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Most people assume that in order to have agreement between GR and the rotation curve of galaxies:

  1. GR isnt correct

  2. There is hidden matter which makes the galaxies rotate faster at their edges.

GR is a set of differential equations with only very few solutions known(Schwarzwild, Kerr metric, linearized GR) however why cant both GR be correct and the rotation of the galaxies be explained at the same time?

Maybe if we COULD have analytical solution of GR for every case , then we would see that the set of differential equations produce the rotation of the galaxies observed by our telescopes...

ZaellixA
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appliedSciences
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We don't need exact analytic solutions to understand a theory's predictions. General relativity in particular reduces to Newtonian gravity in the weak-field (and low-velocity) limit, which is where changes in the Newtonian gravitational potential $\Phi$ are much smaller than $c^2$ (and velocities are much smaller than $c$). General relativistic corrections to Newtonian gravity are suppressed by powers of $\Delta\Phi/c^2$ (and $v/c$, with the lowest velocity correction being at second order).

In galaxies, the potential depth $|\Phi|/c^2\sim 10^{-6}$, so Newtonian gravity is accurate to about one part in a million. This is why we are satisfied with using Newtonian gravity to model galaxies.

See also How can we recover the Newtonian gravitational potential from the metric of general relativity?

Sten
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  • But Netwonian gravity isnt described by the same differential equations with GR so maybe trying to relate these 2 ISNT correct because in differential equations small changes in initial conditions or coefficients or even the structure of the differential equations can cause big changes in the results.That was the whole point of my question... – appliedSciences Mar 16 '23 at 16:51
  • Maybe the approximation isnt good ENOUGH because maybe the analytical solutions of GR produce something entirely different yes it works for the Earth and its low intensity gravitational field but since we are changing the structure of the differential equations describing gravity we change the solutions to all of its domain , maybe by taking the approximation we delete some features of the analytical solution of GR. – appliedSciences Mar 16 '23 at 16:56
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    @appliedSciences it is not necessarily true that small changes in initial conditions lead to large changes in outcomes in every differential equations. – Triatticus Mar 16 '23 at 18:27
  • @Triatticus but we dont change only the initial conditions we even change the structure of the differential equations describing gravity. – appliedSciences Mar 16 '23 at 19:12