I'm doing some revision for a General relativity exam, and came across this question:(adsbygoogle = window.adsbygoogle || []).push({});

A Flat Earth space-time has co-ordinates (t, x, y, z), where z > 0, and a metric

ds^{2}= ((1 + gz)^{2})dt^{2}− dx^{2}− dy^{2}− dz^{2}

where g is a positive constant.

Write down the geodesic equations in this space-time.

Hence, or otherwise, show that A Flat Earth physicist, stationary at a point with z = h, will measure a ‘gravitational acceleration’ of magnitude g(1 + gh)^{−1}

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I've solved the geodesics, but can't get the 'show that' at the end:

Using the Euler-Lagrange equation:

[tex] \stackrel{d}{ds} \stackrel{\partial L}{\partial \dot{x^{\mu}}} - \stackrel{\partial L}{\partial x^{\mu}} =0[/tex]

I get

[tex]\ddot{z} = 2g(1+gz)\dot{t}^2[/tex]

And

[tex]\dot{t} = E/(1+gz)^2[/tex] ( where E is a constant )

So

[tex]\ddot{z} = g{E^2}/(1+gz)^3[/tex]

which isn't right. Alternatively,

[tex]\ddot{z}= \dot{t}^2 z'' + \ddot{t}z'[/tex] ( where ' is (d/dt) )

it's stationary, so z' = 0 thus [tex]z''=\ddot{z}/\dot{t}^2=2g(1+gz)[/tex]

which still isn't right. Any takers?

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# Homework Help: General relativity question - geodesics.

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