Trying to teach myself general relativity. I sort of understand the derivation of the geodesic equation $$\frac{d^{2}x^{\alpha}}{d\tau^{2}}+\Gamma_{\gamma\beta}^{\alpha}\frac{dx^{\beta}}{d\tau}\frac{dx^{\gamma}}{d\tau}=0.$$ which describes "how" objects move through spacetime. But I've no idea "why" they move along geodesics.
Is this similar to asking why Newton's first law works? I seem to remember reading Richard Feynman saying no one knows why this is, so maybe that's the answer to my geodesic question?
Best Answer
You could think of it this way:
1) Take a free particle, put it at some spacetime point, and leave it evolve.
2) Imagine the motion is not geodesic, that is $a_\mu\equiv v^\nu v_{\mu;\nu}\neq 0$, or in other words the acceleration is not zero. Note: We know that $a_\mu v ^\mu = 0$, or the 4-acceleration is normal to 4-velocity.
3) Imagine you are that very particle, that is you are in the reference frame where $v^\mu=(0,0,0,1)$. Because 4-acceleration and 4-velocity are orthogonal, you shall still "see" non-zero 3-vector of acceleration in this frame. I shall not elaborate much on this see, but if you write the equations of motion of test particles located around you, you shall see them accelerating in the direction of $\bf{a}$. I refer to the chapter on comoving reference frames.
Now the punchline. As inertial mass is equivalent to passive gravitational mass, you may never distinguish whether you are standing still or moving in a gravitational field. But if you can see an appearing 3-acceleration, then you actually can distinguish by realising that you are not standing still. Hence the contradiction.
To conclude, the fact that everything moves along the geodesics is closely related to the equivalence principle.