ambda=(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 $(M,\;g_{\mu\nu})$
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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 geode