Re: value of g in black holes

[Vic Decarlo <vdecarlo@DEPAUW.EDU>]

    I'm teaching a course on black holes this semester using
the new excellent new intro text by Taylor and Wheeler
("Exploring Black Holes").  One of the exercises involves
calculating the local acceleration of gravity for someone
standing on a shell of reduced circumference R surrounding
a nonspinning, spherical black hole of mass M.  The answer
turns out to be

                g = (M/R^2) / sqrt(1 - 2M/R)

with M and R expressed in meters and g in units of inverse
meters.  (For reference, the sun's mass in meters is 1470 m
and the local acceleration of gravity at the earth's surface
is about 10^-16 inverse meters).
    Anyway, you can see from the formula that in the limit
R --> 2M (at the event horizon) the local value of g
approaches infinity, irrespective of the mass of the black
hole.
    Perhaps John M., in speaking of a "completely
negligible" value of g at the event horizon of a very
massive black hole, is referring to relative tidal
accelerations for an  observer radially plunging into
the black hole.

Vic DeCarlo
DePauw University

John Mallinckrodt wrote:

> On Wed, 17 Oct 2001, Carl E. Mungan wrote:
>
> > Since no one else answered, I toss out the following. It's not GR so
> > it's probably wrong (I'm no expert), but at least it's digestible by
> > first-year students:
> >
> > escape velocity c = sqrt(2GM/R) where M and R are mass and radius of black hole
> >
> > g = GM/R^2
> >
> > Solve first equation for R, plug into second.
> > Minimum mass of black hole is 3 solar masses. Maximum mass of known
> > stars is something like 30 solar masses.
> >
> > Hence, we find that g can range from about 0.5 to 5 in units of
> > Tm/s^2. This is the value of g at the event horizon *after* the black
> > hole has formed.
>
> It is thought that 10^8 or more solar mass black holes exist in
> the center of some galaxies.  A 10^8 solar mass black hole would
> have R = 1000 light seconds and g = 150 km/s^2 at the event
> horizon.
>
> The universe might have something like 10^22 to 10^24 (give or
> take a few factors of ten) star's worth of mass.  If 10^24 solar
> masses were to coalesce into a black hole (or simply already and
> for all time have been a black hole), R would be billions of light
> years (i.e., something like the size of the universe) and g would
> be 15 pm/s^2 (i.e., completely negliglbe) at the "event horizon."
> Interesting?
>
> John Mallinckrodt      mailto:ajm@csupomona.edu
> Cal Poly Pomona          http://www.csupomona.edu/~ajm