 use these estimates to find some $(s_\varepsilon, t_\varepsilon)\in\{0\le s\le c_0\}\bigcap\{0\le\ub\le\varepsilon\}$ such that the opposite of holds for some particular $(s,t)=(s_\varepsilon,t_\varepsilon)$, which is a contradiction.

However, in order to extend this result beyond spherical symmetry, much more things need to be concerned. First of all, we need to derive suitable a priori estimates in order to establish the *existence* of the solution. Second, we may not benefit from the assumptions like which is from proof by contradiction because the mass, which is essentially the $L^2$ integral of $L\phi$ (together with the outgoing shear) over $C_u$, can no longer govern the whole system without symmetries. In addition, the estimates derived by Christodoulou is so sharp that it is not easy to extend them beyond spherical symmetry.

In this paper, we are able to derive a priori $L^\infty$ bounds of the geometric quantities, including $\frac{\partial}{\partial\ub}\phi$, $\frac{\partial}{\partial u}\phi$, and the derivatives of $r$ and $\Omega$ defined by $-2\Omega^2=g\left(\frac{\partial}{\partial\ub},\frac{\partial}{\partial u}\right)$. These a priori estimates are proved by bootstrap argument to hold in a region deep enough to the future. It then follows easily that the condition eventually holds before these estimates fail. These estimates are robust and analogues of them may hold when no symmetries are imposed. The generalization of these estimates without symmetries will also be used in proving the *existence*. There is a simple way to understand the difference between two arguments, that is, the a priori estimates we derive will in particular imply that some analogue of the assumption really holds with $c_1$ replaced by a larger constant depending on the $L^\infty$ bounds of the data on $C_{o}$ and this condition with a larger constant still implies that holds by Christodoulou’s argument. Nevertheless, we do not need to repeat Christodoulou’s argument because from the bootstrap argument, all estimate