UPDATE March 31, 2024: After correcting the 1 / g₁₁ factor there no longer is a sign error!
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UPDATE March 31, 2024: We added a missing 1 / g₁₁ factor to Γ¹₁₁ and Γ¹₂₂.
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UPDATE March 26, 2024: The error seems to be in the sign of the cross term Γ¹₁₁ Γ¹₂₂ in Wikipedia. The Wikipedia formula is present already in Albert Einstein's 1915 papers. Is the sign convention different there?
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(Pictures: Wikipedia)
An obvious difference between a ball and a cylinder is in the z coordinate. It does not affect anything for a cylinder, but has a prominent effect in the case of a ball.
In our previous blog posts we have used cartesian coordinates. Let us now use polar or cylindrical coordinates, in order to check the calculations that we have made in the past two weeks.
Stretched radial metric in a 2D plane
We treat the radius r as the first coordinate and the angle φ as the second coordinate. The metric in polar coordinates is
ds² = g₁₁(r) dr² + r² dφ²,
that is, g₂₂(r) = r². We assume that g₁₁(r) is very close to 1.
We denote the derivative with respect to r by the prime '. The cross terms Γ * Γ are now important because g₂₂' is very large. In cartesian coordinates, the derivatives of each gij would be very small for weak fields.
Γ¹₁₁ = 1/2 * 1 / g₁₁ * g₁₁',
Γ²₁₁ = 1/2 * 1 / r² * -dg₁₁ / dφ
= 0,
Γ¹₂₁ = Γ¹₁₂ = 0,
Γ²₁₂ = Γ²₂₁
= 1/2 * 1 / r² * g₂₂'
= 1 / r,
Γ¹₂₂ = -1/2 * 1 / g₁₁ * g₂₂'
= -r / g₁₁,
Γ²₂₂ = 0,
R₁₁ = dΓ²₁₁ / dφ - dΓ²₁₂ / dr
+ Γ¹₁₁ Γ¹₁₁ + Γ²₂₁ Γ¹₁₁
- Γ¹₁₁ Γ¹₁₁ - Γ²₁₂ Γ²₂₁
= 0 + 1 / r²
+ 1 / r * 1 / 2 * g₁₁' / g₁₁
- 1 / r²
= 1/2 g₁₁' / g₁₁ * 1 / r.
The value agrees with our March 20, 2024 blog post.
R₂₂ = dΓ¹₂₂ / dr
+ Γ¹₁₁ Γ¹₂₂ + Γ²₂₁ Γ¹₂₂
- Γ²₂₁ Γ¹₂₂ - Γ¹₂₂ Γ²₂₁
= -1 / g₁₁ + r / (g₁₁)² * g₁₁'
+ 1/2 g₁₁' / g₁₁ * -r / g₁₁
+ r / g₁₁ * 1 / r
= 1/2 g₁₁' / (g₁₁)² * r.
The sign is not flipped relative to our March 20, 2024 post.
The formulae for the Christoffel symbols and the Ricci curvatures are incorrect above?
In the above calculations, if we replace the angle φ with a new variable θ:
φ = C θ,
then the constant C is absorbed into the value of the metric g₂₂(r): g₂₂ gets larger by a factor C².
In the above calculations, Γ¹₂₂ and R₂₂ grow by the factor C². The value of R₂₂ is coordinate-dependent!
What is going on? Are the formulae incorrect?
Our earlier Ricci curvature calculations in the past two weeks concerned perturbations of the metric: raising of indexes might not be too relevant.
Let us analyze.
The crucial term above in R₂₂ is
Γ¹₁₁ Γ¹₂₂,
especially
Γ¹₂₂ = 1/2 g¹¹ * -∂₁ g₂₂
= -1/2 * 1 / g₁₁ * g₂₂'.
The derivative is with respect to r, whose metric g₁₁ is close to 1. Raising the index on the infitesimal dx¹ only changes the value a little. Thus, the problem is not about raising indices in the derivatives.
The problem is that changing the variable from φ to θ inflates g₂₂ by a factor C². This inflation is nowhere compensated for.
We should "normalize" g₂₂ by dividing:
g₂₂(x) / g₂₂(x₀)
if we want to study Ricci curvature at a point x₀? This may correspond to the basis vector
ei
in the formula above? The metric is defined:
g₂₂ = e₂ • e₂.
Suppose that we do the change of the coordinate variable φ = C θ. Then the basis vector e₂ becomes C times longer. Thus, the basis vector is not a unit vector in terms of the proper length, as measured by the metric. Rather, the basis vector depends on the coordinates which we choose.
This is strange. The natural way to define basis vectors would be to make their proper length 1.
Christoffel symbols are not tensors. Their value depends on the choice of the coordinates.
The Wikipedia article cites Bishop and Goldberg (1968): Tensor analysis on manifolds when it gives the formula:
Let us check the proofs that the Ricci tensor is coordinate-independent, even though Christoffel symbols are heavily coordinate-dependent.
Solution of the mystery. Tensors are not coordinate-independent! They transform with simple rules into new coordinate systems. The stress-energy tensor T in the Einstein field equations contains, e.g., the mass density relative to the coordinates, not relative to proper distances.
In cylindrical coordinates, a uniform cylinder does not have a uniform density in those coordinates! We should not have R₁₁ = R₂₂ when using polar coordinates, even though the equality holds in cartesian coordinates.
Set the metric of the plane to the surface of a sphere: this conclusively proves that the result above is wrong
The metric is
ds² = g₁₁(r) dr² + r² dφ².
Let us set g₁₁(r) such that this describes the "polar cap" of a spherical surface:
g₁₁(r) = 1 / cos( arc sin(r / R) ),
where R is the radius of the sphere. As r grows, so does arc sin(r / R), and the cosine decreases. Thus, g₁₁'(r) > 0, if r > 0.
The Ricci curvature to every direction is positive on a spherical surface. But the calculation above claims that
R₂₂ = -1/2 g₁₁' / (g₁₁)² * r
< 0 !
Analysis of R₁₁ and R₂₂ on a spherical surface
Let us analyze how the formulae "detect" that the metric g is that of a polar cap. The detection should lead them to compute the curvature to be positive to every direction.
(Picture by Hellerick
https://commons.m.wikimedia.org/wiki/File:Division_of_the_Earth_into_Gauss-Krueger_zones_-_Globe.svg )
The polar cap is projected into a plane. Tangential distances around the origin are just like for a plane:
r dφ.
The meridians in the picture of the globe are the coordinate lines of φ. The latitude lines are some coordinate lines of r, but note that in the projection in the plane, the lines are not evenly spaced. They are evenly spaced in the proper distance, but not in the coordinate distance. The variable r is the coordinate in the plane.
The formulae cannot tell apart the polar cap from the plane looking at g₂₂(r) only. The difference is in g₁₁(r). The radial metric is stretched ever more when we move away from the origin:
dg₁₁(r) / dr > 0.
By looking at this derivative should the formulae realize that we are on a sphere.
Let us now analyze the formulae in detail.
1. Γ¹₁₁ = 1/2 * g₁₁' / g₁₁.
This symbol is the crucial one. It tells us how much the basis vector e to the positive direction of r becomes longer when we move along r to the positive direction. For a polar cap, this is positive. The surface of the sphere becomes ever steeper as we walk away from the North Pole.
2. Γ²₁₂ = Γ²₂₁ = 1/2 * 1 / r² * g₂₂' = 1 / r.
This symbol is only about polar coordinates, not about the spherical surface. It measures the "speed" at which meridian lines move away from each other as r grows. The "speed" is a relative speed: how many percents per a unit of r. The meridian lines projected to the plane are the usual coordinate lines of φ in polar coordinates.
3. Γ¹₂₂ = -1/2 * 1 / g₁₁ * g₂₂' = -r.
This symbol measures the "speed" in a different way: the sign is flipped and there is no "scaling" factor 1 / r². This is again only about polar coordinates, not about the spherical surface at all.
4. R₁₁ = dΓ²₁₁ / dφ - dΓ²₁₂ / dr
+ Γ¹₁₁ Γ¹₁₁ + Γ²₂₁ Γ¹₁₁
- Γ¹₁₁ Γ¹₁₁ - Γ²₁₂ Γ²₂₁.
The symbols which are only associated with polar coordinates must yield a zero sum: a plane has zero curvature. The crucial non-zero term is
Γ²₂₁ Γ¹₁₁ = 1 / r * 1 / 2 * g₁₁' / g₁₁.
Qualitatively, the term says that the coordinate lines associated with the variable 2, or φ, become more distant from each other as r grows, and that the radial metric is stretched as r grows.
\ / coordinate lines of φ
\ /
\/
North Pole
stretch r when r is large =>
| |
| |
\ /
\/
North Pole
We see that the lines start to bend toward each other. This is just what happens when two people walk down from the North Pole along meridians. At the equator, they will walk in parallel.
We see that stretching the metric of r far away from the origin adds positive Ricci curvature. The formula for R₁₁ makes sense.
5. R₂₂ = dΓ¹₂₂ / dr
+ Γ¹₁₁ Γ¹₂₂ + Γ²₂₁ Γ¹₂₂
- Γ²₂₁ Γ¹₂₂ - Γ¹₂₂ Γ²₂₁.
The only term associated with the spherical surface is
Γ¹₁₁ Γ¹₂₂ = 1/2 g₁₁' / (g₁₁)² * -r.
The term is not logical. The symbol Γ¹₂₂ tells us that the meridian lines become more distant when r grows, and Γ¹₁₁ says that the metric of r is stretched when r becomes larger. The term should add positive curvature. But because of the minus sign in front of r, it adds negative curvature!
Why is there a minus sign in Γ¹₂₂?
Recall that Γ¹₂₂ measures the "speed" how fast coordinate lines of φ distance themselves from each other.
The term
∂₁ Γ¹₂₂
in Rij checks if the coordinate lines bend toward each other as r grows. If the speed slows down, that means positive curvature, as we explained above. The minus sign is required to flag this as positive curvature.
Conclusions
The formula for Ricci curvature in Wikipedia seems to be in error. The same formula is found in many sources. Maybe sign conventions save it there?
The problem is that in Rij, the derivative requires a different sign for Γ¹₂₂ than the cross term Γ¹₁₁ Γ¹₂₂ requires.
We are not sure about how to correct the formula.
Our own method of calculating perturbations of metric in cartesian coordinates is immune to the error, because we can ignore the cross terms Γ * Γ.
We will check if other methods of calculating the Christoffel symbols and Ricci curvature are correct. People on the Internet have calculated Ricci curvature for a polar cap using various methods – and did get a positive curvature to all directions. Thus, these various methods probably work correctly.
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