Parametric Equation of a Sphere in General relativity

[Reedeegi]

What parametric equation would one use to describe a sphere in General Relativity in order to calculate curvature? My main problem is what to do with the time dimension... And no, this is not a homework question.

 

[atyy]

Eq 1, Burko et al, http://arxiv.org/abs/gr-qc/0008065
Eq 3.9, 3.10, 4.1, 4.7, Ehlers et al, http://arxiv.org/abs/gr-qc/0510041

 

[Entropia]

In General Relativity, the parametric equation for a sphere can be described using the Schwarzschild metric, which is a solution to Einstein's field equations. This metric takes into account the effects of gravity on the curvature of spacetime.

The equation for a sphere in this metric is given by:

r(t) = R sinθ cos(φ - ωt)

where r(t) represents the radial distance from the center of the sphere, R is the radius of the sphere, θ is the polar angle, φ is the azimuthal angle, and ω is the angular velocity. This equation takes into account the time dimension by incorporating the angular velocity, which represents the rotation of the sphere.

To calculate the curvature of the sphere, we can use the Ricci tensor, which is a measure of the curvature of spacetime. The Ricci tensor is calculated by taking the second derivative of the metric with respect to the coordinates. In this case, the coordinates are time (t) and the spatial coordinates (r, θ, φ).

Overall, the parametric equation for a sphere in General Relativity takes into account the effects of gravity and incorporates the time dimension through the use of the Schwarzschild metric and the Ricci tensor. This equation can be used to accurately calculate the curvature of a sphere in the context of General Relativity.