- #1
Mentz114
- 5,432
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This thread is spawned from an earlier one
https://www.physicsforums.com/showthread.php?t=647147&page=7
For the stationary ( ie comoving ) frame in the Schwarzschild spacetime the co-basis of the frame field is
[tex]
s_0= \sqrt{\frac{r-2m}{r}}dt,\ \ s_1=\sqrt{\frac{r}{r-2m}}\ dr,\ \ s_2=r\ d\theta,\ \ s_3=r\sin(\theta)\ d\phi
[/tex]
The 4-velocity of the stationary observer in the local frame is [itex]u^\mu=\partial_t[/itex] (or [itex](1,0,0,0)[/itex]).
This, [itex]u^\nu\nabla_\mu u_\nu[/itex], which looks like a covariant vector is
[tex]
a_\mu= \frac{m}{{r}^{\frac{3}{2}}\,\sqrt{r-2\,m}} \ dr
[/tex]
All this is well known. Now we boost the co-basis covectors in the r-direction with velocity [itex]-\sqrt{2m/r}[/itex]. The metric is unchanged in form by this. This gives a new basis h
[tex]
h_0= -dt - \frac{\sqrt{2mr}}{r-2m}\ dr,\ \ h_1=\sqrt{\frac{2m}{r}}\ dt + \frac{r}{r-2m} \ dr,\ \ h_2=s_2,\ \ h_3=s_3
[/tex]
According to my calculations, in this basis the acceleration aµ is now zero. Readers are bound to be sceptical about my calculations but I'm very confident there's been no gross error. If it is an error I'll be happy to have found it.
https://www.physicsforums.com/showthread.php?t=647147&page=7
For the stationary ( ie comoving ) frame in the Schwarzschild spacetime the co-basis of the frame field is
[tex]
s_0= \sqrt{\frac{r-2m}{r}}dt,\ \ s_1=\sqrt{\frac{r}{r-2m}}\ dr,\ \ s_2=r\ d\theta,\ \ s_3=r\sin(\theta)\ d\phi
[/tex]
The 4-velocity of the stationary observer in the local frame is [itex]u^\mu=\partial_t[/itex] (or [itex](1,0,0,0)[/itex]).
This, [itex]u^\nu\nabla_\mu u_\nu[/itex], which looks like a covariant vector is
[tex]
a_\mu= \frac{m}{{r}^{\frac{3}{2}}\,\sqrt{r-2\,m}} \ dr
[/tex]
All this is well known. Now we boost the co-basis covectors in the r-direction with velocity [itex]-\sqrt{2m/r}[/itex]. The metric is unchanged in form by this. This gives a new basis h
[tex]
h_0= -dt - \frac{\sqrt{2mr}}{r-2m}\ dr,\ \ h_1=\sqrt{\frac{2m}{r}}\ dt + \frac{r}{r-2m} \ dr,\ \ h_2=s_2,\ \ h_3=s_3
[/tex]
According to my calculations, in this basis the acceleration aµ is now zero. Readers are bound to be sceptical about my calculations but I'm very confident there's been no gross error. If it is an error I'll be happy to have found it.