Is it true that the predictions from the General Theory of Relativity don’t match the observations of galaxy rotation speeds, and that this then started the search for dark matter? Is it only the rotation speeds of galaxies that are in conflict with GR or are there other observations that that don’t match the predictions as well?
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General Relativity has nothing to do with the galaxy orbital speed. General relativity describes strong gravitational fields and the global geometry of spacetime. However, in the limit of the weak field, General Relativity simply matches the Newtonian gravity since the Newtonian gravity already matches the observation for the weak field.
Because gravity on the galactic scale is extremely weak, the orbital speed contradiction is with the Newtonian gravity. There is no direct contradiction with General Relativity, but only through the Newtonian gravity. It is not conceptually impossible that one day someone finds some additional initial or border conditions or some other conditions that would make General Relativity at the galactic scale different from the Newtonian gravity thus resolving the contradiction.
Currently there are two main approaches to resolve this issue. One is to assume the existence of the Dark Matter. The pros are that this assumption explains the observations, including Gravitational Lensing. The cons are that the dark matter itself has not been observed and currently is not a part of the Standard Model of elementary particles. Another approach is modifying the Newtonian gravity on the galactic scale. One such an attempt is known as MOND or Modified Newtonian Dynamics. The pros are that the unobserved dark matter is not required. The cons are that the observational evidence is not fully explained and no obvious fundamental theoretical basis for the modification.
safesphere
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Um, if a galaxy can bend light around it, then a galaxies mass has to be taken as a whole when accounting for galactic rotational speeds at its edges. Because a galaxy as a whole is an extremely large mass, gravity and its effect on orbital velocities on the outer edge of a galaxy would be quite large. As general relativity describes the inertial effects of mass expanding outwards (the force we feel as gravity), that outward force of an entire galaxy taken as a whole should cause a mass on the outer edges of it to "pushed" to higher orbital velocities as space/time bends outward - the same as light being bent around a galaxy.
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General Relativity, in the week field regime and for low velocities (the so-called Post Newtonian expansion, to be distinguished from the case of week fields but arbitrary velocities, the so called Post Minkowskian expansion) reduces to newtonian gravity, so that Dark Matter is required by GR as by Newtonian gravity. This can be shown by using the gravitomagnetic equations (see arXiv:2207.09736 and references therein).The velocities of stars in a disk galaxy are of the order of 0.001c, so the resulting GR corrections to the rotation curves are of the order of 10^-6. It should be recalled that the rotation curve of an exponential disk (as observed in real disk galaxies) is NOT keplerian inside the disk, as well known since the '80. Dark Matter is only required from the rotation curve of HI well outside the optical (stellar) disk.
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The galactic disk is moving with some angular momentum. The local stars in the outer disk will be pulled along because of the neighbor stars. So the outer rim of the galactic disk is not a weak field in GR. Newtonian gravity will not work in this case and the full GR equations are needed to model the mass distribution with the angular momentum. How can the experts get this so wrong?
