Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: saspinski on 28/10/2017 19:54:00

In the so called vacuum solutions, the Einstein tensor G^{μν} = R^{μν}  1/2*g^{μν}*R = 0. I saw a proof that the former equation implies in R = 0 and R^{μν} = 0. It follows from multiplying by g_{μν}, given R  1/2*4*R = 0.
But when deriving the Schwarzschild solution, after calculating the Ricci tensor (R^{μν}) and the Ricci scalar (R), from the spherical symmetries of the problem, both are non zero. And the equation R^{μν}  1/2*g^{μν}*R = 0 is used anyway to derive 4 differential equations that eventually result in the Schwarzschild metric.
How is it possible that the field equation are zero while R^{μν} and R are non zero?

But when deriving the Schwarzschild solution, after calculating the Ricci tensor (R^{μν}) and the Ricci scalar (R), from the spherical symmetries of the problem, both are non zero.
If so then that's news to me. Can you find a source on the internet which shows that to be true? Clearly from Einstein's equations in vacuo R = 0 so I can't see how it can not be zero.
How is it possible that the field equation are zero while R^{μν} and R are non zero?
They can't be. It seems to me that there has to be problem with your assertion above where you state that the Ricci scalar and tensor as zero.
Note: caution should be used when using the "/" operator in textual form. The expression 1/ab is defined as 1/(ab). I say this because you wrote 1/2*4 where you meant to write (1/2)*4. It really means 1/(2*4).

The web site is: https://web.stanford.edu/~oas/SI/SRGR/notes/SchwarzschildSolution.pdf
It is not written there really that the Ricci tensor and the Ricci scalar are different from zero. But, after getting an expression for R as a function of U, V, and its derivatives, (g_{00} = U and g_{11} = V); instead of using for example R_{00} = 0 and R = 0 separately, the equation used was: R_{00}  (1/2)*g_{00}*R = 0, suggesting (for me) that each term was not zero.
But after your answer, I tested the obtained value of U, V and its derivatives on R_{00} and it is really zero.
So, I think that R_{00}  (1/2)*g_{00}*R = 0 was used because the expression was greatly simplified, leading to a differential equation only for V and V', that could be easily integrated.