“Interstellar” and Black Holes
Makers of the movie “Interstellar” boasted how much they tried to be faithful to
physics of gravity. One interesting part of the plot is centered on the Miller planet
entirely covered by ocean, orbiting just outside of a giant black hole called “Gargan-
tua.” Those who visited its surface, in a small landing ship, finds upon their return
to the mother ship that 23 earth-years has passed during their few hour of absence
due to the time-dilating effect of gravity. Some journalists criticized the movie for
not making much sense, referring to “extremely strong gravitational force, necessary
for the extreme time dilation.” To see the potential fallacies committed by the film
and also by these journalists, and also to understand other important physics aspects
of the movie, you may consider the following:
1) Motion of an object of mass maround a non-rotating black hole of mass M
(Gargantua is supposed to be a fast-rotating black hole but, here, for simplicity we
will assume otherwise) follows Lagrange equation of motion with the action
−mc2Zdss1−2GM
c2r˙
t2−1−2GM
c2r−1
˙r2/c2−r2˙
θ2+ (sin θ)2˙
φ2/c2
cis the speed of light and Gis the Newton’s gravitational constant. sis an arbi-
trary parametrization of the motion, t(s), r(s), . . . , with respect to which the “time”
derivative as in ˙
t, ˙r, . . . is taken. You may wish to confine your attention to r > rH≡
2GM/c2, below which things become really weird. The name “Gargantua” refers to
the movie fact that the black hole is so heavy that rHis extremely large, about a
billion kilometers.
Equivalently, one may choose to fix the “time” parameter sto be of special type,
upon which one is allowed to solve for Lagrange equation of
−1
2Zds 1−rH
r˙
t2−1−rH
r−1
˙r2/c2−r2˙
θ2+ (sin θ)2˙
φ2/c2
instead. This should be a little easier to handle. Show why solving this action
principle is equivalent to solving the first action principle. Describe what is the
residual freedom on s, which used to be an arbitrary parameter in the first action
principle.
2) List all symmetries and conserved quantities and solve for the orbit. For generic
orbits, there is no analytic solutions, so qualitative characterization or numerical
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