- #1

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- Summary:
- Schutz derives the expression for p0 , but it's unclear to me why it should be conserved.

I'm looking at Schutz 7.4 where first he obtains the following expression for a geodesic:

$$ m \frac {dp_\beta} {d\tau} = \frac 1 2 g_{\nu\alpha,\beta } p^\nu p^\alpha $$

This means that if all the components of ##g_{\nu\alpha }## are constant for a given ##\beta##, then ##p_\beta## is also constant along the geodesic.

Then, he gives an example for a lab on earth, using the weak gravity metric ##ds^2 = -(1+2\phi)dt^2 + (1-2\phi)(dx^2 + dy^2 +dz^2) ##. After some work and approximation he arrives at

$$ - p_0 \approx m + m\phi + \frac {(p^x)^2 + (p^y)^2 + (p^z)^2} {2m} $$

Nice. We get the mass, potential energy and kinetic energy.

But how is this an example of the geodesic expression above for ##\beta = 0## as the text suggests? A lab on earth is surely not following a geodesic. I suppose that ##g_{\nu\alpha }## are constant on the lab, but the expression that ensures that ##p_0## would then be constant was derived only for geodesics. So that would not work for a lab on Earth, right?

I am confused

$$ m \frac {dp_\beta} {d\tau} = \frac 1 2 g_{\nu\alpha,\beta } p^\nu p^\alpha $$

This means that if all the components of ##g_{\nu\alpha }## are constant for a given ##\beta##, then ##p_\beta## is also constant along the geodesic.

Then, he gives an example for a lab on earth, using the weak gravity metric ##ds^2 = -(1+2\phi)dt^2 + (1-2\phi)(dx^2 + dy^2 +dz^2) ##. After some work and approximation he arrives at

$$ - p_0 \approx m + m\phi + \frac {(p^x)^2 + (p^y)^2 + (p^z)^2} {2m} $$

Nice. We get the mass, potential energy and kinetic energy.

But how is this an example of the geodesic expression above for ##\beta = 0## as the text suggests? A lab on earth is surely not following a geodesic. I suppose that ##g_{\nu\alpha }## are constant on the lab, but the expression that ensures that ##p_0## would then be constant was derived only for geodesics. So that would not work for a lab on Earth, right?

I am confused

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