#### Mentz114

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- 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.

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.

### Born coordinates - Wikipedia

en.wikipedia.org