Time Dilation at Center of a Neutron Star

[PatrickPowers]

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

That formula is only valid for the region exterior to the earth. Inside the Earth the corresponding formula would be
$$t = \frac{\tau}{\sqrt{1 + \frac{2\Phi}{c^2}}}$$
$$\Phi = \frac{1}{2}\frac{GM_{tot}r^{2}}{R^{3}} - \frac{3}{2}\frac{GM_{tot}}{R}$$
where $$M_{tot}$$ is the mass of the planet, R is its radius, r is the distance of the clock inside the Earth from the center with $$\tau$$ as its time and t is the time for a far remote clock. According to this gravitational time dilation is greatest at the center.

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

radius(rough estimate) = 10000m

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.

 

[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.

 

[pervect]

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

The equation of state for a neutron star is still not known. It is assumed that it differs significantly from that of a white dwarf, whose EOS is that of a degenerate gas which can be described in close agreement with special relativity. However, with a neutron star the increased effects of general relativity can no longer be ignored. Several EOS have been proposed (FPS, UU, APR, L, SLy, and others) and current research is still attempting to constrain the theories to make predictions of neutron star matter

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.