UPDATE November 1, 2021: Of course, the Ricci curvature tensor is coordinate dependent. A tensor is something that you have to transform to new coordinates!
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Christoffel symbols are cumbersome to calculate. Let us take the definition of the Einstein tensor from literature for certain metrics.
The Oppenheimer and Snyder 1939 paper contains the Einstein tensor which is calculated for the metric above. The component T₁⁴ contains the time derivative with respect to the coordinate time of the radial metric λ. It is marked with the dot on top of λ. If we define a new time coordinate
T = 2 t,
then the derivative against T has a different value than against t, at the same spacetime point. The value of the Einstein tensor changes through a simple rescaling of time.
Sean Carroll has calculated the Ricci tensor for the FLRW metric given above. Let us define a new radial coordinate
R = 2 r.
Then dr = 1/2 dR and r = 1/2 R. The metric, written in terms of the R coordinate is (we take k = 0):
ds² = -dt² + a²(t) / 4 [ dR² + R² (dθ² + sin²θ dφ²) ]
We see that the metric written in terms of the R coordinate has to be
A(t) = a(t) / 2.
The Ricci R₁₁ component with the R coordinate then has a value which is only 1/4 of the value it had with the r coordinate.
A solution for the Einstein equations is not coordinate independent
If we had a solution for the FLRW metric, we showed above how to break the solution: simply rescale the radial coordinate r.
The coordinate dependency of the Einstein equations is much worse than for newtonian mechanics. One cannot break a solution of newtonian mechanics by defining new coordinates.
Browsing the internet, one cannot find anyone explicitly claiming that the Ricci curvature tensor is coordinate independent. However, it is easy to get such an impression, since Riemann curvature is sometimes defined in a coordinate free way, with a parallel transport of a vector.
What coordinates we can choose in order to satisfy the Einstein equations?
Karl Schwarzschild chose the standard Minkowski coordinates where a distant observer defines the time coordinate t, and the radial coordinate is determined by the circumference of a circle as measured by observers. He was able to solve the Einstein equations - both externally and also inside the uniform spherical mass.
Question. If one tries to use proper time as the time coordinate, can one find a solution for the metric of a spherical uniform mass?
The time inside the spherical mass lags constantly. The time coordinate lines become more and more bent. Can one solve the Einstein equations then?
A week ago we were suspecting that there might be an error in the M. W. Cook (1975) equations because distorting the time coordinate brought a new cross term which breaks the Friedmann equations. Now we understand that the problem is in the coordinate dependency of solutions for the Einstein equations.
We will investigate what consequences does the coordinate dependency have.
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