- #1
Mentz114
- 5,432
- 292
A certain metric gives an Einstein tensor that has the form below. The coordinate labelling is
[itex]x^0=t,\ x^1=r,\ x^2=\theta,\ x^3=\phi[/itex]
[tex]
G_{\mu\nu}= \left[ \begin{array}{cccc}
A & B & 0 & 0\\
B & p1 & 0 & 0\\
0 & 0 & p2 & 0\\
0 & 0 & 0 & p3
\end{array} \right]
[/tex]
where [itex]A,B,C,p1,p2,p3[/itex] are functions of t and r. A transformation [itex]\Lambda[/itex] so [itex]\Lambda^\mu_\rho\ \Lambda^\nu_\sigma\ G_{\mu\nu}[/itex] is diagonal is easily found,
[tex]
\Lambda^\mu_\rho=\left[ \begin{array}{cccc}
1 & -\frac{B}{p1} & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{array} \right]
[/tex]
This seems to be transforming [itex]t[/itex] into [itex]T=t-h\ r[/itex] where [itex]h=B/p1[/itex]. This can be used to give the differential transformation
[tex]
dT=dt -hdr-rdh=dt-hdr-r(\partial_t h\ dt + \partial_r h\ dr)
[/tex]
so we can find [itex]dt^2[/itex] and substitute into the original metric to get a transformed one written in coordinates [itex]T,r,\theta,\phi[/itex].
Question: will the Einstein tensor obtained from the transformed metric be [itex]\Lambda^\mu_\rho\ \Lambda^\nu_\sigma\ G_{\mu\nu}[/itex] ?
I think it will be but I haven't convinced myself.
[itex]x^0=t,\ x^1=r,\ x^2=\theta,\ x^3=\phi[/itex]
[tex]
G_{\mu\nu}= \left[ \begin{array}{cccc}
A & B & 0 & 0\\
B & p1 & 0 & 0\\
0 & 0 & p2 & 0\\
0 & 0 & 0 & p3
\end{array} \right]
[/tex]
where [itex]A,B,C,p1,p2,p3[/itex] are functions of t and r. A transformation [itex]\Lambda[/itex] so [itex]\Lambda^\mu_\rho\ \Lambda^\nu_\sigma\ G_{\mu\nu}[/itex] is diagonal is easily found,
[tex]
\Lambda^\mu_\rho=\left[ \begin{array}{cccc}
1 & -\frac{B}{p1} & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{array} \right]
[/tex]
This seems to be transforming [itex]t[/itex] into [itex]T=t-h\ r[/itex] where [itex]h=B/p1[/itex]. This can be used to give the differential transformation
[tex]
dT=dt -hdr-rdh=dt-hdr-r(\partial_t h\ dt + \partial_r h\ dr)
[/tex]
so we can find [itex]dt^2[/itex] and substitute into the original metric to get a transformed one written in coordinates [itex]T,r,\theta,\phi[/itex].
Question: will the Einstein tensor obtained from the transformed metric be [itex]\Lambda^\mu_\rho\ \Lambda^\nu_\sigma\ G_{\mu\nu}[/itex] ?
I think it will be but I haven't convinced myself.