UPDATE March 15, 2024: The Einstein approximation formula (see the post on August 15, 2023) produces roughly the Levi-Civita metric.
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Tullio Levi-Civita (1919) was able to find a cylindrical metric where Ricci curvature is zero.
In the formula above, D and σ are constants. Let us set D = 1.
In newtonian gravity,
dg₀₀ / dr ~ -1 / r
around a cylinder. In the Levi-Civita metric,
dg₀₀ / dr ~ -4 σ / r * r^(4 σ).
If we set σ to a small positive constant, then the Levi-Civita metric of time mimics quite accurately the newtonian solution. The Levi-Civita metric stretches the radial metric and the metric to the z direction equally.
y axis points
out of screen cylinder
●--------> z ==============
|
| r
|
| • m
v x
Let us set
g₀₀ = -1 - ln(r),
grr = 1 - ln(r),
gnn = 1 - ln(r),
g₃₃ = 1 - ln(r),
where gnn is the metric normal to the radius r and the z axis. We assume that the value of r is very close to 1.
We assume that r points approximately to the x direction, that is, |y| / |x| is small. Let us write
ds² = grr dr² + gnn dn² + g₃₃ dz²,
= (1 - ln(r)) * (dx² + dy² + dz²)
The metric of time distorted, spatial metric flat
Γ⁰₀₀ = 0,
Γ¹₀₀ = 1/2 * -dg₀₀ / dx
= 1/2 * 1 / x
= -Γ⁰₁₀ = -Γ⁰₀₁,
Γ²₀₀ = 1/2 * -dg₀₀ / dy
= 1/2 * y / x²,
R₀₀ = dΓ¹₀₀ / dx + dΓ²₀₀ / dy,
= -1/2 * 1 / x² + 1/2 * 1 / x²
= 0,
R₁₁ = -dΓ⁰₁₀ / dx
= 1/2 * 1 / x²,
Γ⁰₂₀ = 1/2 * -1 * dg₀₀ / dy
= 1/2 * y / x * 1 / x
= 1/2 * y / x²,
R₂₂ = -Γ⁰₂₀ / dy
= -1/2 * 1 / x²,
R₃₃ = 0.
The metric of time -1, the spatial metric stretched
R₀₀ = dΓ¹₀₀ / dx + dΓ²₀₀ / dy + dΓ³₀₀ / dz
= 0,
Γ²₁₁ = -1/2 dg₁₁ / dy
= -1/2 d(1 - ln(r)) / dr * dr / dy
= 1/2 * 1 / r * y / x
= 1/2 * y / x²,
Γ³₁₁ = dg₁₃ / dx - 1/2 dg₁₁ / dz
= 0,
Γ²₁₂ = 1/2 dg₂₂ / dx
= 1/2 * -1 / x,
Γ³₁₃ = 1/2 dg₃₃ / dx
= 1/2 * -1 / x,
R₁₁ = dΓ²₁₁ / dy + dΓ³₁₁ / dz
- dΓ²₁₂ / dx - dΓ³₁₃ / dx
= 1/2 * 1 / x² + 0
- 1/2 * 1 / x² - 1/2 * 1 / x²
= -1/2 * 1 / x².
Let us calculate R₂₂:
Γ¹₂₂ = -1/2 dg₂₂ / dx
= 1/2 * 1 / x,
Γ³₂₂ = dg₂₃ / dy - 1/2 dg₂₂ / dz
= 0,
Γ⁰₂₀ = -1/2 dg₀₀ / dx
= 0,
Γ¹₂₁ = 1/2 dg₁₁ / dy
= -1/2 * y / x²,
Γ³₂₃ = 1/2 dg₃₃ / dy
= -1/2 * y / x²
R₂₂ = dΓ¹₂₂ / dx + dΓ³₂₂ / dz
- dΓ⁰₂₀ / dy - dΓ¹₂₁ / dy - dΓ³₂₃ / dy
= -1/2 * 1 / x² + 0
- 0 + 1/2 * 1 / x² + 1/2 * 1 / x²
= 1/2 * 1 / x².
Then R₃₃:
Γ¹₃₃ = -1/2 dg₃₃ / dx
= 1/2 * 1 / x,
Γ²₃₃ = -1/2 dg₃₃ / dy
= 1/2 * y / x²,
R₃₃ = dΓ¹₃₃ / dx + dΓ²₃₃ / dy
- dΓ⁰₃₀ / dz - dΓ¹₃₁ / dz - dΓ²₃₂ / dz
= -1/2 * 1 / x² + 1/2 * 1 / x²
- 0 - 0 - 0
= 0.
Conclusions
The Levi-Civita metric is very simple, if we ignore the exponent 8 σ² of r in the first term. The spatial metric is stretched uniformly to x, y, and z directions, where the stretching depends on r.
We were able to confirm that the Ricci tensor is approximately 0.
The notion of uniform stretching of the spatial metric is strange. If our physical system consists of point particles moving at various speeds and bumping into each other, we can either interpret that local time has slowed down, or that all the distances have become longer.
Maybe we can measure "true" proper local distances through force fields, e.g., through the Coulomb field? If we double the distance, then the force is 1/4, while slowing down local time by a factor 1/2 only reduces the impulse given by the force by a factor 1/2. But this does not reveal a difference if we assume that the inertia of a charged particle doubled.
We here have a new relativity principle: relativity of time versus distances. In the cylinder case, if we interpret that the spatial metric is flat, and that local time has slowed down and the inertia of particles has increased, them the Ricci tensor is not zero.
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