How Does Acceleration Affect the Geodesic Equation in General Relativity?

[Karlisbad]

Whenever there's no acceleration then: (Geodesic equation)

$$\nabla _{u} u =0$$

where the covariant derivative includes "Christoffel symbols" so $$\Gamma_{kl}^{i} =0$$ for an Euclidean space-time..however when generalizing to a system under an acceleration then Gedesic equation becomes:

$$\nabla _{u} u -a^{\mu}=0$$ (foruth dimensional acceleration)

for a test particle $$\acute{{R}^{\mu}}_{\alpha \nu \beta} \acute{u}^{\alpha} \acute{x}^{\nu} \acute{u}^{\beta} = - \acute{f}^{\mu}$$

where does this 4-dimensional force come from?? but if you use "Weak field approximation" then the potential of the particle (and force) is: $$\Gamma^{i}_{00}$$

and using all that how you derive Einstein field equation so $$R_{ab}=0$$

 

[Chris Hillman]

What causes acceleration?

where does this 4-dimensional force come from??

From some non-gravitational physical interaction or process.

Some typical examples:

1. In a static spherically symmetric perfect fluid solution (often used as an idealized stellar model), in the interior of the fluid ball, bits of fluid are accelerated radially outward by the differential pressure of the fluid, so that their world lines are timelike but nongeodesic curves. This is simply "hydrostatic equilibrium".

2. In the vacuum region outside such a fluid ball (modeled by a portion of the famous Scharzschild vacuum solution), we might consider "static test particles". These would also have world lines with outward pointing radial acceleration vector. We can think of such test particles as tiny rocket ships firing idealized rocket engines directly inward with just the right thrust to oppose the gravitational attraction of the "star". We would not of course generally attempt to model such rocketry using gtr! Fortunately, in the test model approximation, we don't have to.

3. In an "electrovacuum solution" (say the region outside an isolated charged object), a charged particle will in general be accelerated by the Lorentz force, just as in ordinary electromagnetism, but fixed up to work in a curved spacetime.

A more exotic example: according to gtr, nonspinning uncharged test particles (in an electrovacuum region outside a rotating magnetized neutron star, say) will have world lines which are timelike geodesics. However, if the test particle is spinning, in principle it will experience tiny spin-spin forces which will force its world line off a geodesic. In terms of the Bel decomposition of the Riemann curvature tensor, these spin-spin forces can be computed from the magnetogravitic tensor in a manner similar to how one can compute tidal forces from the electrogravitic tensor. They work a bit like the electromagnetic interaction of two dipole magnets, although such analogies are potentially misleading.

I should also stress that when you crunch the numbers, in most scenarios spin-spin forces turn out be very weak. Nonethelss, you might be wondering: could two spinning black holes, with coaxial and opposite spins, perhaps, in principle, be held in static equilibrium by the spin-spin force counteracting their mutual "mass monopole" gravitational attraction? This question was the focus of a protracted debate in the research literature some years ago, but this appears to have been settled in the negative.

Chris Hillman