- #1
Bishal Banjara
- 90
- 3
- TL;DR Summary
- I want to know if the metrics in Schwarzschild is inversed, then Einstein's tensor components get modified or not.
Assuming the line element ##ds^s=e^{2\alpha}dt^2-e^{2\beta}dr^2-r^2{d\Omega}^2 ##as usual into the form ##ds^s=e^{-2\alpha}dt^2-e^{-2\beta}dr^2-r^2{d\Omega}^2##, I found that the ##G_{tt}## tensor component of first expression do not reconcile with the second one though, it fits for ##G_{rr} ##component. I followed the way as it is guided in https://web.stanford.edu/~oas/SI/SRGR/notes/SchwarzschildSolution.pdf for all calculations of Christoffels, Ricci tensor components and Einstein's tensor components. But I applied the final result for both exterior and an interior solution treating full Einstein equation following Carroll's book at pp.231. I obtained ##G_{tt}=\frac{1}{r^2}e^{2\beta-2\alpha}[2r\delta_r\beta+1-e^{-2\beta}]## instead of ##G_{tt}=\frac{1}{r^2}e^{-2\beta+2\alpha}[2r\delta_r\beta-1+e^{2\beta}].##I want to know whether I am doing wrong though metric is independent to the final result or it doesn't reconcile, naturally.
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