26 The International Space Station orbits at an altitude of about
350 km and a speed of about 8000 m/s relative to the ground. Com-
pare the gravitational and kinematic time dilations. Over all, does
time run faster on the ISS than on the ground, or more slowly?
27 Section 7.4.3 presented a Newtonian estimate of how compact
an object would have to be in order to be a black hole. Although
this estimate is not really right, it turns out to give the right answer
to within about a factor of 2. To roughly what size would the earth
have to be compressed in order to become a black hole?
28 Clock A sits on a desk. Clock B is tossed up in the air from
the same height as the desk and then comes back down. Compare
the elapsed times. .Hint, p. 1032 .Solution, p. 1042
29 The angular defectdof a triangle (measured in radians)
is defined ass−π, wheresis the sum of the interior angles. The
angular defect is proportional to the areaAof the triangle. Consider
the geometry measured by a two-dimensional being who lives on the
surface of a sphere of radiusR. First find some triangle on the sphere
whose area and angular defect are easy to calculate. Then determine
the general equation fordin terms ofAandR.
√30 (a) In this chapter we’ve represented Lorentz transformations
as distortions of a square into various parallelograms, with the de-
gree of distortion depending on the velocity of one frame of reference
relative to another. Suppose that one frame of reference was mov-
ing atcrelative to another. Discuss what would happen in terms
of distortion of a square, and show that this is impossible by using
an argument similar to the one used to rule out transformations like
the one in figure e on page 403.
(b) Resolve the following paradox. Two pen-pointer lasers are placed
side by side and aimed in parallel directions. Their beams both
travel atcrelative to the hardware, but each beam has a velocity of
zero relative to the neighboring beam. But the speed of light can’t
be zero; it’s supposed to be the same in all frames of reference.
31 The products of a certain radioactive decay are a massive
particle and a gamma ray, which is massless. See example 24 on
p. 438 for a discussion of the energy and momentum of a gamma
ray. (a) Show that, in the center of mass frame, the energy of the
gamma is less than the mass-energy of the massive particle.
(b) Show that the opposite inequality holds if we compare thekinetic
energy of the massive particle to the energy of the gamma. [Problem
by B. Shotwell.]Problems 463