Retardation presents a grave problem to this approach: the behavior of the field at a location A must only depend on what happened inside the light cone of A. If a location B is outside the light cone of A, then nothing that happens at B is allowed to affect A.
But an stationary point of the action is a global minimum, maximum, or an inflection point of the action integral. The value of the global stationary point at A may depend on things which happened outside the light cone of A. That will produce solutions in which faster-than-light communication is possible. And we must not allow faster-than-light communication.
FLRW models
A spherical collapse or expansion history can often be derived as a global stationary point of an integral. The FLRW solutions are (probably) stationary points of the Einstein-Hilbert action.
Does the FLRW solution at a location A depend solely on things that happened inside the light cone of A?
The FLRW universe is perfectly uniform. But suppose that there is a mechanical device which will break the uniformity at a location B at a time t₀. The stationary point of the action integral at a location and time A, t₁ may depend on the events at B, t₀, even if B, t₀ is outside the light cone of A. The time t₀ might even be in the future of A, t₁.
On May 21, 2024 we showed that the Einstein-Hilbert action does not have a solution for any "dynamic" system. That is another problem in the action, but it is different from the retardation problem.
Expansion of a spherical shell of electric charges
On January 3, 2025 we discusses the expansion of a uniform shell of charged particles. The naive solution, which ignores retardation, probably is an stationary point of an action integral. But it is a wrong solution because it allows faster-than-light communication.
A "global field" has to be replaced with "private" interactions between particles?
The way to enforce retardation is to assume that the system consists of particles in Minkowski space, and that the interaction of the particles respects the light speed limit.
We would abandon the concept of a global field.
The self-force of the field of an electron on the electron itself probably cannot be explained with a global field. The concept of a field must be fragmented into the individual fields of each charge carrier. An individual field interacts with another individual particle. There is not much "global" in this.
Since general relativity depends on the existence of a "global" spacetime geometry, it is doomed.
Loss of information in a sum global field
Our arguments above suggest that the global field actually is the collection of the individual fields of charge carriers.
If we try to reduce the global field into a simple sum of individual field strengths, then we lose information about individual fields – and we would need that information in calculating the behavior of the system.
Claim. A "global field", given as the sum of fields of individual charge carriers, is an approximation which simplifies calculations in many cases, but does not handle retardation correctly.
The information loss allows faster-than-light communication
Let us again look at the expanding shell of charges in the January 3, 2025 blog post.
If we want to construct an action integral which prevents faster-than-light signals, we must penalize such signals harshly, so that a history containing such signals cannot be a stationary point of the action integral. Maybe the action integral is not defined at all, if such rogue signals happen. An example is a faster-than-light particle m in a typical action integral. Its contribution would be imaginary:
m / sqrt(1 - v² / c²).
But a global field defined as a sum of individual fields loses information. The sum global field can look benign, with no harsh penalty, even though individual fields change faster-than-light!
Thus, a sum global field often allows faster-than-light signals to happen. This is a major shortcoming in the concept of a global field. We must replace it with individual fields of the particles, to avoid loss of information.
A problem with FLRW solutions of general relativity: adjusting the metric using superluminal information
Let us again look at the development of an (approximately) FLRW model at a location A. The solution is not allowed to "know" that the expansion of the universe will slow down uniformly as the time t passes. We have to look at the solution for various possible decelerations of the expansion far away from A. Let a family of possible expansion rates be S(n).
There is a problem in this approach, though. On May 21, 2024 we proved that the Einstein-Hilbert action does not have a stationary point for a "dynamic" system. Thus, there is no solution, unless the expansion rate is the same everywhere! Let us for a while assume that we have been able to correct the action formula, and can find a stationary point.
Setting the metric close to A to some special (different) value for each S(n) may optimize the action, unless the action somehow recognizes that we are using superluminal information, and harshly penalizes such a break of rules. But how could the action recognize that? We are not sending gravitational waves whose energy would be infinite or imaginary. We are simply adjusting the metric in some seemingly innocent way.
An innocent adjustment may amount to a superluminal signal.
Here we again bump into the problem that general relativity does not have canonical coordinates. In Minkowski space, it would be easier to recognize superluminal signals. Though, we still would have to look at the individual field of each particle.
An individual field for each particle is in the spirit of quantum field theory
In quantum field theory, individual particles interact with each other, without any reference to a "global electromagnetic field". It makes sense to introduce an individual force field for each particle.
Conservation of energy and momentum in quantum field theory is implemented through particles exchanging (virtual) quanta. This is a possible solution to the conservation problem in macroscopic fields, though this does not tell us in detail what a macroscopic field does, and how does a macroscopic field implement conservation laws.
Conclusions
We have discovered strong evidence against the traditional global field concept, where the field is understood as the sum of the fields of the individual charges (sum global field). The sum loses information. It cannot recognize and ban faster-than-light signals in some cases.
The self-force of the field on the electron may be hard to describe through a sum global field.
The simple solution to the problem is to split the sum global field into individual fields of each elementary particle. Quantum mechanics likes this solution.
General relativity has major problems, though: there it is not clear what is the field of an individual particle. The mass of the particle acts as a source of spacetime "curvature" at the location of the particle. It affects the curvature also elsewhere, but what is the individual field of a single particle is a fuzzy concept. Nonlinearity of gravity makes this inevitable: how do we assign nonlinear effects to each component field?
Anyway, the individual field of each particle is a useful concept in gravity, too.
The FLRW model is an unusual application of the field concept because the spatial topology is that of a 3-sphere. Can we define the electric field of a single charge in such a topology? Where would the lines of force end? We are not sure if such a topology makes sense at all as a physical model. Is it so that the universe must be a flat Minkowski space?
We will investigate what retardation means in the case of the FLRW model. Does retardation affect the deceleration of the expansion of the universe? Does retardation explain dark energy?
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