UPDATE October 29, 2021: We updated the text to reflect the fact that the theorem in the October 16, 2021 blog post is erroneous.
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We got an astonishing result: by rescaling the time coordinate t and the metric of time b(t) we can always add a new term to the Friedmann equations.
The Einstein tensor is made from the Riemann curvature tensor, and the basic idea of the Riemann tensor is that it is coordinate independent. Simply rescaling the time coordinate should have no effect.
One can derive the Friedmann equations from symmetry arguments in a uniform universe. A solution of Einstein field equations must have a uniform metric of space.
Similarly, we can argue that a solution must have a uniform metric of time.
Assume that we have found such a uniform solution for the Einstein equations, and the global coordinate time is T. Then we can replace our global time coordinate T with the proper time coordinate t of an observer in the universe. We get a solution which has the Friedmann metric:
Because of coordinate independence, the solution with the new time coordinate t is a solution of the Einstein equations. It also a solution of the Friedmann equations.
From where does the cross term originate?
Alan R. Parry (2014) has written in the link a very detailed derivation of the Ricci tensor for a varying metric of time.
The cross term comes from the first product term of two Christoffel symbols. It is the term V_t * M_t on the right. V represents the metric of time and M is the spatial metric.
Suppose that we take a history of the universe, and replace the global time coordinate t with
t' = t for t < 0,
t' = 2 t for t >= 0.
The metric b(t) of time suddenly drops to a half when t' = 0. Obviously the change of the time coordinate should not affect the Riemann tensor or the Ricci tensor.
However, if proper time slows down just in a part of the universe, that does introduce extra curvature.
The cross term probably should be something like
d² b(t, r) / (dt dr)
to reveal the curvature which comes from non-uniform rate of proper time in the universe. That is, instead of a product of partial derivatives with respect to t and r, we should have a second derivative.
The error probably is in the use of Christoffel symbols. Instead of a product there should be a second derivative.
Riemann curvature is defined through a parallel transport of a vector. Literature claims that the curvature can be calculated through Christoffel symbols using the formula above, but we did not find any proof for this claim.
Albert Einstein in his paper on the foundations of general relativity in 1916 writes that the curvature can be calculated with that formula from Christoffel symbols. He uses the curvy bracket notation {...} for the symbols.
Almost all known solutions for Einstein equations have the cross term zero, because the metric of time does not change in them. The cross term does appear in the Oppenheimer-Snyder 1939 paper, but it is not relevant there.
Hermann Weyl's 1918 book "Raum. Zeit. Materie." states that the Einstein tensor of general relativity is defined through the above formula of Christoffel symbols. It might be that Albert Einstein introduced the formula, and no one ever checked if it is coordinate independent.
Does the cross term say anything about the Riemann curvature?
If we have an arbitrary function
f(x, y),
we may imagine that it describes the height of each point on a hill. Then
d²f / dx², d²f / dy²
might measure "curvature", while
d²f / (dx dy), d²f / (dy dx)
might measure "torsion". On the other hand, the cross term
df / dx * df / dy
can be non-zero even if the surface is a plane.
If the hill is circularly symmetric, then
d²f / dr²
tells us the curvature. If that is zero everywhere but on the top, then the hill is a cone.
On a cone,
d²f / dx², d²f / dy²
are != 0.
Let us imagine observers living on the surface of a Riemannian manifold
(M, g).
The observers obey the metric g and measure the angles of triangles, whose sides have to be straight (geodesic). They deduce the curvature of space from the sum of angles, if it is 180 degrees or something else.
This setup is self-evidently coordinate independent. The metric determines what the observers see or measure.
Correcting the Christoffel symbol formula makes general relativity more sensible?
The correct formula for Riemann or Ricci curvature has to be coordinate independent. We will try to correct the Christoffel symbol formula.
A basic idea of general relativity is that mass focuses straight lines drawn to the direction of time, and pressure focuses straight lines drawn to spatial directions. How can we focus lines to spatial directions without stretching the spatial metric? It might be that a corrected Christoffel symbol formula changes general relativity to a "rubber membrane" model where pressure does affect the spatial metric. Then there might exist a solution for a sudden pressure change.
In this blog we have been perplexed by the interior Schwarzschild metric, where pressure does not affect the spatial metric at all. That is very different from a rubber membrane model. We tried to design a perpetuum mobile which utilizes the infinite rigidity of space. If the corrected formula removes this strange feature, then general relativity looks better.
Friedmann equations might have solutions after we have corrected the Christoffel symbols. If pressure can affect the spatial metric, then there is more freedom, and a solution may exist. It is like a rubber membrane which can adapt to all kinds of stresses.
Wikipedia says that the Ricci tensor is defined in the (unique?) way, such that calculating the Ricci tensor for new coordinates defined by a mapping y, is surprisingly easy, requiring only the Jacobian matrix of first derivatives for the mapping y.
However, this is not what coordinate independence should be. It should be that observers living on the manifold, under the metric, measure triangles, and calculate the curvature. The curvature should not depend on an arbitrary choice of coordinates y at all. It is not enough that the Ricci curvature is easy to calculate for the new coordinates y.
The obvious way to get coordinate independence is to use locally proper distances and proper time at a given point p as the framework. Essentially, it is a local observer on the manifold who measures the curvature. Then the choice of coordinates has no effect whatsoever on the calculated curvature.
If proper time and proper distances are approximately the same as coordinate time and coordinate distances, then the Ricci curvature calculated from the coordinates is approximately the same as the Ricci curvature measured by a local observer.
Unfortunately, using proper distances and times as the framework does not work in a cosmological FLRW model where k = 0. In such a universe all observers measure that curvature is zero, which makes it impossible to satisfy the Einstein field equations.
It may be that we need to introduce a preferred frame where curvature is measured from global coordinates. This is in the spirit of our Minkowski & newtonian model, where inertial Minkowski coordinates are the preferred frame.
Conclusions
Moving to a preferred frame would prevent us from changing coordinates in a way where the rogue cross term can attain arbitrary values.
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