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Derivation of geodesic equation in curved space using variational principle

by Octavian on Tuesday, June 16, 2009

The real path of the particle who moves in space under gravitational field is equal to the real path of particle in curved space without that field, since the gravitational field is just a manifestation of the curved space.

The action of this particle, then, depends on the line element of its path, since in the curved space there will be no external force acting on this particle, therefore its path must be a minimum-length. Define its action as S = -mc \int ds, with constants m and c are needed to match the dimension of action, and ds is the interval between two points in the curved space, ds^2 = g_{ab} dx^a dx^b. We will vary this action in order to find the real path, since for the real path the action must be minimum.

S = -mc \int ds

\delta S = -mc \int \delta(ds)

But we also have \delta(ds^2) = 2ds \delta(ds) = \delta g_{ab} dx^a dx^b + 2 g_{ab} \delta(dx^a) dx^b

Hence, \delta(ds) = \frac{1}{2} \delta g_{ab} \frac{dx^a}{ds} dx^b + g_{ab} \delta(dx^a) \frac{dx^b}{ds}

Then, \delta S = -mc \int \Big( \frac{1}{2} \delta g_{ab} \frac{dx^a}{ds} \frac{dx^b}{ds} ds + g_{ab} \frac{dx^b}{ds} \delta(dx^a) \Big)

\delta S = -mc \int \Big( \frac{1}{2} \frac{\partial g_{ab}}{\partial x^c} \frac{dx^a}{ds} \frac{dx^b}{ds} ds \delta x^c+ g_{ab} \frac{dx^b}{ds} \delta(dx^a) \Big)

Now use partial integral, we will have

\delta S = -mc \int \Big( \frac{1}{2} \frac{\partial g_{ab}}{\partial x^c} \frac{dx^a}{ds} \frac{dx^b}{ds} \delta x^c - \frac{d}{ds} \big( g_{ab} \frac{dx^b}{ds} \big) \delta(x^a) \Big) ds

Since the interval ds and the proper time d \tau differ by a constant, i.e. c, the speed of light, then we can interchange ds with d \tau,

\delta S = -mc \int \Big( \frac{1}{2} \frac{\partial g_{ab}}{\partial x^c} \frac{dx^a}{d \tau} \frac{dx^b}{d \tau} \delta x^c - \frac{d}{d \tau} \big( g_{ab} \frac{dx^b}{d \tau} \big) \delta(x^a) \Big) d \tau

\delta S = -mc \int \Big( \frac{1}{2} \frac{\partial g_{ab}}{\partial x^c} \frac{dx^a}{d \tau} \frac{dx^b}{d \tau} \delta x^c - \frac{\partial g_{ab}}{\partial x^c} \frac{dx^c}{d \tau} \frac{dx^b}{d \tau} \delta x^a - g_{ab} \frac{d^2 x^b}{d \tau^2} \delta x^a \Big) d \tau

0 = -mc \int \Big( \frac{1}{2} \frac{\partial g_{bc}}{\partial x^a} \frac{dx^b}{d \tau} \frac{dx^c}{d \tau} - \frac{1}{2} \frac{\partial g_{ab}}{\partial x^c} \frac{dx^b}{d \tau} \frac{dx^c}{d \tau} - \frac{1}{2} \frac{\partial g_{ac}}{\partial x^b} \frac{dx^b}{d \tau} \frac{dx^c}{d \tau} - g_{ab} \frac{d^2 x^b}{d \tau^2} \Big) \delta x^a d \tau

Therefore,

g_{ab} \frac{d^2 x^b}{d \tau^2} = \frac{1}{2} \Big( \frac{\partial g_{bc}}{\partial x^a} - \frac{\partial g_{ab}}{\partial x^c} - \frac{\partial g_{ac}}{\partial x^b} \Big) \frac{dx^b}{d \tau} \frac{dx^c}{d \tau}

g^{da} g_{ab} \frac{d^2 x^b}{d \tau^2} = \frac{1}{2} g^{da} \Big( \frac{\partial g_{bc}}{\partial x^a} - \frac{\partial g_{ab}}{\partial x^c} - \frac{\partial g_{ac}}{\partial x^b} \Big) \frac{dx^b}{d \tau} \frac{dx^c}{d \tau}

\frac{d^2 x^a}{d \tau^2} = \frac{1}{2} g^{ad} \Big( \frac{\partial g_{bc}}{\partial x^d} - \frac{\partial g_{bd}}{\partial x^c} - \frac{\partial g_{cd}}{\partial x^b} \Big) \frac{dx^b}{d \tau} \frac{dx^c}{d \tau}

This is the geodesic equation we want to find, an equation of path of the particle moving under gravitational field.

From → General Relativity

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