Gravitational time dilation for a spherical body of finite radius

[GeorgeDishman]

I am considering the gravitational time dilation at the centre of a spherical, non-rotating body (such as the Earth). The usual formula for gravitational time dilation is √(1-r_s/r) where r_s is the Schwarzschild Radius and r is the radius of the clock compared to one at infinity, however, this can't be used for r=0.

I believe the gravitational potential can also be used and the Newtonian approximation gives

V = -GM/r for r>a

V = -GM/2a(3-(r/a)^2) for r<a

where a is the radius of the body

Assuming a >> r_s, can that formula be used to find the time dilation at any radius or is the use of the potential only valid outside the sphere? Is there a more accurate fully relativistic method?

 

[WannabeNewton]

You can certainly use the Newtonian approximation if you wish, assuming that the fluid making up the source is a Newtonian fluid, but you can also very easily do this relativistically for a static spherically symmetric fluid star such that the fluid has no viscosity or heat conduction between fluid elements. You can find derivations and analyses of such static spherically symmetric perfect fluid solutions to Einstein's equation in just about every general relativity text. Once you have such an interior solution (i.e. the interior metric) and use the fact that Birkhoff's theorem forces the exterior space-time geometry to be Schwarzschild, you can straightforwardly derive gravitational time dilation using the usual technique.

See chapter 10 of Schutz "A First Course in General Relativity", section 6.2 of Wald "General Relativity", chapter 7 of Straumann "General Relativity", chapter 10 of Hobson et al. "General Relativity: An Introduction for Physicists", and chapter 16 of Stephani et al. "Exact Solutions of Einstein's Field Equations".

 

[A.T.]

Is there a more accurate fully relativistic method?

Schwarzschild's interior solution:
https://www.physicsforums.com/showthread.php?p=1543402#post1543402

The gravitational time dilation factor inside a uniform sphere is given by the term inside the first brackets.

 

[GeorgeDishman]

Schwarzschild's interior solution:
https://www.physicsforums.com/showthread.php?p=1543402#post1543402

The gravitational time dilation factor inside a uniform sphere is given by the term inside the first brackets.

Thanks, that's just what I needed. Merry Christmas everyone.

George

 

[sardar]

Thank you for your question. I can confirm that the formula for gravitational time dilation at the center of a spherical body of finite radius can indeed be derived using the gravitational potential, as you have mentioned. This approach is known as the Newtonian approximation and is valid for distances much larger than the Schwarzschild radius. However, for a more accurate and fully relativistic method, we need to use the Schwarzschild metric, which takes into account the curvature of spacetime caused by the mass of the sphere.

Using this metric, we can calculate the time dilation at any radius, including at the center of the sphere. The formula for gravitational time dilation in this case is given by √(1-r_s/r_c), where r_c is the distance from the center of the sphere to the clock. This formula is valid for any value of r_c, including r_c=0. Therefore, the use of the potential is not limited to outside the sphere, but the Schwarzschild metric provides a more accurate and comprehensive approach for calculating time dilation.

In conclusion, while the Newtonian approximation using the gravitational potential can be used to calculate time dilation at the center of a spherical body, a more accurate and fully relativistic approach is to use the Schwarzschild metric. I hope this helps clarify your understanding of gravitational time dilation.