Ngô Quốc Anh

June 5, 2010

Why the conformal method is useful in studying the Einstein equations?

Filed under: Nghiên Cứu Khoa Học, PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 19:20

I presume you have some notions about general relativity, especially the Einstein equations

{\rm Eins}_{\alpha\beta}=T_{\alpha\beta}.

As these equations are basically hyperbolic for a suitable metric, it is reasonable to study the Cauchy problems for them. Under the Gauss and Codazzi conditions, we have two constraints called Hamiltonian and Momentum constrains. Cauchy problem is to determine the solvable of these constrains of variables K-the extrinsic curvature and g-the spatial metric. Interestingly, the conformal method says that we can start with an arbitrary metric then we recast the constrain equations into a form which is more amenable to analysis by splitting the Cauchy data. In this method, we try to solve \gamma within the conformal class represented by the initial metric. So, in general, the conformal factor is chosen so that we eventually have a simplest model.

This idea is given via the following theorem.

Theorem. Let \mathcal D =(\gamma, \sigma, \tau,\psi,\pi) be a conformal initial data set for the Einstein-scalar field constraint equations on \Sigma. If

\displaystyle \widetilde \gamma =\theta^\frac{4}{n-2}\gamma

for a smooth positive function \theta, then we define the corresponding conformally transformed initial data set by

\displaystyle\widetilde{\mathcal D} =(\widetilde\gamma, \widetilde \sigma, \widetilde \tau,\widetilde\psi,\widetilde \pi)=(\theta^\frac{4}{n-2}\gamma, \theta^{-2}\sigma, \tau,\psi,\theta^\frac{-2n}{n-2}\pi).

Let W be the solution to the conformal form of the momentum constrain equation w.r.t. the conformal initial data set \mathcal D and let \widetilde W be the solution to the conformal form of the momentum constrain equation w.r.t. the conformal initial data set \widetilde{\mathcal D} (we just assume both exist). Then \varphi is a solution to the Einstein scalar field Lichnerowicz equation for the conformal data \mathcal D with W

\displaystyle \Delta_\gamma \varphi - \mathcal R_{\gamma, \psi}\varphi +\mathcal A_{\gamma, W, \pi}\varphi^{-\frac{3n-2}{n-2}}-\mathcal B_{\tau, \psi}\varphi^\frac{n+2}{n-2}=0

if and only if \theta^{-1}\varphi is a solution to the Einstein scalar field Lichnerowicz equation for the conformal data \widetilde{\mathcal D} with \widetilde W

\displaystyle \Delta_{\widetilde\gamma} (\theta^{-1}\varphi) - \mathcal R_{\widetilde\gamma, \widetilde\psi}(\theta^{-1}\varphi) +\mathcal A_{\widetilde\gamma, \widetilde W, \widetilde\pi}(\theta^{-1}\varphi)^{-\frac{3n-2}{n-2}}-\mathcal B_{\widetilde\tau, \widetilde\psi}(\theta^{-1}\varphi)^\frac{n+2}{n-2}=0.

We refer the reader to a paper due to Yvonne Choquet-Bruhat et al. [here] published in Class. Quantum Grav. in 2007 for details. We adopt this theorem from that paper, however, there is no proof there.

Outline of the proof. Let

\overline \gamma=\varphi^\frac{4}{n-2}\gamma.

Obviously we have (go to this entry for further information)

\displaystyle \frac{n-2}{4(n-1)} \overline{{\rm Scal}}_{\overline\gamma}=\varphi^{-\frac{n+2}{n-2}}\left(-\Delta_\gamma\varphi + \frac{n-2}{4(n-1)}{\rm Scal}_\gamma \varphi\right).

However, it is clear to see that

\overline \gamma=(\theta^{-1}\varphi)^\frac{4}{n-2}\widetilde \gamma

which also yields

\displaystyle \frac{n-2}{4(n-1)} \overline{{\rm Scal}}_{\overline\gamma}=(\theta^{-1}\varphi)^{-\frac{n+2}{n-2}}\left(-\Delta_{\widetilde\gamma}(\theta^{-1}\varphi) + \frac{n-2}{4(n-1)}{\rm Scal}_{\widetilde\gamma} (\theta^{-1}\varphi)\right).

Thus,

\displaystyle \boxed{-\Delta_{\widetilde\gamma}(\theta^{-1}\varphi) + \frac{n-2}{4(n-2)}{\rm Scal}_{\widetilde\gamma} (\theta^{-1}\varphi)=\theta^{-\frac{n+2}{n-2}}\left(-\Delta_\gamma\varphi + \frac{n-2}{4(n-1)}{\rm Scal}_\gamma \varphi\right)}.

Keep in mind that

\displaystyle \mathcal R_{\gamma, \psi}=\frac{n-2}{4(n-1)}\left({\rm Scal}_\gamma - |\nabla \psi|_\gamma^2\right).

Next we have

\displaystyle |\nabla \psi|_\gamma^2 =\gamma^{ij}\nabla_i \psi\nabla_j\psi=\theta^{\frac{4}{n-2}}{\widetilde\gamma}_{ij}\nabla_i \psi\nabla_j\psi=\theta^{\frac{4}{n-2}}|\nabla \widetilde\psi|_{\widetilde\gamma}^2,

i.e.

\displaystyle\boxed{|\nabla \widetilde\psi|_{\widetilde\gamma}^2=\theta^{-\frac{4}{n-2}}|\nabla \psi|_\gamma^2}.

We refer the reader to this entry showing that how to calculate the norm of tensors. It is also clear to see that

\displaystyle \boxed{\mathcal B_{\widetilde\tau, \widetilde\psi} = \mathcal B_{\tau, \psi}}

since \tau and \psi remain unchanged under the conformal change. Concerning \mathcal A, from its definition

\displaystyle \mathcal A_{\gamma, W, \pi}=\frac{n-2}{4(n-1)}\left(|\sigma+\mathbb DW|_\gamma^2+\pi^2\right)

we see that

\displaystyle \boxed{\mathcal A_{\widetilde\gamma, \widetilde W, \widetilde\pi}=\theta^{-\frac{4n}{n-2}}\mathcal A_{\gamma, W, \pi}}.

Thus adding all gives

\displaystyle\begin{gathered} - {\Delta _{\widetilde\gamma }}({\theta ^{ - 1}}\varphi ) + \frac{{n - 2}}{{4(n - 2)}}{\text{Sca}}{{\text{l}}_{\widetilde\gamma }}({\theta ^{ - 1}}\varphi ) \hfill \\ \qquad\qquad- \frac{{n - 2}}{{4(n - 2)}}|\nabla \widetilde\psi |_{\widetilde\gamma }^2({\theta ^{ - 1}}\varphi ) + {\mathcal{A}_{\widetilde\gamma ,\widetilde W,\widetilde\pi }}{({\theta ^{ - 1}}\varphi )^{ - \frac{{3n - 2}}{{n - 2}}}} + {\mathcal{B}_{\widetilde\tau ,\widetilde\psi }}{({\theta ^{ - 1}}\varphi )^{\frac{{n + 2}}{{n - 2}}}} \hfill \\ \qquad= {\theta ^{ - \frac{{n + 2}}{{n - 2}}}}\left( { - {\Delta _\gamma }\varphi + \frac{{n - 2}}{{4(n - 1)}}{\text{Sca}}{{\text{l}}_\gamma }\varphi } \right) \hfill \\ \qquad\qquad- \frac{{n - 2}}{{4(n - 2)}}{\theta ^{ - \frac{4}{{n - 2}}}}|\nabla \psi |_\gamma ^2({\theta ^{ - 1}}\varphi ) + {\theta ^{ - \frac{{4n}}{{n - 2}}}}{\mathcal{A}_{\gamma ,W,\pi }}{({\theta ^{ - 1}}\varphi )^{ - \frac{{3n - 2}}{{n - 2}}}} + {\mathcal{B}_{\tau ,\psi }}{({\theta ^{ - 1}}\varphi )^{\frac{{n + 2}}{{n - 2}}}} \hfill \\ \qquad= {\theta ^{ - \frac{{n + 2}}{{n - 2}}}}\left[ { - {\Delta _\gamma }\varphi + \frac{{n - 2}}{{4(n - 1)}}{\text{Sca}}{{\text{l}}_\gamma }\varphi - \frac{{n - 2}}{{4(n - 2)}}|\nabla \psi |_\gamma ^2 + {\mathcal{A}_{\gamma ,W,\pi }}{\varphi ^{ - \frac{{3n - 2}}{{n - 2}}}} + {\mathcal{B}_{\tau ,\psi }}{\varphi ^{\frac{{n + 2}}{{n - 2}}}}} \right]. \hfill \\ \end{gathered}

The proof now follows.

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