# Slowly add mass to a neutron star till it collapses.

1. Nov 6, 2009

### Spinnor

Say we slowly add mass to a neutron star till it collapses to a black hole. Just before it does has time in the neutron star almost come to a stop? Is the passage of time different for different locations in the neutron star just before it collapses? Would the strings of string theory come to a near halt?

Thanks for any help.

2. Nov 7, 2009

### yuiop

Say we have a neutron star that is that is sufficiently large that it can take the mass of two observers without collapsing but is also nevertheless close to its Schwarzschild radius. The two observers are high up and are twins. One descends to the surface and spends a period of time on the surface and later his twin descends to join him on the surface. The one that has been on the surface the longest, has aged the least, so it can be seen that proper time slows down in strongly curved spacetime in a very real (measurable) sense. When the radius of the neutron star is 9/8 of its Schwarzschild radius the proper time of a clock at the centre of the neutron star becomes negative, i.e. it starts ticking backwards. The exact radius that this effect occurs at, is dependent on the density distribution of mass within the neutron star. The 9/8 RS figure assumes even density everywhere but the more likely situation of the core having higher density than the shell makes the situation worse. To avoid clocks ticking backwards inside the near collapse neutron star, the mass would have to distributed with higher density at the shell and lower density at the core. This analysis is based on a simple application of the interior Schwarzschild solution and it is quite possible that pressure and stress effects on spacetime curvature, will also conspire to prevent proper time reversing.

3. Nov 7, 2009

### Spinnor

Clocks ticking backwards? !@#%&*

Would you point to a reference where backwards time is studied.

Thank you!

4. Nov 7, 2009

### yuiop

I can point you to an equation posted by George here https://www.physicsforums.com/showthread.php?p=1543402#post1543402

$$\left( \frac{d\tau }{dt}\right) ^{2}=\left( \frac{3}{2}\sqrt{1-\frac{2M}{R}}-\frac{1}{2}\sqrt{1-\frac{2Mr^{2}}{R^{3}}}\right) ^{2}-v^{2},$$

If we consider the simple case of of a non rotating spherical body, then the -v^2 term on the end can be ignored and the equation simplifies to:

$$\left( \frac{d\tau }{dt}\right) =\left( \frac{3}{2}\sqrt{1-\frac{2M}{R}}-\frac{1}{2}\sqrt{1-\frac{2Mr^{2}}{R^{3}}}\right)$$

where $\tau$ is the proper time of a clock located at a radius r inside the body of radius R and t is the clock rate at infinity.

Now if the clock is situated at the centre of the massive body, then r=0 and if the radius of the body is just slightly larger than Schwarzschild radius (9/8*2M) then:

$$\left( \frac{d\tau }{dt}\right) =\left( \frac{3}{2}\sqrt{1-\frac{2M}{(2M*9/8)}}-\frac{1}{2}\right)$$

$$\left( \frac{d\tau }{dt}\right) =\left( \frac{3}{2}\sqrt{\frac{1}{9}}-\frac{1}{2}\right)$$

$$\left( \frac{d\tau }{dt}\right) =\left( \frac{3}{6}-\frac{1}{2}\right) = 0$$

So there you have it. Frozen proper time, and the neutron star has not even formed a black hole yet. For radii less than 9/8 Rs but still greater than Rs, the equation gives negative proper time rates for the clock at the centre.

5. Nov 7, 2009

### Spinnor

and get:

Abstract. We refine the Buchdahl 9/8ths stability theorem for stars by describing quantitatively the behavior of solutions to the Oppenheimer–Volkoff equations when the star surface lies inside 9/8ths of the Schwarzschild radius. For such solutions we prove that the density and pressure always have smooth profiles that decrease to zero as the radius r→ 0, and this implies that the gravitational field becomes repulsive near r= 0 whenever the star surface lies within 9/8ths of its Schwarzschild radius.
Received: 19 June 1996 / Accepted: 13 September 1996

from:

Wow!

6. Nov 7, 2009

7. Nov 9, 2009

### Spinnor

Fantastic! Time can stop, and go backwards. Stop the presses %^)

Does backwards time violate any laws of physics?

Thank you.