Our analysis on May 21, 2024 about "rogue" variations in the Einstein-Hilbert action finally uncovered what is the problem in the paper.
The authors define the metric of time as identically -1 during the collapse. In this text we think in terms of the metric signature (- + + +), while the authors use (+ - - -).
But time, as measured by an observer far away, progresses slower at the center of the star. Setting the metric of time a constant -1 makes the time coordinate unnatural: a particle can travel backward in the time coordinate if it approaches the center of the star. Clocks at the center show a significantly earlier time than clocks at the surface.
When we work with the matter lagrangian LM in the Einstein-Hilbert action, we subconsciously assume that the coordinate time always progresses for a particle. If a particle can zigzag in the time coordinate, its mass will contribute in a surprising way to the integral of LM in the Einstein-Hilbert action. The particle can appear as many particles at certain coordinate times.
Thus, variations of the particle path in LM must include paths which travel backward in the coordinate time.
A "very rogue" metric variation
In our May 21, 2024 blog post we defined a rogue variation of a metric: a variation which keeps the spacetime geometry constant, but moves coordinate lines spatially relative to the spacetime geometry (or relative to the old coordinate lines if we interpret the old coordinate lines fixed to the spacetime geometry).
Such variations have surprising results.
Let us define a very rogue metric variation: it moves coordinate lines in such a way that a particle moves back in the coordinate time of the old coordinate lines. The particle will zigzag in the time coordinate of the old coordinates.
The collapse in ordinary static coordinates: a rogue variation
Let us have a spherical uniform cloud of dust (pressure = zero) at the start of the collapse.
• •
• • • •
• • • • ---> move coordinates
• • m (t, x, y, z)
M
Let us assume that we have a solution g for the Einstein field equations for the collapse, for a coordinate time interval T.
We assume that a particle m in the cloud at a coordinate time t is at certain spatial coordinates x, y, z.
Let us keep the overall spacetime geometry intact, but vary the metric g in such a way that the spatial location x, y, z moves to a larger radial distance r at the time t. The variation takes the particle m along with it.
Another variation will bring m back to its original path at a later coordinate time.
Since we never changed anything in the original spacetime geometry, the Einstein-Hilbert action integral S of the Ricci scalar R did not change at all.
But the mass m was at a higher gravity potential for some coordinate time: the value of the volume element sqrt(-det(g)) was larger for it, because the metric of time was closer to -1 at a larger r.
Therefore the action integral S of LM is smaller than it originally was.
If we temporarily move m to a smaller radius r, we can make the action integral S larger.
We conclude that the collapsing dust cloud has no stationary point in the Einstein-Hilbert action.
We still have to analyze if this proves that the Einstein equations cannot have a solution for a collapsing dust cloud. Probably it does.
The collapse in the Tolman (1934) comoving coordinates: a very rogue variation
The comoving coordinates used by Oppenheimer and Snyder in 1939 are from a 1934 paper by Richard C. Tolman.
The coordinates co-move which each particle of dust. The particles stay at a fixed space coordinates, but their spatial metric distance shrinks as the the coordinate time progresses.
The metric of time is a fixed -1.
The rogue variation of the preceding section refuted any attempt to find a stationary point for the Einstein-Hilbert action. But our analysis required that the metric of time is closer to -1 at a larger r. In the Tolman coordinates, the metric is a constant -1. Is it so that the analysis is not correct in the new coordinates?
In the Tolman coordinates, we can make the particle m to go backward in the coordinate time.
^
|
| __
| / /
|/ /
/
|
|
• m particle
^ t (coordinate)
|
-----> r
The path can zigzag like above. Then the contribution of the mass m makes LM smaller than originally, since the particle m is counted as three particles for some time. The action integral S is smaller.
The problem in the paper of Oppenheimer and Snyder seems to be the following: if a particle m can zigzag in the time coordinate, then the variation of LM against changes in the metric of time, g₀₀, becomes complicated.
A small change in the metric of g₀₀ can make three copies of the particle to appear, if we use the old coordinates in the calculation of the S integral of LM.
But Oppenheimer and Snyder assume that the variation of the S integral of LM can be calculated from the simple form of the stress-energy tensor T:
ρ 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0.
That simple form cannot handle the appearance of several copies of m.
Our result probably means that there is no solution for the Einstein field equations of the dust cloud, even if we use the Tolman coordinates.
The problems stem from the missing kinetic energy when the metric is stretched
In a lagrangian and action of standard newtonian mechanics, rogue paths like above are not a problem because the kinetic energy of the particle increases if it tries to deviate from its calculated path.
In general relativity, one can make a particle to move by manipulating the metric.
We could maybe fix the problem in general relativity if we had canonical coordinates against which kinetic energy is measured. But the spirit of general relativity is that such canonical coordinates do not exist.
In our own Minkowski & newtonian gravity model, standard Minkowski coordinates are the canonical coordinates.
The big picture: an action integral S must be able to determine the kinetic energy, in order to work
The lagrangian density, and the action, of newtonian mechanics:
kinetic energy - potential energy
works correctly. But general relativity has made it difficult, or impossible, to determine the kinetic energy. That is one of the reasons why the Einstein-Hilbert action cannot work for dynamic systems. The other reason is non-conservation of pressure.
The Oppenheimer-Snyder collapse might be the only known dynamic solution of the Einstein field equations in an asymptotic Minkowski space. Our analysis suggests that it is incorrect. In our blog we have speculated that the Einstein equations have no dynamic solutions at all. This result provides more evidence that the hypothesis is true.
The Vaidya metric
What about the Vaidya metric where a shell of light expands around a star?
A rogue variation of the metric can move the shell of light farther from the star without changing the overall spacetime geometry. A photon rises to a higher gravity potential, which makes the integral of LM smaller. The action integral S grows smaller. This suggests that the Vaidya metric is incorrect. We have to check it.
The photon moves "faster than light" in the rogue variation. Superluminal movement is not prohibited in general relativity. A change in the spatial metric can cause distances to grow faster than light.
Cosmological models
If a cosmological model contains variations in mass density, and these variations are not supported by a pressure, then a rogue metric variation will change the value of the Einstein-Hilbert action S.
Let us have an expanding universe whose mass density is uniform. The metric of time is the same at every location. A rogue metric variation then cannot change the value of the Einstein-Hilbert action S.
"annihilation"
/ \
/ \
m₁ m₂
^ t
|
------> x
What about very rogue variations? We can ban metric variations where an observer could see two particles m₁ and m₂ to "annihilate" each other, without emitting any energy. Then a particle cannot turn back in time. Neither can it turn back in the time coordinate of the expanding universe.
Above we wrote that one could maybe correct the Einstein-Hilbert action by introducing canonical coordinates, against which the kinetic energy of a particle is measured. In the case of a cosmological model, defining these canonical coordinates is challenging. There is no asymptotic Minkowski metric far away, into which we could fix the coordinates. Also, an expanding universe can expand "faster than light", in which case it is difficult to define canonical coordinates.
Conclusions
In this blog we have speculated that the Einstein field equations do not have a solution for any dynamic system.
A counterexample to that claim was the Oppenheimer-Snyder collapse (1939). Our analysis above suggests that their solution is erroneous. The reason is that in the Tolman coordinates, particles can move backward in the time coordinate. Then varying LM is complicated, but Oppenheimer and Snyder assumed that it is straightforward.
Our analysis about rogue variations suggest that the Einstein equations cannot have a solution for any dynamic system, with the exception of a totally uniform expanding FLRW universe. Such a universe is unrealistic.
We have to check very carefully how the Einstein equations handle a rogue variation. Above we assumed that they handle it correctly.
It has been an open problem if the Einstein equations have a solution for a planet orbiting a star. Our analysis answers this question: there is no solution.
The fundamental reason for the failure of the Einstein equations is that general relativity does not contain "canonical coordinates", or an "absolute spacetime", against which we could measure the kinetic energy. The problem does not appear in newtonian mechanics or in Minkowski space, because they are built on canonical coordinates.
Our own Minkowski & newtonian gravity model probably handles this without problems because we use Minkowski space.
Our observation provides further evidence for the claim that a spatially finite, expanding or contracting, universe is not a reasonable physical model. That is because defining canonical coordinates is problematic in it.
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