UPDATE January 9, 2025: We maybe have to replace the classical law of Gauss with a spacetime version: the average flux of a the electric field E through a closed surface S is a constant times the charge Q enclosed inside S. The instantaneous flux may vary. Conservation of energy requires that the average flu, in some sense, must be constant. Otherwise, we could construct a perpetuum mobile.
Note that in special relativity, time and space cannot be separated. Gauss's law is suspicious because it thinks that space is a separate entity from time. Our retardation argument uncovers the problem.
----
E <------- total electric field
E₀ <-- electric field of cap
______
/ \
• <-- | |
q v \_______/
test charge "cap" charged shell
Let us have a charged spherical shell which initially is static. Suddenly, the shell starts expanding at a constant speed v. A test charge at some distance learns about the expansion for the nearest part of the shell first. The remaining part of the sphere q still "sees" as static.
The test charge q "sees" the electric field E₀ of the nearest part as it would emanate from a "cap" which is closer to q than the rest of the sphere.
The test charge q sees the total electric field E of the charged sphere now stronger than it was a while ago.
The integral of the electric field E over a spherical surface at the distance of q increased: we broke Gauss's law!
The analysis above uses the retardation law for the electric field: if a charge Q moves at a speed v relative to the laboratory frame, then a test charge which "knows" of the movement of Q, at a laboratory time t, will see Q at the location where Q is at that same laboratory time t. That is, q will see the cap closer.
The cap is moving at a speed v to the left. Let us Lorentz transform its field, in order to make sure that the field E₀ of the cap is now larger than it was when q still saw the cap static.
electric
field
meter
O s ruler
-----------------------
• <-- |
q v Q cap
-------> x
At a laboratory time t, q's x coordinate is 0 and the x coordinate of the cap is s.
The cap is carrying a ruler which extends over the laboratory distance s to the test charge q. To the ruler there is an electric field meter attached. In the moving frame of the cap, the distance is longer:
s' = s / sqrt(1 - v² / c²).
The electric field of Q in the moving cap frame is
E₀' = 1 / (4 π ε₀) * Q / s'²
= 1 / (4 π ε₀) * Q / s² * (1 - v² / c²).
The Lorentz transformation of E₀' is the identical mapping:
E₀ = E₀'.
If the velocity v = c / 10, then q sees the cap 10% closer than the rest of the sphere, and q sees the field E₀ about 20% stronger than when the cap still was static. The correction coefficient in this case is 1 - v² / c² = 0.99, very close to 1. For small v, we can usually ignore Lorentz corrections, since they are ~ v² / c².
The magnetic field of the expanding sphere is zero, because of the symmetry.
What is the effect of the electromagnetic wave which q may "see" from charges in in the sphere suddenly accelerating to the velocity v?
In the analysis above we ignored a possible electromagnetic wave which q may see.
In the laboratory frame, there certainly is no electromagnetic wave from the sudden acceleration. There are no longitudinal waves. Thus, q is not expected to see anything.
But let us analyze in more detail. The test charge q sees that a ring of charges in the shell suddenly accelerates toward q. Does the symmetry eliminate any magnetic field?
× Be
•
q
v <-- • e
Let e be an elementary charge in the ring. It suddenly accelerates to the left. A magnetic field Be is born at q. In the diagram, the field Be is normal to the screen (hence the symbol ×). When we add the fields Be for each e in the ring, their sum is zero.
It looks like q will not see any electromagnetic wave.
The effect of the extra inertia of q in the field of the sphere
If q and the sphere have a charge of the same sign, then the extra inertia of q is expected to push q to the left. Inertia will not save Gauss's law.
Could there exist a mysterious force which saves Gauss's law? No
The mysterious force should be able to differentiate between the following configurations, drawn in the laboratory frame:
E
<---
______
/ \
<-- | |
v \_______/
only the "cap" moves, shell static
^ v
|
__________
/ \
/ \
| |
\ /
\___________/
|
v v
entire shell expands
The mysterious force would compensate the difference of E to what Gauss's law predicts. The force would understand that it must not do anything in the first diagram above, but must do compensation in the second diagram.
For the force to do immediate compensation, it should get information faster than light. Such a force cannot exist if we believe special relativity.
Energy and momentum conservation: an accelerating shell of charges e
We firmly believe that energy and momentum are conserved. The expanding shell creates an electric field E which is larger than what Gauss's law predicts. The impulse to q must somehow eventually have an opposite impulse in the shell.
This is the general problem of conservation of momentum: if particles A and B interact, how does the field store the impulse and eventually deliver it to the other intracting particle?
If we have an accelerating, expanding shell of elementary charges e, then, apparently, any elementary charge e in the shell sees an electric field E which differs from the one given by Gauss's law.
But energy conservation requires that, eventually, the elementary charges e must possess the kinetic energy which we can calculate by integrating E in the Gauss's law solution.
There has to be a compensation mechanism which eventually sets the velocities of elementary charges e to correct ones.
If the acceleration is "too fast" initially, it must be "too slow" later.
How would this compensation mechanism behave? Does it oscillate somehow? In our December 30, 2024 post we speculated that the oscillation between a "too fast" expansion of the universe recently (= dark energy), and "too slow" earlier is due to the compensation mechanism.
Discussion
Gauss's law is one of Maxwell's laws. Let us check the literature, if anyone has realized that retardation spoils Gauss's law.
In our blog we have suspected that Gauss's law fails. We have claimed that the concept of a "field" is too simple to describe complex interactions between particles. The interaction should be calculated individually for each pair of particles. The interaction is "private" between particles. It cannot be simplified into one common "field".
In the link people discuss retardation effects in the case where the current through a wire changes. It is much more complicated than our expanding spherical shell.
Is this a miracle that we found a very simple counterexample to Gauss's law? No. Nobody really has claimed that Maxwell's equations are consistent. The self-force of the field on the electron is an open problem. Besides that, there are several well-known, persistent paradoxes.
Birkhoff's theorem fails for gravity: general relativity contradicts special relativity
Weak gravity fields in general relativity should behave much like electric fields. Our argumemt shows that the gravity of a uniform mass shell can vary. The result refutes Birkhoff's theorem which claims that the gravity field of a spherically symmetric object stays constant outside the object.
The refutation uses principles of special relativity. Birkhoff's theorem is derived from the Einstein field equations. We have a proof that the Einstein field equations are incompatible with special relativity.
See our blog post on January 4, 2025 for an analysis why this happens.
Conclusions
We realized on December 30, 2024 that retardation affects collapses and expansions which happen under a force field. A simple consequence is that Gauss's law fails for the electric field.
The speed of light is very large compared to the speed of charge carriers in everyday laboratory experiments. We do not know if we are able to measure the electric field retardation effect in a laboratory.
By far the largest expansion in nature happens in the universe, and the velocities are close to the speed of light. Retardation effects might be important in this huge scale. We will try to calculate an estimate for them in the ΛCDM model. They might explain the peculiarities in the cosmic expansion.
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