- #1
FunkyDwarf
- 489
- 0
Hello,
I am curious as to how one appropriately matches an interior and exterior solution in GR, i.e. where the interior corresponds to the field of some finite spherical mass (perfect fluid sphere, for the Schwarzschild interior solution). Specifically, looking at both the Schwarzschild interior, and the metrics given here
http://iopscience.iop.org/article/10.1088/0305-4470/10/4/017/meta
http://www.jstor.org/stable/78530?seq=1#page_scan_tab_contents
it seems that not only is the dt^2 coefficient continuous at the boundary of the object (r=R), but it is also differentiable (in the schwarzschild coordinates, i think?). This is in contrast to the dr^2 coefficient.
Is there some deeper requirement that forces the dt^2 to be smooth and continuous whereas the dr^2 coefficient does not? How to Israel's junction conditions translate into conditions on simple metrics? That is to say, if you know that you are matching an interior to the vacuum solution, what can you say/conditions can you impose on the value of the dr^2 and dt^2 interior coefficients (and their derivatives) at the boundary?
Thanks!
-FD
I am curious as to how one appropriately matches an interior and exterior solution in GR, i.e. where the interior corresponds to the field of some finite spherical mass (perfect fluid sphere, for the Schwarzschild interior solution). Specifically, looking at both the Schwarzschild interior, and the metrics given here
http://iopscience.iop.org/article/10.1088/0305-4470/10/4/017/meta
http://www.jstor.org/stable/78530?seq=1#page_scan_tab_contents
it seems that not only is the dt^2 coefficient continuous at the boundary of the object (r=R), but it is also differentiable (in the schwarzschild coordinates, i think?). This is in contrast to the dr^2 coefficient.
Is there some deeper requirement that forces the dt^2 to be smooth and continuous whereas the dr^2 coefficient does not? How to Israel's junction conditions translate into conditions on simple metrics? That is to say, if you know that you are matching an interior to the vacuum solution, what can you say/conditions can you impose on the value of the dr^2 and dt^2 interior coefficients (and their derivatives) at the boundary?
Thanks!
-FD