Some Evolution Problems in the Vacuum Einstein Equations

Abstract

We discuss two problems in the evolution of the vacuum Einstein equations. The first one is about the formation of trapped surface, and the second one is about the characteristic problems with initial data on complete null cones.

Appendix

We will give a precise geometric description on the characteristic initial data, which is originally formulated in Chap. 2 of [8]. One can see [23, 25] for the reduction from the geometric description to a harmonic gauge. We consider double characteristic initial data problem, which means that the initial data is given on two intersecting null hypersurfaces $C_0$ and $\underline{C}_0$, with $S_0=C_0\,\bigcap \,\underline{C}_0$. The Cauchy-characteristic mixed problem can always reduce to the double characteristic problem locally. Let g be the solution of the initial data problem to the VEE.¹⁰ Choose affine functions s and $\underline{s}$ ¹¹ on $C_0$ and $\underline{C}_0$. Now we restrict ourselves to $C_0$ and then $\underline{C}_0$ can be handled in a similar way. Let $(e_1,e_2,e_3,e_4)$ be a null frame on $C_0$, where $e_A, A=1,2$ are tangent to the affine sections $S_s$, $e_4$ be the tangent vector field of the null generators of $C_0$ (and therefore is null and perpendicular to $S_s$), and $e_3$ be another null vector that is also perpendicular to $S_s$ (and therefore transversal to $C_0$), with $g(e_3,e_4)=-2$. We define the following tensors that are tangential covariant tensors on $S_s$:

$$\begin{aligned} \chi _{AB}=g(\nabla _{e_A}e_4,e_B),\ \underline{\chi }_{AB}=g(\nabla _{e_A}e_3,e_B),\ \zeta =\frac{1}{2}g(\nabla _{e_A}e_4,e_3). \end{aligned}$$

(9)

Here $\nabla$ is the Levi-Civita connection relative to g, and the tensors above are components of the connection. Let $\mathrm {tr}\chi$ and $\mathrm {tr}\underline{\chi }$ be the traces of $\chi$ and $\underline{\chi }$, with the trace being taken on $S_s$, and $\widehat{\chi }$, $\widehat{\underline{\chi }}$ be the trace-free part. Let also $g \!\! /$ be the metric induced on $S_s$ by g, and be the Levi-Civita connection relative to $g \!\! /$. Then the initial data given on $C_0$ consists the following quantities:

• 1. The metric $g \!\! /$ on $S_0$, the tensor $\zeta$ on $S_0$, and two functions $\mathrm {tr}\chi$, $\mathrm {tr}\underline{\chi }$ on $S_0$ ¹²;

• 2. The conformal geometry on $C_0$.

Assigning the conformal geometry on $C_0$ means we should assign on each affine section $S_s$ a conformal class $[\widehat{g \!\! /}]$, by specifying some representative $\widehat{g \!\! /}$. The representative $\widehat{g \!\! /}$ can be assigned in a special way to simplify the calculation. Let $\varPhi _{s}$ be the one parameter diffeomorphism group generated by $e_4$. We require that $\widehat{g \!\! /}(s)\triangleq \varPhi _{s}^*\widehat{g \!\! /}|_{S_s}$, which is a metric on $S_0$, has the same volume form as $g \!\! /|_{S_{0}}$, which is given. Let $\phi (s)>0$ be the conformal factor, which means $g \!\! /(s)\triangleq \varPhi _s^*g \!\! /|_{S_s}=\phi (s)^2\widehat{g \!\! /}(s)$. Thanks to the above special choice of $\widehat{g \!\! /}$, we can compute

$$\begin{aligned} \widehat{\chi }_{AB}=\frac{1}{2}\phi ^2\frac{\partial \widehat{g \!\! /}_{AB}}{\partial s},\ \mathrm {tr}\chi =\frac{2}{\phi }\frac{\partial \phi }{\partial s}, \end{aligned}$$

provided that $e_A$ are chosen such that $[e_4,e_A]=0$. The Raychaudhuri equation

$$\begin{aligned} \frac{\partial }{\partial s}\mathrm {tr}\chi =-\frac{1}{2}(\mathrm {tr}\chi )^2-|\widehat{\chi }|_{g \!\! /}^2 \end{aligned}$$

then reduces to

$$\begin{aligned} \frac{\partial ^2}{\partial s^2}\phi =-e\phi \end{aligned}$$

(10)

where $e=\frac{1}{2}|\widehat{\chi }|_{g \!\! /}^2=\frac{1}{8}\widehat{g \!\! /}^{AC}\widehat{g \!\! /}^{BD}\frac{\partial \widehat{g \!\! /}_{AB}}{\partial s}\frac{\partial \widehat{g \!\! /}_{CD}}{\partial s}$ is completely determined by the conformal geometry $[\widehat{g \!\! /}]$. Once $\phi$ is solved, then $\widehat{\chi }$ and $\mathrm {tr}\chi$ are known, and the other geometric quantities, in (9) and (6), are solved by the following equations¹³ $^,$ ¹⁴, with the initial values¹⁵ given on $S_0$: (where K is the Gauss curvature of $S_s$ and $\varepsilon \!\! /$ is the volume form relative to $g \!\! /$ of $S_s$)

Now we turn to the initial data of Theorem 5. We only discuss the part of the initial data given on $C_0$ because the initial data on $\underline{C}_0$ is only required to be finite and has nothing to do with the asymptotic behavior.¹⁶ Now because $C_0$ is complete, then s is allowed to tend to $+\infty$. The initial conditions on $C_0$ imposed in [20] is the following.¹⁷ We require there exists a constant C such that $C^{-1}(1+s)\le r\le C(1+s)$. Let $\lambda (s)$ and $\varLambda (s)$ be the smaller and larger eigenvalues of $r(s)^{-2}g \!\! /(s)$ relative to $r(0)^{-2}g \!\! /(0)$, we require that there exists a constant C such that $C^{-1}<\lambda (s)\le \varLambda (s)<C$ for all s. We require that the following norms¹⁸

(11)

are finite. We also require the following norms¹⁹

(12)

are finite. Notice that we do not require any conditions on $\alpha$, which is not the same as compared to (7) or (8). But the existence result stated in Theorem 6 still holds due to a smart observation due to Luk and Rodnianski in [21]. Although they worked on finite region, we have showed that their observation still works in our case.

It seems that it is not natural to impose the (11) and (12) on the initial data. However, these conditions are natural in the sense that they will propagate along the evolution, at least locally in retarded time, which are exactly what was proven in [20]. But it is also interesting to illustrate that the conditions above are reasonable, by using a more direct way to formulate the asymptotic conditions on the initial data. One possible way to impose the asymptotic conditions on the initial data on $C_0$, is that only imposing a condition on $\widehat{\chi }$, together with some geometric assumptions. This is because, by the above argument, we can see that once we have solved $\widehat{\chi }$ and $\mathrm {tr}\chi$ (by solving $\phi$), we can deduce suitable estimates for all other components of the connection and curvature directly. Imposing conditions on the conformal representative $\widehat{g \!\! /}$ seems to be the most natural way, since $\widehat{g \!\! /}$ is completely free but $\widehat{\chi }$ is not, but it requires a detailed study on the quantitative behavior of the Eq. (10) and we do not pursue this.²⁰

The following conditions will imply the asymptotic conditions (11) and (12). We require that there exists a constant C such that

$$\begin{aligned} C^{-1}(1+s)\le r\le C(1+s),\ C^{-1}<\lambda (s)\le \varLambda (s)<C,\ |\mathrm {tr}\chi -\overline{\mathrm {tr}\chi }|\le Cr^{-2}, \end{aligned}$$

here $\overline{\mathrm {tr}\chi }$ means the average of $\mathrm {tr}\chi$ on $S_s$. These are the geometric assumptions on the complete null cone $C_0$. The last assumption describes the shape of the sections of $C_0$ near infinity. One possible condition on $\widehat{\chi }$ is the following:

(13)

where $\delta >0$ and m is chosen sufficiently large ($m=10$ is sufficient). It is not hard to derive (11) and (12) by assuming (13). One can refer to [20] in the case that $\widehat{\chi }$ is assumed to be compactly supported, and it is similar when (13) is assumed.

Finally, we make several remarks. First, when $\widehat{\chi }$ is assumed to be compactly supported, then in the region where $\widehat{\chi }$ vanishes, the Eq. (10) reduces to $\frac{\partial ^2}{\partial s^2}\phi =0$ and is much easier to understand. Therefore, simply assuming the conformal representative $\widehat{g \!\! /}$ does not change from some s is sufficient to guarantee (11) and (12). Second, although the form of the condition (13) is quite concise, but it is not satisfactory in the following sense. The decay rate imposed on $\widehat{\chi }$ in (13) is stronger than that in (11), which is necessary to ensure the decay rate on $\beta$ in (13). In fact, the decay rate on sufficiently high derivatives of $\widehat{\chi }$ is not necessarily such strong, or we can impose a weaker decay rate as in (11) on every order derivatives of $\widehat{\chi }$ and then impose a decay rate on $\beta$ individually. On the other hand, it is obvious that the regularity required on $\widehat{\chi }$ in (13) is much more than that in (11). This is due to the nature of losing derivatives in characteristic problems. As we mention above, what are propagated in evolution are (11) and (12).