- #1
Mentz114
- 5,432
- 292
- TL;DR Summary
- Starting with a rotating frame field (spherical Born coordinates) and setting ##\omega\equiv \omega(r)## then solving the differential equation ##\vec{a}=0## , ##\vec{a}## being the proper acceleration gives the frame field of a circular geodesic.
The Born frame field (see ref below) describes a rotating system and the proper acceleration ##\vec{a}=\nabla _{{{\vec {p}}_{0}}}\,{\vec {p}}_{0}={\frac {-\omega ^{2}\,r}{1-\omega ^{2}\,r^{2}}}\,{\vec {p}}_{2}##. If ##\omega## depends on coordinate ##r## then ##\vec{a}=\frac{{r}^{2}\,w\,\left( \frac{d}{d\,r}\,w\right) +r\,{w}^{2}}{{r}^{2}\,{w}^{2}-1}## and solving the ODE ##\vec{a}=0## gives ##\omega(r)=M/r## where ##M>0## is a constant.
Obviously there must be a source now and sure enough the Ricci and Einstein tensors are not zero. The metric is transformed to
[tex]
g_{\mu\nu}=\begin{pmatrix}
-1 & 0 & 0 & -\frac{M}{r}\\
0 & 1 & 0 & 0\\
0 & 0 & {r}^{2} & 0\\
-\frac{M}{r} & 0 & 0 & -\frac{{M}^{2}-1}{{r}^{2}}
\end{pmatrix}
[/tex]
and clearly ##M<1## is a constraint.
The Einstein tensor in the local frame is
[tex]
E_{mn}=\begin{pmatrix}
-\frac{r\,{M}^{2}-4\,{M}^{2}+4\,r}{4\,{r}^{3}} & 0 & 0 & \frac{M\,\left( 3\,{M}^{2}-4\,r\right) }{4\,{r}^{3}}\\
0 & \frac{\left( M-2\right) \,\left( M+2\right) }{4\,{r}^{2}} & 0 & 0\\
0 & 0 & -\frac{{M}^{2}-8}{4} & 0\\
\frac{M\,\left( 3\,{M}^{2}-4\,r\right) }{4\,{r}^{3}} & 0 & 0 & \frac{{M}^{2}\,\left( 3\,{M}^{2}-4\,r-3\right) }{4\,{r}^{4}}
\end{pmatrix}
[/tex]
I don't know what to make of this so any comments welcomed.https://en.wikipedia.org/wiki/Born_coordinates
Obviously there must be a source now and sure enough the Ricci and Einstein tensors are not zero. The metric is transformed to
[tex]
g_{\mu\nu}=\begin{pmatrix}
-1 & 0 & 0 & -\frac{M}{r}\\
0 & 1 & 0 & 0\\
0 & 0 & {r}^{2} & 0\\
-\frac{M}{r} & 0 & 0 & -\frac{{M}^{2}-1}{{r}^{2}}
\end{pmatrix}
[/tex]
and clearly ##M<1## is a constraint.
The Einstein tensor in the local frame is
[tex]
E_{mn}=\begin{pmatrix}
-\frac{r\,{M}^{2}-4\,{M}^{2}+4\,r}{4\,{r}^{3}} & 0 & 0 & \frac{M\,\left( 3\,{M}^{2}-4\,r\right) }{4\,{r}^{3}}\\
0 & \frac{\left( M-2\right) \,\left( M+2\right) }{4\,{r}^{2}} & 0 & 0\\
0 & 0 & -\frac{{M}^{2}-8}{4} & 0\\
\frac{M\,\left( 3\,{M}^{2}-4\,r\right) }{4\,{r}^{3}} & 0 & 0 & \frac{{M}^{2}\,\left( 3\,{M}^{2}-4\,r-3\right) }{4\,{r}^{4}}
\end{pmatrix}
[/tex]
I don't know what to make of this so any comments welcomed.https://en.wikipedia.org/wiki/Born_coordinates