Orbital Period In General Relativity

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Discussion Overview

The discussion revolves around the calculation of the orbital period in General Relativity using the Schwarzschild metric. Participants explore the relationship between classical mechanics and relativistic effects, particularly focusing on circular orbits and the implications of the Schwarzschild radius.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents a formula for the orbital period in classical mechanics and expresses uncertainty about its application in General Relativity.
  • Another participant asserts that for a circular orbit, the orbital period can be derived from Kepler's Third Law by substituting the Schwarzschild radius as the orbital radius, noting the distinction between this radius and the physical distance from the central mass.
  • A further contribution reiterates the use of the Schwarzschild radius in the context of the areal radius, suggesting an integration approach over the radial coordinate.
  • Another participant challenges the previous claim about the areal radius, providing a definition based on the surface area of a sphere and indicating that the earlier description was incorrect.

Areas of Agreement / Disagreement

The discussion features multiple competing views regarding the definition and calculation of the areal radius and its implications for the orbital period, indicating that consensus has not been reached.

Contextual Notes

Participants express varying interpretations of the Schwarzschild metric and its application to orbital mechanics, highlighting potential misunderstandings about the definitions involved, particularly concerning the areal radius.

dsaun777
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What is the orbital period in General Relativity using the Schwarzschild metric? In classical mechanics, it is something like
T=2pi(GnM/a3). Where a is the semi-major axis, this is for a small body orbiting a larger one. I think I have an idea but I am not 100% sure. I am interested in an outside observer far away viewing a small particle m in orbit of some mass M.
 
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dsaun777 said:
What is the orbital period in General Relativity using the Schwarzschild metric?
For a circular orbit, it's the Kepler's Third Law expression with the Schwarzschild ##r## plugged in as the orbital radius. Note that this is the case even though ##r## is not the same as the physical distance from the center of mass of the central body.
 
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PeterDonis said:
For a circular orbit, it's the Kepler's Third Law expression with the Schwarzschild ##r## plugged in as the orbital radius. Note that this is the case even though ##r## is not the same as the physical distance from the center of mass of the central body.
Yeah, its the areal radius found by integrating over the radial coordinate from r to rs dr using the metric components related to radial coordinates.
 
dsaun777 said:
Yeah, its the areal radius
Yes, but...

dsaun777 said:
found by integrating over the radial coordinate from r to rs dr using the metric components related to radial coordinates.
...no, that's not what the areal radius is. The areal radius is ##r = \sqrt{A / 4 \pi}##, where ##A## is the surface area of the 2-sphere labeled by ##r## that is centered on the central mass.
 

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