---
abstract: 'The Gao-Wald theorem related to time delay [@gaowald] assumes that the Null Energy Condition and the Null Generic Condition are satisfied, and that the underlying gravity theory is General Relativity. In the present work it is shown that the Gao-Wald theorem is true if the space time is null geodesically complete, if the curvature satisfy some reasonable properties stated along the text, and if every null geodesic contains at least two conjugate points. This result may apply to modified theory of gravities and to violating Null Energy Condition models as well.'
author:
- 'Juliana Osorio Morales [^1] and Osvaldo P. Santillán [^2]'
title: Variations of a theorem due to Gao and Wald
---

Introduction
============

Since the introduction of the *Alcubierre bubble* [@alcubierre] or the *Krasnikov tube* [@krasnikov], there has been a growing interest in the concept of time delay in General Relativity as well as in modified theories of gravity. The definition of time delay is indeed very subtle [@olum]. The Alcubierre bubble is a space time in which it is possible to make a round trip from two stars $A$ and $B$ separated by a proper distance $D$ in such a way that a fixed observer at the star $A$ measures the proper time for the trip as less than $2D/c$. In fact, this time can be made arbitrary small. This fact does not indicate that the observers travel faster than light, as they are traveling inside their light cone. The Alcubierre constructions employ the fact that, for two comoving observers in an expanding universe, the rate of change of the proper distance to the proper time may be larger than $c$ or much more smaller, if there is contraction instead of expansion. The Alcubierre space time is Minkowski almost everywhere, except at a bubble around the traveler which endures only for a finite time, designed for making the round trip proper time measured by an observer at the star $A$ as small as possible. Details can be found in [@alcubierre].

The examples given above are interesting, however a more careful definition of time delay was introduced in [@olum]. In this reference, a space time which appears to allow time advance was constructed, but it was proven that it is in fact the flat Minkowski metric in unusual coordinates. This suggests that to analyze time advance by simple inspection of the metric may be misleading. However, given a space time that is Minkowski outside a tube or a bubble such that the Alcubierre or Krasnikov space times, the notion of time delay is well defined. By use of some results due to Tipler and Hawking [@tipler1]-[@tipler3], it can be shown that all these examples violate the Null Energy Conditions at least in some region of the manifold. Further issues related to time delay and quantum gravity can be found in the works [@otras]-[@otras4] and references therein.

Recall that the Null Energy Condition states that the matter content energy momentum tensor satisfies $T_{\mu\nu}k^\mu k^\nu\geq 0$ for every null vector $k^\mu$ tangent to any null geodesic $\gamma$. This implies, in the context of General Relativity, that $R_{\mu\nu}k^\mu k^\nu\geq 0$ [@Wald]. On the other hand, the Null Generic Condition means that $k_{[\alpha} R_{\beta]\sigma\delta[\epsilon}k_{\gamma]}k^\sigma k^\delta\neq 0$ for some point in the geodesic $\gamma$. Both conditions automatically imply that any null geodesic $\gamma(\lambda)$ possesses at least a pair of conjugate points $p$ and $q$, if it is past and future inextendible, see [@Wald Proposition 9.3.7]. These results hold in the context of General Relativity, and should not be extrapolated to modified gravity theories without further analysis.

The results just described raise the question of whether time delay could hold in theories which do not violate the Null Energy Conditions. In this context, a theorem due to Gao and Wald [@gaowald] may be relevant. Its statement is the following.\
*Gao-Wald theorem:* Consider a null geodesically complete space time ($M$, $g_{\mu\nu}$) such that the Null Energy and Null Generic Conditions are satisfied. Then, given a compact region $K$, there exists a compact $K'$ containing $K$ such that for any pairs of points $p, q\notin K'$ and $q$ belonging to $J_+(p)-I_+(p)$, no causal curve $\gamma$ connecting both points intersects $K$.\
The Gao-Wald theorem stated above is related to time advance hypothesis as follows. If there were possible to deform the geometry in a region $K$, similar perhaps to a bubble, in order to produce a time advance, then a fastest null geodesic would enter in the region $K$ in order to minimize this time. The theorem states that this is not possible if the Null Energy Conditions and Null Generic Conditions are satisfied in the space time in consideration. This may constitute a no go theorem. However, there is no control over the size of the region $K'$, thus this theorem should be considered only as a weak version of a time advance hypothesis.

The aim of the present work is two folded. The first purpose is to show that the Null Energy and Null Generic conditions are not mandatory, neither is to work in the context of General Relativity, for the Gao-Wald theorem to be true. It will be shown that the Gao-Wald theorem holds when the following three requirements are satisfied.\
- *First requirement:* The space time ($M$, $g_{\mu\nu}$) is null geodesically complete.\
- *Second requirement:* Every null geodesic possesses at least two conjugate points.\
-*Third requirement:* Consider the set $S$ of pairs $\Lambda_0=$($p_0$, $k_0^\mu$) with $p$ a point in $M$ and $k_\mu$ a null vector in $TM_{p_0}$ properly normalized (see formula (\[norma\]) below) and defining a null geodesic $\gamma_0$. Then there exists an open set $O$ in $S$ containing $\Lambda_0$ for which the following two properties hold. For every pair $\Lambda=$($p$, $k^\mu$) in $O$, the corresponding geodesic $\gamma_\Lambda(\gamma)$ will posses a conjugate point $q$ to $p$, $q \in J_+(p)-I_+(p)$. Furthermore the map $h: O\to M$ such that $h(\Lambda)=q$ is continuous at $\Lambda_0$.\
The two properties described in the *third requirement* look a bit technical, but the intuition behind is the following. The first implies that, for a geodesic with two conjugate points $p_0$ and $q_0$, there exist an open set around $p_0$ such all the points $p$ in the open set will have a conjugate point $q$ with respect to some null geodesic emanating from them. The second part states that the conjugate point $q$ to $p$ will be very close to $q_0$ when $p$ is close to $p_0$ and when the geodesics are, in a very rough sense, “pointing in similar directions”.

The second and main purpose of the present work is to prove that the *second requirement* implies the *third one* under some more or less reasonable hypotheses about the curvature of the space time. We feel that this statement may be relevant for extending the Gao-Wald results to more general gravity theories or to models violating the Null Energy Conditions.

The organization of the present work is as follows. In section 2 some generalities about conjugate points in generic space times are discussed. In addition, certain topological issues related to the light cones in space times are also presented. The presentation is not exhaustive, but focused in the aspects more relevant for our purposes. At the end, a proof of the *third requirement* when the underlying model is General Relativity with Null Energy and Null Generic Conditions outlined. This is included by completeness, as this is one of the results to be generalized here. In section 3, some properties for the curvature of the space time are presented, which are not related neither to General Relativity nor to the weak and strong energy conditions, ensuring that the *second requirement* implies the *third requirement*. In section 4 the aforementioned implication is proved explicitly by the means of some propositions described in section 3. This section is rather technical. In section 5, the modified Gao-Wald theorem is proved explicitly, and the possible application of the obtained results is discussed.

The *third requirement* in GR with Null Energy and Null Generic Conditions
==========================================================================

As discussed above, the Gao-Wald theorem relies on the notion of conjugate points. Thus, it is convenient to recall some basic but important concepts about them, taking into account some standard references [@Wald]-[@penrose]. In addition, at the end of this section, a sketch of the proof of the *third requirement* in the context of GR and with Null Energy and Null Generic Conditions [@gaowald] is included. The next sections are devoted to generalize this proof to more general gravity models.

Null geodesics and conjugate points
-----------------------------------

In the present discussion, the space time ($M$, $g_{\mu\nu}$) is assumed to be null geodesically complete such that there exists a globally defined time like future pointing vector $t_\mu$ on it. Given a point $p$ in ($M$, $g_{\mu\nu}$), a point $q$ in $J_+(p)-I_+(p)$ is said to be conjugated to $p$ if the following holds. Consider a null geodesic $\gamma(\lambda)$ emanating from $p$, together with the associated differential equation =-R\^\_k\^k\^A\^\_, supplemented with the following initial conditions $$A^\mu_\nu|_p=0,\qquad \frac{dA^\mu_\nu}{d\lambda}\bigg|_p=\delta^\mu_\nu.$$ Here $\lambda$ is the affine parameter describing $\gamma(\lambda)$ and $k^\mu$ is a vector tangent to the curve $\gamma(\lambda)$, normalized by the following conditions k\^k\_=0, k\^t\_=-1. The point $q=\gamma(\lambda_0)$ is said to be conjugated to $p$ if and only if $$A_\mu^\nu(\lambda_0)=0.$$ The matrix $A^\mu_\nu(\lambda)$ has the following interpretation: the number $A^{\mu}_{\nu}$ are the coefficients of the Jacobi field $\eta^{\mu}$ along $\gamma$, i.e, $$\eta^\mu(\lambda)=A^\mu_\nu(\lambda) \frac{d\eta^\nu}{d\lambda}\bigg|_0,\qquad  \eta(0)|_p=0,$$ then (\[smile\]) implies that $\eta(\lambda)$ satisfies the Jacobi equation (hence the name) on $\gamma$ given by =-R\^\_k\^k\^\^. The classical definition of a conjugate point $q$ to $p$ is the existence of a solution $\eta^\mu(\lambda)$ of the Jacobi equation such that $\eta^\mu(0)=0$ and $\eta^\mu(q)=0$. Clearly, the fact that $A^\mu_\nu(\lambda_0)=0$ implies that $\eta^\mu(q)=0$, thus $q$ is a conjugate point to $p$ in the usual sense. For further details see [@Wald Section 9.3].

There is no warrantee that there exists a point $q$ conjugate to a generic point $p$ for a given space time ($M$, $g_{\mu\nu}$). In addition, there might exist two or more different points $q$ and $s$ conjugate to $p$, joined to $p$ by different geodesics.

The study of conjugate points has been proven to have many applications in Riemannian and Minkowski geometry. It is well known that, in Riemannian geometry, a geodesic $\gamma(\lambda)$ starting at a point $p=\gamma(0)$ and ending at a point $r=\gamma(\lambda_0)$ is not necessarily length minimizing if there is a conjugate point $q=\gamma(\lambda_1)$ to $p$ such that $\lambda_1<\lambda_0$. The presence of a conjugate point in the middle usually spoil the minimizing property. For time like geodesics in Minkowski geometries, the proper time elapsed to travel between $p$ and $r$ is not maximal if there is a conjugate point in the middle. For null geodesics, there is an important result which will be used below, see [@Wald Theorem 9.3.8].

Let $\gamma$ a smooth causal curve and let $p, r\in \gamma$. Then there does not exist a smooth one parameter family of causal curves $\gamma_s$ connecting both points, such that $\gamma_0=\gamma$ and such that $\gamma_s$ are time like for $s>0$ if and only if there is no conjugate point $q$ to $p$ in $\gamma$.

By reading this statement as a positive affirmation, it is found that if a null curve connecting $p$ and $r$ can be deformed to a time like curve, then there is a pair of conjugate points in between and, conversely, if there is such pair, the curve can be deformed to a time like one.

The matrix $A_\mu^\nu(\lambda)$ defined by equation (\[smile\]) takes values which depend on the choice of the null geodesic $\gamma$. For this reason it may be convenient to denote it as $(A_\gamma)^\mu_\nu$. The same follows for the quantity $$G_\gamma(\lambda)=\sqrt{\det A_\gamma(\lambda)},$$ which also vanish at both $p$ and $q$. Note that the initial conditions below (\[smile\]) imply that $\det A_\gamma>0$ until the point $q$ is reached, thus the square root in this definition does not pose a problem. The equation (\[smile\]) implies that $G_\gamma(\lambda)$ satisfies the following second order equation [@gaowald] =-\[\_\^+R\_ k\^k\^\]G\_, and that $G_\gamma(0)=0$ and $G_\gamma(\lambda_0)=0$. These two values correspond to the points $p$ and $q$. Here $\sigma_{\mu\nu}$ is the shear of the null geodesics emanating from $p$. The last is an equation of the form $$\frac{d^2 G_\gamma}{d\lambda^2}=-p_\gamma(\lambda) G_\gamma.$$ As near the point $p$ the initial conditions in (\[smile\]) imply that $A_\mu^\nu\sim \delta_\mu^\nu \lambda$ it follows that, at $\lambda=0$ one has that $$G_\gamma(0)=0, \qquad \frac{dG_\gamma(0)}{d\lambda}=0.$$ Then, if $p_\gamma(\lambda)$ is $C^{\infty}$, by taking derivatives of equation (\[smile2\]) with respect to $\lambda$ it may be shown that $$\frac{d^n G_\gamma(0)}{d\lambda^n}=0.$$ This suggest that $G_\gamma(\lambda)$ may not analytical at the point $\lambda=0$.

Another typical equation appearing in the literature [@marolf] is given in terms of the expansion parameter $\theta_\gamma(\lambda)$, which is related to $G_\gamma(\lambda)$ by the formula G\_()=G\_i\_[\_i]{}\^\_() d, with $G_i=G(\lambda_i)$ the value of $\sqrt{\det A_\gamma(\lambda)}$ at generic parameter value $\lambda_i>0$. In terms of $\theta_\gamma$ the equation (\[smile2\]) becomes the well known Raychaudhuri equation +=-\_\^-R\_k\^k\^. The definition (\[folio\]) implies that \_= Thus $\theta_\gamma(\lambda)\to-\infty$ when $\lambda\to \lambda_0$, since $G_\gamma(\lambda)$ approaches to zero from positive values at $q$. Analogously, $\theta_\gamma(\lambda)\to\infty$ when $\lambda\to 0$, since $G_\gamma(\lambda)$ grows from the zero value when starting at $p$.

On the other hand, the fact that $\theta_\gamma\to-\infty$ at $q$ itself does not imply that $G_\gamma(\lambda)\to 0$ when $\lambda\to\lambda_0$. This can be seen from (\[folio\]), as the integral of the divergent quantity $\theta_\gamma$ may be still convergent. By an elementary analysis of improper integrals it follows that, at the conjugate point $q=\gamma(\lambda_0)$, the expansion parameter $\theta_\gamma(\lambda)$ is divergent with degree \_()\~,0, up to multiplicative constant. The behavior (\[asin\]) will play an important role in the next sections.

Some further remarks are in order. In a generic case, there is a possibility that $\theta=G'/2G$ may be divergent at a point $r$ non conjugate to $p$. In this case $G\neq 0$ but then $G'$ should be divergent. However, the equation (\[smile2\]) implies that $G''$ exists unless there is a singularity of $p_\gamma(\lambda)$. As an example, this may happen due to a some sort of singularity of the scalar $R_{\mu\nu}k^\mu k^\nu$ at $r$. The space times considered in the present work are assumed to be free of these pathologies. This means that $\theta_\gamma(\lambda)$ is well defined everywhere except at $p$ and $q$, where it takes the values $\pm \infty$. In other words, between the points $p$ and $q$ or, what is the same, when $\lambda$ varies in the interval $(0,\;\lambda_0)$, the expansion parameter $\theta_\gamma(\lambda)$ takes every real value. If instead $p$ does not have a conjugate point along $\gamma$, then $\theta_\gamma(\lambda)$ is expected to be finite and continuous for every finite value of $\lambda$.

In addition, note that the quantity (\[folio\]) is not well defined when $\lambda_i \to 0$, that is, when the initial point is $p$. This reflects the expansion parameter is singular at $p$.

Each of the equations (\[raychaudhuri\]) and (\[smile2\]) have their own advantages. In the following, both versions will play an important role, and will be employed in each situation by convenience.

Future light cones in curved space times
----------------------------------------

In addition to conjugate points, another important concept is the future light cone emanating from a point $p$ in the space time ($M$, $g_{\mu\nu}$). Given the point $p$ one has to consider all the future directed null vectors $k^\mu$ in $TM_p$ which satisfy the normalization (\[norma\]). Far away from $p$ it is likely that these geodesics may form a congruence $\gamma_\sigma(\lambda)$, but for $\lambda=0$, the congruence is singular since $\gamma_\sigma(0)=p$ for every value of $\sigma$. In other words, $p$ is the tip of the cone.

Close to the point $p$ there is an open set $U$ composed by points $p'$, with their respective set of future directed null vectors $k'^\mu$ in $TM_{p'}$ which satisfy the normalization (\[norma\]). When comparing geodesics emanating from different points $p$ and $p'$, one should compare not only both points but also the corresponding null vectors $k_\mu$ and $k'_\mu$. In some vague sense, two null geodesics $\gamma$ and $\gamma'$ are “close’ when $p$ and $p'$ are at close and the corresponding vectors $k_\mu$ and $k'_\mu$ ”point in similar directions". In order to put this comparison in more formal terms, it is convenient to introduce the set $S$ defined as follows [@gaowald] $$S=\{\Lambda=(p, k^\mu)~|~ p \in M, \quad k^\mu \in TM_p,\quad k^\mu k_\mu=0, \quad k^\mu t_\mu=-1\}.$$ This set has an appropriate topology which allows to compare a pair $\Lambda=(p, k^\mu)$ with another one $\Lambda'=(p', k'^\mu)$ and to determine if they are “close”. The definition implies that the vectors $k^\mu$ are all null and satisfying the normalization (\[norma\]).

The null geodesic corresponding to the element $\Lambda=(p, k^\mu)$ will be denoted as $\gamma_\Lambda(\lambda)$ in the following. All the quantities depending on this curve such as $G_\gamma(\lambda)$ will be subsequently denoted as $G_\Lambda(\lambda)$ and so on. The reason for this notational change is the desire to study continuity properties of these quantities as functions on $S$.

Proof of the *third requirement* for GR with Null Conditions
------------------------------------------------------------

The proof for the *third requirement* when the underlying theory is GR and the Null Energy and Null Generic Conditions are fulfilled is given in [@gaowald]. As this is one of the theorems to be generalized here, it is convenient to sketch the original argument. Consider a null geodesic $\gamma_0(\lambda)$ with $p_0=\gamma_0(0)$ and $q_0=\gamma_0(\lambda_0)$ conjugate points along it, with $\lambda_0>0$. Then $G_0(0)=G_0(\lambda_0)=0$ and $G_0(\lambda)>0$ for all $\lambda$ in the interval $0<\lambda<\lambda_0$. The Null Energy Condition $T_{\mu\nu}k^\mu k^\nu\geq 0$ implies, in the context of General Relativity, that $R_{\mu\nu}k^\mu k^\nu\geq 0$. This, together with (\[smile2\]) shows that $G_0''(\lambda)<0$ in the interval $0<\lambda<\lambda_0$. The mean value theorem applied to $G_0$ shows that $G_0'(\lambda_1)=-C^2$ for some value $\lambda_1$ in the interval and furthermore $G_0'(\lambda_1)<-C^2$ for $\lambda_1<\lambda<\lambda_0$, with $C^2$ a positive constant. By choosing $\lambda_0-\delta<\lambda<\lambda_0$ one has that $$\frac{G_0(\lambda_1)}{|G'_0(\lambda_1)|}<\delta,$$ since $|G_0'(\lambda)|$ is larger than $C^2$ if $\delta$ is small enough.

Consider now a small open $O\subset S$ around the point $\Lambda_0=(p_0, k^\mu)$ generating $\gamma_0(\lambda)$. As $G_\Lambda(\lambda)$ and its derivatives are continuous when moving in this open, then $G'_\Lambda(\lambda)<0$ and $$\frac{G_\Lambda(\lambda_1)}{|G'_\Lambda(\lambda_1)|}<\delta,$$ for all the $\Lambda=(p, k^\mu)\in O$ if $O$ is small enough. As $G_\Lambda(\lambda)>0$ and $G_\Lambda''(\lambda)>0$ due to the Null Energy Condition, it can be shown that for all the points $\Lambda=(p, k^\mu)$ in $O$ one has $G_\Lambda(\lambda')=0$ for some point $\lambda'$ such that $|\lambda'-\lambda_0|<\delta$, see [@gaowald] for further details. This shows that there exists a conjugate point $q$ to $p$, which is close to $q_0$ when $O$ is small enough. This is basically the statement of the *third requirement*.

For the purposes of the present work however, the Null Energy and Null Generic conditions are not assumed to hold. This means that it can not be assumed that $G_\Lambda''(\lambda)<0$ neither that mean value theorem applied to $G_\Lambda(\lambda)$, in the form presented above, is true. The following part is devoted to sort out the technical complications arising by relaxing these two important conditions. This effort may be useful, as it may allow to extend the Gao-Wald theorem to models with Averaged Null Energy Conditions or Quantum Null Energy Conditions [@averaged1]-[@averaged25] or to modified gravity theories [@odintsov]-[@odintsov3]. The interest in these conditions arises when considering quantum effects in gravitational models.

The assumed properties for the space time $(M,\;g_{\mu\nu})$
============================================================

The next step is to specify a new set of properties for the null geodesically complete space time ($M$, $g_{\mu\nu}$) which, together with the *second requirement*, lead directly to the *third requirement*. The implication is shown in the next sections. The postulated properties are the following.\
*Property 1:* $(M, \; g_{\mu\nu})$ is null geodesically complete. Every null geodesic $\gamma_\Lambda(\lambda)$ in $(M, \; g_{\mu\nu})$ will contain at least a pair of conjugate points $p$ and $q$, with $q$ in $J_+(p)-I_+(p)$.\
*Property 2:* For any constant $c>0$ the integral I\_(\_i)=\_ \_[\_i]{}\^e\^[-c ]{} \[R\_k\^k\^+\_\^\]\_()d, corresponding to a generic geodesic $\gamma_\Lambda$, is always finite, for every finite value of the initial affine parameter $\lambda_i$. In addition, the integrand is continuous and derivable everywhere in the space time $(M, \;g_{\mu\nu})$.\
The *Property 1* is equivalent to the *first* and *second requirements* stated in the introduction. Note that space time manifold ($M$, $g_{\mu\nu}$) is *not* assumed to be a solution of General Relativity neither matter is assumed to satisfy the Null Energy and Null Generic Conditions. It should be emphasized that these properties are not necessarily the unique properties leading to a Gao-Wald theorem. The task of finding variation may be a relevant one.

The *Property 2* looks a bit technical, but it may clarified as follows. It basically suggests that a light traveller measures the quantity $[R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_\Lambda(\lambda)$, which involves the curvature, and does not find an asymptotic grow of an exponential type, with negative sign. In other words, the behavior $$[R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_\Lambda(\lambda) \sim- e^{c\lambda},$$ when $\lambda\to \infty$ is forbidden. This is a ad hoc hypothesis, but in authors’s opinion, it is physically reasonable and may cover part of the spectrum of null energy violating models. The main task is to show that properties 1 and 2 imply that ($M$, $g_{\mu\nu}$) satisfy the three requirements described in the previous section and consequently that they lead to a Gao-Wald theorem.

There is another reason for considering the quantity (\[tang\]), which is related to a mathematical property of the Raychaudhuri equation. This property is summarized in the propositions of the next subsection.

Some mathematical consequences of the Properties 1 and 2
--------------------------------------------------------

The following proposition was used extensively in the works [@galloway1]-[@galloway2]. For the origins of the relation between Riccati inequality and Raychaudhuri equation see [@erlich].

\[prop1\] Given a non linear differential equation of the Riccati form +=p\_(), with an initial condition $\theta(0)=\theta_i$ then, if there exists a constant $c_\Lambda>0$ such that \_i+&lt; \_ \_[0]{}\^e\^[-c\_]{} p\_() d, it follows that a solution $\theta_\Lambda$ does not extend beyond a finite value of the affine parameter $\lambda=\lambda_e$.

The equation (\[u\]) may be converted, by the change of variables y()=-(\_+c\_)e\^[-c\_]{},into the following one =+r(),y(0)=y\_0. Here r()=e\^[-c\_]{}(p\_()+),q()=e\^[-c\_]{},and $y_0=-\theta_i-c_\Lambda$. From the definition of $q(\lambda)$ given in (\[erre\]) it is clear that \_[+]{}\_0\^+. Assume that the initial condition is such that \_[+]{} \^\_0 r()d&gt;-y\_0, and that $y(\lambda)$ extend to the whole interval $[0,\infty)$. This will imply a contradiction which will show that this affirmation is false thus, $y(\lambda)$ does not extend beyond a finite affine parameter value $\lambda=\lambda_e$. The same observation will hold for $\theta_\Lambda$, as it is defined in terms of $y(\lambda)$ by (\[y\]).

In order to show the aforementioned contradiction, note that (\[cudo\]) implies the existence of affine parameter $\lambda_1$ for which $$\int^\lambda_0 r(\xi)d\xi>-y_0, \qquad \textrm{for}\qquad \lambda>\lambda_1.$$ By integrating the equation (\[teor\]) and taking into account the last inequality, it follows that y()=\_0\^ d+\_0\^r() d+y\_0&gt;\_0\^ d. It is convenient to introduce the quantity given by R()=\_0\^ d. As $q(\lambda)$ is positive and $\lambda>0$, it can directly be seen that that $R(\lambda)\geq 0$, the equality holds only for $\lambda=0$. This definition and the inequality (\[teor2\]) shows that &lt;=, for $\lambda>\lambda_1$. From here it is concluded, for every $\lambda_2>\lambda_1$, that $$\int_{\lambda_2}^{\lambda}\frac{d\xi}{q(\xi)}<\int_{\lambda_2}^{\lambda}\frac{1}{R^2}\frac{dR}{d\lambda}d\xi=\frac{1}{R(\lambda_2)}-\frac{1}{R(\lambda)}<\frac{1}{R(\lambda_2)}.$$ However, by condition (\[cudon\]) it follows that the left hand is not bounded when $\lambda\to \infty$. Thus, the last inequality makes sense only for times $\lambda<\lambda_e$, with $\lambda_e$ a fixed time. This shows that $y(\lambda)$ can not extend to the whole interval $ [0,\infty)$, but only to an interval inside $[0, \lambda_e)$. The same applies for the quantity $\theta_\Lambda$ related to $y(\lambda)$ by $y(\lambda)=-(\theta_\Lambda+c_\Lambda)e^{-c_\Lambda\lambda}$. In terms of $\theta_\Lambda$ the condition (\[cudo\]) becomes $$\theta_0+c_\Lambda< \lim_{\lambda\to\infty} \textrm{inf}\int^\lambda_0 e^{-c_\Lambda\lambda}\bigg(p_\Lambda(\lambda)+\frac{c_\Lambda^2}{2}\bigg)d\xi<\lim_{\lambda\to\infty} \textrm{inf}\int^\lambda_0 e^{-c_\Lambda\lambda}p_\Lambda(\lambda)d\xi+\frac{c_\Lambda}{2},$$ or, equivalently $$\theta_0+\frac{c_\Lambda}{2}<\lim_{\lambda\to\infty} \textrm{inf}\int^\lambda_0 e^{-c_\Lambda\lambda}p_\Lambda(\lambda)d\xi.$$ This is precisely the condition (\[tang2\]), which shows the desired result.

Proposition \[prop1\] may apply to the Raychaudhuri equation for $\theta_\Lambda$ with $$p_\Lambda(\lambda)=-[R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_\Lambda(\lambda).$$ In fact, the right hand of (\[tang2\]) becomes (\[tang\]) under this identification. This point will be elaborated in more detail below.

\[theta\_i\] Consider again a null geodesically complete space-time $(M,g_{\mu\nu})$ satisfying Properties 1 and 2. There are a pair of conjugate points along any null geodesic $\gamma_\Lambda(\lambda)$, generically denoted as $p$ and $q$. Then for any positive real number $c_\Lambda>0$, there exists a point $s=\gamma_\Lambda(\lambda_i)$ such that $\theta_i:=\theta_\Lambda(\lambda_i)$ satisfies the following inequality \_i+&lt; \_[+]{}\_[\_i]{}\^e\^[-c\_]{} p\_(). d, Here $p=\gamma_\Lambda(0)$, the value of $\lambda_i$ in the integral in (\[tang3\]) is such that $0<\lambda_{i}<\lambda_{e}$, where $\lambda_e$ is the parameter defined by $q=\gamma_\Lambda(\lambda_e)$.

This remark is a direct consequence of the fact that $\theta_{\Lambda}((0,\lambda_{e}))=\mathbb R$, the existence of such $\lambda_{i}$ is guaranteed by continuity. This can be seen as follows. Recall that the scalar expansion $\theta_\Lambda$, is such that $\theta_\Lambda(\lambda)\to -\infty$ when $\lambda\to \lambda_e$, which corresponds to the point $q$. On the other hand, as the integrand function in is continuous and by the Property 2 it is seen that all the integrals $$I(\lambda_i)=\liminf_{\lambda\to+\infty} \int_{\lambda_i}^\lambda e^{-c_\Lambda\xi} [R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_\Lambda(\xi) d\xi,$$ are bounded from below for every $c_\Lambda>0$ and $\lambda_i$. By considering $I(\lambda_{i})$ as a function of $\lambda_{i}$ with $c_\Lambda>0$ fixed, it is seen that $I(\lambda_{i})$ attains a global minima $I_M$ when $\lambda_i$ varies in the compact interval $[0,\lambda_e]$. This follows from the fact that the function $I(\lambda_i)$ by Property 2 is bounded and the interval just defined is compact. Choose a special value of the parameter $\lambda_i$ such that $$\theta_\Lambda(\lambda_i)+\frac{c_\Lambda}{2}<I_M.$$ This parameter exists as $\theta_\Lambda(\lambda)$ takes every real value continuously in $(0, \lambda_e)$. By the minimality of $I_M$ it is seen that this condition implies (\[tang3\]), and this clarifies the Remark \[theta\_i\].[^3]

The next task is to apply the content of Proposition \[prop1\] to the study of conjugate points in the null geodesically complete space time ($M$, $g_{\mu\nu}$).

\[prop 2\] Let $(M,g_{\mu\nu})$ be a null geodesically complete space time satisfying the Property 1 and Property 2. Consider a geodesic $\gamma_\Lambda(\lambda)$. Let $q$ be conjugate to $p$ through $\gamma_{\Lambda}$ such that $q\in J_+(p)-I_+(p)$. Then the value of the affine parameter $\lambda_e$ such that $\gamma_\Lambda(\lambda_e)=q$ is given implicitly by the following formula: =\_[\_1]{}\^[\_e]{}{1+}\^2e\^[c]{}d. Here $\lambda_1$ is any value of the affine parameter such that $\lambda_i<\lambda_1<\lambda_e$ and the quantity $R_\Lambda(\lambda)$ is defined in (\[dji\]). The definition of $R_\Lambda(\lambda)$ involves $c_\Lambda$ but the formula (\[explota\]) is universal, that is, does not depends on the choice of $c_\Lambda$, neither on the choice of $\lambda_1$.

Without loss of generality, it may be assumed that $\gamma_\Lambda(0)=p$ and $\gamma_\Lambda(\lambda_e)=q$. First, it is convenient to re-write the Raychaudhuri equation in the form . This gives $$y(\lambda)=\int_{\lambda_i}^\lambda \frac{y^2(\xi)}{q(\xi)} d\xi +\int_{\lambda_1}^\lambda r(\xi) d\xi+y_0,$$ with $y(\lambda)$ defined in (\[y\]). From here it is seen that $$\frac{e^{c_\Lambda \lambda}y^2(\lambda)}{2}=\bigg[\int_{\lambda_i}^\lambda \frac{y^2(\xi)}{q(\xi)} d\xi +I(\lambda)\bigg]^2\frac{ e^{c_\Lambda \lambda}}{2} ,$$ with $I(\lambda)=\int_{\lambda_i}^\lambda r(\xi) d\xi+y_0$. The last equation can be expressed in terms of the quantity $R_\Lambda(\lambda)$ defined in (\[dji\]), the result is $$\frac{dR}{d\lambda}=[R_\Lambda(\lambda)+I(\lambda)]^2\frac{e^{c_\Lambda \lambda}}{2}.$$ By dividing this result by $R^2_\Lambda(\lambda)$ and by integrating with respect to $\lambda$ leads to $$\frac{1}{R_\Lambda(\lambda_1)}-\frac{1}{R_\Lambda(\lambda_2)}=\int_{\lambda_1}^{\lambda_2}\bigg[1+\frac{I(\xi)}{R_\Lambda(\xi)}\bigg]^2\frac{e^{c_\Lambda \xi}}{2}d\xi.$$ Here $\lambda_1<\lambda_2$ and both values are in $(0, \lambda_e)$ but otherwise arbitrary. Now, under the working hypothesis, the value $\theta_\Lambda(\lambda)\to-\infty$ as $\lambda\to \lambda_e$ signals the presence of a conjugate point. Then the asymptotic condition (\[asin\]) applies and it is seen from (\[dji\]) that $R_\Lambda(\lambda)\to \infty$ when $\lambda\to \lambda_e$. Then, at $\lambda_2=\lambda_e$, the last expression becomes $$\frac{1}{R_\Lambda(\lambda_1)}=\int_{\lambda_1}^{\lambda_e}\bigg[1+\frac{I(\xi)}{R_\Lambda(\xi)}\bigg]^2\frac{e^{c_\Lambda \xi}}{2}d\xi.$$ Here $\lambda_i<\lambda_1<\lambda_e$ is arbitrary. By expressing the last formula in terms of $\theta_\Lambda(\lambda)$ from (\[y\]) and by taking into account the first (\[erre\]) the expression (\[explota\]) is obtained.

Proof of the *third requirement* as a consequence of Properties 1 and 2
=======================================================================

The purpose of this section is to show that Property 1 and Property 2 given above imply the *third requirement*. This leads to the conclusion that the Gao-Wald theorem holds for space times satisfying these properties.

Before making formal statements, it is worth to give the intuition behind the proof of *third requirement* given below. One considers a reference null geodesic $\gamma_0(\lambda)$ which, by Property 1, has two conjugate points let’s say $p_0$ and $q_0$. Remark \[theta\_i\] shows that (\[tang2\]) is satisfied for some point in the middle $s_0$. That is, for this point $s_0=\gamma_0(\lambda_i)$ one has $$\theta_0(\lambda_i)+\frac{c_\Lambda}{2}< \lim_{\lambda\to\infty} \text{inf}\int_{\lambda_i}^\lambda e^{-c_\Lambda \xi} p_0(\xi) 
d\xi.$$ The strategy is to show that, for some geodesics emanating from points $p$ close to $p_0$, the condition (\[tang2\]) is also satisfied for the points $s=\gamma_\Lambda(\lambda_i)$, which are close to $s_0$. That is $$\theta_\Lambda(\lambda_i)+\frac{c_\Lambda}{2}< \lim_{\lambda\to\infty} \text{inf}\int_{\lambda_i}^\lambda e^{-c_\Lambda \xi} p_\Lambda(\xi) 
d\xi,$$ for all these geodesics. The Proposition \[prop1\] will allow to conclude that $\theta_\Lambda(\lambda)$ also is going to tend to $-\infty$ at a finite value of the affine parameter $\lambda$. Therefore, these points $p$ will have a conjugate point $q$ joined by the null geodesic $\gamma_\Lambda(\lambda)$. This leads to the first part of the *third requirement*.

The second part is more subtle. The point is that, by assumption, the reference geodesic $\gamma_0(\lambda)$ has two conjugate points $p_0$ and $q_0$. At these points the determinant $G_0(\lambda)$ defined in (\[smile2\]) vanishes. Thus $G_0(0)=0$ and $G_0(\lambda_e)=0$. By continuity in $S$, one may work by analogy with section 2.3 and try to show that $|G_\Lambda(\lambda_e)|<\epsilon$ if $\Lambda\in O$ with $O$ an open in $S$ small enough. But even taking into account that $G_\Lambda(\lambda)$ has a very small modulus, this does not ensure that it is going to vanish for $|\lambda-\lambda_e|<\delta$. In the present case, as the Null Energy Condition is not assumed valid, it follows that $G''_\Lambda(\lambda)$ is not always negative, and the mean value result of section 2.3 does not work. It may be the case that the value of $G_\Lambda(\lambda)$ is close to zero at $\lambda_e$, then grows very rapidly and only at a value $\lambda=\lambda'_e$ which is very far from $\lambda_e$ will vanish. This would spoil the continuity property of the *third requirement*. Below, it will be proven that this is not the case. The proof relies heavily on the formula (\[explota\]) for the explosion value of the affine parameter $\lambda=\lambda_e$, by considering how it varies when moving along “close geodesics”.

In view of this discussion, it is convenient to divide the proof of the *third requirement* in two parts.

Proof of the first part of the third requirement
------------------------------------------------

Consider an arbitrary null reference geodesic $\gamma_0$ in the space time ($M$, $g_{\mu\nu}$). The *first requirement* implies that it has two conjugate points $p_0$ and $q_0$, where $q_0$ belongs to $J_+(p_0)-I_+(p_0)$. Without losing generality one may assume that the affine parameter is such that $p_0=\gamma_0(0)$, the value at $q_0$ will be denoted as $\lambda=\lambda_0$. The expansion parameter $\theta_0\to \infty$ at $\lambda\to 0^+$ and $\theta_0\to -\infty$ at $q_0$. As stated before, this leads to the important conclusion that this expansion parameter $\theta_0$ takes every real value when $\lambda$ moves in the interval ($0$, $\lambda_0$), if $q$ is the first conjugate point to $p$.

The reference null geodesic $\gamma_0$ is generated by a point $\Lambda_0=$($p_0$, $k_0^\mu$) in $S$. At a given value $\lambda_i$ such $0<\lambda_i<\lambda_0$, a generic value of $\theta(\lambda_i)=\theta_i$ is achieved. Denote the corresponding point in the space time by $s_0=\gamma_0(\lambda_i)$. Consider the map given by[^4] $H(\Lambda, \lambda)=\gamma_{\Lambda}(\lambda)$. For any small enough neighborhood $U$ in $M$ containing $s_0$ there exists an open $O$ in $S$ containing $\Lambda_0$ such that $H(\Lambda, \lambda)$ belongs to $U$ for every point $\Lambda=(p, \;k^\mu)$ in $O$ if $|\lambda-\lambda_i|<\delta$. Here $\gamma_\Lambda(0)=p$. In these terms, one may show the following proposition.\

\[1stpart3rdrequire\] Given a null geodesically complete space time ($M$, $g_{\mu\nu}$) with the Property 1 and 2, consider a geodesic $\gamma_0(\lambda)$ with pair of conjugate points $p_0$ and $q_0$, corresponding to a point $\Lambda_0=(p_0$, $k_0^\mu)$ in $S$. Then there exists an open $O$ in $S$ containing $\Lambda_0$ such that every geodesic $\gamma_\Lambda(\lambda)$ generated by points $\Lambda=(p$, $k^\mu)$ in $O$ possess a conjugate point $q$ to $p$.

Before going to the proof, note that the Property 1 already states that every null geodesic contains a pair of conjugate points. But it does not specify where these points are located along the geodesic. The new information this proposition gives is that, once $p_0$ has a conjugate point, then all the points $p$ in a neighbourhood of $p_0$ will have a conjugate point along to some geodesic emanating from them.

Choose a compact set $O'$ containing the point $\Lambda_0$ defined in the previous paragraph. Consider a curve generated by a point $\Lambda$ inside the compact set $O'$. By Property 2, the integral $$I(\lambda_s)=\lim_{\lambda\to\infty}\text{inf}\int_{\lambda_s}^\lambda e^{-c_\Lambda\xi} [R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_\Lambda(\xi) d\xi,$$ is bounded from below for any fixed value $\lambda_s$. Vary the value of $\lambda_s$ in the interval $[0,\;\lambda_0]$, which is compact, and find the smallest value $I_i$, which will be achieved for a specific value $\lambda^\ast_s$. At this point, this procedure mimics the proof of Proposition 2. The strategy however is not doing this with one particular null geodesic, but with all the geodesics $\gamma_\Lambda(\lambda)$ generated by the points $\Lambda$ in $O'$. The resulting minimum for a given geodesic, denoted by $I_{\Lambda}$, is not necessarily a continuous function in $O'$ but, by Property 2, is bounded by below. As $O'$ is compact, there will exists an infimum value at a point $\Lambda_m\in O'$, with its corresponding smallest value $I_{\Lambda_m}$. By construction this value is smaller or equal than the corresponding to any other generic point $\Lambda$ in $O'$. Below, for notational simplicity, this value will be denoted by $I_m$ instead of $I_{\Lambda_m}$.

Once the value $I_m$ has ben found, choose a value $\lambda_i$ such that $\theta_0(\lambda_i)+c/2<I_m-\eta$ with $\eta> C>0$. The required value of $\lambda_i$ in $(0,\lambda_0)$ exists since, as discussed above, the expansion parameter $\theta_0(\lambda)$ for the reference geodesic $\gamma_0$ takes every real value when $\lambda$ varies in that closed interval. Then, from the minimality of $I_m$ it is clear that $\theta_0(\lambda_i)+c/2<I_m-\eta$ implies that $$\theta_0(\lambda_i)+\frac{c}{2}< \lim_{\lambda\to\infty} \text{inf}\int_{\lambda_i}^\lambda e^{-c \xi} [R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_0(\xi) 
d\xi.$$ In these terms, one may choose an open $O$ inside $O'$ containing $\Lambda_0$, such that for every $\theta_\Lambda(\lambda)$ determined by a point $\Lambda$ in $O$ the inequality $|\theta_\Lambda(\lambda_i)-\theta_0(\lambda_i)|<\epsilon$ holds. The validity of this statement may be seen from the continuity of $G(\lambda, \Lambda)$ and $G'(\lambda, \Lambda)$ in $O$, which implies the continuity of $\theta_\Lambda(\lambda)=G'(\Lambda,\lambda)/G(\Lambda, \lambda)$ with respect to $\Lambda$ if $G(\lambda, \Lambda)\neq 0$, that is, outside a conjugate point. This continuity property follows from the fact that $(A_\Lambda)_\mu^\nu$ satisfies the ordinary equation (\[smile\]), and thus $(A_\Lambda)_\mu^\nu$ and $G_\Lambda(\lambda)$ vary continuously with respect to $\Lambda$ and $\lambda$.

Now, for $\epsilon<\eta$, the minimality of $I_m$, together with the fact that $|\theta_\Lambda(\lambda_i)-\theta_0(\lambda_i)|<\epsilon$ and $\theta_0(\lambda_i)+c/2<I_m-\eta$, imply that \_(\_i)+&lt; \_ \_[\_i]{}\^e\^[-c ]{} \[R\_k\^k\^+\_\^\]\_()d, for all the points $\Lambda$ of $O\subset S$. A direct application of this fact and the Proposition 1 implies that for any point $\Lambda=$($p$, $k^\mu$) in the open set $O$, if $\gamma_\Lambda(0)=p$, the solution $\theta_\Lambda(\lambda)$ does not extend beyond a finite value $\lambda_e$ if $\epsilon<\eta$. As discussed below formula (\[asin\]), for the space time in consideration the expansion parameter $\theta_\Lambda(\lambda)$ is continuous everywhere except at the conjugate point. Therefore, the only possibility is that $\theta_\Lambda(\lambda)\to -\infty$ when $\lambda\to \lambda_e$. This implies that at $\lambda_e$ there appears a conjugate point $q$ to $p$. This concludes the proof.

Proof of the second part of the third requirement
-------------------------------------------------

Before proving the second part, the following observation is needed. In the proof of Proposition \[1stpart3rdrequire\], the initial value $\theta_0(\lambda_i)+c/2<I_m-\eta$ with $\eta> C>0$ has been selected. Choose $\eta>c/2+\epsilon$. From the minimality $I_m$ and from (\[plat\]) it follows that \_(\_i)+c&lt;\_[\_i]{}\^e\^[-c ]{} \[R\_k\^k\^+\_\^\]\_()d, for every value of $\lambda\geq \lambda_i$ and $\Lambda$ in $O$. This particular choice will be useful below. In these terms, the proposition to be proved is the following.

Consider a generic point $\Lambda=(p, k^\mu)$ in an open $O\subset S$ containing $\Lambda_0=(p_0, k_0^\mu)$. Under the conditions of the Proposition \[1stpart3rdrequire\], the map $h: O\to M$ defined by $h(\Lambda)=q$, with $q$ the first conjugate point to $p$, is continuous at $\Lambda_0$.

As in the proof given before, the geodesic $\Lambda_0$ joins two conjugate points $p_0$ and $q_0$ and, by Proposition \[1stpart3rdrequire\], there is an open $O$ in $S$ containing $\Lambda_0$ such that, for all $\Lambda=(p, k^\mu)$ in $O$, there is a conjugate point $q$. By the proof of Proposition \[1stpart3rdrequire\] and the discussion above, one may find a value $\lambda_i$ such that for all the geodesics defined by every $\Lambda$ in $O$, the inequality (\[inicial\]) holds. Note that the value $\lambda_i$ is strictly larger than zero if the origin is defined such that $\gamma_\Lambda(0)=p$.

The length parameter $\lambda_e$ that defines the conjugate point $q_0$ in the geodesic $\gamma_0$ is given by (\[explota\]) which, adapted to the present situation, is given by =\_[\_1]{}\^[\_e]{}{1+}\^2e\^[c]{}d. Here $\lambda_i<\lambda_1<\lambda_e$ is an initial parameter, and p\_0()=\[R\_k\^k\^+\_\^\](),R\_0(\_1)=\_[\_i]{}\^[\_1]{}e\^[c]{}(\_0()+c)\^2 d. The length parameter defining the conjugate point $q$ for another geodesic $\gamma_\Lambda(\lambda)$ is $\lambda_e+\Delta\lambda_e$, and is given by =\_[\_1]{}\^[\_e+\_e]{}{1+}\^2e\^[c]{}d, with p\_()=\[R\_k\^k\^+\_\^\]\_(),R\_(\_1)=\_[\_i]{}\^[\_1]{}e\^[c]{}(\_()+c)\^2 d. Note that $\Delta \lambda_e$ depends on the choice of the geodesic, that is, $\Delta \lambda_e=f(\Lambda)$. This dependence would be implicitly understood in the following reasoning. The task is to show that $|\Delta \lambda_e|=|f(\Lambda))|<\epsilon$ when $\Lambda$ is in an open $O$ of $S$ small enough, containing $\Lambda_0$.

A point that might cause confusion is that, in principle, $\lambda_e+\Delta\lambda_e$ may be such that $\lambda_e+\Delta \lambda_e<\lambda_1$, as $\Delta\lambda_e$ may be negative. But the formula (\[mat2\]) is true for the opposite case. Thus, $\lambda_1$ should not be chosen so arbitrary. However, it has been mentioned that, from the continuity of $\theta_\Lambda(\lambda)$ in $S$ outside a conjugate point, one has that $|\theta_\Lambda(\lambda_i)-\theta_0(\lambda_i)|<\epsilon$ if $O$ is small enough. In fact $|\theta_\Lambda(\lambda)-\theta_0(\lambda_i)|<\epsilon$ if $O$ is small and $|\lambda-\lambda_i|\leq \delta$. This means that, up to a point $\lambda=\lambda_i+\delta'$ with $\delta'<\delta$, the value of $\theta_\Lambda(\lambda)$ does not explode. Choose $\lambda_i<\lambda_1<\lambda_i+\delta'$, then $\lambda_e+\Delta\lambda_e>\lambda_1$ for all the curves parameterized by $\Lambda$. Then the mentioned problem does not arise. The actual value of $\lambda_1$ is not known, but it exists, and this is the only thing needed in the following reasoning.

The subtraction of both expressions (\[mat\]) and (\[mat2\]) obtained above gives that $$\frac{1}{R_\Lambda(\lambda_1)}-\frac{1}{R_0(\lambda_1)}=\int_{\lambda_1}^{\lambda_e+\Delta \lambda_e}\bigg\{1+\frac{1}{R_\Lambda(\beta)}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \beta})+\int_{\lambda_1}^\beta e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]\bigg\}^2e^{c\beta}d\beta$$ $$-\int_{\lambda_1}^{\lambda_e}\bigg\{1+\frac{1}{R_0(\beta)}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \beta})+\int_{\lambda_i}^\beta e^{-c\xi} p_0(\xi)\bigg] \bigg\}^2e^{c\beta}d\beta.$$ The last expression can be cast in the following form $$\frac{1}{R_\Lambda(\lambda_1)}-\frac{1}{R_0(\lambda_1)}=\int_{\lambda_1}^{\lambda_e}\bigg\{1+\frac{1}{R_\Lambda(\beta)}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \beta})+\int_{\lambda_i}^\beta e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]\bigg\}^2e^{c\beta}d\beta$$ $$-\int_{\lambda_1}^{\lambda_e}\bigg\{1+\frac{1}{R_0(\beta)}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \beta})+\int_{\lambda_i}^\beta e^{-c\xi} p_0(\xi) \bigg]\bigg\}^2e^{c\beta}d\beta$$ +\_e {1+}\^2e\^[c’]{}, where in the last integral the mean value theorem has been employed and $\lambda'$ is some value in the interval $[\lambda_e, \; \lambda_e+\Delta \lambda_e]$.

The last formula (\[artdeco\]) already gives an intuition about the intended proof. The condition (\[inicial\]) implies that the term multiplying $\Delta \lambda_e$ in is strictly positive (as $R_\Lambda>0$). It vanish for $\lambda=\lambda_i$ but $\lambda'>\lambda_1>\lambda_i$. Thus, $R_\Lambda(\lambda')$ is never vanishing. On the other hand, one may show by use of analysis methods that the remaining terms are as small as possible by restricting $O$ to be is small enough. Thus $\Delta \lambda_e$ will be also very small, and this will prove the continuity property stated in the Proposition. Roughly speaking, it will imply that a point $q$ conjugated to $p$ is close to a point $q_0$ conjugated to $p_0$.

A method to prove this intuition goes as follows. First note that, from the definitions (\[prima\]) and (\[prima2\]) and by use of a mean value theorem, one has that $$\bigg|\frac{1}{R_\Lambda(\lambda_1)}-\frac{1}{R_0(\lambda_1)}\bigg|=\bigg|\frac{R_0(\lambda_1)-R_\Lambda(\lambda_1)}{R_\Lambda(\lambda_1)R_0(\lambda_1)}\bigg|=\frac{1}{R_\Lambda(\lambda_1)R_0(\lambda_1)}\bigg|\int_{\lambda_{i}}^{\lambda_{1}}e^{c\xi}[(\theta_0(\xi)+c)^2-(\theta_\Lambda(\xi)+c)^2]\,d\xi\bigg|$$ =|\_0(’)-\_(’)||\_0(’)+\_(’)+2c|(\_1-\_i). where in the last expression the mean value theorem was applied and the resulting expression was factored. Here the value $\lambda'$ depends on the geodesic, that is, $\lambda'=g(\Lambda)$. The quantity (\[cito\]) can be as small as possible since $\delta\theta(\lambda')=\theta_0(\lambda')-\theta_\Lambda(\lambda')$ goes to zero and the other quantities are under control. To see this clearly, given the open $O'$ in $S$ consider a compact $O_c\subset O'$. Find the minimum value of $R_\Lambda(\lambda_1)$ in this compact, denoted as $R_m$, and the maximum value of $\theta_\Lambda(\lambda)$ in $O_c\times [\lambda_1,\lambda_i]$, denoted as $\theta_m$. Then choose another open $O\subset O_c$. The last expression implies that |-| |\_m||2\_m+2c|(\_1-\_i), for any $\Lambda$ in $O$. Here $\delta\theta_m$ is the maximum value of $\delta \theta(\lambda')$ in $O_c\times [\lambda_i,\lambda_1]$. As this set is compact, the Cantor-Heine theorem allows to conclude that $|\theta_0(\lambda)-\theta_\Lambda(\lambda')|$ is small when $|\lambda-\lambda'|<\delta$ and $O_c$ is small enough, independently on the value of $\lambda'(\Lambda)$. This implies in particular that, by making $O'$ and consequently $O_c$ and $O$ small enough one may chose $\delta \theta_m$ such that $$|\delta \theta_m|< \frac{R_m^2\epsilon}{(\lambda_1-\lambda_i)|2\theta_m+2c|}.$$ in $O$, which implies (\[socia\]) that |-| , for all $\Lambda \in O$. This prove that the left side of (\[artdeco\]) can be made arbitrarily small.

On the other hand, the absolute value of the sum of those terms of the right side of (\[artdeco\]) which are independent on $\Delta \lambda_e$ can be converted by use of a mean value theorem into $$I=\bigg|\int_{\lambda_1}^{\lambda_e}\bigg\{1+\frac{1}{R_\Lambda(\beta)}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \beta})+\int_{\lambda_i}^\beta e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]\bigg\}^2e^{c\beta}d\beta$$ $$-\int_{\lambda_1}^{\lambda_e}\bigg\{1+\frac{1}{R_0(\beta)}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \beta})+\int_{\lambda_i}^\beta e^{-c\xi} p_0(\xi) \bigg]\bigg\}^2e^{c\beta}d\beta\bigg|$$ $$=\bigg|(\lambda_e-\lambda_1)\bigg\{1+\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]\bigg\}^2e^{c\lambda'}$$ $$-(\lambda_e-\lambda_1)\bigg\{1+\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]\bigg\}^2e^{c\lambda'}\bigg|$$ $$=(\lambda_e-\lambda_1)e^{2c\lambda'}\bigg|\bigg\{\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$-\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]\bigg\}$$ $$\times \bigg\{2+\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$+\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]\bigg\}\bigg|.$$ $$\leq (\lambda_e-\lambda_1)e^{2c\lambda_e} M \bigg|\bigg\{\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ -}| In the last expression $\lambda'$ is some value in the interval $[\lambda_1,\lambda_e]$, which comes from the mean value theorem for integrals, and $M$ denotes the maximum defined by $$M=\text{Max}\bigg|\bigg\{2+\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$+\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]\bigg\}\bigg|_{O_1},$$ in an initial compact set $O_1\times [\lambda_1,\lambda_e]$. This maximum exists since the function $R_\lambda(\lambda)$ is never zero for $\lambda$ in $[\lambda_1,\lambda_e]$ and the integrals are convergent since the space time $(M, g_{\mu\nu})$ in consideration is such that $p_\Lambda(\lambda)=[R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_\Lambda(\lambda)$ is not divergent at finite values of $\lambda$. The expression inside the brackets in the last step in (\[last\]) is then $$D_\Lambda=\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$-\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg],$$ where the denomination $D_\Lambda$ is chosen in order to emphasize that it represents a difference. Write this quantity as $$D_\Lambda=\bigg(\frac{1}{R_\Lambda(\lambda')}-\frac{1}{R_0(\lambda')}\bigg)\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$+\frac{1}{R_0(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$-\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]$$ By taking into account (\[andros\]), it is already seen that the last quantity is bounded since the integral is finite. In any case, by simplifying some terms, the last expression becomes $$D_\Lambda=\frac{1}{R_0(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)+\theta_0(\lambda_i)+\int_{\lambda_1}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi-\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]\bigg\}$$ $$+\bigg(\frac{1}{R_\Lambda(\lambda')}-\frac{1}{R_0(\lambda')}\bigg)\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg],$$ From the continuity of $\theta_\Lambda(\lambda)$ it follows that $|\theta_\Lambda(\lambda_i)-\theta_0(\lambda_i)|\leq \epsilon'/4$ by choosing a suitable open $O\subset O_1$ containing $\Lambda_0$. In addition, one has that $$\bigg|\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi-\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi)d\xi \bigg|\leq (\lambda_e-\lambda_i)\delta p^M_\Lambda,$$ where $\delta p^M_\Lambda$ denote the maximum value of $\delta p_\Lambda=e^{-c\lambda_i}(p_\Lambda(\lambda)-p_0(\lambda))$. Again, by a Cantor-Heine argument it can be seen that this maximum can be made arbitrarily small by making $O_1$ small enough. In particular, it can be made smaller than $\epsilon'/4(\lambda_e-\lambda_1)$. The minimum value of $R_0(\lambda')$ is $R_0(\lambda_1)$, as (\[prima2\]) shows that this quantity is monotone increasingly with $\lambda$. Thus by selecting $\epsilon'=\epsilon R_0(\lambda_1)$ one has that |D\_|+|(-)|. As the integrand in the last expression is finite for every curve $\gamma_\Lambda(\lambda)$ then it has a maximum $M'$ in the factors in parenthesis in the right side in the compact $C=O_1\times[\lambda_1,\lambda_e] $. By choosing $O_1$ and consequently $O$ small enough one may use (\[andros\]) in order to prove that $$\bigg|\frac{1}{R_\Lambda(\lambda_1)}-\frac{1}{R_0(\lambda_1)}\bigg| \leq\widetilde{\epsilon}.$$ By choosing $\widetilde{\epsilon}=\epsilon/2 M'$ it is seen from (\[last2\]) that |D\_|, for $\Lambda$ inside $O$. Now, the inequality (\[last\]) implies that $$I\leq (\lambda_e-\lambda_1)e^{2c\lambda_e} M |D_\Lambda|.$$ By use of (\[epsi\]) it is clear that, by choosing $O_1$ and thus $O$ small enough then $|D_\Lambda|\leq \epsilon/M(\lambda_e-\lambda_1)e^{2c\lambda_e} $, and this implies that I. In view of this discussion, consider again (\[artdeco\]). It is clear that it can be written as $$\frac{1}{R_\Lambda(\lambda_1)}-\frac{1}{R_0(\lambda_1)}=\pm I+\Delta \lambda_e \bigg\{1+\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\lambda')d\xi\bigg]\bigg\}^2e^{c\lambda'},$$ where the $\pm$ is to indicate that in the definition (\[last\]) of $I$, the absolute value has been taken. The sign of the quantity inside the modulus is not known, thus the identity holds with a plus or a minus sign, which will be not relevant in the following discussion. From the last expression and by taking into account (\[ep\]) and (\[andros\]), it is concluded that, by choosing a very small open $O$ around $\Lambda_0$, one has $$\bigg|\Delta \lambda_e \bigg\{1+\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\lambda')d\xi\bigg]\bigg\}^2e^{c\lambda'}\bigg|\leq \epsilon_1+\epsilon_2,$$ where both $\epsilon_i$ comes either from $I$ in (\[ep\]) or either from (\[andros\]). The quantity multiplying $\Delta \lambda_e$ is ensured to be positive by the choice (\[inicial\]) employed. In addition, it has a maximum in a compact $O'$ inside $O$ containing $\Lambda_0$, called $M_\Delta$. Find an open $O_2$ inside $O'$ containing $\Lambda_0$ such that $\epsilon_i\leq \epsilon/2 M_\Delta e^{c\lambda_e}$. Then $$|\Delta \lambda_e(\Lambda)|\leq \epsilon,$$ for every $\Lambda$ in $O_2$. Thus $\Delta \lambda_e(\Lambda)\to 0$ continuously as $\Lambda\to \Lambda_0$, which shows the desired result.

A modified Gao-Wald theorem
===========================

After proving that the Properties 1 and 2 described in section 3 imply that the *second requirement* and the *third requirement* stated in section 1 are satisfied, the next task is to show that the Gao-Wald theorem [@gaowald] is true when these properties are satisfied.

Let $(M,g_{\mu\nu})$ be a null geodesically complete space time such that the Property 1 and Property 2 (given in section 3) are satisfied. Then, given a compact region $K$ in $M$ there exists a compact $K'$ containing $K$ such that, for any two points $p, q\notin K'$ and $q$ belonging to $J_+(p)-I_+(p)$, no causal curve $\gamma$ joining $p$ with $q$ can intersect $K$.

As the space time manifold $M$ is assumed to be paracompact, it can be made into a Riemannian manifold with Riemannian metric $q_{\mu\nu}$. This metric can be assumed to be complete by multiplying it by a conformal factor if necessary [@Hicks]. Fix a point $r \in M$ and let $d_{r}:M\to\mathbb R$, $d_{r}(s)$ denotes the geodesic distance between $r$ and $s$ using the metric $q_{\mu\nu}$. This function is continuos in $M$ and for all $R>0$ the set $B_{R}=\{p\in M:~ d_{r}(p)\leq R\}$ is compact (see [@Hicks Theorem 15]). In these terms, given $\Lambda \in S$ let $\gamma_{\Lambda}$ a null geodesic determined by $\Lambda$, let’s define the function $f:S\to \mathbb R$ by: $$\begin{aligned}
f(\Lambda) =& \inf_{R}\{B_{R}\text{ contains a connected segment of $\gamma_{\Lambda}$ that includes the initial point }\\
&\quad \quad \text{determined by $\Lambda$ together with a pair of conjugate points of $\gamma_{\Lambda}$}\}\end{aligned}$$ The function $f(\Lambda)$ is upper semicontinuous [@gaowald], when the *second requirement* and the *third requirement* are satisfied, the proof has been given in the reference [@gaowald]. Let $K\subset M$ be a compact set. Let $S_{K}=\{(p,k^{\mu}\in S \text{ with } p \in K)\}$, since the tangent bundle has the product topology, $K$ is compact and $k^{\mu}$ is of bounded norm, then $S_{K}$ is compact. Since $f$ is upper-semicontinuous, it must achieve a maximum in $S_{K}$, let’s denote it by $\bar R$. Let $K' = B_{\bar R}$. Let $p,q\notin K'$ and $q\in J_+(p)-I_+(p)$ and let $\gamma$ a causal curve joining $p$ with $q$, then $\gamma$ must be a null geodesic since $q\in J_+(p)-I_+(p)$. However, the Proposition \[temporal\] insures that $\gamma$ does not contain a pair of conjugate points between $p$ and $q$. If $\gamma\cap K\neq \emptyset$ then by the definition of $K'$, $\gamma$ must have a pair of conjugate points lying in $K'$ and in between $p$ and $q$. This contradiction completes the proof.

Note that the theorem given above is true if Properties 1 and 2 are replaced by the *first*, *second* and *third requirements* given in the introduction, as the proof would be unchanged.

In brief, in the present work it has been shown that the Gao-Wald theorem holds when the space time ($M$, $g_{\mu\nu}$) is null geodesically complete, every null geodesic posses at least a pair of conjugate points, and the curvature is such that a the quantity $R_{\mu\nu}k^\mu k^\nu\neq -e^{c\lambda}$ for large $\lambda$ values, when evaluated on a null geodesic. Therefore this result may apply to models which violate the Null Energy Condition [@averaged1]-[@averaged25] or to modified gravity theories such as the ones described in [@odintsov]-[@odintsov3]. This deserves further attention. In our opinion, the curvature condition introduced in the text may hold for several interesting solutions for these models. On the other hand, the condition that every null geodesic contains a pair of conjugate points sounds a bit stringent, and it may be of interest to relax it if possible. Another relevant task is to understand it is possible to control the size of the region $K'$ of the Gao-Wald theorem. We leave this for a future investigation.

Acknowledgments {#acknowledgments .unnumbered}
===============

Both authors are supported by CONICET, Argentina.

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[^1]: Instituto de Matemática Luis Santaló (IMAS), UBA CONICET, Buenos Aires, Argentina juli.osorio@gmail.com.

[^2]: Instituto de Matemática Luis Santaló (IMAS), UBA CONICET, Buenos Aires, Argentina firenzecita@hotmail.com and osantil@dm.uba.ar.

[^3]: Note that the remark \[theta\_i\] is in agreement with Proposition \[prop1\], as $\theta_\Lambda(\lambda)$ is expected to explode at affine parameter value $\lambda_e$.

[^4]: This map is defined such that $H: O\times [0,\lambda_0]\to M$ where $O$ is an open in $S$.
