estimate.

The required mean normal curvature for this toy model is extremely small, scarcely more restrictive than being a trapped surface. Of course a model of a massless scalar should not be taken seriously as a model of astrophysical black hole formation which involves multiple species of interacting particles. However, it shows that a model where the NEC is be violated can still lead to geodesic incompleteness with very weak restrictions on the initial conditions.