- #1
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- TL;DR
- Seeking feedback on whether and how Novikov coordinates on Schwarzschild spacetime can be reworked to use the areal radius at ##\tau = 0## directly, and whether the cycloidal time coordinate ##\eta## can be used instead of ##\tau##.
The general intention of Novikov coordinates on Schwarzschild spacetime is to construct a "comoving" coordinate chart for purely radial timelike geodesics, i.e., every such geodesic should have a constant radial coordinate, and the time coordinate should be the same as proper time for observers following the geodesics. The straightforward way to assign the radial coordinate is to look at the "maximum expansion" surface (the surface at which all of the geodesics are momentarily "at rest", i.e., orthogonal to the surface of constant proper time ##\tau##), standardly labeled ##\tau = 0##, and which corresponds to the ##T = 0## hypersurface in Kruskal coordinates (i.e., the "x-axis"). Each radial timelike geodesic intersects this surface at a unique point (we are ignoring the angular coordinates, so "point" here really means "2-sphere" in the full spacetime), which has an areal radius ##R## associated with it.
In the usual form of Novikov coordinates that I have seen, the radial coordinate is not ##R## directly, but is a rescaled coordinate that MTW calls ##R^*## (what I am calling ##R## here, MTW calls ##r_{\text{max}}##, defined as follows:
$$
R^* = \sqrt{ \frac{R}{2M} - 1 }
$$
This fixes the range of ##R^*## to ##0 \le R^* \lt \infty## (whereas ##R## has a minimum at ##2M## on the hypersurface in question). When working with the maximally extended Schwarzschild spacetime, this makes sense. However, it also makes the ##g_{R^* R^*}## term in the metric look quite complicated.
That leads to my first question: I have tried to rework these coordinates to use ##R## (i.e., the areal radius of each radial timelike geodesic at ##\tau = 0##) directly (because I want to use them in the exterior region of the Oppenheimer-Snyder collapse spacetime, for which it will make it easier to match to the coordinates of the interior matter region at the boundary). I would like any feedback that others can give on whether the following transformation of the metric is correct, or any references to existing treatments along these lines in the literature.
The metric in the usual Novikov coordinates, using ##R^*##, is:
$$
ds^2 = - d\tau^2 + \frac{{R^*}^2 + 1}{{R^*}^2} \left( \frac{\partial r}{\partial R^*} \right)^2 d{R^*}^2 + r^2 d\Omega^2
$$
Here ##r## is the areal radius at the given event (i.e., not ##R##, the areal radius at ##\tau = 0##, but the actual areal radius on the geodesic labeled by ##R^*## as a function of ##\tau## and ##R^*##) and ##d \Omega^2## is the standard metric on a unit 2-sphere in the usual angular coordinates.
Using the equation for ##R^*## in terms of ##R## above, I obtain:
$$
R = 2M \left( {R^*}^2 + 1 \right)
$$
$$
\frac{\partial r}{\partial R^*} = \frac{\partial r}{\partial R} \frac{dR}{dR^*} = \frac{\partial r}{\partial R} 4 M R^* = \frac{\partial r}{\partial R} 4 M \sqrt{ \frac{R}{2M} - 1 }
$$
$$
dR^* = \frac{1}{4 M \sqrt{\frac{R}{2M} - 1}} dR = \frac{1}{\frac{dR}{dR^*}} dR
$$
Substituting these into the metric gives
$$
ds^2 = - d\tau^2 + \frac{1}{1 - \frac{2M}{R}} \left( \frac{\partial r}{\partial R} \right)^2 dR^2 + r^2 d\Omega^2
$$
where ##r## is now a function of ##R## and ##\tau##.
This form of the metric is interesting because the first factor in ##g_{RR}## now looks very similar to the factor in Schwarzschild coordinates, but with ##R## instead of ##r##--i.e., that factor is constant along every radial timelike geodesic.
That observation leads to my second question: we know that we can define a cycloidal time coordinate ##\eta## such that, for every radial timelike geodesic labeled by ##R## (its areal radius at "maximum expansion"), we have
$$
r = \frac{1}{2} R \left( 1 + \cos \eta \right)
$$
$$
\tau = \frac{1}{2} \sqrt{\frac{R^3}{2M}} \left( \eta + \sin \eta \right)
$$
I note that the above equations imply
$$
\frac{\partial r}{\partial R} = \sqrt{\frac{2M}{R^3}} \frac{\partial \tau}{\partial \eta}
$$
This would seem to indicate that we should be able to rewrite the metric above to use ##\eta## as the time coordinate instead of ##\tau##. However, I have not been able to find any treatment along these lines. Does anyone know of a reference where this is done?
In the usual form of Novikov coordinates that I have seen, the radial coordinate is not ##R## directly, but is a rescaled coordinate that MTW calls ##R^*## (what I am calling ##R## here, MTW calls ##r_{\text{max}}##, defined as follows:
$$
R^* = \sqrt{ \frac{R}{2M} - 1 }
$$
This fixes the range of ##R^*## to ##0 \le R^* \lt \infty## (whereas ##R## has a minimum at ##2M## on the hypersurface in question). When working with the maximally extended Schwarzschild spacetime, this makes sense. However, it also makes the ##g_{R^* R^*}## term in the metric look quite complicated.
That leads to my first question: I have tried to rework these coordinates to use ##R## (i.e., the areal radius of each radial timelike geodesic at ##\tau = 0##) directly (because I want to use them in the exterior region of the Oppenheimer-Snyder collapse spacetime, for which it will make it easier to match to the coordinates of the interior matter region at the boundary). I would like any feedback that others can give on whether the following transformation of the metric is correct, or any references to existing treatments along these lines in the literature.
The metric in the usual Novikov coordinates, using ##R^*##, is:
$$
ds^2 = - d\tau^2 + \frac{{R^*}^2 + 1}{{R^*}^2} \left( \frac{\partial r}{\partial R^*} \right)^2 d{R^*}^2 + r^2 d\Omega^2
$$
Here ##r## is the areal radius at the given event (i.e., not ##R##, the areal radius at ##\tau = 0##, but the actual areal radius on the geodesic labeled by ##R^*## as a function of ##\tau## and ##R^*##) and ##d \Omega^2## is the standard metric on a unit 2-sphere in the usual angular coordinates.
Using the equation for ##R^*## in terms of ##R## above, I obtain:
$$
R = 2M \left( {R^*}^2 + 1 \right)
$$
$$
\frac{\partial r}{\partial R^*} = \frac{\partial r}{\partial R} \frac{dR}{dR^*} = \frac{\partial r}{\partial R} 4 M R^* = \frac{\partial r}{\partial R} 4 M \sqrt{ \frac{R}{2M} - 1 }
$$
$$
dR^* = \frac{1}{4 M \sqrt{\frac{R}{2M} - 1}} dR = \frac{1}{\frac{dR}{dR^*}} dR
$$
Substituting these into the metric gives
$$
ds^2 = - d\tau^2 + \frac{1}{1 - \frac{2M}{R}} \left( \frac{\partial r}{\partial R} \right)^2 dR^2 + r^2 d\Omega^2
$$
where ##r## is now a function of ##R## and ##\tau##.
This form of the metric is interesting because the first factor in ##g_{RR}## now looks very similar to the factor in Schwarzschild coordinates, but with ##R## instead of ##r##--i.e., that factor is constant along every radial timelike geodesic.
That observation leads to my second question: we know that we can define a cycloidal time coordinate ##\eta## such that, for every radial timelike geodesic labeled by ##R## (its areal radius at "maximum expansion"), we have
$$
r = \frac{1}{2} R \left( 1 + \cos \eta \right)
$$
$$
\tau = \frac{1}{2} \sqrt{\frac{R^3}{2M}} \left( \eta + \sin \eta \right)
$$
I note that the above equations imply
$$
\frac{\partial r}{\partial R} = \sqrt{\frac{2M}{R^3}} \frac{\partial \tau}{\partial \eta}
$$
This would seem to indicate that we should be able to rewrite the metric above to use ##\eta## as the time coordinate instead of ##\tau##. However, I have not been able to find any treatment along these lines. Does anyone know of a reference where this is done?
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