Update on cylinder metrics

Once we removed (?) calculation errors in the cylindrical coordinates, it turned out that finding an approximate static metric g for any cylindrically symmetric static mass distribution is surprisingly easy. We can proceed like this:

1.   Assume that the metric is orthogonal. Make g₂₂ fixed r². Assume that fields are weak.

2.   Determine the metric g₀₀ from the newtonian gravity potential V of the mass.

3.   Solve g₁₁' from four equations.

4.   Solve g₃₃.

We ignore all terms which are a product of two "perturbations".

Thus, general relativity, in its approximate form, does have quite a lot of "freedom" to allow a solution.

Treating the z coordinate as a "time" coordinate in a static configuration: shear

On November 5, 2023 we tentatively proved that a time-dependent pressure breaks the Einstein field equations.

          ^  r

          |

          |

           -------------------------------> z "time"

Currently, we are studying if we can treat the z axis like a "time" coordinate in a static cylindrical setup. Can the Einstein field equations handle "changes" in pressure and shear stresses when we travel along the z axis?

Having just radial pressure does not require any "interaction" between different values of z. An approximate metric is found easily. Shear stresses add an "interaction" on the z axis. Can the Einstein equations handle that?

Gravitational lensing in the sky and the metric around a cylinder

Our formulae suggest that the vacuum spatial metric around a cylinder is strange: the spatial metric is stretched in the z direction, but not in the radial r direction. Though, this has to be checked carefully. In the Levi-Civita metric, also the r and φ dimensions are stretched. We are interested in gravitational lensing, and have to determine proper distances for rays of light. Different coordinate systems can confuse the "true" perturbation of the spatial metric, for an actual ray of light which travels through the perturbed metric.

For a point mass, a half of the gravitational lensing effect comes from the stretching of the r metric. Should we see surprisingly little gravitational lensing around an elongated object when viewed from the side? We are now looking at the literature and at actual astronomical observations.

https://link.springer.com/article/10.1007/s11214-024-01051-8

P. Natarajan et al. (2024) calculated the dark matter distribution of the galaxy cluster MACS 0416 using the Lenstool software (the picture above is from Image 3 in their paper).

Questions:

1.   Is Lenstool aware of the metric which the Einstein equations give around a cylinder?

2.   If yes, do the actual observations match that metric?

The "true" perturbation of the spatial metric

Suppose that we have an approximate solution for the Einstein field equations inside or outside a finite cylinder.

In our calculations we have defined r as 1 / (2 π) of the proper length of a circle drawn with r as the radius.

It could still be that the spatial metric is stretched in both the φ dimension and the r dimension. In that case, there is gravitational lensing, even though, superficially, the metric appears flat close to the cylinder.

How to detect such stretching in the metric? Far away from the cylinder, the metric approaches the Schwarzschild metric, in which there is no stretching in the φ dimension. Does this rule away the possibility that φ is stretched close to the cylinder?

Conclusions

We will check if our formulae give the correct gravitational lensing around a finite cylinder. Far away, the metric is almost Schwarzschild. How does that affect the lensing?

We will check if astronomical observations match general relativity.

We will check if shear stresses break the Einstein equations inside a long cylinder.