Gravitational lensing outside a finite cylinder

Our April 21, 2024 post brought up a very important observation: we can test general relativity by looking at gravitational lenses in the sky.

Gravitational lenses in the sky

Our calculations suggest that the approximate spatial metric perturbation around a finite cylinder is very different from the perturbation which we obtain by summing Schwarzschild metric perturbations for the mass elements which make up the cylinder.

But in the literature no one mentions that the gravitational lens of an elongated object would be anything special. There are filament-like structures in the sky. These should act like cylinders as gravitational lenses.

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

Above, in the image, we have the inferred dark matter distribution in the 2024 paper by N. Natarajan et al. (Figure 3, galaxy cluster MACS 0416). The dark matter is shown as blue haze.

https://arxiv.org/abs/1711.02109

M. Annunziatella et al. (2017) present another mass density profile for MACS 0416 (Figure 2 in the paper).

Our calculations on April 13, 2024 suggest that the lens effect near a finite cylinder comes solely from the newtonian gravity effect on time, g₀₀, not from a distorted radial spatial metric.

In the Schwarzschild metric, the lens effect is twice the newtonian gravity effect, because of the stretched radial metric.

If the authors use a sum of Schwarzschild metric perturbations, they will think that the mass of the filament is only 1/2 of the mass predicted by general relativity.

At first sight, the image above does not "look like" that the mass of the filament between the two large mass concentrations would be underestimated.

It could well be that the Schwarzschild metric is the correct way to model cylinder lens effects! Then general relativity is wrong.

https://arxiv.org/abs/1010.3829

Matthias Bartelmann (2010), in section 1.4 of the paper in the link, writes that one uses a "newtonian metric", whose formula (28 in  the paper) seems to be the Schwarzschild metric.

https://arxiv.org/abs/astro-ph/0608407

Douglas Clowe et al. (2006) analyzed the lensing "strength" in the Bullet Cluster (Figure 1 in their paper). The lensing looks a lot like two Schwarzschild metric perturbations.

If the mass distribution is two spherical masses, like in the Bullet Cluster, then we get a very good approximation for the metric by summing the Schwarzschild perturbations. The cylinder metric requires that there is a continuous filament of mass.

Most papers about lensing ignore the discussion of general relativity.

The entire metric around a finite cylinder

Let us use our formulae to approximate the vacuum metric around a finite cylinder.

2 R₁₃  =  dg' / dz  +  1 / r * dg₁₁ / dz              = 0,

                                                        

2 R₀₀  =  -g''  -  g' / r   -  d²g / dz²                    = 0,

                                                         

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.

We will delay this calculation to a later time.

Conclusions

The observations about the gravitational lens effect of a filament in the sky are not accurate enough, yet. They do not reveal if general relativity calculates correctly the metric inside a cylinder and around it.