Problem Set #2 1. Problem 4.15 Suppose a constant force of ...

Problem Set #2
1. Problem 4.15 Suppose a constant force of magnitude F acts on a particle of mass
m initially at rest.
(a) Integrate the formula for the acceleration found in Problem 4.14:
⃗a =
F ⃗
γm − ⃗v
γmc 2 ( F ⃗ · ⃗v) (1)
to show that the speed of the particle after time t is given by:
v
c =
(F/m)t
√
(F/m) 2 t 2 + c 2 (2)
(b) Rearrange this equation to express the time t as a function of v/c. If the particle’s
initial acceleration at time t = 0 is a = g = 9.80 m/s 2 , how much time is
required for the particle to reach a speed of v/c = 0.9? v/c = 0.99? v/c = 0.999?
v/c = 0.9999? v/c = 1?
2. Problem 17.4 Leadville, Colorado, is at an altitude of 3.1 km above sea level. If a
person there lives for 75 years (as measured by an observer at a great distance from
Earth), how much longer would gravitational time dilation have allowed that person
to live if he or she had moved at birth from Leadville to a city at sea level?
Figure 1: Local inertial frames for measuring the deflection of light near the Sun
3. Problem 17.6 Imagive a series of rectangular inertial reference frames suspended by
cables in a line near the Sun’s surface, as shown in Fig. 1. The frames are carefully
lined up so that the tops and sides of neighboring fames are parallel, and the tops of
the frames lie along the z-axis. A photon travels unhindered through the frames. As
the photon enters each frame, the frame is released from rest and falls freely toward
the center of the Sun.
(a) Show that as it passes through the frame located at angle α (shown in the figure),
the angular deflection of the photon’s path is
dφ = g 0 cos 3 α
c 3 dz, (3)
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where dz is the width of the reference frame and g 0 is the Newtonian gravitational
acceleration at the point of closest approach, O. The angular deflection is small,
so assume that the photon is initially traveling in the z-direction as it enters the
frame. (Hint: The width of the frame in the z-direction is dz, so the time for
the photon to cross the frame can be taken to be dz/c)
(b) Integrate the result you found in part (a) from α = −π/2 to +π/2 and so find the
total angular deflection of the photon as it passes through the curved spacetime
near the Sun.
(c) Your answer (which is also the answer obtained by Einstein in 1911 before he
arrived at his field equations) is only half the correct value of 1.75”’. Can you
qualitatively account for the missing factor of two?
4. Problem 17.9 Consider a spherical blackbody of constant temperature and mass M
whose surface lies at radial coordinate r = R. An observer located at the surface of
the sphere and a distant observer both measure the blackbody radiation given off by
the sphere.
(a) If the observer at the surface of the sphere measures the luminosity of the blackbody
to be L, use the gravitational time dilation formula:
∆t 0
= v (
∞
= 1 − 2GM ) 1/2
∆t ∞ v 0 r 0 c 2 (4)
to show that the observer at infinity measures
(
L ∞ = L 1 − 2GM )
Rc 2 . (5)
(b) Both observers use Wien’s law
λmaxT = 0.002897755 m K, (6)
to determine the radius of the spherical blackbody. Show that
T ∞ = T
(c) Both observers use the Stefan-Boltzmann law,
√
1 − 2GM
Rc 2 . (7)
L = 4πR 2 σT 4 e , (8)
to determine the radius of the spherical blackbody. Show that
R ∞ =
R
√
1 − 2GM/Rc 2 . (9)
Thus, using the Stefan-Boltzmann law without including the effects of general
relativity will lead to an overestimate of the size of a compact blackbody.
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5. A flashbulb is mounted midway between two mirrors (the mirror at smaller x’ we’ll
call ”‘left”’ and the one at larger x’ we’ll call ”‘right”’) which are separated by a
proper distance L along the x’ axis. At time t’ = 0, two flashes are emitted, one
traveling toward each mirror. The mirrors reflect the light back to the bulb and the
bulb reabsorbs the photons.
(a) Draw the Minkowski diagram used by the observers traveling with this apparatus.
Be sure to show:
i. The worldlines of the two mirrors, the flashbulbs and the photons (use dashed
lines for the photons).
ii. Put a label at the following events:
• events A - emission of the flashes
• event B - reflection of the front flash
• event C - reflection of the rear flash
• event D - reabsorption of the front flash
• event E - reabsorption of the rear flash
• event F - the right end (larger x) mirror at time t = 0
iii. Which pairs of the following events are simultaneous?
iv. Find the time (t’) for each of these events (in units of L and c)
(b) Draw the Minkowski diagram for these same events using the station (S) coordinates
(x and t). These observers see the apparatus traveling to the right (towards
larger x) at speed u and they draw the x and t axes at right angles.
i. Show the worldlines mentioned in ai.
ii. Show the events A, B, C, D, E, and F.
iii. Are the same pairs of events simultaneous?
iv. What is the separation (∆x) of the mirrors? Label this separation on the
diagram.
(c) Find the spacetime interval (∆s) 2 between the events
i. A and B
ii. B and C
iii. B and D
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