- #1

- 9,947

- 1,127

Consider the equatorial plane of a spherically symmetric space-time. Then we can write the metric in the equatorial plane (theta=pi/2) in terms of three coordinates - [t, r, phi]

ds^2 = -f(r) dt^2 + g(r) dr^2 + h(r) dphi^2

For the Schwarzschild metric we can write:

[tex]f = c^2(1 -\frac{2 G M}{c^2 r}) \quad g = 1 / (1 -\frac{2 G M}{c^2 r}) \quad h = r^2[/tex]

For the isotropic Schwarzschild metric we can write:

[tex]f = c^2 \left( \frac{1-\frac{GM}{2 c^2 r}}{1+\frac{GM}{2 c^2 r}} \right)^2 \quad g = (1 + \frac{GM}{2 c^2 r})^4 \quad h = r^2 (1 + \frac{GM}{2 c^2 r})^4 [/tex]

For the non-cartesian PPN apprxomiation to isotropic Schwarzschild we can write:

[tex]f = c^2 \left( 1 - \frac{2GM}{c^2 r} + \frac{2 G^2 M^2}{c^4 r^2} \right) \quad g = \left( 1 + \frac{2GM}{c^2 r} \right) \quad h = r^2 \left( 1 + \frac{2GM}{c^2 r} \right)

[/tex]

The geodesic equation for r is

[tex]

\frac{d^2r}{d \tau^2} + \frac{(\frac{df}{dr})}{ 2g} (\frac{d^2t}{d \tau^2}) + \frac{(\frac{dg}{dr})}{ 2g} (\frac{d^2 r}{d \tau^2}) + \frac{(\frac{dh}{dr})}{ 2g} (\frac{d^2\phi }{d \tau^2}) = 0

[/tex]

Various useful quantities are (the answers all appear to be sensible to me in Schwarzschild coordinates):

The magnitude of the proper acceleration of a static observer (constant r) as measured with an onboard accelerometer

[tex]c^2 \frac{(\frac{d f}{d r})}{2 f \sqrt{g}}[/tex]

The coordinate acceleration (d^2 r / dt^2) of a geodesic observer who starts out at rest, (dr/dt=0, dphi/dt=0):

[tex]-\frac{(\frac{d f}{d r})}{2g}[/tex]

The coordinate angular velocity [itex]d \phi / dt[/itex]

[tex]\sqrt{\frac{(\frac{d f}{d r})}{(\frac{d h}{d r})}}[/tex]

The proper angular velocity [itex]d \phi / d \tau[/itex]

[tex]c \frac{(\frac{d f}{d r}) } { f (\frac{d h}{d r}) - h (\frac{d f}{d r})} [/tex]

coordinate and proper orbital periods - 2 pi over the respective angular velociteis

The orbital period as measured by a co-located static observer

*correction*

[tex] \left( \frac{2 \pi}{c} \right){ \sqrt{f (\frac{d h}{d r}) / (\frac{d f}{d r}) } }[/tex]

circumference of the circle with coordinate values r

[tex]2 \pi \sqrt{h} [/tex]

orbital velocity measured by a colocated static observer

[tex]c \sqrt{ \frac{h (\frac{d f}{d r}) }{ f (\frac{d h}{d r})}}[/tex]

The concept of velocity from "remote points of view" is not well-defined in GR, due to the problems of parallel tranpsorting velocities.

http://math.ucr.edu/home/baez/einstein/node2.html

though various ad-hoc approaches are possible.

Note: The factors of c are annoyig to keep tract of - I took the simple approach of making sure the units come out correctly.

ds^2 = -f(r) dt^2 + g(r) dr^2 + h(r) dphi^2

For the Schwarzschild metric we can write:

[tex]f = c^2(1 -\frac{2 G M}{c^2 r}) \quad g = 1 / (1 -\frac{2 G M}{c^2 r}) \quad h = r^2[/tex]

For the isotropic Schwarzschild metric we can write:

[tex]f = c^2 \left( \frac{1-\frac{GM}{2 c^2 r}}{1+\frac{GM}{2 c^2 r}} \right)^2 \quad g = (1 + \frac{GM}{2 c^2 r})^4 \quad h = r^2 (1 + \frac{GM}{2 c^2 r})^4 [/tex]

For the non-cartesian PPN apprxomiation to isotropic Schwarzschild we can write:

[tex]f = c^2 \left( 1 - \frac{2GM}{c^2 r} + \frac{2 G^2 M^2}{c^4 r^2} \right) \quad g = \left( 1 + \frac{2GM}{c^2 r} \right) \quad h = r^2 \left( 1 + \frac{2GM}{c^2 r} \right)

[/tex]

The geodesic equation for r is

[tex]

\frac{d^2r}{d \tau^2} + \frac{(\frac{df}{dr})}{ 2g} (\frac{d^2t}{d \tau^2}) + \frac{(\frac{dg}{dr})}{ 2g} (\frac{d^2 r}{d \tau^2}) + \frac{(\frac{dh}{dr})}{ 2g} (\frac{d^2\phi }{d \tau^2}) = 0

[/tex]

Various useful quantities are (the answers all appear to be sensible to me in Schwarzschild coordinates):

The magnitude of the proper acceleration of a static observer (constant r) as measured with an onboard accelerometer

[tex]c^2 \frac{(\frac{d f}{d r})}{2 f \sqrt{g}}[/tex]

The coordinate acceleration (d^2 r / dt^2) of a geodesic observer who starts out at rest, (dr/dt=0, dphi/dt=0):

[tex]-\frac{(\frac{d f}{d r})}{2g}[/tex]

The coordinate angular velocity [itex]d \phi / dt[/itex]

[tex]\sqrt{\frac{(\frac{d f}{d r})}{(\frac{d h}{d r})}}[/tex]

The proper angular velocity [itex]d \phi / d \tau[/itex]

[tex]c \frac{(\frac{d f}{d r}) } { f (\frac{d h}{d r}) - h (\frac{d f}{d r})} [/tex]

coordinate and proper orbital periods - 2 pi over the respective angular velociteis

The orbital period as measured by a co-located static observer

*correction*

[tex] \left( \frac{2 \pi}{c} \right){ \sqrt{f (\frac{d h}{d r}) / (\frac{d f}{d r}) } }[/tex]

circumference of the circle with coordinate values r

[tex]2 \pi \sqrt{h} [/tex]

orbital velocity measured by a colocated static observer

[tex]c \sqrt{ \frac{h (\frac{d f}{d r}) }{ f (\frac{d h}{d r})}}[/tex]

The concept of velocity from "remote points of view" is not well-defined in GR, due to the problems of parallel tranpsorting velocities.

http://math.ucr.edu/home/baez/einstein/node2.html

though various ad-hoc approaches are possible.

Note: The factors of c are annoyig to keep tract of - I took the simple approach of making sure the units come out correctly.

Last edited: