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.(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Gravitational time dilation for a spherical body of finite radius

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