P8.6 Consider a 2D spacetime whose metric is
ds^(2)=-e^(-(x)/(a))dt^(2)+dx^(2)
where a is a constant with units of meters. (Unlike the
spacetime in problem P8.5, the underlying geometry of
this spacetime is not flat.)
a. Show that the t component of the geodesic equation
implies that d(t)/(d au) =ce^((v)/(a)), where c is some constant.
b. The x component of the geodesic equation is hard to
integrate, but show that requiring u*u=-1 yields
(dx)/(d au)=+-sqrt(c^(2)e^((x)/(a))-1)
c. We can calculate the trajectory x(t) for a geodesic in
this spacetime by calculating
(dx)/(dt)=(d(x)/(d au))/(d(t)/(d au))
and integrating. Your integral will yield t(x): you have
to invert this function to find x(t). Express your result
in terms of a, c, and the initial conditions x=x_(0) and
d(x)/(d au)=u_(0) at time t=0. You should find that
x(t)=aln[((t)/(2a)+(u_(0))/(c))^(2)+(1)/(c^(2))]
Also verify that x(0)=x_(0).
d. Find the trajectory x(t) for a freely-falling particle
starting from rest at the origin (i.e., x_(0)=0 and u_(0)=0).
Plot a graph of (x)/(a) versus (t)/(a) for this geodesic world-
line, making the t axis vertical if possible. (Hint: Note
that although I left the constant c in equation 8.70 for
the sake of simplicity, it is completely determined by
the initial conditions.)
e. We often also describe a worldline by specifying
x^(mu)( au). Find expressions for x( au) and t( au) for the case
of a particle starting from rest at the origin (make sure
that you set au=0 when t=0). Argue that the proper
time au measured along this worldline asymptotically
approaches a maximum as t->infty and calculate that
maximum. (The constant a should be the only undetermined constant in your final expression.)