Metrics for the Sun & Earth: Rs & Assumptions

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SUMMARY

The discussion focuses on the Schwarzschild Metric and its application to the Sun and Earth, specifically addressing the Schwarzschild Radius (Rs) for both celestial bodies. The Schwarzschild Radius for Earth is calculated as 8.87 x 10^-3 meters, while for the Sun, it is 2954 meters. The conversation emphasizes that these radii are located within the physical bodies of the Earth and Sun, yet the exterior Schwarzschild solution remains valid for spherically symmetric static spacetimes. Additionally, it highlights the assumptions required to transition from General Relativity (GR) to Newtonian physics.

PREREQUISITES
  • Understanding of the Schwarzschild Metric in General Relativity
  • Familiarity with the concept of Schwarzschild Radius
  • Knowledge of Newtonian physics and its equations
  • Basic grasp of spacetime and curvature in physics
NEXT STEPS
  • Research the implications of the Schwarzschild Metric on black hole physics
  • Explore the transition from General Relativity to Newtonian physics
  • Study the properties of spherically symmetric static spacetimes
  • Investigate the significance of coordinate singularities in metrics
USEFUL FOR

Physicists, astrophysicists, and students studying General Relativity and cosmology, particularly those interested in the mathematical modeling of celestial bodies and the transition between GR and Newtonian frameworks.

Philosophaie
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Is there a Metric for the Sun and Earth other than the Schwarzschild Metric who has the Schwarzschild Radius on the interior of the body?

What are the assumptions that turn that from GR to Newtonian?

Rs(Earth)=8.87*10^-3m
Rs(Sun)=2954m
 
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The Schwarzschild radius is outside the body pretty much by definition... though you can see where the Schwarzschild radius would be if all the matter were concentrated at the center ... but that is always inside the body for objects less massive than black holes. So I think you need to be a bit more precise about what you mean.

It is probably possible to produce a metric which has a coordinate singularity anywhere you like.
 
Philosophaie said:
Is there a Metric for the Sun and Earth other than the Schwarzschild Metric who has the Schwarzschild Radius on the interior of the body?

What are the assumptions that turn that from GR to Newtonian?

Rs(Earth)=8.87*10^-3m
Rs(Sun)=2954m
As you point out, the Schwarzschild radii for the Earth and Sun are inside the physical bodies. But that is not a problem because the exterior Schwarzschild solution applies to any spherically symmetric static( stationary?) spacetime.

If the spatial curvature is ignored and all velocities << c, and r>>m one can recover the Newtonian equations.
 
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