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Escape velocity in GR

  1. Aug 28, 2008 #1
    (This was originally posted in Why mass causes space to bend 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)-

    [tex]g=G\frac{m}{r^2}=G\frac{4\pi r_0^3}{3r^2} \rho[/tex]

    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-

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

    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-

    [tex]v_e=\sqrt{\frac{2Gm}{r}}=\sqrt{2gr}[/tex]

    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.'
    The Meaning of Einstein's Equations 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: Aug 28, 2008
  2. jcsd
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