In the following we usually assume that if a metric g in vacuum has the Einstein tensor value (residual value ΔT) very close to a zero tensor, then g is a good "approximation" of a solution. However, it might be that g is not a good approximation! The nonlinearity of the equations means that g could be very far from the correct exact solution. Any small deviation ΔT ≠ 0 can cause a drastic change to g. This is similar to paper bending.
On March 14, 2024 we showed that summing Schwarzschild metric perturbations for each atom in a spherical mass shell produces a drastically wrong metric.
Let us call approximate general relativity a (fuzzy) assumption that
1. the metric g₀₀ which we obtain from the newtonian gravity potential V is very close to the solution of general relativity;
2. we get a good approximate metric g in vacuum if ΔT is very small for g, except in the case of:
3. the metric around a spherically symmetric mass, where we must use the Schwarzschild metric; even if ΔT is very small, the metric may be totally wrong.
^ r
|
| cylinder
----------> z =========
Mechanical clocks are slowed down in our model because gravity gives more inertia to the parts of the clock, and at the same time, the energy in the potential in force fields is diminished because of the low gravity potential. Two effects which we would expect to happen, even if we would not know general relativity.
Slowdown of clocks is what is meant by "time slowing down".
A photon of energy E in a low newtonian gravity potential -V carries an extra inertia of V / c². That is why the speed of light is slower. One may imagine that moving a test mass m for a distance s involves a "field energy" flow of m V for the distance s. This explains the slow speed of light.
^ r
|
| L
--------> z ============ cylinder
^
|
• m test mass
|
| rope
The stretching of the spatial metric is an additional process. When a test mass m approaches the cylinder, energy is shipped from the field to the kinetic energy of m, or if we have a rope which does not allow m to speed up, then to the work done on the rope. Light moves slower to the direction of r. We interpret this that the "metric of space" has been stretched in the direction of r.
Let us have a cylinder of a length L. The force on m is
F = m G ρ / r,
where ρ is the mass of the cylinder per a unit length.
Exact general relativity and a cylinder
Suppose that exact general relativity has a solution g for the vacuum metric around a cylinder, and g is a small perturbation from the flat metric.
The linear equations that we derived on April 13, 2024 must then be approximately true for g. Our formulae give Ricci curvatures which are not zero, because we omitted the products of two Christoffel symbols Γ * Γ. But the deviations from zero are necessarily very small.
Suppose that we derive from our formulae an equation for g. Is it so that g must approximately satisfy that equation? Probably yes, but we should do a proper error analysis for each derived formula.
We may in many cases derive equations which the solution of the Einstein equations, if any, must approximately satisfy.
We tentatively showed on April 22, 24, and 26, 2024 that there is no solution for our formulae if a shear is present inside the cylinder. The contradiction may be numerically substantial. There cannot exist any metric g which approximately satisfies our equations.
If the Einstein equations would have a solution for that configuration, then we can probably prove that that solution would approximately satisfy our formulae. We have a proof then that the Einstein equations cannot have a solution.
The reasoning above tells us that we can use our formulae as a guide which tells us what a solution in exact general relativity would look like, if such a solution exists.
This implies that we can compare Minkowski & newtonian gravity to our formulae. The formulae can guide us.
The radial metric g₁₁ agrees between approximate general relativity and Minkowski & newtonian
Let us guess that when we lower m closer to the cylinder, the field energy gained by m is shipped over the distance r. The spatial metric of r is then
g₁₁ ≈ 1 + 2 m G ρ / (m c²)
= 1 + 2 G ρ / c².
Generally, we have
g₁₁ = 1 + 2 F r / (m c²)
= 1 - g' m c² r / (m c²)
= 1 - r g'
=>
g₁₁' = -g' - r g''.
Let us compare this to what we calculated using approximate general relativity. On April 22, 2024 we derived the formula:
g₁₁' = r d²g / dz².
We also derived:
2 R₀₀ = -g'' - g' / r - d²g / dz² = 0
=>
d²g / dz² = -g'' - g' / r
=>
g₁₁' = -r g'' - g'.
The result agrees with the Minkowski & newtonian prediction!
M
● --------------------> z axis
R r
• m
How does this match with what we know about the Schwarzschild metric? Our formulae above are for a general rotationally symmetric mass distribution. It does not need to look like a cylinder.
In the Schwarzschild metric, Minkowski & newtonian predicts the radial metric gR from the energy shipment from the central mass M to the test mass m. The distance R from M to m is generally larger than the distance r from the symmetry axis z. How can the shorter distance r work above? The explanation has to be that the Schwarzschild metric in cylindrical coordinates is not orthogonal. The value of g₁₁ does not tell the whole story about the metric in cylindrical coordinates. The skew, g₁₃ plays a role, too.
Generalizing Birkhoff's theorem to two spherical masses
Approximate general relativity says:
2 R₂₂ / r² = g' / r - g₃₃' / r + g₁₁' / r
+ 2 / r * dg₁₃ / dz = 0,
2 R₀₀ = -g'' - g' / r - d²g / dz² = 0.
In Minkowski & newtonian, the reason for the skew g₁₃ ≠ 0 might be that the spatial metric is stretched in an oblique direction. The Schwarzschild metric has the barrel distortion, if presented in cartesian coordinates. The radial spatial metric is stretched around the central mass M.
Let us find out what approximate equations a solution of general relativity for two masses M must satisfy. Our April 13, 2024 equations may get us quite far in the endeavour.
^ r
|
| L L
---------- ● --------|-------- ● ----------> z
M 0 M
The masses are located symmetrically at positions -L and L on the z axis.
We know that the Schwarzschild perturbation for each M satisfies our formulae, and our formulae are linear. Therefore, the sum of the Schwarzschild perturbations satisfies the formulae. Can we prove that any solution of the Einstein equations where g₀₀ is very close to the metric derived from the newtonian potential V, is very close to this sum of two Schwarzschild perturbations?
Birkhoff's theorem states that the only static metric around a spherical M is the Schwarzschild metric. Thus, for a single small mass M, we know that the solution g is very close to the metric derived from the newtonian g₀₀. What about two masses M?
On April 22, 2024 we solved g₁₁' from g₀₀. If we assume that g₁₁ is asymptotically 1 far away, we can determine the value of g₁₁ uniquely at any point, from the function g₀₀. From now on, we can assume that g₀₀ and g₁₁ are known (approximately).
2 R₃₃ = -g₃₃'' - g₃₃' / r + d²g / dz²
- d²g₁₁ / dz²
+ 2 dg₁₃' / dz + 2 / r * dg₁₃ / dz = 0,
2 R₁₁ = g'' - g₃₃'' + g₁₁' / r - d²g₁₁ / dz²
+ 2 dg₁₃' / dz = 0,
2 R₂₂ / r² = g' / r - g₃₃' / r + g₁₁' / r
+ 2 / r * dg₁₃ / dz = 0.
The three formulae above concern g₃₃ and g₁₃. Other functions are known. Can we determine unique values for g₃₃ and g₁₃?
There is some flexibility in drawing the z = constant coordinate lines. The lines must match with g₀₀. A displacement of the z = constant lines is a perturbation. The change to the value of g₀₀ in the displacement is a second order perturbation, and thus negligible.
We conclude that we are able to perturb the z metric freely, as long as the three equations above are satisfied. We cannot distill unique values for g₃₃ and g₁₃ separately from the three equations. Any attempt to eliminate g₃₃ eliminates also g₁₃, and vice versa. The individual values of g₃₃ and g₁₃ depend on the choice of coordinates.
Let us assume that g₁₃ is identically zero everywhere: we have defined the z coordinate in the way where it is always orthogonal to r. Then we can solve g₃₃' explicitly from g₀₀. Let us check if the solution satisfies all the three equations for g₃₃.
We calculated on April 22, 2024:
d²g₁₁ / dz² = -r d²g' / dz²,
g₁₁' / r = d²g / dz².
The equation for R₂₂ gives:
g₃₃' = g' + g₁₁'
= g' + r d²g / dz².
Then
g₃₃'' = g'' + d²g / dz² + r d²g' / dz².
The equation for R₁₁ says:
g₃₃'' = g'' + g₁₁' / r - d²g₁₁ / dz²
= g'' + d²g / dz² + r d²g' / dz².
Thus, our solution for g₃₃ agrees with that equation. The equation about R₃₃ says:
g₃₃'' + g₃₃' / r - d²g / dz² + d²g₁₁ / dz² = 0
<=>
g'' + d²g / dz² + r d²g' / dz²
+ g' / r + d²g / dz²
- d²g / dz²
- r d²g' / dz² = 0
<=>
g'' + g' / r + d²g / dz² = 0.
The equation for R₀₀ implies the above:
2 R₀₀ = -g'' - g' / r - d²g / dz² = 0.
We showed that out choice of g₃₃ together with g₁₃ satisfies all equations of g₃₃.
We showed that we are allowed to make the skew g₁₃ zero and we can still find g₃₃ which satisfies the formulae of April 13, 2024.
The metric g₃₃ is strictly determined by g₀₀, once we have "gauge fixed" g₁₃ = 0. The metric g₁₁ is strictly determined even without gauge fixing.
Let us then consider the sum of the Schwarzschild perturbations for each M. The sum satisfies the April 13, 2024 formulae, but g₁₃ is not identically zero in the standard Schwarzschild coordinates.
Orthogonalization. The way to orthogonalize a rotationally symmetric metric is to start from r = 0, and draw the new coordinate lines Z = constant in such a way that they are always orthogonal to the r = constant coordinate lines. We obtain new coordinates, in which the reformulated metric G has the skew component G₁₃ zero.
Generalized Birkhoff's theorem for two masses. If the Einstein field equations have a rotationally symmetric solution for two spherically symmetric masses centered on the z axis, then the solution is very close to the sum of the Schwarzschild metric perturbations for each mass. Here we assume that the metric of time, g₀₀, is very close to the one derived from the newtonian gravity potential. We also assume weak fields.
Sketch of proof. Let us have fixed cylindrical coordinates r, φ, z where we place the two masses.
Let us assume that (r, φ, z, g) solves the Einstein equations exactly for the two masses. Using the orthogonalization above, we orthogonalize the z coordinate, yielding Z and G. Then (r, φ, Z, G) solves the Einstein equations exactly, and consequently, is an approximate solution of our April 13, 2024 formulae.
Let (r, φ, z, g') be the sum of the Schwarzschild perturbations for each mass. It is an approximate solution of the Einstein field equations, and is rotationally symmetric. We orthogonalize the z coordinate, yielding Z' and G'. Then (r, φ, Z', G') is an approximate solution of the Einstein equations and of the April 13, 2024 formulae.
We assumed that the metric of time, g₀₀, is very close to the one derived from the newtonian gravity potential. We have above shown that we can uniquely solve the April 13, 2024 formulae, starting from g₀₀. Let the solution be G''.
Both G and G' are approximate solutions of the April 13, 2024 formulae. If, for example, G₁₁ differs substantially from G₁₁'', then some intermediate result in our calculation of G'' would have a substantially different value for the metric G. This means that G would not be not an approximate solution of the April 13, 2024 formulae. A contradiction. The same applies to G₃₃.
Every component in G and G' must be very close to the corresponding component in G''. This means that G is very close to G'. Q.E.D.
We can further generalize Birkhoff's theorem to any number of spherically symmetric masses centered on the z axis.
Minkowski & newtonian agrees with general relativity for spherically symmetric masses placed on the z axis
Let us have a test mass m close to such a configuration of (small) masses M on the z axis. We define that the energy shipping of gravity field energy in this case happens individually from the center of each mass M to m. Then the metric which Minkowski & newtonian simulates is the sum of the Schwarzschild perturbations for each M.
If general relativity has a solution for this configuration, then generalized Birkhoff's theorem implies that the solution is very close to what Minkowski & newtonian says.
A thin disk in Minkowski & newtonian: g₃₃ differs a lot from general relativity
Let us try to figure out what Minkowski & newtonian says about a thin disk.
On March 14, 2024 we showed that general relativity is horribly nonlinear for a spherical mass shell. We cannot obtain a reasonable metric by summing the Schwarzschild perturbations for each atom in the shell.
r
^
|
| | m test mass
-----|------- • -------> z
|
thin disk
^
|
• m test mass
Above, we have already calculated that Minkowski & newtonian agrees with general relativity in the g₁₁ metric, if we assume that the field energy is carried over the distance r when we lower the test mass m closer to a rotationally symmetric system.
Our April 13, 2024 formulae give, assuming that the skew g₁₃ is identically zero:
g₃₃' = g' + g₁₁'
= g' - r d²g / dz²
= -r g''.
The integral function:
d(-r g' + g) / dr = -g' - r g'' + g'
= -r g''.
Then
∞
g₃₃(r) = 1 - / (-r g' + g)
r
= 1 - (-1 + r g' - g)
= 2 + g - r g'.
If we have a test mass m on the z axis, and we move it closer to the thin disk, then the gravity field does work on m.
^ r
|
| thin disk
| |
| |
0 -- | <---- • m test mass
| |
| |
|
------|----------------------> z
0
Let m be at a location (0, z) close to the disk, and let m move toward the disk. The gravity force on m is almost constant.
Most of the gravity field pull energy for m is shipped from a short distance, ~ 2 z. As m comes closer to the disk, the metric g₃₃ should approach 1, according to Minkowski & newtonian.
This is in contradiction to general relativity, which says that on the z axis:
g₃₃ = 2 + g₀₀.
As m goes to a lower potential, g₀₀ < 0 increases, and g₃₃ > 1 grows.
Here Minkowski & newtonian differs a lot from general relativity. If we have a disk-like gravitational lens, and we view it from a flat side, the gravity lens effect in Minkowski & newtonian is much less than in general relativity.
● M
<-- • m
-----------------------------> z
General relativity claims that the edges of the disk stretch the g₃₃ metric surprisingly much when m is close to the disk. This is in contrast to what general relativity says about a single point mass M: if M would be almost directly above m, then the metric component g₃₃ would almost exactly be 1. It is counterintuitive to claim that if we put many such masses M in a ring around m, then g₃₃ somehow becomes stretched.
We have to look at gravitational lenses in the sky. Is there any data about a flat lens viewed from the flat side?
Astronomers (wrongly) seem to use Schwarzschild perturbations to calculate the lens effect. But for a flat lens, the g₃₃ metric differs greatly from the sum of Schwarzschild perturbations for each mass element in the lens.
Discussion of general relativity and a thin disk: one can fool general relativity by imitating a certain gravity field
Question. Can we derive something clearly contradictory or strange from the g₃₃ metric of general relativity for a thin disk?
The stretched metric g₃₃ is like if general relativity would think that the flat disk is a part of a large spherical shell of mass, and the gravity energy is shipped from the center of this large shell, over a very large distance. Consequently, a test mass m has hard time moving in the z direction: the g₃₃ metric appears stretched!
The newtonian gravity field of the disk close to surface of the disk is almost homogeneous. Locally, the field does look like that of a large spherical shell. The formulae of April 13, 2024 determine the metric quite locally from the newtonian gravity potential V. They are not aware of the big picture. This explains why general relativity believes that g₃₃ is substantially stretched.
It is as if we could fool general relativity by constructing an imitation of a certain gravity field. One cannot fool Minkowski & newtonian because it looks at the genuine energy shipping distance. It cannot be fooled with the local field.
Conclusions
Let us close this very long blog post. We saw that Minkowski & newtonian and general relativity agree for spherical masses placed on the z axis. For most astronomical objects, the two gravity models produce essentially identical results.
Our generalized Birkhoff's theorem is a new way to prove exact results about general relativity. We have to assume that the metric of time g₀₀ closely imitates the newtonian gravity potential, and that the fields are weak. Then we can prove results of the type:
if general relativity has a solution, the solution necessarily has a certain property.
Proving the existence of a solution is almost impossible in general relativity. Our technique circumvents that problem.
We also showed that Minkowski & newtonian and general relativity differ greatly in their predictions of the z metric of a thin disk facing to the z direction. General relativity only looks at the local field and thinks that it is the field of a large spherical shell of mass, while Minkowski & newtonian considers the whole system. We will investigate this peculiar feature of general relativity in another blog post. Does it make sense?
