# Escape velocity in GR

1. Aug 28, 2008

### stevebd1

(This was originally posted in https://www.physicsforums.com/showthread.php?t=248625&page=2" but I started a new thread as it was going slightly off topic)

Einsteins equation for gravity is g=ρc2+3P (basically energy density plus 3 times the pressure)-

The equation for Newtonian gravity being (incorporating m=Vρ)-

$$g=G\frac{m}{r^2}=G\frac{4\pi r_0^3}{3r^2} \rho$$

where G is the gravitational constant, r0 is the radius of the object of mass, r is the distance between the center of the object and the point at which gravity is being calculated and ρ is the density of the object of mass in kg/m^3 (Note when calculating gravity at the surface, r0^3/r^2 can be reduced to simply r0)

and the equation for GR gravity being-

$$g=G\frac{4\pi r_0^3}{3r^2} \left(\rho+\frac{3P}{c^2}\right)$$

as above but where P is pressure in N/m^3 and c is the speed of light

The escape velocity of an object relates directly to g-

$$v_e=\sqrt{\frac{2Gm}{r}}=\sqrt{2gr}$$

Which raises the issue of the effects of pressure on the collapse of a neutron star into a black hole as the collapse of the star is relative to the escape velocity exceeding c-

'There are a number of important situations in which ρ does not dominate P. In a neutron star, for example, which is held up by degeneracy pressure of the neutronium it consists of, pressure and energy density contribute comparably to the right-hand side of Einstein's equation. Moreover, above a mass of about 2 solar masses a nonrotating neutron star will inevitably collapse to form a black hole, thanks in part to the gravitational attraction caused by pressure.'
http://math.ucr.edu/home/baez/einstein/einstein.pdf" [Broken] by J C Baez and E F Bunn, page 7

GR escape velocity might result in a temporarily different Schwarzschild radius as the pressure would increase the gravity, therefore increasing the escape velocity and increasing the radius which ve=c.

Considering the escape velocity of a static 3 sol mass neutron star on the brink of collapse with a radius of 11 km and an average equation of state of ~1/7 -

Newtonian gravity and escape velocity-
gN = 3.291x10^12 m/s^2, ve = 2.691x10^8 m/s (0.898c)

GR gravity and escape velocity-
gGR = 4.625x10^12 m/s^2, ve = 3.190x10^8 m/s (>c)

So if pressure is included the star would collapse sooner into a black hole. But would the 'new' Schwarzschild radius be temporary as the black hole settled down to 2Gm/c^2?

Last edited by a moderator: May 3, 2017