[SOLVED] Physics 396 homework set 8

1. In geometrized units, the Schwarzschild line element takes the form
ds2 =

1 2M
r

dt2 +

1 2M
r
1
dr2 + r2
(d2 + sin2 d2
), (1)
where M is the total mass of the central object. Consider two massless shells concentric
with the central object. The inner shell has a circumference of 6M and the outer shell
has a circumference of 20M.a) Calculate the physical radial distance between these two shells.
b) Calculate the spatial volume between the two spherical shells.2. In class, we arrived at an equation of motion describing the radial coordinate of a freelyfalling particle about a spherically-symmetric massive object of the form
E = 1
2
dr
d
2
+ Vef f (r), (2)
where M is the mass of the central object, E (e2 1)/2, and
Vef f (r) = M
r
+
`2
2r2 M`2
r3 . (3)a) Setting r Mx and ` M `, show that Eq. (3) can be written in the form
Vef f (x) = 1
x
+ `2
22 `2
x3 . (4)b) Using your favorite plotting software (i.e. Maple, Matlab, Wolframalpha, etc.), plot
Vef f (x) vs x for ` = 3, ` = p12, ` = 4, and ` = 4.5. Your x-axis should have a range
of 2 <x< 50 to properly show the curves. These four curves should be positioned
on the same plot.3. Consider the spacetime geometry exterior to a spherically symmetric massive object, in
the presence of a constant vacuum energy. The line element takes the form
ds2 =

1 2M
r
3
r2

dt2 +

1 2M
r
3
r2
1
dr2 + r2
(d2 + sin2 d2
), (5)
where M is the mass of the central object and is the cosmological constant. It is noted
that the can take on both positive and negative values.Here, were interested in
obtaining the radial equation of motion for a massive particle freely-falling in this curved
spacetime, which is analogous to Eq. (2).a) The line element of Eq. (5) contains two symmetries associated with the t and
coordinates. Construct the two first integrals associated with these symmetries of
the form
e

u (6)
`

u, (7)
where e and ` are constants,
u is the four-velocity of the massive particle, and ,
are the Killing vectors associated with the t, coordinates, respectively.b) Another first integral can be constructed from the fact that freely-falling massive
particles follow timelike four-velocities, namely,

u

u = 1. (8)
Using Eq. (5), construct this first integral.c) As the angular momentum of this freely-falling massive particle is conserved, one can
choose the motion to occur in the = /2 plane. Using this fact and Eqs. (6) and
(7), show that Eq. (8) can be written in the form
E = 1
2
dr
d
2
+ Vef f (r), (9)
where E (e2 1)/2 and
Vef f (r) = M
r
+
`2
2r2 M`2
r3
6
(`
2 + r2
) (10)4. Reconsider the spacetime geometry of the previous problem where the line element is given
in Eq. (5).
a) Construct the equation that determines the radii corresponding to the extrema of the
eective potential. Write the expression as a polynomial and state its order.b) Construct the equation that determines the radii corresponding to the turning points
of the motion.
c) Setting r Mx, ` M `, and /M2, show that Eq. (10) can be written in the
form
Vef f (x) = 1
x
+ `2
22 `2
x3
6
(`
2 + x2
). (11)d) Using your favorite plotting software (i.e. Maple, Matlab, Wolframalpha, etc.), plot
Vef f (x) vs x for ` = 4.5 and = +1.0 105, = 0, and = 1.0 105. Your
x-axis should have a range of 2 <x< 80 to properly show the curves. These three
curves should be positioned on the same plot.5. We now wish to consider light ray orbits in the spacetime described by the line element of
Eq. (5).
a) Construct the analogous expressions to Eqs. (6) and (7) in terms of the ane
parameter .b) Construct the first integral corresponding to the fact that light rays have null
four-velocites, namely,

u

u = 0. (12)c) One can choose the motion to occur in the = /2 plane. Using this fact and Eqs.
(6) and (7), show that Eq. (12) can be written in the form
e2
`2 = 1
`2
dr
d
2
+ Wef f (r), (13)
where
Wef f (r) = 1
r2

1 2M
r
3
r2

. (14)d) Calculate the radius or radii corresponding to the extrema of this eective potential.
Calculate the value(s) of this eective potential at this radius (or these radii).e) Construct the equation that determines the radii corresponding to the turning points
of the motion.