UPDATE November 1, 2021: We do not need to make Ricci curvature coordinate independent. A tensor - by definition - is coordinate dependent!
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On the other hand, Gauss curvature of a manifold is usually defined by the observers living inside the manifold measuring the circumference and the radius of a circle. That definition is, of course, coordinate independent.
Wikipedia gives the formula for Ricci curvature for new coordinates defined with a mapping:
y: ℝ⁴ → U,
where U is an open subset of ℝ⁴ and (U, g) is the manifold (M, g).
J(q) is the matrix of the first derivatives of y at the point q ∊ ℝ⁴. That is, J(q) is the Jacobian matrix. In the formula, the matrix on the right side is multiplied by J(q) and the transpose of J(q). The Ricci curvature matrix on the left side is not the same as on the right side. The value of Ricci curvature changes in the coordinate transformation.
Can we make the Einstein Ricci curvature coordinate independent?
How to make the Einstein formula for Ricci curvature coordinate independent?
It is not enough to take the derivatives against the proper distance and proper time. In the M. W. Cook case we had the derivatives against proper time, but the rogue cross term still appeared.
What about using proper time and the proper distance as coordinates? If the surface is curved, one cannot map the measured distances to a euclidean plane. It is not possible to use such coordinates.
Let us define a function:
s(p, q) = the proper distance of points p ∊ U and q ∊ U.
Ricci curvature should be calculated from the function s. Then it would be coordinate independent since it would be something that observers living inside the manifold measure themselves.
We may introduce polar coordinates around each spacetime point p. Curvature would mean that tangential distances are surprisingly short or long. The radial coordinate would be the proper distance. We will calculate what this is in terms of the metric.
Curvature in polar coordinates is like fitting a sphere to the curvature, where proper distances are the framework. We need to look at theorema egregium by Carl Gauss (1827).
Let us have a 2D surface. Let us assume that the metric is
g_x(x, y) 0
0 g_y(x, y)
Furthermore, we assume that
g_x(0, 0) = g_y(0, 0) = 1
and the first derivatives of g_x and g_y are zero at (0, 0).
y
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----------------------------------------> x
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We want to calculate the proper circumference of a square
(s, s) -- (s, -s) -- (-s, -s) -- (-s, s)
around the origin, where s > 0 is a small real number.
To get an estimate for the proper length of the upper line (-s, s) -- (s, s) of the square we need the second derivative
D_yy g_x.
To estimate the proper length of the "radius" (0, 0) - (0, s) we need the second derivative
D_yy g_y.
If there is some distortion in the proper circumference of the square relative to its proper "radius", then the following simple formula calculates it:
D_yy g_x - D_yy g_y + D_xx g_y - D_xx g_x. (1)
If g_x(0, 0) is arbitrary, we need to "normalize" it to 1 and its first derivatives to 0:
G_x(x, y) = [ g_x(x, y) - x D_x g_x(0, 0) - y D_y g_x(0, 0) ]
/ g_x(0, 0).
Furthermore, we need to normalize the second derivatives D_yy and D_xx.
Introduce a preferred coordinate system?
Another option is to take a preferred coordinate system. The obvious candidate is an inertial frame in the Minkowski space which is asymptotic around a mass system. The Schwarzschild external and internal solutions seem to be sensible. Their coordinate system fixes the time to the proper time of a distant observer. The radial coordinate r is defined as the circumference measured by observers divided by 2 π.
In the waterfall analogy of a black hole, the coordinates of the river banks are the natural ones that we should use. Observers swimming in the water measure the geometry with sound signals. Their view is very different from someone standing on the river bank. Using the syrup model of gravity, the water is the syrup which is flowing and taking all frames of internal observers along with it.
A rubber membrane model of gravity is naturally coordinate independent
If we treat spacetime as rubber which is bent and stretched, we will have a coordinate independent theory. The problem with the Einstein equations may be that they do not allow pressure to change the spatial metric.
The name of the stress-energy tensor suggests that it is rubber which is being stretched.
In rubber, it is not curvature itself, but the stretching of the material which takes energy. The Einstein tensor, in some sense, describes the "force" with which spacetime resists being curved. In a rubber model, the force should be calculated by varying the metric and determining the energy it takes to vary the metric.
Calculate the Ricci curvature in the coordinates which minimize it?
Another option to remove coordinate dependency is to demand that the Ricci curvature must be calculated in global coordinates which, in some sense, minimize the curvature.
This would make calculations very difficult. Besides optimizing the metric, we should also optimize the coordinate system.
The covariant derivative
Around the year 1900 Gregorio Ricci-Curbastro and Tullio Levi-Civita introduced the concept of a covariant derivative.
Wikipedia states:
"This new derivative was covariant in the sense that it satisfied Riemann's requirement that objects in geometry should be independent of their description in a particular coordinate system."
But when we look at the "Informal definition", the formula of the derivative is:
The derivative is against xⁱ, which is a coordinate. The derivative is not against an infinitesimal proper length on the manifold. The derivative does depend on the coordinate system. It is not "intrinsic" in the manifold.
The value of the derivative is easy to transform to new coordinates, though.
Conclusions
The fact that the Einstein equations are coordinate dependent suggests that there might be something wrong with them. In a reasonable physical theory the choice of coordinates has no effect whatsoever.
For a rubber membrane model we have:
1. preferred coordinates: the Minkowski coordinates of the surrounding space;
2. naturally coordinate independent formulation.
We will look at making the Einstein equations coordinate independent.
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