On April 13 and April 22, 2024 we derived two equations for g₁₁ in vacuum:
dg₁₁ / dz = -r dg₀₀' / dz,
and
dg₁₁ / dr = r d²g₀₀ / dz².
Since
d²g₁₁ / (dr dz) = d²g₁₁ / (dz dr),
we obtain an equation for g₀₀. Do all gravity potentials satisfy that equation?
"NinjaDarth" (2023) gives the Schwarzschild metric in cylindrical coordinates:
There, ρ corresponds to our r coordinate, and rs is the Schwarzschild radius. We recently checked that the Schwarzschild metric in cylindrical coordinates has the Ricci tensor R = 0, when the curvatures are calculated with our formulae of April 13, 2024. Maybe the formulae are finally correct after a month of polishing?
---- ●●●●---●●● ----> z axis
spheres
Thus, we know that the Schwarzschild metric satisfies all our equations. Since the equations are linear, any combination of spheres centered on the z axis satisfies all the equations, even if the spheres overlap.
But is it possible to create all cylindrically symmetric mass distributions from overlapping spheres centered on the z axis?
There exists a metric in vacuum around a thin disk?
z = 0
|
--|---------> z axis
|
thin circular disk
A thin uniform circular disk might be one which cannot be combined from centered spheres.
Let us check if its gravity potential V satisfies
d²g₁₁ / (dr dz) = d²g₁₁ / (dz dr).
The metric of time g₀₀ we, for simplicity, denote just by g. We calculate in vacuum around the thin disk.
-dg' / dz - r dg'' / dz = r d³g / dz³
<=>
-g'' / dz - 1 / r * dg' / dz - d³g / dz³ = 0.
We have
-g'' - g' / r - d²g / dz² = 0
in vacuum for all newtonian gravity potentials. It implies the above equation!
What about higher partial derivatives? They do match, too. We have to look up a mathematical theorem which proves that g₁₁ has a well defined value then.
The asymptotic condition g₁₁ = 1, g₃₃ = 1 at infinity
The equations
dg₁₁ / dz = -r dg₀₀' / dz,
and
dg₁₁ / dr = r d²g₀₀ / dz²
define the value of g₁₁(r, z) in two ways: assume that g₁₁(r, z) = 1 very far away. Integrate the equations above to calculate g₁₁(r, z) in two ways. Do the values match?
The gravity potential V of a point mass m at the origin, obviously, satisfies this condition. The same holds for a point mass at any location. The equations above are linear. Any mass distribution can be formed as a sum of point masses. Hence, the condition holds for any mass distribution.
In our equations of April 13, 2024, only the r derivative of g₃₃ appears. The z behavior of g₃₃ has to be determined by integrating from infinity.
Varying pressure along z leads to a contradiction inside a cylinder?
Let u(r, z) be the pressure in the z direction inside a cylinder and s(r, z) the shear between r and z.
The stress-energy tensor is
T =
ρ 0 0 0
0 0 0 s
0 0 0 0
0 s 0 u.
The trace is
Tr = -ρ + u,
and the Ricci tensor
R = T - 1/2 Tr g
=
1/2 (ρ + u) 0 0 0
0 1/2 (ρ - u) 0 s
0 0 1/2 r² (ρ - u) 0
0 s 0 1/2 (ρ + u).
The equations are:
2 R₁₃ = dg' / dz + 1 / r * dg₁₁ / dz = 2 s,
2 R₀₀ = -g'' - g' / r - d²g / dz² = ρ + u,
2 R₃₃ = -g₃₃'' - g₃₃' / r + d²g / dz²
- d²g₁₁ / dz²
+ 2 dg₁₃' / dz
+ 2 / r * dg₁₃ / dz = ρ + u,
2 R₁₁ = g'' - g₃₃'' + g₁₁' / r - d²g₁₁ / dz²
+ 2 dg₁₃' / dz = ρ - u,
2 R₂₂ / r²
= g' / r - g₃₃' / r + g₁₁' / r
+ 2 / r * dg₁₃ / dz = ρ - u.
Summing the equations for R₀₀, R₁₁, and R₂₂, and subtracting the equation for R₃₃ gives:
2 g₁₁' / r - 2 d²g / dz² = 2 ρ - 2 u
<=>
g₁₁' / r = d²g / dz² + ρ - u
<=>
dg₁₁ / dr = r d²g / dz² + ρ r - u r.
The first equation gives:
dg₁₁ / dz = -r dg' / dz + 2 r s.
Then,
d²g₁₁ / (dz dr) = -dg' / dz - r dg'' / dz
2 s + 2 r s'
=
d²g₁₁ / (dr dz) = r d³g / dz³
+ r dρ / dz - r du / dz.
<=>
-dg'' / dz - dg' / dz - d³g / dz³
= dρ / dz - du / dz - 2 s / r - 2 s'
= dρ / dz + du / dz,
where the last line comes from the equation for R₀₀.
We obtain:
du / dz = -s / r - s'.
Is this equation reasonable? It looks strange.
<----- F₁
--------
u₁ -----> | box | <-- u₂
--------
F₂ -->
^ r
|
---------------------------------> z
The shear force F grows as r grows. The unit box 1 × 1 × 1 in the diagram feels a horizontal net force from the shear. The net force causes the pressure to be different on the left side and the right side of the box. We have omitted the shear forces in the r direction.
Thus, the term -s' makes sense. But the term -s / r is not reasonable.
In this example we did not need to assume anything about g₁₃.
Discussion about "dynamic" systems and the Einstein field equations
On November 5, 2023 we tentatively proved that the Einstein field equations cannot handle a changing pressure. There exists no solution for the equations.
In March and April 2024, we tried to solve an approximate metric for a rotationally symmetric system, often a cylinder. It looks like that we do get a very good approximate solution in the cases where there is no shear inside the cylinder. For a mass distribution with no pressure, the solution can be directly calculated from the newtonian gravity potential V of the mass. Solving the equations succeeds inside the cylinder, and in the vacuum surrounding it.
But a shear, even in the very simple form of a tangential pressure along the φ dimension, seems to prevent solutions of the Einstein field equations.
If we interpret a spatial dimension as a "time" coordinate, a shear seems to introduce a "dynamic" behavior to the system. The problem with the Einstein equations may be that they cannot handle a "dynamic" behavior.
For a century, people have, in vain, tried to prove that solutions do exist for the Einstein field equations. The best results so far, by Demetrios Christodoulou and Sergiu Klainerman, and others, show that a small perturbation of the Minkowski metric in vacuum is allowed. There exists a solution for the Einstein field equations in that case.
If there were an efficient iterative method which solves the Einstein field equations, then it might be relatively simple to prove that exact solutions exist. Just iterate the method and prove that it (rapidly) converges to an exact solution.
On March 19, 2024 we wrote that if one would always be able to find a small metric perturbation h which approximately solves the Einstein equations for a small metric tensor ΔT, then we would probably have a very fast converging solution of the Einstein field equations.
Let us try to solve the metric g in vacuum through an iterative method. Let g be an approximate solution. Let
R - 1/2 R g = ΔT.
We then look up a small perturbation h such that
R - 1/2 R h ≈ ΔT.
We get a better approximate solution from the metric
g - h.
Our calculations in the past two months have shown that:
1. if ΔT is physically impossible – does not respect conservation laws, and so on – then there probably is no h;
2. if ΔT contains shear, then there probably is no h.
If our iterative method produces such ΔT, then the method does not work.
Conclusions
We now have evidence that general relativity fails in "dynamic" systems. The "dynamics" can be spatial as well as temporal.
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