Ngô Quốc Anh

January 19, 2010

Why the Einstein field equations are hyperbolic equations?

Filed under: Nghiên Cứu Khoa Học, PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 1:24

Today we discuss a little about the Einstein field equations, that is, {\rm Eins}_{\alpha\beta} := {\rm Ric}_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}R(g)=T_{\alpha\beta}. We shall prove that in the Minkowski spaces, the Einstein field equations are nothing but hyperbolic equations.

We start with the Riemann curvature tensor R, an (3,1)-tensor, defined to be

\displaystyle R: (X,Y,Z) \mapsto R(X,Y)Z

where

\displaystyle R(X,Y)Z = {\nabla _X}{\nabla _Y}Z - {\nabla _Y}{\nabla _X}Z - {\nabla _{\left[ {X,Y} \right]}}Z.

Note that, the metric that we are using is Riemannian metric, therefore \left[ {\frac{\partial }{{\partial {x^i}}},\frac{\partial }{{\partial {x^j}}}} \right] = 0. Therefore, in local coordinates,

\displaystyle\begin{gathered}R_{ijk}^\alpha \frac{\partial }{{\partial {x^\alpha }}} = {\nabla _i}{\nabla _j}\frac{\partial }{{\partial {x^k}}} - {\nabla _j}{\nabla _i}\frac{\partial }{{\partial {x^k}}} \hfill \\ \qquad= {\nabla _i}\left( {\Gamma _{jk}^l\frac{\partial }{{\partial {x^l}}}} \right) - {\nabla _j}\left( {\Gamma _{ik}^l\frac{\partial }{{\partial {x^l}}}} \right) \hfill \\ \qquad= \Gamma _{jk}^l{\nabla _i}\left( {\frac{\partial }{{\partial {x^l}}}} \right) + \frac{{\partial \Gamma _{jk}^l}}{{\partial {x^i}}}\frac{\partial }{{\partial {x^l}}} - \Gamma _{ik}^l{\nabla _j}\left( {\frac{\partial }{{\partial {x^l}}}} \right) - \frac{{\partial \Gamma _{ik}^l}}{{\partial {x^j}}}\frac{\partial }{{\partial {x^l}}} \hfill \\ \qquad= \Gamma _{jk}^l\Gamma _{il}^\beta \frac{\partial }{{\partial {x^\beta }}} + \frac{{\partial \Gamma _{jk}^l}}{{\partial {x^i}}}\frac{\partial }{{\partial {x^l}}} - \Gamma _{ik}^l\Gamma _{jl}^\beta \frac{\partial }{{\partial {x^\beta }}} - \frac{{\partial \Gamma _{ik}^l}}{{\partial {x^j}}}\frac{\partial }{{\partial {x^l}}} \hfill \\ \qquad= \left( {\Gamma _{jk}^l\Gamma _{il}^\alpha+ \frac{{\partial \Gamma _{jk}^\alpha }}{{\partial {x^i}}} - \Gamma _{ik}^l\Gamma _{jl}^\alpha- \frac{{\partial \Gamma _{ik}^\alpha }}{{\partial {x^j}}}} \right)\frac{\partial }{{\partial {x^\alpha }}}. \hfill \\ \end{gathered}

the \alpha-component of the Riemann curvature tensor is

\displaystyle R_{ijk}^\alpha= \Gamma _{jk,i}^\alpha- \Gamma _{ik,j}^\alpha+ \left\{ {\rm no \; derivative} \right\}.

We now compute Ricci tensor, to this purpose, we get

\displaystyle {R_{\alpha \beta }} = {g^{ij}}{R_{\alpha i\beta j}} = {g^{ij}}{g_{\alpha \gamma }}R_{i\beta j}^\gamma= R_{\alpha \beta \gamma }^\gamma = \Gamma _{\beta \gamma ,\alpha }^\gamma- \Gamma _{\alpha \gamma ,\beta }^\gamma+ \left\{ {\rm no \; derivative} \right\} .

In terms of metric tensor, one has

\displaystyle\Gamma _{ij}^k = \frac{1}{2}{g^{kl}}\left( {\frac{{\partial {g_{jl}}}}{{\partial {x^i}}} + \frac{{\partial {g_{il}}}}{{\partial {x^j}}} - \frac{{\partial {g_{ij}}}}{{\partial {x^l}}}} \right)

which implies

\displaystyle\Gamma _{\beta \gamma ,\alpha }^\gamma- \Gamma _{\alpha \gamma ,\beta }^\gamma= \frac{1}{2}{g^{\gamma l}}\left( {{g_{\gamma l,\beta \alpha }} + {g_{\beta l,\gamma \alpha }} - {g_{\beta \gamma ,l\alpha }} - {g_{\gamma l,\alpha \beta }} - {g_{\alpha l,\gamma \beta }} + {g_{\alpha \gamma ,l\beta }}} \right).

Thus

\displaystyle\Gamma _{\beta \gamma ,\alpha }^\gamma- \Gamma _{\alpha \gamma ,\beta }^\gamma= \frac{1}{2}\left( {g_{\beta ,\gamma \alpha }^\gamma+ g_{\alpha ,\gamma \beta }^\gamma- {g_{,\beta \alpha }} - \square {g_{\alpha \beta }}} \right).

where \square= {g^{ij}}{\nabla _i}{\nabla _j} is the d’Alembert wave operator.To see this, clearly

\displaystyle\begin{gathered}\square ({g_{\alpha \beta }}d{x^\alpha } \otimes d{x^\beta }) = {g^{ij}}{\nabla _i}{\nabla _j}({g_{\alpha \beta }}d{x^\alpha } \otimes d{x^\beta }) \hfill \\ \qquad= {g^{ij}}\left[ {{\nabla _i}{\nabla _j}({g_{\alpha \beta }}d{x^\alpha }) \otimes d{x^\beta } + {\nabla _i}(d{x^\alpha } \otimes {\nabla _j}({g_{\alpha \beta }}d{x^\beta }))} +\cdots\right] \hfill \\ \qquad= {g^{ij}}\left[ {{g_{\alpha \beta }}{\nabla _i}{\nabla _j}(d{x^\alpha }) \otimes d{x^\beta } + {g_{\alpha \beta }}{\nabla _i}(d{x^\alpha } \otimes {\nabla _j}(d{x^\beta }))} +\cdots\right] \hfill \\ \qquad= {g^{ij}}\left[ { - {g_{\alpha \beta }}{\nabla _i}(\Gamma _{jk}^\alpha d{x^k} \otimes d{x^\beta }) - {g_{\alpha \beta }}{\nabla _i}(d{x^\alpha } \otimes \Gamma _{jk}^\beta d{x^k})} +\cdots\right] \hfill \\ \qquad= {g^{ij}}\left[ { - {g_{\alpha \beta }}\Gamma _{jk,i}^\alpha d{x^k} \otimes d{x^\beta } - {g_{\alpha \beta }}\Gamma _{jk,i}^\beta d{x^\alpha } \otimes d{x^k} +\cdots } \right] \hfill \\ \qquad= {g^{ij}}\left[ { - {g_{\alpha \beta }}\Gamma _{jk,i}^\alpha d{x^k} \otimes d{x^\beta } - {g_{\alpha \beta }}\Gamma _{jk,i}^\beta d{x^\alpha } \otimes d{x^k} +\cdots } \right] \hfill \\ \qquad=- \frac{1}{2}{g^{ij}}\left[ {({g_{j\beta ,ki}} + {g_{k\beta ,ji}} - {g_{jk,\beta i}})d{x^k} \otimes d{x^\beta } + ({g_{j\alpha ,ki}} + {g_{k\alpha ,ji}} - {g_{jk,\alpha i}})d{x^\alpha } \otimes d{x^k} +\cdots } \right] \hfill \\ \end{gathered}

which gives

\displaystyle\square {g_{\alpha \beta }} =- \frac{1}{2}{g^{ij}}\left[ {({g_{j\beta ,\alpha i}} + {g_{\alpha \beta ,ji}} - {g_{j\alpha ,\beta i}}) + ({g_{j\alpha ,\beta i}} + {g_{\beta \alpha ,ji}} - {g_{j\beta ,\alpha i}})} \right] +\cdots.

Thus

\displaystyle\square {g_{\alpha \beta }} =-{g^{ij}}{g_{\beta \alpha ,ji}} +\cdots.

Contracting once more with g^{\alpha\beta} the scalar curvature is obtained as

\displaystyle R = {g^{\alpha \beta }}{R_{\alpha \beta }} = \frac{1}{2}{g^{\alpha \beta }}\left( {g_{\beta ,\gamma \alpha }^\gamma+ g_{\alpha ,\gamma \beta }^\gamma- {g_{,\beta \alpha }} - \square {g_{\alpha \beta }}} \right) = g_{,\alpha \beta }^{\alpha \beta } - {g^{\alpha \beta }}\square {g_{\alpha \beta }}.

Thus, the Einstein tensor is

\displaystyle\begin{gathered}{\rm Ein}{s_{\alpha \beta }} = {\rm Ric}_{\alpha \beta } - \frac{1}{2}{g_{\alpha \beta }}R \hfill \\ \qquad= \frac{1}{2}\left[ {\left( {g_{\beta ,\gamma \alpha }^\gamma+ g_{\alpha ,\gamma \beta }^\gamma- {g_{,\beta \alpha }} - \square {g_{\alpha \beta }}} \right) - {g_{\alpha \beta }}\left( {g_{,ij}^{ij} - {g^{ij}}\square {g_{ij}}} \right)} \right]. \hfill \\ \end{gathered}

If we introduce the following notation

\displaystyle {\overline g _{\alpha \beta }} = {g_{\alpha \beta }} - \frac{1}{2}{g_{\alpha \beta }}{g^{ij}}\square {g_{ij}}

then

\displaystyle\overline g _{\beta ,\gamma \alpha }^\gamma+ \overline g _{\alpha ,\gamma \beta }^\gamma- \square {\overline g _{\alpha \beta }} - {g_{\alpha \beta }}\overline g _{,ij}^{ij} = {T_{\alpha \beta }}.

In Minkowski space, if we require the Lorenz condition, or the Lorenz gauge, we then have

\displaystyle\square {g _{\alpha \beta }} = - {T_{\alpha \beta }}.

This is what we need. Coordinates that obey the Lorenz condition are called harmonic.

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