# Time Dilation at Center of a Neutron Star

1. Dec 7, 2011

### PatrickPowers

From https://www.physicsforums.com/showthread.php?t=40391 in gravitational time dilation.

At the center of neutron star PSR J1614-2230 at 1.97 SM

mass of Sun 1.9891×10^30 kg

mass of PSR J1614-2230. 4 ×10^30

gravitational constant = 6.67300 × 10-11 m3 kg-1 s-2

phi = - 3/2 * 6.67300 × 10-11 * 4 ×10^30 / 10000

phi = - 4 × 10^20 / 10000

phi = - 4 × 10^16

sqrt( 1 + 2phi/c^2 )

sqrt( 1 - 2 * 4 × 10^16 / (3^10^8)^2

sqrt( 1 - 8 / 9 )

sqrt( 1/9) = .33

so time runs perhaps three times as slowly at the center than in an area of weak gravity.

I have read that the gravity of a neutron star owes more to its pressure than its mass. I don't know whether these figures take that into account.

2. Dec 7, 2011

### Matterwave

Presumably the M_tot includes all the contributions from pressure/energy. I'm not sure about interior solutions, but this is at least the case for the exterior Schwarzschild solution.

3. Dec 7, 2011

### pervect

Staff Emeritus
The general form of the interior metric of a neutron star is not currently known, as far as I know, because the equation of state of the neutron degenerate matter, rho(P), is not known.

http://en.wikipedia.org/w/index.php?title=Neutron_star&oldid=464462435

If you did know (or assume you knew) the equation of state, for a static star solving Einstein's equations (the interior Schwarzschild solution) arising from that equation of state would give you the interior metric, and hence the time dilation. If you wanted to go beyond the static approximation (i.e. to include the effects of radiation on the stellar structure), you'd need an even more sophisticated model.