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Kostik

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- TL;DR Summary
- Dirac shows that, along a world line, ##\frac{d\rho}{ds} = -\rho {v^\mu}_{;\mu}.## He states "this is a condition that fixes how ##\rho## varies along the world line of an element of matter."

Dirac ("GTR" p. 47) makes an interesting observation immediately after obtaining Einstein's field equations with the simple energy-momentum tensor ##T^{\mu\nu}=\rho v^\mu v^\nu##. (##v^\mu## is the four-velocity.)

First, the conservation of matter ##\left( \rho v^\mu \sqrt{-g} \right)_{,\mu}=0## implies ##\left( \rho v^\mu \right)_{;\mu}=0##. Hence $$\rho_{;\mu}v^\mu + \rho {v^\mu}_{;\mu}=0. \quad(*)$$ On a geodesic, Dirac previously showed that ##{v^\mu}_{;\nu}=0## so the second term in ##(*)## vanishes, hence ##\rho_{;\mu}=\rho_{,\mu}=0##. This doesn't seem surprising: a small element of matter in free-fall shouldn't experience a change in its CMFR density ##=\rho##.

However, Dirac takes ##(*)## one step further. We have $$\frac{d\rho}{ds} = \frac{\partial\rho}{\partial x^\mu}v^\mu = -\rho {v^\mu}_{;\mu}.$$ He states "this is a condition that fixes how ##\rho## varies along the world line of an element of matter."

This is intriguing.

I suppose that if a small element of matter is

First, the conservation of matter ##\left( \rho v^\mu \sqrt{-g} \right)_{,\mu}=0## implies ##\left( \rho v^\mu \right)_{;\mu}=0##. Hence $$\rho_{;\mu}v^\mu + \rho {v^\mu}_{;\mu}=0. \quad(*)$$ On a geodesic, Dirac previously showed that ##{v^\mu}_{;\nu}=0## so the second term in ##(*)## vanishes, hence ##\rho_{;\mu}=\rho_{,\mu}=0##. This doesn't seem surprising: a small element of matter in free-fall shouldn't experience a change in its CMFR density ##=\rho##.

However, Dirac takes ##(*)## one step further. We have $$\frac{d\rho}{ds} = \frac{\partial\rho}{\partial x^\mu}v^\mu = -\rho {v^\mu}_{;\mu}.$$ He states "this is a condition that fixes how ##\rho## varies along the world line of an element of matter."

This is intriguing.

**Does this mean that along a world line***other*than a geodesic, the CMFR density ##\rho## is changing by a factor of ##{v^\mu}_{;\mu}##?*Why?*I suppose that if a small element of matter is

*not*moving on a geodesic then it must be subject to some non-gravitational force, but the factor ##{v^\mu}_{;\mu}## doesn't contain any information about such a force.*Can someone shed a little light here?*
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