When does the nonlinearity occur? If we have a very large static mass M, then general relativity claims that its gravity is surpringly strong close to M. Let us simulate M with two particles flying to opposite directions at almost the speed of light. The rest mass of the particles is very small, but they have lots of kinetic energy.
M
v <-- ● 1
2 ● --> v
• m test mass
Let us switch to a comoving frame of the rightmost particle 2. In that frame, the particle 2 is very light and, intuitively, very insignificant. The particle 1 and the test mass m move at almost the speed of light to the left and 1 possesses a lot of energy. We can imagine that m has an infinitesimal rest mass which is infinitesimal even when m moves almost at the speed of light.
But if we would remove the very light particle 2, the acceleration of m would drop to a half. That sounds strange, but is a result of special relativity.
● 2
v ≅ c <-- • m
Let us then remove the particle 1. We double the rest mass of 2 and look at the configuration in a comoving frame of 2. In the comoving frame of 2, the particle 2 is very light. Doubling its rest mass certainly cannot have any nonlinear effects.
But in the comoving frame of m, the particle 2 does carry a lot of mass-energy, and there might be nonlinear effects. Do we have a contradiction here?
Conservation of momentum and nonlinearity
How momentum is conserved when fields are retarded? That is an open problem in field theories. Maybe nature performs "transactions" which ensure momentum conservation? Let us check how general relativity is supposed to handle momentum conservation.
The ADM formalism implies conservation of momentum. But we in this blog have tentatively shown that the Einstein field equations do not have any solutions for a typical dynamic system. The ADM formalism assumes that a solution exists.
F' F
● --> <----- •
M m
Suppose that we have a very large mass M, such that its gravity is significantly larger than the linear (newtonian) gravity at a test mass m. Then the field of the mass M seems to pull m with a force
F > F',
which is larger than the force that the field of m exerts on M!
This is suspicious. Newton's law of action and reaction is broken? The Einstein-Hilbert action is translation independent. By Noether's theorem, we expect it to conserve momentum. Maybe the existence of the strong field of M makes the field of m non-linear?
Let us use canonical Minkowski space coordinates. General relativity recognizes "proper" momentum, which is measured by a local observer. That differs from coordinate momentum.
Gravitational waves will take away some momentum. The Ricci tensor is zero in them. The Einstein-Hilbert action is not aware of their existence. This implies that momentum is not conserved, unless we define the momentum through some pseudotensor.
Relativity of simultaneousness
M
v <-- ●
● --> v
M
• m
Let v be very close to c. In the frame of m, the two masses M overlap. But in the comoving frame of the right-moving M, there is very little overlap. Is this compatible with nonlinear gravity?
Nonlinearity from polarization
Imagine that a very strong gravity field pulls positive mass-energy from empty space toward itself, and repels negative mass-energy. That would make gravity steeper than the newtonian 1 / r² gravity.
However, the gravity field close to a huge black hole is not very strong. The polarization hypothesis requires that it is a low potential which produces polarization and not the field strength.
In quantum electrodynamics, vacuum polarization makes the interaction stronger at very short distances, or at very high energies. But that requires energies which are much larger than the mass of the electron, 511 keV. In our blog we hold the view that the vacuum itself is not polarized, but a pair which is born from the energy of a particle collision, simulates vacuum polarization in a scattering experiment.
Can we somehow calculate how much positive/negative mass-energy might pop up if we have a very low gravity potential? That assumes that the vacuum itself can become polarized.
Nonlinearity is needed to satisfy an equivalence principle?
On August 23, 2023 we showed that general relativity breaks the weak equivalence principle. Why should we use nonlinearity to satisfy a certain equivalence principle, when general relativity breaks the most fundamental equivalence principle?
Nonlinearity makes everything complicated
_____ _____ rubber sheet plane
\ ● /
M
In a rubber sheet model of gravity, nonlinearity means that the rubber becomes weaker against stretching if it is depressed below the sheet plane more than a certain distance. The weakening is not from the stress on the rubber but from the depth of the depression – that is, from a low potential.
Why should it become weaker? Is there some kind of a heater which heats the rubber when it is pressed down a lot?
It is obvious that having a rubber sheet whose strength depends in such a way on the depth of the depression makes calculations complicated if the system is dynamic.
Empirical data
We have to check what is known about accretion disks. Is there any empirical proof that the gravity potential must be steeper than the newtonian one?
In September and October 2023 we showed that the Kerr metric probably is wrong for a rotating black hole. We have to calculate a new solution for a rotating mass and compare empirical data against it.
The paper by Cosimo Bambi (2013) leaves an impression that empirical data does not tell much about the structure of black holes, besides the fact that matter does not hit a solid surface as it is devoured by a black hole.
If gravity is linear, then the Schwarzschild radius is replaced by a Newton radius, which is 1/2 of the Schwarzschild radius. Once we get more accurate measurements of gravitational waves, we may be able to determine what the correct radius is.
The M87 central black hole (photo Wikipedia)
The Event Horizon Telescope measured the radius of the "photon ring" around the M87 black hole. The margin of uncertainty is given as approximately 10%.
In a Schwarzschild black hole, the lowest circular orbit for a photon is at
3 G M / c².
In a newtonian black hole, the lowest photon orbit is at
(1 + sqrt(5))² / 4 * G M / c²
= 2.62 G M / c².
The data from the Event Horizon Telescope cannot decide which is correct.
Conclusions
There probably is nothing which prevents an interaction from being nonlinear in the sense that a large aggregate charge interacts stronger than the sum of the interactions of its components. We can imagine that the components "help" each other to gain more strength. This certainly is possible in newtonian physics, and we did not find any reason why special relativity would prohibit it.
Nonlinearity creates more "effective charge" for a large mass, when the charge is measured from a short distance. This creation of more charge probably breaks the Einstein equations.
Nonlinearity makes everything complicated and there is no obvious need for nonlinearity. Our own Minkowski-newtonian gravity model works without nonlinearity, though one can add there nonlinearity.
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