Kostik
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- TL;DR Summary
- Does the (strong) Equivalence Principle imply that a free-falling observer follows a geodesic?
From https://www.astro.gla.ac.uk/users/norman/lectures/GR/part4-screen.pdf:
"It is possibly not obvious, but the Strong EP also tells us how matter is affected by spacetime. In SR, a particle at rest in an inertial frame moves along the time axis of the Minkowski diagram – that is, along the timelike coordinate direction of the local inertial frame, which is a geodesic. The Strong EP tells us that the same must be true in GR, so that this picks out the curves generated by the timelike coordinate of a local inertial frame, which is to say: Space tells matter how to move: Free-falling particles move on timelike geodesics of the local spacetime."
I am finding this explanation a little murky. Where does the geodesic equation appear? In the spirit of the "comma-goes-to-semicolon rule", the way to apply the EP should be:
1. Make a statement about a physical law in an inertial frame (flat spacetime);
2. Write it as a tensor equation, using the fact that the Christoffel symbols vanish in flat space with rectilinear coordinates;
3. Argue that a tensor equation is the same in all coordinate systems, so the covariant equation (this should be the geodesic equation!) is correct in any coordinate system.
Can anyone take a stab at a better explanation?
EDIT: I'm guessing the geodesic equation should be written as a tensor equation: $${v^\mu}_{;\sigma} \, v^\sigma = 0$$ which is the same as $$\frac{dv^\mu}{ds} +\Gamma^\mu_{\nu\sigma}v^\nu v^\sigma \, .$$
"It is possibly not obvious, but the Strong EP also tells us how matter is affected by spacetime. In SR, a particle at rest in an inertial frame moves along the time axis of the Minkowski diagram – that is, along the timelike coordinate direction of the local inertial frame, which is a geodesic. The Strong EP tells us that the same must be true in GR, so that this picks out the curves generated by the timelike coordinate of a local inertial frame, which is to say: Space tells matter how to move: Free-falling particles move on timelike geodesics of the local spacetime."
I am finding this explanation a little murky. Where does the geodesic equation appear? In the spirit of the "comma-goes-to-semicolon rule", the way to apply the EP should be:
1. Make a statement about a physical law in an inertial frame (flat spacetime);
2. Write it as a tensor equation, using the fact that the Christoffel symbols vanish in flat space with rectilinear coordinates;
3. Argue that a tensor equation is the same in all coordinate systems, so the covariant equation (this should be the geodesic equation!) is correct in any coordinate system.
Can anyone take a stab at a better explanation?
EDIT: I'm guessing the geodesic equation should be written as a tensor equation: $${v^\mu}_{;\sigma} \, v^\sigma = 0$$ which is the same as $$\frac{dv^\mu}{ds} +\Gamma^\mu_{\nu\sigma}v^\nu v^\sigma \, .$$
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