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Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Posted:
Apr 8, 2013 8:18 PM


On Apr 7, 7:14 pm, Hetware <hatt...@speakyeasy.net> wrote: > This is the geodesic equation under discussion: > d^2(r)/dt^2 = r(dp/dt)^2 > d^2(p)/dt^2 = (2/r)(dp/dt)(dr/dt).
Where the problem comes from is not important, since all you're asking about is how this is solved.
Notice that it's independent of p and depends only on do/dt. So, define v = dr/dt, f = dp/dt and write the system as dv/dt = rf^2, df/dt = 2fv/r (along with dr/dt = v, dp/dt = f).
The second equation can be rewritten as 0 = 1/f df/dt + 2/r dr/dt = 1/(f r^2) d(f r^2)/dt Therefore, f = K/r^2, for some constant K.
The first equation then becomes, dv/dt = K^2/r^3. The standard trick is to turn this into a conservation of energy integral: v dv/dt = K^2/r^3 v = K^2/r^3 dr/dt, from which it follows that d/dt (v^2/2) = d/dt (K^2/2 1/r^2). The solution is v^2 = v_0^2  K^2/r^2 for some constant v_0.
You can take it on from here; solving for r as a function of t, and then putting this into the equation for f (i.e. dp/dt) to get p as a function of t.
If you go back to the original problem, the following properties hold true. The gravitational source (which I assume is the Schwarzschild solution) has rotational symmetry and time translation symmetry. This corresponds (by way of the Noether Theorem) to conserved quantities: Angular Momentum for rotational symmetry and Energy for time translation symmetry. Therefore, the first things to look for in the geodesic law are to extract out the integrals for angular momentum and energy. The angular momentum part was already removed from the problem by the time you brought the matter here in sci.math, so that left the energy integral to take care of. That was the missing step.


Date

Subject

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4/7/13


Guest

4/7/13


Lord Androcles, Zeroth Earl of Medway

4/8/13


Lord Androcles, Zeroth Earl of Medway

4/8/13


Hetware

4/8/13


Lord Androcles, Zeroth Earl of Medway

4/8/13


Hetware

4/9/13


Lord Androcles, Zeroth Earl of Medway

4/9/13


Hetware

4/8/13


Hetware

4/8/13


Lord Androcles, Zeroth Earl of Medway

4/9/13


Hetware

4/9/13


Lord Androcles, Zeroth Earl of Medway

4/9/13


Hetware

4/9/13


Lord Androcles, Zeroth Earl of Medway

4/9/13


Hetware

4/8/13


Dirk Van de moortel

4/8/13


Lord Androcles, Zeroth Earl of Medway

4/8/13


rotchm@gmail.com

4/9/13


Dirk Van de moortel

4/13/13


Dono

4/13/13


rotchm@gmail.com

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Dono

4/13/13


Dono

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Dono

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rotchm@gmail.com

4/13/13


Dono

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rotchm@gmail.com

4/13/13


Dono

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Dono

4/13/13


rotchm@gmail.com

4/13/13


Dono

4/13/13


rotchm@gmail.com

4/13/13


Dono

4/9/13


Guest

4/9/13


Dirk Van de moortel

4/9/13


Lord Androcles, Zeroth Earl of Medway

4/8/13


Rock Brentwood

4/8/13


Hetware

4/8/13


Lord Androcles, Zeroth Earl of Medway


