UPDATE August 15, 2023: for the perturbation method to work well, we must use the Schwarzschild coordinates around a massive body. See our post today about how other coordinate systems are error-prone.
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Linearity of solutions conjecture
Let us denote the Minkowski metric by η. Let us have two metrics
η + h₁
η + h₂,
where the tensors h₁ and h₂ differ from zero very little. The sources of the perturbations h₁ and h₂ are moving or static masses, which are either smallish or very far away.
Let us sum the perturbations to obtain
η + h₁ + h₂.
How good an approximation is this for a solution of the Einstein equations for the combined system of masses?
Empirically, we do know from astronomy that one is allowed to sum newtonian gravity fields, and one obtains a very good approximation for the combined system.
Linearity of solutions conjecture. Summing weak perturbations (gravity fields) gives an excellent approximation for the combined system.
In the usual approach to gravitoelectromagnetism, people linearize the Einstein equations. That is risky, because gravitomagnetic effects are weak, and linearization may alter them substantially. Their conjecture is that linearized equations give an excellent approximation of the system.
Our conjecture above goes to the opposite direction. We claim that solutions to the Einstein equations can be summed linearly, and we get an excellent approximation of the system.
Linearized Einstein equations do work with gravitational waves. We know empirically that binary pulsars radiate power according to linearized equations, within an uncertainty margin of 0.3%.
Gravitoelectromagnetism is not about waves but sources of gravity. It is not immediately clear why linearized Einstein equations would work well in the context of sources.
It is not possible to prove that the Einstein equations have a solution for any realistic mass distribution. An exact proof of our conjecture probably is too difficult, but one may be able to give heuristic proofs.
Linearity of the geodesic equation
Suppose that we have summed two perturbations. How does a test mass move in the new metric? In what sense is the geodesic equation linear?
In the Minkowski metric η, a test mass moves at a constant velocity, and in canonical coordinates all the second derivatives above are zero.
The geodesic equation depends on the Christoffel symbols Γ. Are those linear in perturbations to the Minkowski metric?
Yes. The definition of Γ_cab clearly is linear in small perturbations of the metric g.
The geodesic equation has the raised indices μ and 0. We can approximate the metric g as the Minkowski metric η, and raising an index does not spoil the linearity in small perturbations.
In the Schwarzschild metric, a test mass can move along a curved orbit because the metric of time varies by the spatial location. That corresponds to a newtonian force. The test mass may also take a curved orbit because the radial spatial metric is stretched. Gravitational lensing is an example of this. One half of the deflection of a photon comes from the newtonian force, and the other half from the stretching of the radial metric.
It is obvious that we can sum the perturbations in the metric of time to calculate the orbit of a test mass. It is like summing newtonian forces.
Since Christoffel symbols are linear in the metric, linearity holds for perturbations of the spatial metric, too. For example, if a ray of light would curve to the left because the spatial metric on the left is "shorter", we can compensate by adding a perturbation which makes the metric on the right shorter.
Lorentz covariance of solutions
Covariance means that one can beautifully transform the description of a physical system between different reference frames.
The Lorentz transformation formulae tell us how one transforms between frames whose relative velocity is constant.
Covariance was one of the fundamental principles on which general relativity was built. We believe that the Einstein equations respect Lorentz covariance.
t
^
| t'
| ^
| | -----> v new frame
| |
| --------------> x'
|
| m • --->
|
| M1 ● -->
| <-- ● M2
|
------------------------------> x
Canonical Minkowski coordinates
Suppose that we have sources M1, M2, ... of a weak gravity field in an almost Minkowski metric. We use canonical Minkowski coordinates. We switch to a frame which moves at a constant velocity v relative to the old frame, in the canonical Minkowski coordinates.
We claim that for weak fields, the Einstein equations in the new frame must accept the Lorentz-transformed solution (transformed using the Minkowski coordinates) of the old frame. Also, an orbit of a test mass m in the new frame must be obtained through a Lorentz transformation of the orbit in the old frame.
For strong gravity fields, a global Lorentz transformation may be a more complicated question.
Linearity of the geodesic equation relative to a newtonian gravity orbit
We may want to calculate an orbit in newtonian gravity, and make general relativistic corrections to it. This is called a post-newtonian approximation.
Is the geodesic equation linear in the sense that we can add these corrections for each perturbation of the Minkowski metric η?
metric η + hⱼ
^ newtonian acceleration
|
|
m • --------------> v
|
v correction
Let us use the frame of canonical coordinates of the Minkowski metric η. We calculate accelerations with respect to these canonical spatial coordinates and the canonical time coordinate.
By the linearity of solutions conjecture, we can sum the accelerations for each perturbation hⱼ, to obtain the acceleration for the sum of perturbations.
We can also sum the newtonian accelerations calculated for each perturbation, to obtain the newtonian acceleration for the sum of perturbations.
This implies that we can sum the corrections for each perturbation, to obtain the correction for the sum of perturbations.
If the corrections to newtonian gravity are very small, then the corrections could be smaller than the error from nonlinearity that we get when we sum the accelerations for each perturbation hⱼ. In that case, summing corrections does not help much.
It would be useful to have an estimate on what is the error in summing perturbation accelerations.
What does "newtonian gravity" actually mean?
The concept is fuzzy. In most cases we want to add the kinetic energy to the gravitating mass, though that is not what Isaac Newton in 1687 thought.
Thermal energy is kinetic energy, and we certainly want to include thermal energy into the gravitating mass.
It is best to specify in each individual case what exactly is included in "newtonian gravity".
Conclusions
If the linearity of solutions conjecture is true, then we can sum weak gravity fields, and also sum corrections to the newtonian orbit.
This allows us to calculate the effect of moving masses on the orbit of a test particle. We obtain a correct treatment of gravitomagnetic and other corrections on the orbit.
Gravitoelectromagnetism wants to call all these corrections "magnetic". But the examples in our previous blog posts suggest that there is no way to make all the corrections analogous to magnetic corrections in electromagnetism. The central idea of gravitoelectromagnetism fails.
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