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Topic: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Replies: 38   Last Post: Apr 13, 2013 11:57 PM

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Rock Brentwood

Posts: 121
Registered: 6/18/10
Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Posted: Apr 8, 2013 8:18 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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 Author
4/7/13
Read Misner, Thorne and Wheeler, Exercise 8.5 (c)
Guest
4/7/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Lord Androcles, Zeroth Earl of Medway
4/8/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Lord Androcles, Zeroth Earl of Medway
4/8/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Hetware
4/8/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Lord Androcles, Zeroth Earl of Medway
4/8/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Hetware
4/9/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Lord Androcles, Zeroth Earl of Medway
4/9/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Hetware
4/8/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Hetware
4/8/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Lord Androcles, Zeroth Earl of Medway
4/9/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Hetware
4/9/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Lord Androcles, Zeroth Earl of Medway
4/9/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Hetware
4/9/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Lord Androcles, Zeroth Earl of Medway
4/9/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Hetware
4/8/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Dirk Van de moortel
4/8/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Lord Androcles, Zeroth Earl of Medway
4/8/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
rotchm@gmail.com
4/9/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Dirk Van de moortel
4/13/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Dono
4/13/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
rotchm@gmail.com
4/13/13
Read MTW, Exercise 8.5 (c), Crotchm reasserts his imbecility
Dono
4/13/13
Read MTW Exercise 8.5 (c) - Crotchm the poser puts his foot in his mouth,
big time
Dono
4/13/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Dono
4/13/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
rotchm@gmail.com
4/13/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Dono
4/13/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
rotchm@gmail.com
4/13/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Dono
4/13/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Dono
4/13/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
rotchm@gmail.com
4/13/13
Read MTW Exercise 8.5 (c) - Crotchm the poser puts his foot in his mouth,
big time
Dono
4/13/13
Read Re: MTW Exercise 8.5 (c) - Crotchm the poser puts his foot in his
mouth, big time
rotchm@gmail.com
4/13/13
Read Re: MTW Exercise 8.5 (c) - Crotchm the poser puts his foot in his
mouth, big time
Dono
4/9/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Guest
4/9/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Dirk Van de moortel
4/9/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Lord Androcles, Zeroth Earl of Medway
4/8/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Rock Brentwood
4/8/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Hetware
4/8/13
Read Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Lord Androcles, Zeroth Earl of Medway

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