[SOLVED] Physics 396 homework set 6

1. Consider the 4D spacetime, spanned by coordinates (v, r, , ), with a line element
ds2 =

1 Rs
r

dv2 + 2dvdr + r2
(d2 + sin2 d2
), (1)
where Rs 2GM/c2 is called the event horizon (a.k.a. Schwarzschild radius). It is noted
that the above line element describes the 4D spacetime outside a static, spherically
symmetric object of mass M in Eddington-Finkelstein coordinates.Consider a radial null curve ( = const., = const., ds2 = 0).
a) Calculate the slopes of the light cone at a point (v, r).b) Evaluate the slopes of the light cones at points (v, r) = (0, Rs/2), (0, Rs), (0, 2Rs),
(0, 3Rs), and (0, 4Rs). Draw the corresponding (v, r) spacetime diagram with
the apex of the light cones positioned at these locations. This diagram should fill
a standard piece of paper. Use a ruler to make sure that your drawn curves are
consistent with your calculated values.c) How do the slopes of the light cones change with v for a given value of r?2. In lecture, we studied the 4D wormhole spacetime with a line element of the form
ds2 = c2
dt2 + dr2 + (b2 + r2
)(d2 + sin2 d2
). (2)We found the embedding diagram for a 2D equatorial slice of the 4D wormhole and that
this surface has two asymptotically flat regions connected by a throat of circumference
2b. We also found that this spacetime is spherically symmetric since a surface with
constant r and t has the geometry of a sphere.Consider a t = const. slice of the wormhole geometry bounded by two spheres of
coordinate radius R centered on r = 0, which reside on each side of the throat.a) Calculate the circumference of the equator of these spheres.
b) Calculate the distance S from one sphere to the other, through the throat, along a
= const., = const. line.c) Calculate the area of the two-sphere of coordinate radius r = R.
d) Calculate the 3D volume bounded by the two spheres of coordinate radius R.3. In lecture, we arrived at the 2D non-Euclidean line element of a two-sphere of radius R
from 3D Euclidean space by performing the coordinate transformation
x = R sin cos
y = R sin sin
z = R cos , (3)
where 0 < and 0 < 2.It is noted that the above transformation obeys the
equation of constraint
x2 + y2 + z2 = R2
, (4)
which eectively constrains the radial coordinate to take on a constant value, r = R.a) Here we wish to arrive at a 3D non-Euclidean line element of a three-sphere of radius
R from a fictitious 4D Euclidean space. Consider the coordinate transformation
x = R sin sin cos
y = R sin sin sin
z = R sin cos
w = R cos , (5)
where 0 < , 0 < , and 0 < 2.Show that the above transformation
obeys the equation of constraint
x2 + y2 + z2 + w2 = R2
, (6)
which eectively constrains the radial coordinate to take on a constant value, r = R.b) Calculate dx, dy, dz, dw.
c) Calculate dx2 + dy2.
d) Calculate dx2 + dy2 + dz2.e) Show that the 4D Euclidean line element
dS2 = dx2 + dy2 + dz2 + dw2 (7)
becomes that of a 3D non-Euclidean line element of a three-sphere of radius R of
the form
dS2 = R2
d2 + sin2 (d2 + sin2 d2
)
(8)
under the above coordinate transformation.f) By defining r sin , show that the above line element takes the form
dS2 = R2
dr2
1 r2 + r2
(d2 + sin2 d2
). (9)Notice that the line element of Eqs. (8) and (9) equate to a t = const. slice of a
homogeneous closed universe, which was analyzed in Example 7.6.
g) What is the range of this r coordinate?4. In lecture, we found an embedding diagram for the 4D wormhole. This equated to finding
a curved 2D surface in 3D Euclidean space with the same intrinsic geometry as a 2D
equatorial slice (t = const., = /2) of the 4D wormhole line element.a) Consider the t = const. slice of a homogeneous closed universe given by Eq. (9).
Construct the corresponding 2D equatorial slice ( = /2) for this homogenous
closed universe, analogous to Eq. (7.41) of your text.b) Following a procedure similar to that of Section 7.7 of your text, show that the curved
2D surface obeys the dierential equation
dz
dr = R r
p1 r2
. (10)c) By integrating Eq. (10), show that the solution obeys the expression
z2
R2 + r2 = 1. (11)
d) Plot Eq. (11) on a z(r)/R vs. r set of axes.