[SOLVED] Physics 396 homework set 5

1. Consider the line element of the form
ds2 =

1 + gx0
c2
2
c2
dt02 + dx02 + dy02 + dz02
, (1)
where g is a constant with units of acceleration. This line element describes an observer
in an accelerating frame of reference with coordinates (t
0
, x0
, y0
, z0
).a) Consider an observer in this frame of reference, Alyce, who remains at a fixed
location x0 = h. Alyce emits two light signals and therefore measures the proper
time interval between these two events. Obtain an expression for this proper time,
A, in terms of the coordinate time t
0
.b) Consider a second observer in this same frame of reference, Bob, who remains at a
fixed location x0 = 0. Bob receives the two light signals at his location and therefore
measures the proper time interval between these two events. Obtain an expression
for this proper time interval, B, in terms of the coordinate time t
0
.c) As the coordinate times in a) and b) are the same, obtain an expression connecting
A and B. Who measures the longer time interval?2. Consider the line element in Eq. (1) and the coordinate transformations
ct0 = c2
g
tanh1
ct
x + c2/g
,
x0 =

x +
c2
g
2
c2
t
2
#1/2
c2
g ,
y0 = y, z0 = z. (2)By inserting the coordinate transformations of Eqs. (2) into the line element of Eq. (1),
show that Eq. (1) takes the form of that of flat Minkowski spacetime. Hint: Keep the
combination (x + c2/g) as one when performing the calculation.Since this coordinate transformation brings the line element to the form of Minkowski
spacetime, and since the line element is an invariant quantity, this implies that Eq. (1) is
also that of Minkowski spacetime, but in the coordinates of an accelerating frame.3. Consider the Minkowski line element of the form
ds2 = c2
dt2 + dx2 + dy2 + dz2
, (3)
in (t, x, y, z) coordinates. Also consider the coordinate transformations
ct =
c2
g
+ x0

sinh gt0
c

,
x =
c2
g
+ x0

cosh gt0
c

c2
g ,
y = y0 , z = z0
, (4)
where g is a constant with units of acceleration. By inserting the coordinate
transformations of Eqs. (4) into the line element of Eq. (3), show that Eq. (3) takes the
form of Eq. (1).4. a) Show that the coordinate transformations of Eqs. (4) equate to the inverse coordinate
transformations of Eqs. (2).b) Show that in the limit when (ct0
, x0
) c2/g, the coordinate transformations of
Eqs. (4) reduce to the approximate form
ct = ct0
1 + o(gx0
/c2
) ct0
x = x0 +
1
2
gt02
1 + o(gx0
/c2
) x0 +
1
2
gt02
. (5)Notice that this corresponds to the coordinate transformations of a uniformly
accelerating frame relative to an inertial frame in Newtonian mechanics.c) Notice that the ratio c2/g defines a length scale (and therefore a time scale) where
Eqs. (5) are valid. Consider an accelerating frame where g is set equal to the value
of acceleration for a freely-falling object. Calculate the length scale (in light years)
and time scale (in years) of this ratio of constants.5. The coordinate transformations given in Eq. (4) of Problem 3 equate to going from
the inertial coordinates of Minkowski spacetime (t, x, y, z) to the coordinates of an
accelerating observer (t
0
, x0
, y0
, z0
) relative to this Minkowski spacetime.a) Using Eqs. (1) and (4) and the chain rule, compute the velocity four-vector of a point
in this accelerating frame, namely,
u = dx
d . (6)b) Compute the acceleration four-vector of a point in this accelerating frame, namely,
a = d2x
d 2 . (7)c) Compute the invariant magnitude of the acceleration four vector, namely,
a
aa1/2
. (8)
d) How does this invariant quantity dier for an observer at x0 = h compared to one at
x0 = 0?