Let us continue our analysis of the assumption that a clock cannot "know" faster than light how it should tick. It can only receive information of the gravity potential at the local speed of light.
What empirical evidence do we have for the correctness of the Einstein field equations?
We know that the Schwarzschild metric describes gravity phenomena very accurately within the Solar system.
We know that binary pulsars orbit in the way predicted by the Schwarzschild metrics of the components. Also, the power of the produced gravitational waves match calculations where linearized Einstein equations are used, to an accuracy < 1%.
Gravitational waves observed by LIGO match numerical calculations made using numerical programs. However, we have not checked the heuristics used in the numerical programs.
On June 20, 2024 we showed that the Einstein fields equations do not have a solution at all for a two-body system. How do LIGO numerical models handle the nonexistence of solutions? Also, the LIGO measured results have a large margin of error, something like 10%.
The Schwarzschild metric is a static system. Gravitational waves are produced by a quadrupole system. These are configurations which are quite different from a collapse or an expansion of a spherically symmetric dust ball.
The only empirical data which we have about a large collapse or an expansion is what we know about the expansion of the observable universe.
We do know that a star can collapse into a neutron star or a black hole. But we do not have any detailed measured data about the process.
The only data which we have about a large expansion does not match the Einstein field equations. The equations do not predict dark energy, or the Hubble tension. They may also fail to predict the things seen by the James Webb telescope.
Since the Einstein equations fail for our (sparse) empirical data, there is a good chance that the equations describe a collapse or an expansion incorrectly.
The June 20, 2024 result about the nonexistence of "dynamic" solutions to the Einstein equations
The problem in that result seemed to be that general relativity does not possess "canonical coordinates" where we could determine kinetic energy in an unambiguous way. The corresponding problem does not occur in Minkowski space, where any inertial frame can be taken as the canonical coordinates.
Canonical coordinates require that there must be retardation in the rate of clocks?
Suppose that we can determine against a canonical time coordinate how fast a clock ticks. How does the clock know how fast it should tick? Empirically, we know that clocks tick slower in a low gravity potential. But how does the clock know that it is in a low potential?
If we can use canonical coordinates similar to Minkowski space, it is natural to assume that the information about the allowed clock rate can only spread at the speed of light. If the clock does not know that it has fallen into a lower gravity potential, then the clock maintains its rate.
How does general relativity handle this?
How does general relativity decide at which rate a clock should tick?
Empirically, we know that a clock in a static, low gravity potential ticks slower than far away in space. This has been demonstrated with atomic clocks on Earth, as well as in satellites. The phenomenon is also present in the redshift of light when it rises up from the surface of Earth. The redshift is approximately one billionth.
If we have a clock which ticks once per second, then in general relativity, the metric of time determines how many times it will tick in a second of coordinate time. There should be one tick in a second of proper time. The rate of the proper time is
sqrt(-g₀₀)
times the rate of the coordinate time.
General relativity does not allow the metric to change "faster than light"?
This is a question which we have touched several times in this blog. How fast can a change in the metric propagate in general relativity?
People often seem to assume that it cannot propagate faster than the local speed of light. Some changes in the metric can be detected by an observer. It would open a channel of faster-than-light communication, if changes in the metric can propagate faster than light.
Our example of a collapsing dust shell in the previous blog post seems to contradict this principle. People usually assume that the metric of time slows down instantaneously inside the shell, as the shell contracts.
# --> × <-- #
dust shell center dust shell
In the link, Ajay Mohan cites a book by E. Poisson. The metric inside a thin shell is assumed to be Minkowski.
However, the assumption may not be sound. Suppose that the shell is not exactly symmetric. Then the metric inside the shell should change in a complicated way as the shell contracts. An observer inside the shell can measure the metric close to him. The metric is not flat, but has a complicated form.
If we let the changes in the metric happen instantaneously inside the shell, that can open a faster-than-light communication channel.
If the shell is perfectly spherically symmetric, then slowing down the metric of time inside the shell instantaneously does not enable communication – but that is not a realistic physical configuration.
Maybe we should adopt the rule that the metric cannot change faster than light?
Then we encounter another problem. Inside a spherically symmetric collapsing shell there is no matter inside. Does that require that the metric inside is flat? If yes, then the metric of time will propagate faster than light inside the shell.
Can we find a curved metric inside the shell, such that its Ricci tensor is zero? The Schwarzschild solution is an example of a curved metric for which the Ricci tensor is zero.
Birkhoff's theorem may imply that the metric inside the shell must be flat.
The singularity theorems of Roger Penrose assume that an empty volume of space cannot focus or defocus a beam of light. But if the speed of light is faster at the center of the collapsing shell, then there is defocusing.
Hypothesis. A realistic collapsing shell does not have a solution in general relativity, such that the solution would not allow faster-than-light communication.
Conjecture. Any solution for a spherically symmetric collapsing shell in general relativity requires the metric of time to change faster than light.
Note that we already proved on June 20, 2024 that general relativity probably does not have a solution for any realistic dynamic problem at all. The hypothesis above is probably void. But it might be that any attempt to find a solution will also lead to faster-than-light communication.
In the case of the conjecture above, there may exist a solution where the thickness of the shell is the Dirac delta function. It is not physically realistic.
We may have uncovered yet another fundamental problem in general relativity: it would allow faster-than-light communication if it would have solutions at all!
What implications does faster-than-light communication have?
If we can change the undulating metric inside a shell instantaneously, that probably enables us to transfer energy faster than light. A faster-than-light energy transfer is forbidden in an energy condition.
The dominant energy condition demands that energy can never flow faster than light.
What would happen if we could send signals inside a shell faster than light? Then the physics inside the shell cannot be analogous to Minkowski space, because such signals cannot happen in Minkowski space. This would probably break an equivalence principle.
Can general relativity correctly handle the collapse of a dust ball?
On May 26, 2024 we showed that the Oppenheimer-Snyder 1939 solution is incorrect, since the comoving Tolman coordinates allow one to travel to an "earlier" time coordinate. But maybe there exists a correct solution to the problem?
The Einstein field equations are
where the cosmological constant Λ is zero and the stress-energy tensor is denoted by T. Let us use the standard Schwarzschild coordinates.
Zhang and Yi (2012) write about Birkhoff's theorem.
Willem van Oosterhuit (2019) gives Birkhoff's theorem in the following form:
Birkhoff's theorem. Any C² solution of the vacuum Einstein equations, which is spherically symmetric in an open set U, is locally isometric to the maximally extended Schwarzschild solution in U.
# • • • • #
shell S ball D shell S
We interpret the theorem in this way: let us have a collapsing spherically symmetric dust ball D and a spherically symmetric shell S enclosing D. Then the vacuum solution between S and D stays isometric (= isomorphic) to the Schwarzschild solution for a fixed mass M.
Whatever we do with S, the metric between S and D stays isometric (= isomorphic) to the Schwarzschild metric associated with a fixed M.
This means that D cannot "know" if we let S descend lower or not. The vacuum between S and D prevents any flow of information between S and D.
This implies that if we measure things with proper distances and proper time intervals, the collapse of D happens in the exact same way, regardless of what we do with S.
Let us then compare two histories:
- in history A, the shell S and D form one, almost uniform, dust ball, with an infinitesimal gap between them and we let them collapse freely;
- in history B, we use a tangential pressure within S to slow down its collapse; D collapses freely.
In history B, the metric of time, g₀₀, in the vacuum between S and D, will eventually differ from history A. The metric there will still be isomorphic to the fixed Schwarzschild metric M, but the absolute value of g₀₀ will be different in A and B.
The rate of clocks (i.e., g₀₀) in the vacuum below S depends on how low we let S descend. The gravity potential of a clock depends on how high S is.
The collapse of D happens in the exact same way, measured in proper times and proper lengths, regardless of how high S is. This implies that g₀₀ must change immediately throughout the dust ball D, if we manipulate the shell S.
We proved that the metric of time, g₀₀, changes instantaneously in the vacuum between S and D, and within D. The change in the metric propagates faster than light.
That is, the problem of the infinitely fast metric change remains if we have a dust ball enclosed in a dust shell S.
Discussion
If general relativity has solutions at all for a collapse of a uniform or a slightly nonuniform dust ball, it seems to require infinitely fast changes in the metric of time within the ball. This may even enable faster-than-light communication within the ball.
General relativity seems to break a fundamental principle of special relativity. We conclude that general relativity probably is a wrong model for a dust ball collapse.
The FLRW model of the expanding universe looks very much like a dust ball expansion in general relativity. If general relativity cannot handle a dust ball correctly, why would it handle an expanding universe correctly?
Dark energy is an indication that general relativity fails to treat an expanding universe correctly. If the expansion is accelerated, that seriously contradicts the general relativity model.
What aspects of the FLRW model have been verified empirically?
A. Nucleosynthesis fits the FLRW model.
B. The expansion of the universe by a factor 1,100 since the last scattering (cosmic microwave background, CMB) fits FLRW.
C. Baryon acoustic oscillations (BAO) fit the model where the age of the universe at the last scattering was as in FLRW.
Deviations from FLRW are:
1. the Hubble constant derived (in a complicated way) from the CMB differs from standard candle observations by 7%;
2. the James Webb telescope sees "too many" mature galaxies when the age of the universe was just 300 million years;
3. dark energy seems to be accelerating the expansion of the universe, while the expansion should slow down.
If retardation in the rate of clocks makes the expansion of the universe to oscillate, that might explain items 1, 2, and 3. The average speed of the expansion is correctly predicted by FLRW (or a newtonian gravity model), but an oscillation in the speed of the expansion can produce even large anomalies to the smooth process.
Question. Can retardation explain cosmic inflation?
Retardation when the dust ball approaches its Schwarschild radius
The matter and dark matter density of the observable universe is estimated to be 30% of the "critical density", Ω = 0.30.
The "current" radius of the observable universe is 46 billion light-years and its Schwarzschild radius is 14 billion light-years. Their ratio is approximately 0.30.
Maybe the accelerating expansion is associated with the (dark) matter density falling to 0.3X the critical density?
On the right side is a constant. The density ρ ~ 1 / a³, where a is the scale factor. Thus,
ρ a² ~ 1 / a.
The value of 1 / Ω - 1 is now roughly 2. When a was 1/2, its value must have been roughly 1, or Ω = 0.5. When a was 1/4, Ω = 0.67. When a was 1/1,000, then Ω = 0.998.
The fine-tuning, or flatness, problem is why Ω was so close to 1 in the early stages of the universe.
Let us try to calculate the effect of retardation when a dust ball collapses close to its Schwarzschild radius. The gravity potential of the edges falls fast. We expect to see a large repulsive force which arises from the retardation of clocks near the center. That is, clocks at the center tick significantly faster than at the edges. A ray of light is bent from the center toward the edge of the ball.
Let us first use newtonian gravity to calculate the retardation potential. The mass of the dust ball is M and the radius R. The gravity potential is
-G M / r for r > R,
-3/2 G M / R + 1/2 G M r² / R³ for r < R.
The potential at the center is
V = -3/2 G M / R(t).
Let dR(t) / dt = -v. Then
dV / dt = -3/2 G M * -1 / R(t)² * -v
= -3/2 G M v / R(t)².
The delay for the center of the ball to know about the decline in the potential is very crudely a half of the radius R(t) divided by the speed of light c:
1/2 R(t) / c.
The retardation then would mean that the potential at the center is higher than calculated in newtonian gravity, very roughly by the amount:
ΔV = 3/2 G M v / R(t)² * 1/2 R(t) / c
= 3/4 G M / R(t) * v / c.
We can compare this to the newtonian potential difference between r = 1/2 R(r) and the center:
1/8 G M / R(t).
We see that if the velocity v = c / 6, then the "retardation force" would approximately cancel the newtonian gravity force when r < R(t) / 2.
When the dust ball is approaching its Schwarzschild radius, the speed of its surface dR(t) / dt is relativistic. We conclude that the retardation force can easily cancel the newtonian gravity force inside the dust ball. The order of magnitude is large enough.
However the retardation potential close to the center is linear in r, while the newtonian gravity potential is ~ r². This would cause the uniform density of the dust ball to be compromised. Would that make the cosmic microwave background (CMB) in the sky nonuniform?
Our dust ball model has hard time explaining the uniformity of the CMB, anyway. Any phenomenon which is associated with the edges of the ball, can easily break the uniformity of the CMB.
The uniformity of the CMB
The cosmic microwave background is uniform in every direction to one part in 100,000. A cosmological model must be able to account for this phenomenon. In ΛCDM, two ad hoc assumptions are introduced in order to explain this:
1. the spatial topology of the universe is 3D surface of a 4-dimensional sphere, and
2. inflation.
It is not economical if we have to explain one observed fact with two ad hoc hypotheses. There is no evidence that the spatial topology can differ from a 3D plane, besides the hypothesized FLRW model of the universe. Inflation creates energy from nothing. It runs counter to all the observations we have about nature: energy is conserved.
In our blog we have tried to build a model where the spatial topology is a 3D plane and the observable universe is an explosion of a dust ball. The uniformity should be explained by some mechanism which makes a uniform dust ball to stay uniform when it collapses or expands.
An ad hoc solution would be to claim that in a large dust ball, we can calculate the contraction or expansion speed at a location x simply by looking at some environment of x, and ignoring the rest of the ball. This principle seems to hold for the gravitational attraction: locally, the expansion of the universe seems to obey newtonian gravity (with the exception of dark energy).
But why would retardation obey such a locality principle? And if it obeys that, why should we calculate the retardation based on the radius of the observable universe?
Dark energy is weakening?
Lodha et al. published their results from the Dark Energy Spectroscopic Instrument on March 18, 2025. Dark energy seems to be weakening recently.
If that really is the case, it is consistent with our retardation hypothesis: the expansion rate may even accelerate at times, but on the average, it should obey the formulae of the FLRW model.
Note that if ΛCDM is augmented with an "evolving" dark energy, the model becomes even more ad hoc than it was before. We can explain any deviation from the FLRW expansion rate by adding an evolving dark energy!
Retardation generates "negative mass" inside a collapsing spherical shell
Retardation makes light to bend away from the central volume of a collapsing shell. This is equivalent to putting some negative mass to the central volume
Could it be that this negative mass is relatively uniform throughout the collapsing dust ball? This could explain the uniformity of the CMB.
Let us use comoving coordinates of the dust in a collapsing dust ball. On January 18, 2025 we argued that gravity in those coordinates may look newtonian. We can draw lines of force for the gravity field in a familiar way.
Let us imagine that the collapsing dust ball consists of concentric collapsing dust shells. Could it be that, in the comoving coordinates, these shells create a fairly uniform density of "negative mass" inside the dust ball?
The density of negative mass is zero at the edge of the dust ball, but may be relatively uniform inside the ball. Then in a small subvolume of the ball, it may look almost exactly uniform.
Let us have a contracting uniform shell whose radius is R(t). The first guess for the "retardation potential" for the shell is something like
V ~ -r
for r < R(t). That is the, potential is the highest at the center of the shell. However, this does not seem like a good guess, since the negative mass density for this potential is
ρ ~ 1 / r.
There would be a singularity at the center, which does not look nice.
In a rubber sheet model of gravity, a collapsing shell corresponds to a ring of weights moving toward a center. The rubber sheet in this case can "anticipate" linear processes: the sheet moves downward at a constant speed. But it cannot anticipate an accelerating motion of the weights.
Maybe we should only make a retardation potential based on the acceleration of the masses in the collapsing dust ball?
Conclusions
We have many reasons to believe that there is retardation when a gravity potential adjusts the rate of clocks. It would be strange if a clock at the center of a collapsing spherical shell would immediately know how fast it should tick.
Retardation causes a potential which resists the collapse of the dust ball. A very naive calculation shows that retardation may at times create a repulsive force which is stronger than the attraction of gravity.
The naive retardation model is awkward since the negative mass which would create that potential would have an infinite density at the center.
We will next look at a more sophisticated model in which only the acceleration of the collapse creates retardation. Is retardation large enough to explain dark energy?
The result published on March 18, 2025 makes ΛCMD even more awkward than it was before. Dark energy density can change as time progresses. The predictive power of such a physical model is zero!
