- #1

Dukon

- 73

- 3

- TL;DR Summary
- convert a differential version to absolute version of time dilation formula

Just as the time dilation formula for the Schwarzschild metric in terms of the position ##r## away from center of mass for a gravitational body and the Schwarzschild radius ##r_s = {2GM}/{c^2}## is given by

$$ \tau = t \sqrt{1 - \frac{r_s}{r} } $$

so I'd like to know the corresponding absolute (non-differential) form for the Kerr metric but all I could find for the Kerr metric time dilation formula was some differential ratio subtracted from unity. In terms of the modified distance ##\rho = f(r)## which reduces to ##r## if ##J=0## the Kerr time dilation is apparently given by (if this can be trusted)

$$ \eqalign{

\rho^2 &= r^2 + \left( \frac{J}{Mc} \right)^2 \cos^2 \theta \cr

1 - \left( \frac{{\rm d}\tau}{{\rm d}t} \right)^2 &= \frac{2 GM r}{c^2 \rho^2}

}

$$

How can I a) get rid of the subtraction from unity, and b) not have it in the differential form but rather just the absolute form like above for the Schwarzschild metric: ##\tau##, time on a moving frame relative to a reference frame on which is the reference time ##t##?

If anyone has derived these other absolute time dilation formulae for the Kerr-Newman and Reissner-Nordstrom cases as well please do share.

$$ \tau = t \sqrt{1 - \frac{r_s}{r} } $$

so I'd like to know the corresponding absolute (non-differential) form for the Kerr metric but all I could find for the Kerr metric time dilation formula was some differential ratio subtracted from unity. In terms of the modified distance ##\rho = f(r)## which reduces to ##r## if ##J=0## the Kerr time dilation is apparently given by (if this can be trusted)

$$ \eqalign{

\rho^2 &= r^2 + \left( \frac{J}{Mc} \right)^2 \cos^2 \theta \cr

1 - \left( \frac{{\rm d}\tau}{{\rm d}t} \right)^2 &= \frac{2 GM r}{c^2 \rho^2}

}

$$

How can I a) get rid of the subtraction from unity, and b) not have it in the differential form but rather just the absolute form like above for the Schwarzschild metric: ##\tau##, time on a moving frame relative to a reference frame on which is the reference time ##t##?

If anyone has derived these other absolute time dilation formulae for the Kerr-Newman and Reissner-Nordstrom cases as well please do share.