- #1
Dema
- 12
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Hello,
I’d have a question regarding two apparently different formulas for the time reduction factors for observers in orbit around a rotating black hole, as reported in this nice thread: Equation for time dilation of body in orbit around Kerr black hole?
The first one is:
$$A=\sqrt{g_{tt} + 2\Omega g_{\phi t}+\Omega^2 g_{\phi \phi}}$$
While the second one is:
$$A=\alpha\cdot\sqrt{1-v_{\pm}^2}$$
where ##\alpha## is the time reduction factor for a steady observer and ##v_{\pm} ## the velocity with respect this observer.
I guess the first one is a direct consequence of the general relativistic formula for proper time ##
d \tau = \sqrt{g_{\mu \nu} \dot{x^{\mu}} \dot{x^{\nu}}}dt##, where the dot refers to derivative with respect the coordinate time, and is obtained for the special case of a circular orbit around a rotating black hole in the equatorial plane, where ##\frac{d\phi}{dt} = \Omega##
On the other hand, I have no idea how the second one is derived.
A difference I noticed on the formulas, is that the velocity appearing on the first one is that observed at infinity, while the velocity appearing on the second one is that measured by a stationary observer.
So, I’m interested to know:
Thank you a lot!
D
I’d have a question regarding two apparently different formulas for the time reduction factors for observers in orbit around a rotating black hole, as reported in this nice thread: Equation for time dilation of body in orbit around Kerr black hole?
The first one is:
$$A=\sqrt{g_{tt} + 2\Omega g_{\phi t}+\Omega^2 g_{\phi \phi}}$$
While the second one is:
$$A=\alpha\cdot\sqrt{1-v_{\pm}^2}$$
where ##\alpha## is the time reduction factor for a steady observer and ##v_{\pm} ## the velocity with respect this observer.
I guess the first one is a direct consequence of the general relativistic formula for proper time ##
d \tau = \sqrt{g_{\mu \nu} \dot{x^{\mu}} \dot{x^{\nu}}}dt##, where the dot refers to derivative with respect the coordinate time, and is obtained for the special case of a circular orbit around a rotating black hole in the equatorial plane, where ##\frac{d\phi}{dt} = \Omega##
On the other hand, I have no idea how the second one is derived.
A difference I noticed on the formulas, is that the velocity appearing on the first one is that observed at infinity, while the velocity appearing on the second one is that measured by a stationary observer.
So, I’m interested to know:
- If my derivation of the first formula is correct
- How the second formula is derived in the general case
- How are they related
- Are the general formulas ##\frac{d \tau}{ dt}= \sqrt{g_{\mu \nu} \dot{x^{\mu}} \dot{x^{\nu}}}## and ##A=\alpha\cdot\sqrt{1-v_{\pm}^2}## valid, with suitable substitutions, for any particular motion in a generic spacetime (I mean not only circular orbits in equatiorial planes for back holes)?
Thank you a lot!
D