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quasar987

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## Homework Statement

Find the orbit of a planet in a grav. field corresponding to the metric

[tex]d\tau^2=(1+\alpha^2(x^2+y^2))dt^2-(dx^2+dy^2+dz^2)[/tex]

in the newtonian limit with the initial conditions at t=0, x=R, dx/dt=[itex]\beta[/itex],0=y=z=dy/dt=dz/dt where alpha and beta are cositive constants.

## Homework Equations

The Levi-Civita connexion and the geodesic equations

## The Attempt at a Solution

I calculates the connexion coefficients and wrote down the 4 geodesic equations with the proper time as the parameter:

[tex]\frac{d^2t}{d\tau^2}+\frac{2\alpha^2x}{(1+\alpha^2(x^2+y^2))} \frac{dt}{d\tau}\frac{dx}{d\tau}+\frac{2\alpha^2y}{(1+\alpha^2(x^2+y^2))}\frac{dt}{d\tau}\frac{dy}{d\tau}=0[/tex]

[tex]\frac{d^2x}{d\tau^2}+\alpha^2x\left(\frac{dt}{d\tau}\right)^2=0[/tex]

[tex]\frac{d^2y}{d\tau^2}+\alpha^2y\left(\frac{dt}{d\tau}\right)^2=0[/tex]

[tex]\frac{d^2z}{d\tau^2}=0[/tex]

I can solve the 4th equation of course:

[tex]z(\tau)=a\tau +b[/tex]

and if I understand correctly, the newtonian limit means that

[tex]\frac{dx^i}{d\tau}<<\frac{dt}{d\tau}[/tex]

But I don't see how that can be applied here to simplify the equations.

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