On 8 Oct 1998, Chris Hillman wrote:
Subject : Re: Re[9]: No such thing as BHs?
From : "C. Hillman" hillman@math.washington.edu
Date : Thu, 8 Oct 1998
The geodesic equations for the Schwarzschild coordinates for the Schwarzschild
vacuum are readily computed by the Lagrangian method from
L(t,r,theta,phi, t*, r*, theta*, phi*, s)
= (1-2m/r) t*^2 - r*^2/(1-2m/r) - r^2 (theta*^2 + sin^2 theta phi*^2)
where * denotes differentiation with respect to the parameter s. They are
t** + 2m/(r-2m) t* r* = 0
r** + m(r-2m)/r^3 t*^2 - m/(r(r-2m)) r*^2 + (r-2m) theta*^2
- (r-2m) sin^2 theta phi*^2 = 0
theta** + 2/r r* theta* - sin 2 theta theta* phi* = 0
phi** + 2/r r* phi* -2 cot theta theta* phi* = 0
For theta = pi/2 (orbit in the equatorial plane) and r = R (circular
orbit) all but the third equation drops out, and after dividing by a
common factor it becomes simply
m/R^3 t*^2 - phi*^2 = 0
or (d phi/dt)^2 = m/R^3 which is, of course, Kepler's Law for circular
orbits, which remains valid for gtr (Not so for elliptical orbits, of
course!). Thus, constant speed according to the Kepler law is the
"projection" of a geodesic in the spacetime, which appears as a perfect
helix in Schwarzschild coordinates.
Now, we haven't checked that the tangent vectors to this putative world
line are in fact timelike. Plugging dphi^2 = m/R^3 dt^2, r = R, and theta
= pi/2 into the line element we find
ds^2 = (1-2m/R) dt^2 - R^2 m/R^3 dt^2
= (1-3m/R) dt^2
or ds/dt = sqrt{1-3m/R} when R 3 m. In other words, when R 3m, the
parametrized curve
t(s) = s/sqrt(1-3m/R)
r(s) = R
theta(s) = pi/2
phi(s) = sqrt(m/R^3) s/sqrt(1- 3m/R))
is a timelike geodesic parametrized by arc length. For r = 3m, this
represents a null geodesic (world line of a photon in a circular
orbit--- i.e. a perfect helix in spacetime--- this is not a closed
timelike curve!). For r 3m, this is a spacelike geodesic.
We still haven't checked whether these circular orbits are stable under
small perturbations (see below for a surprise!).
Note that everything up to this point has been a study of circular orbits
of test particles around an isolated nonrotating massive body (analyzed in
the exterior static Schwarzschild coordinate chart).
It is interesting to compare the time kept by a satellite orbiting in a
circular orbit R_o with a clock on the surface of the Earth, which we
idealize as a nonrotating ball with radius R_g 2 m, and with exterior
geometry given by the Schwarzschild vacuum. Then
ds_o/dt = sqrt(1 - 3m/R_o)
ds_g/dt = sqrt(1-2m/R_g)
so
ds_o sqrt(1-3m/R_s)
---- = --------------
ds_g sqrt(1-3m/R_g)
This is increasing, approaches sqrt(R_g-3m)/sqrt(R_g-2m) for R_o - R_g,
approaches 1/sqrt(1-3m/R_g) for R_o - infty, and equals one for
R_o = (3/2) R_g
Thus, a satellite in low orbit moves so rapidly (according to Kepler's
law) that the kinematical time dilation (plus the gravitational time
dilation at radius R_o R_g) of the rapidly moving satellite clock
overwhelms the gravitational time dilation of the ground clock (at the
radius R_g), so that it runs slow relative to the ground clock. Satellites
with very high circular orbits are moving slowly and suffer little
kinematical or gravitational red shift, so they run fast relative to
ground clocks. At R_o = (3/2) R_g, the competing str and gtr effects
exactly balance, so that the clocks run at the same rate.
Chris Hillman
It is not true that the competing str and gtr effects
exactly balance at R_o = (3/2) R_g. Indeed,
the SR time dilation factor is sqrt(1-(GM/rc^2)*r/(r+H)), and
the GR factor is given by sqrt(1+(GM/rc^2)*2H/(r+H)), where
G is the universal gravitational constant, M the mass and r
the radius of the Earth, and H the altitude of the satellite
orbiting the Earth.
Note that
1) sqrt(1 - (GM/rc^2)*r/(r+H)) or sqrt(1 - v^2/c^2)
is the rate of the orbiting clock relative to a _stationary_
clock ** at the same height H **. (cf. Paul B. Andersen)
2) sqrt(1+(GM/rc^2)*2H/(r+H))
is the rate of a _stationary_ clock at the height H compared
to a stationary clock at the surface.
From 1) and 2), it is clear that the rate of the orbiting clock
relative to a stationary clock at the surface of the Earth is
given by the product of the SR clock slowing factor and the
GR factor, i.e.
sqrt((1 - (GM/rc^2)*r/(r+H) * (1+(GM/rc^2)*2H/(r+H)) =
sqrt((1 - (GM/rc^2)*r/(r+H) + (GM/rc^2)*2H/(r+H) - (GM/rc^2)^2*
2Hr/(r+H)^2)) =
sqrt((1 + (GM/rc^2)*(2H-r)/(r+H) - (GM/rc^2)^2 * 2Hr/(r+H)^2))
The GR solution is limited to sqrt((1 + (GM/rc^2)*(2H-r)/(r+H)),
which indeed becomes 1 for 2H=r, because it neglects the higher
order term (GM/rc^2)^2 * 2Hr/(r+H)^2.
Iow, GRT gives an approximate solution.
In order to obtain the correct altitude H at which the SR and GR
effects exactly balance, one has to solve the equation
(GM/rc^2)*(2H-r)/(r+H) - (GM/rc^2)^2 * 2Hr/(r+H)^2) = 0
Then, H = ((Rs-r)+ sqrt((Rs-r)^2+8*r^2))/4, where
Rs is the Schwarzschild radius.
It is only when Rs=0 (!), that the value of H is r/2, as predicted
by GRT.
Marcel Luttgens