F. K. Lamb Fall 2008
PHYSICS 598AST: ASTROPHYSICS
First Homework Assignment
1. (a) Show that the spacetime interval
s2= (t1−t2)2−(x1−x2)2−(y1−y2)2−(z1−z2)2,
between two events (t1,x1,y1,z1) and (t2,x2,y2,z2) is invariant under a Lorentz boost. Here we have
used units in which c= 1. (In general, a Lorentz transformation between two inertial frames requires a
spatial rotation, followed by a boost, followed by a further spatial rotation.)
(b) Deduce from s2the line element for infinitesimally separated events.
(c) What does s2become if t1=t2, and how is this interval related to the spatial separation `between
the two events?
2. The purpose of this problem is to gain insight into the effects of curved spacetime by studying the motions
of two particles in the Newtonian approximation. It is possible to gain insight into the effects of curved
spacetime in this way because trajectories (geodesics) computed using General Relativity (GR) and
trajectories computed using Newtonian dynamics match in the (weak-field) regime where the Newtonian
approximation is valid.
In GR, the frame of a freely falling observer is a local inertial frame, i.e., the spacetime is locally Euclidean,
which means that it appears Euclidean for small spacetime separations. This problem explores how a
freely falling observer can determine whether the spacetime is also globally Euclidean or is instead curved.
One way to do this is to study the relative motion of two free particles that are initially at rest with
respect to one another (i.e., they have zero relative velocity). If the spacetime is globally Euclidean, the
velocities of both particles will be constant for all time, so their relative velocity will remain zero and
their separation will not change. If instead the global spacetime is curved, the velocities of two particles
at different spatial positions will not be constant for all time, their relative velocity will not remain zero,
and their separation will change with time. In the framework of GR, we say that the geodesics followed
by the two particles deviate from one another. Such a deviation is a signature of spacetime curvature.
(a) Suppose that two particles of negligible mass are initially positioned in an otherwise empty space on
a straight line extending from the origin of the coordinate system and that for times t < 0 they are held
at rest relative to the origin at distances rand r+hfrom it. At t= 0 both particles are released and
are afterward free to move. How does the separation hof the two particles evolve with time, i.e., what
is h(t) for t > 0?
(b) Suppose now that the two particles are initially positioned as in part (a), but that there is in addition
a third particle of mass Mpositioned at the origin of the coordinate system. Both particles are again
released at t= 0 and that h(0) r(0). Assume also that r(0) GM/c2, so the effect of gravity on the
initial motion of the two test particles can be computed accurately using the Newtonian approximation.
Show that their relative separation initially increases, i.e., that dh/dt is initially positive.
(c) Suppose now that each of the first two particles mentioned in part (b) has mass m. How small must
m/M be so that the effect of their mutual gravitational attraction does not overwhelm the effect found
in part (b)?
