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Hey folks,

working problems in Hartle's GR book and having trouble with this one. Chapter 9 discusses the simplest physically relavent curved geometry, that of Mr. Swarzschild

[tex] ds^s = -(1 - \frac{2 G M}{r}) dt^2

+ (1 - \frac{2 M}{r})^{-1} dr^2

+r^2(d\theta^2 + sin^2\theta d\phi^2) [/tex]

In this geometry the derivative of the effective potential goes to zero at two places. one is a stable equilibrium and the other is unstable.

[tex]

V_{eff} = -\frac{M}{r} + \frac{l^2}{2 r^2} - \frac{M L^2}{r^3}

[/tex]

the unstable r value for circular orbits occurs at

[tex]

r_{max} = \frac{l^2}{2 M}( 1-\sqrt{1-12 (\frac{M}{l}) ^2})

[/tex]

the goal is to show that when this orbit is perturbed in the radial direction that the perturbation grows exponentialy

[tex]

\delta_{r} \propto e^{\tau / \tau_*}

[/tex]

where [tex] \tau_* [/tex] is some constant

we also have a constant related to the conserved energy per unit rest mass to take advantage of

[tex]

\epsilon=\frac{1}{2}(\frac{dr}{d\tau})^2+V_{eff}(r)

[/tex]

i know that the derivative above will determine how this pertubation will grow and that

[tex] \epsilon - V_{eff} = 0 [/tex]

for circular orbits at the unstable r. I've tried expanding the potential around this unstable equilibrium point which yields derivatives of V multiplied by the deltar under a square root for the derivative but I cant see how to connect this to the exponential form of the proper time.

working problems in Hartle's GR book and having trouble with this one. Chapter 9 discusses the simplest physically relavent curved geometry, that of Mr. Swarzschild

[tex] ds^s = -(1 - \frac{2 G M}{r}) dt^2

+ (1 - \frac{2 M}{r})^{-1} dr^2

+r^2(d\theta^2 + sin^2\theta d\phi^2) [/tex]

In this geometry the derivative of the effective potential goes to zero at two places. one is a stable equilibrium and the other is unstable.

[tex]

V_{eff} = -\frac{M}{r} + \frac{l^2}{2 r^2} - \frac{M L^2}{r^3}

[/tex]

the unstable r value for circular orbits occurs at

[tex]

r_{max} = \frac{l^2}{2 M}( 1-\sqrt{1-12 (\frac{M}{l}) ^2})

[/tex]

the goal is to show that when this orbit is perturbed in the radial direction that the perturbation grows exponentialy

[tex]

\delta_{r} \propto e^{\tau / \tau_*}

[/tex]

where [tex] \tau_* [/tex] is some constant

we also have a constant related to the conserved energy per unit rest mass to take advantage of

[tex]

\epsilon=\frac{1}{2}(\frac{dr}{d\tau})^2+V_{eff}(r)

[/tex]

i know that the derivative above will determine how this pertubation will grow and that

[tex] \epsilon - V_{eff} = 0 [/tex]

for circular orbits at the unstable r. I've tried expanding the potential around this unstable equilibrium point which yields derivatives of V multiplied by the deltar under a square root for the derivative but I cant see how to connect this to the exponential form of the proper time.

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