We would assume that a clock at the center of the shell "calculates" the radius of the shell based on the latests contraction velocity v that the clock knows of.
Is this a reasonable assumption?
Let us look at a single mass M and a test mass m (or a test clock) in the field of M. The usual retardation rule is that if M moves at a constant speed v, then the test mass or test clock will know the gravity field of M as if m would know the current position of M in the laboratory coordinates.
It makes a lot of sense to assume that a test clock m can "calculate" its own gravity potential based on the assumption that masses M continue their movement at a constant velocity. The clock then adjusts its rate according to that gravity potential.
Another way to look at this is to assume that M is static in the laboratory coordinates, and the test clock m moves around. An atomic clock on the surface of Earth can adjust its ticking based on what is its distance from the center of Earth.
In our March 15, 2025 blog post we calculated the retardation inside a shell assuming that the clock is not aware of the contraction speed v of the shell. That yields a very large retardation effect. If the ignorance of the clock only concerns the acceleration of the shell, the retardation effect is much smaller. But is the effect still large enough to explain dark energy?
A crude calculation based on the expansion of the universe
The "current" radius of the observable universe is estimated to be 46 billion light-years. The age of the universe is 13.7 billion years.
The universe was expanding significantly faster than now, say, 6.9 billion years ago.
The scale factor in the matter-dominated phase is
a(t) ~ t^⅔.
The time derivative is
da / dt ~ 1 / t^⅓.
When the age of the universe was a half of the current age, the expansion speed was
2^⅓ = 1.26
times the current speed.
If a clock at the center of the observable universe "calculates" its rate by assuming that the universe would still be expanding at that 1.26X speed, then the clock will overestimate its gravity potential and will tick too fast. The speed of light is "too fast" close to the center, which means a repulsive force from the center.
Could it be that the current value of matter and dark matter Ω = 0.3 has something to do with the observed acceleration of the expansion?
Let us try to estimate the repulsion, using the comoving coordinates of the "dust" (= matter and dark matter in the universe). On January 18, 2025 we argued that gravity looks very much newtonian in comoving coordinates. But we did not consider retardation of clocks then.
Conservation of energy in a rubber sheet model of gravity
We can simulate the collapse of a dust ball by letting small slippery weights to slide toward a central depression in the rubber sheet.
In the rubber sheet model, longitudinal, spherically symmetric waves exist.
A rubber sheet model allows the collapse process to "oscillate". The potential energy of the weights flows in a complicated way into the elastic energy of the sheet, as well as to the kinetic energy of the weights.
We want energy conservation in the collapse process. The rubber sheet guarantees energy conservation.
In a simplistic retardation model, conservation of energy probably would be breached.
We conclude that a satisfactory retardation model must involve something similar to the rubber sheet model of gravity.
In the rubber sheet model, is it possible that the collapsing dust ball could send longitudinal waves also outward?
The weight of the dust ball, the kinetic energy, and the longitudinal waves stays constant. Therefore, during the early stages of the collapse, the shape of the rubber sheet outside the ball stays constant. But could it be that once the longitudinal wave inside the ball has moved past the center, it could come out of the ball?
A simpler case is a circular ring of weights sliding toward the center.
Any "delayed" spherically symmetric process requires the existence of longitudinal waves?
There are no spherically symmetric transverse waves. If we have a spherically symmetric process which alters some local parameter (e.g., slowing down of clocks), the process maybe has to be "relayed" through longitudinal waves?
If we can measure the propagation of the wave, we presumably can extract energy from the wave?
As an example, consider the density of air in a spherically symmetric vessel. Let us suddenly contract the vessel. Eventually, the air pressure inside the vessel must even out. The process happens through sound waves, which are longitudinal waves.
Literature seems to claim that there are no longitudinal waves in general relativity. Let us investigate this.
Move a mass M suddenly closer to a clock: general relativity cannot satisfy Gauss's law
● --> o
M clock
We suddenly move a large mass M closer to a clock. The clock slows down. Can we describe the process as a "longitudinal wave"?
According to our June 20, 2024 blog post, there probably is no solution at all for the process in general relativity: the process is "dynamic". But let us assume that a solution would exist. The metric of time,
g₀₀,
determines the rate of the clock.
If the metric of time could change "faster than light", then we would be able communicate faster than light: simply compare the rate of two adjacent clocks.
It is reasonable to assume that the change in the metric of time cannot propagate faster than the local speed of light.
Let us assume that M is not huge. Then the force of gravity is almost all derived from the metric of time, g₀₀?
Let us assume that instead of M, we have an electric charge Q. An approximate solution for the problem can be derived using Edward M. Purcell's approach, drawing lines of force which do not break. The lines of force can be derived from the 4-potential of the field.
But g₀₀ corresponds to the scalar potential φ only. For electromagnetism, there is gauge freedom. The scalar potential φ can be chosen in many ways. This breaks the analogy between g₀₀ and φ.
Can general relativity describe a "magnetic gravity" field at all? In the familiar Edward M. Purcell diagram above, the lines of force in the circular zone turn a lot. Is it possible to generate such a force by modifying the metric of space?
The figure corresponds to a charge Q which suddenly acquires a constant speed v to the right.
Let us imagine that it is a large mass M which acquires a speed v to the right, and we have test masses m floating around M to test the direction of the force at various locations. Will the force be anything like the corresponding electromagnetic force?
There is a circular "transition zone" where the lines of force turn sharply. Outside that zone, Coulomb's force and the newtonian gravity force are analogous.
The question is what happens in the circular transition zone.
If the test masses m are static, then according to the geodesic equation, only the metric of time can exert a force on m. Just outside the transition zone, the metric of time, g₀₀, is constant.
line of force
|
|
---- × transition zone
|
|
●
M
Let us look at the zone straight up from the mass M in the diagram. In the transition zone, there is a strong force to the right. That requires that the value of g₀₀ must decline steeply when we go to the right. But then, at the location marked with ×, there should be a very strong force downward. We cannot implement the lines of force with the metric g₀₀.
In electromagnetism, the lines of force require the vector potential A, in addition to the scalar potential φ, which is analogous to g₀₀.
We conclude that general relativity cannot satisfy Gauss's law for an accelerating mass M.
Gauss's law in general relativity seems to fail
In general, general relativity does not satisfy Gauss's law. In the Schwarzschild metric, test masses "float" at the event horizon, if we use the standard Schwarzschild coordinates.
For a small mass M, the Schwarzschild metric is approximately equivalent to newtonian gravity, and Gauss's law approximately holds.
In the previous section, we argued that a metric cannot satisfy Gauss's law for an accelerating mass M.
But in the FLRW model of the universe, gravity works in the newtonian way, and Gauss's law does hold.
Let us then have a spherically symmetric mass shell which suddenly starts to contract. If the change in the metric of time, g₀₀, cannot propagate faster than the local speed of light, then |g₀₀| will be larger at the center of the shell than near the shell. The geodesic equation implies that a static test mass m will feel force from the center toward the shell. This breaks Gauss's law because it would be equivalent to having negative mass at the center.
Our blog post on February 12, 2025 showed that Maxwell's equations fail for accelerating systems of charges. But in many cases, Gauss's law seems to hold for electromagnetism, either approximately or even precisely.
For general relativity, Gauss's law seems to fail, except for the FLRW model. This makes the FLRW model suspicious. Why would the universe satisfy something which is not generally true in general relativity?
Conclusions
For a collapsing shell, general relativity seems to require that a change in the metric of time, g₀₀, can propagate infinitely fast. This is a very dubious result.
The FLRW model has an (unrealistic) symmetry, and Gauss's law holds in it, even though Gauss's law does not normally hold in general relativity. This makes FLRW even more dubious than it was before.
Let us assume that the change in the metric of time can only propagate at the local speed of light. Is the propagation process "wavelike"? In physics, most propagating processes are wave phenomena. Or is the propagation "rigid" so that the rate of a clock immediately changes when it "knows" that it is in a low gravity potential?
A rubber sheet model behaves in a wavelike fashion, and the there exist longitudinal waves in it.
We have definitely shown the following: we cannot assume that Gauss's law holds for gravity. The collapse of a massive dust ball is likely to differ from a simple newtonian gravity model.
In a forthcoming blog post we will analyze the collapse of a dust ball further. The magnitude of the retardation seems to be large enough, so that it can explain dark energy, but the details are still very obscure.
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