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The following argument is used to determine the unknown factors (e.g., $A(r)$ and $B(r)$) in the Schwarzschild metric. $$ \lim_{r \to ∞}A(r) = \lim_{r \to ∞}B(r) = 1 \space\space\space\space\space\space \Rightarrow \space\space\space\space\space\space A(r)=1/B(r). $$ [For example, Weinberg, Steven. Gravitation and cosmology. p. 186.]

Because of this step, it seems to me that the geodesic equation is valid only for a motion of a free-falling particle, which starts with zero velocity at infinity in the Schwarzschild space.

For example, a planet bound to a star appears to be an object that does not meet the above condition of "zero velocity at infinity".

Is this my understanding correct?

If so, how is this problem solved in known gravity theories?

Qmechanic
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SOQEH
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2 Answers2

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The geodesic equation in general relativity works to find the trajectory of any free point particle with any initial conditions (as long as the metric is $C^{1,1}$, anyway). While the existence and uniqueness of such geodesics is guaranteed, their solvability is another matter, which is why Schwarzschild geodesics are typically very specific examples.

You may remember that even in classical mechanics, the solution of an orbiting body cannot be found in closed form, that is, $r(t)$ has no closed form solution in general (this is why it's usually solved as $r(\theta)$ instead). The situation is not made any better in GR, and there is no generic solution to the geodesic equation for Schwarzschild. But there are certainly orbital solutions of the form $r(\theta)$ for orbital mechanics, as well as a few solvable orbits as long as you make some assumptions on the initial conditions.

Slereah
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The geodesic equation and its initial conditions are not used here. Firstly,

$$ A(r)B(r)~=~\text{constant} \tag{8.2.4}$$

follows from the vanishing of the Ricci-curvature. Secondly,

$$\lim_{r \to \infty}A(r) ~=~ \lim_{r \to \infty}B(r) ~=~ 1\tag{8.2.5}$$

follow from assuming the metric approaches the Minkowski metric as $r \to \infty$. Together they imply

$$ A(r)B(r)~=~1. \tag{8.2.6}$$

See also this related Phys.SE post.

Qmechanic
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