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## Main Question or Discussion Point

In part, this thread is a tangential extension of an earlier thread entitled

The difference between the 2 effective potential (Veff) plots attached to this post is not really that important to this new thread, the main point to highlight is that the shape of the max/min curve is essentially identical, although the vertical scales differ. Both Veff curves correspond to a given value of angular momentum [L=7.42E12]. The value was selected as suitable based on the assumption the L>3.4642GMm/c. The max and min points of the curve correspond to a quasi-stable inner orbit and a stable outer orbit, while the graphs allude to the approximate value, the actual values are said to be calculated from the equations:

[tex] r_{outer} = \frac {a^2}{Rs}\left(1+\sqrt{1-\frac{3Rs^2}{a^2}\right)[/tex]

[tex] r_{inner} = \frac {a^2}{Rs}\left(1-\sqrt{1-\frac{3Rs^2}{a^2}\right)[/tex]

where [tex] a = \frac{L}{mc}[/tex]

See 3rd attachment shows a graph against different values of normalised [L]

Details of the derivation can be found on the Wikipedia site:

http://en.wikipedia.org/wiki/Kepler_problem_in_general_relativity

For the example value of angular momentum [L=4.2GMm/c] the outer and inner radius, as a ratio to the Schwarzschild radius [Rs], are 6.9 and 1.92 respectively. However, one of the questions being raised to the PF forum concerns the velocity that corresponds to these radii. Clearly, at these radii, the orbit is subject to relativistic effects due to gravity and orbital velocity. How these effects apply is assumed to depend on the observer:

1. Distant [dt]

2. Stationary/shell at [r]

3. Orbiting at [r] [tex]d\tau[/tex]

As a generalisation, the assumption is that distant observer is essentially unaffected by gravity or velocity, while the stationary (shell) observer is affected by gravity. However, the orbiting observer is affected by both gravity and velocity. So the first question is:

The answer to this question is required in order to determine the frame of reference of the angular momentum. For a circular orbit, the normal assumption is that [L=mvr]. Therefore, by rearranging, we can determine a velocity [v=L/mr]. However, if we apply this equation to both radii produced above, the outer orbital velocity [v=9.10E7], while the implication is that the inner velocity would exceed the speed of light [c] at [v=3.28E8]. Taking another approach, we might try solve the Schwarzschild metric for [tex][d\phi/dt][/tex] and [tex][d\phi/d\tau][/tex], which is assumed to correspond to the distant and orbiting observers. The following solutions are assumed, although only the [tex][d\phi/dt][/tex] case has been checked:

[tex]v_o = r\left(\frac {d\phi}{d\tau}\right)=c\sqrt{\frac{Rs}{2r-Rs}} [/tex]

[tex]v_o = r\left(\frac {d\phi}{dt}\right)=c\sqrt{\frac{Rs}{2r}} [/tex]

Again, using the value in the specific example cited leads to a value of v=8.35E7 for the outer orbit, as perceived by the orbiting observer and the shell observer. This assumption is based on the fact that these two observers sit at the same radius, i.e. same gravity effects, and therefore are only separated by velocity. Velocity under special relativity is assumed to be invariant to both observers. The velocity with respect to the distant observer appears to come out at [v=8.05E7]. If these equations are applied with [r=inner radius] then the respective velocities are [1.78E8] and [1.53E8] and so, at least, fall below the speed of light [c]. However, focusing on just the values calculated for the outer orbit by [tex][d\tau][/tex] and [dt] using the original assumption [L=mvr], we then appear to get an inconsistency to the value of [L=7.42E12] derived from the relationship [L=N*GMm/c] where [N=4.2] in this example, i.e.

V=8.35E7; L=mvr=6.82E12

V=8.05E7; L=mvr=6.56E12

Initially, I assumed that although the distant observer is in flat spacetime, other results from the Schwarzschild metric suggest that measurements taken by this observer are distorted by the gravity and velocity effects acting on spacetime at [r]. This leaves the other 2 observers sitting at the same radius [r], although they are separated by the orbital velocity. Now relativity seem to suggest that time with respect to the distant observer will run slower, while distant in the radial direction will be expanded due to gravity. The addition of a relativistic velocity would cause time for the orbiting observer to appear to run even slower with respect to both the distant and shell observer, while the velocity causes the circumference distant to be contracted maintaining the invariance of velocity. However:

Sorry if this might appear as a very convoluted set of questions, but would appreciate any help in trying to understand exactly how these issues are resolved by the standard text. Thanks

*`Questions on Effective Potential`*. However, this thread now moves the discussion in the direction of interpreting the results of the effective potential in terms of the circular orbital velocity, as derived from the Schwarzschild metric.The difference between the 2 effective potential (Veff) plots attached to this post is not really that important to this new thread, the main point to highlight is that the shape of the max/min curve is essentially identical, although the vertical scales differ. Both Veff curves correspond to a given value of angular momentum [L=7.42E12]. The value was selected as suitable based on the assumption the L>3.4642GMm/c. The max and min points of the curve correspond to a quasi-stable inner orbit and a stable outer orbit, while the graphs allude to the approximate value, the actual values are said to be calculated from the equations:

[tex] r_{outer} = \frac {a^2}{Rs}\left(1+\sqrt{1-\frac{3Rs^2}{a^2}\right)[/tex]

[tex] r_{inner} = \frac {a^2}{Rs}\left(1-\sqrt{1-\frac{3Rs^2}{a^2}\right)[/tex]

where [tex] a = \frac{L}{mc}[/tex]

See 3rd attachment shows a graph against different values of normalised [L]

Details of the derivation can be found on the Wikipedia site:

http://en.wikipedia.org/wiki/Kepler_problem_in_general_relativity

For the example value of angular momentum [L=4.2GMm/c] the outer and inner radius, as a ratio to the Schwarzschild radius [Rs], are 6.9 and 1.92 respectively. However, one of the questions being raised to the PF forum concerns the velocity that corresponds to these radii. Clearly, at these radii, the orbit is subject to relativistic effects due to gravity and orbital velocity. How these effects apply is assumed to depend on the observer:

1. Distant [dt]

2. Stationary/shell at [r]

3. Orbiting at [r] [tex]d\tau[/tex]

As a generalisation, the assumption is that distant observer is essentially unaffected by gravity or velocity, while the stationary (shell) observer is affected by gravity. However, the orbiting observer is affected by both gravity and velocity. So the first question is:

*Who measures the radii [r] calculated from the equations?*The answer to this question is required in order to determine the frame of reference of the angular momentum. For a circular orbit, the normal assumption is that [L=mvr]. Therefore, by rearranging, we can determine a velocity [v=L/mr]. However, if we apply this equation to both radii produced above, the outer orbital velocity [v=9.10E7], while the implication is that the inner velocity would exceed the speed of light [c] at [v=3.28E8]. Taking another approach, we might try solve the Schwarzschild metric for [tex][d\phi/dt][/tex] and [tex][d\phi/d\tau][/tex], which is assumed to correspond to the distant and orbiting observers. The following solutions are assumed, although only the [tex][d\phi/dt][/tex] case has been checked:

[tex]v_o = r\left(\frac {d\phi}{d\tau}\right)=c\sqrt{\frac{Rs}{2r-Rs}} [/tex]

[tex]v_o = r\left(\frac {d\phi}{dt}\right)=c\sqrt{\frac{Rs}{2r}} [/tex]

Again, using the value in the specific example cited leads to a value of v=8.35E7 for the outer orbit, as perceived by the orbiting observer and the shell observer. This assumption is based on the fact that these two observers sit at the same radius, i.e. same gravity effects, and therefore are only separated by velocity. Velocity under special relativity is assumed to be invariant to both observers. The velocity with respect to the distant observer appears to come out at [v=8.05E7]. If these equations are applied with [r=inner radius] then the respective velocities are [1.78E8] and [1.53E8] and so, at least, fall below the speed of light [c]. However, focusing on just the values calculated for the outer orbit by [tex][d\tau][/tex] and [dt] using the original assumption [L=mvr], we then appear to get an inconsistency to the value of [L=7.42E12] derived from the relationship [L=N*GMm/c] where [N=4.2] in this example, i.e.

V=8.35E7; L=mvr=6.82E12

V=8.05E7; L=mvr=6.56E12

*So how do you correlate angular moment [L] to velocity and who measures it?*Initially, I assumed that although the distant observer is in flat spacetime, other results from the Schwarzschild metric suggest that measurements taken by this observer are distorted by the gravity and velocity effects acting on spacetime at [r]. This leaves the other 2 observers sitting at the same radius [r], although they are separated by the orbital velocity. Now relativity seem to suggest that time with respect to the distant observer will run slower, while distant in the radial direction will be expanded due to gravity. The addition of a relativistic velocity would cause time for the orbiting observer to appear to run even slower with respect to both the distant and shell observer, while the velocity causes the circumference distant to be contracted maintaining the invariance of velocity. However:

*Would this change the perception of [r] based on [tex]r=C/2\pi[/tex]?*Sorry if this might appear as a very convoluted set of questions, but would appreciate any help in trying to understand exactly how these issues are resolved by the standard text. Thanks