PHYS 5770 Grav Theory Spring 2008. Problem Set 5. Due Tue 15 Apr
1. Tidal forces falling into a Schwarzschild black hole
In the Gullstrand-Painlev´e tetrad, the non-zero components of the tetrad-frame Riemann
tensor are
−1
2Rtrtr =Rtθtθ =Rtφtφ =−Rrθr θ =−Rrφrφ =1
2Rθφθφ =C(1.1)
where
C=−M
r3(1.2)
is the Weyl scalar.
(a) Tidal forces
By construction of the Gullstrand-Painlev´e tetrad, a person who falls radially from zero
velocity at infinity is at rest in the Gullstrand-Painlev´e tetrad, with tetrad-frame 4-velocity
um={1,0,0,0}. From the equation of geodesic deviation
D2δξm
Dτ 2+Rklmn δξkulun= 0 (1.3)
deduce the tidal acceleration on the person in the radial and transverse directions. Does the
tidal acceleration stretch or compress? [Hint: The equation of geodesic deviation gives the
proper acceleration between two points a small distance δξmapart, where ξmare the locally
inertial coordinates of the tetrad frame. Notice that this problem is much easier to solve
with tetrads than with the traditional coordinate approach.]
(b) Choose a black hole to fall into
What is the mass of the black hole for which the tidal acceleration M/r3is 1 gee per meter
at the horizon? If you wanted to fall through the horizon of a black hole without first being
torn apart, what mass of black hole would you choose? [Hint: 1 gee is the gravitational
acceleration at the surface of the Earth.]
(c) Time to die
In problem set 3 you showed that the proper time to free-fall radially from radius rto the
singularity of a Schwarzschild black hole is
τ=√2
3rr3
M.(1.4)
How long, in seconds, does it take to fall to the singularity from the place where the tidal
acceleration is 1 gee per meter?
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