Physics for Scientists and Engineers with Modern Physics
10th Edition
ISBN: 9781337553292
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Textbook Question
Chapter 38.4, Problem 38.3QQ
Suppose the observer O′ on the train in Figure 38.6 aims her flashlight at the far wall of the boxcar and turns it on and off, sending a pulse of light toward the far wall. Both O′ and O measure the time interval between when the pulse leaves the flashlight and when it hits the far wall. Which observer measures the proper time interval between these two events? (a) O′ (b) O (c) both observers (d) neither observer
Figure 38.6 (a) A mirror is fixed to a moving vehicle, and a light pulse is sent out by observer O′ at rest in the vehicle. (b) Relative to a stationary observer O standing alongside the vehicle, the mirror and O′ move with a speed v and the light pulse follows a diagonal path. (c) The right triangle for calculating the relationship between ∆t and ∆tp.
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Question B3
Consider the following FLRW spacetime:
t2
ds² = -dt² +
(dx²
+ dy²+ dz²),
t2
where t is a constant.
a)
State whether this universe is spatially open, closed or flat.
[2 marks]
b) Determine the Hubble factor H(t), and represent it in a (roughly drawn) plot as a function
of time t, starting at t = 0.
[3 marks]
c) Taking galaxy A to be located at (x, y, z) = (0,0,0), determine the proper distance to galaxy
B located at (x, y, z) = (L, 0, 0). Determine the recessional velocity of galaxy B with respect
to galaxy A.
d) The Friedmann equations are
2
k
8πG
а
4πG
+
a²
(p+3p).
3
a
3
[5 marks]
Use these equations to determine the energy density p(t) and the pressure p(t) for the
FLRW spacetime specified at the top of the page.
[5 marks]
e) Given the result of question B3.d, state whether the FLRW universe in question is (i)
radiation-dominated, (ii) matter-dominated, (iii) cosmological-constant-dominated, or (iv)
none of the previous. Justify your answer.
f)
[5 marks]
A conformally…
SECTION B
Answer ONLY TWO questions in Section B
[Expect to use one single-sided A4 page for each Section-B sub question.]
Question B1
Consider the line element
where w is a constant.
ds²=-dt²+e2wt dx²,
a) Determine the components of the metric and of the inverse metric.
[2 marks]
b) Determine the Christoffel symbols. [See the Appendix of this document.]
[10 marks]
c)
Write down the geodesic equations.
[5 marks]
d) Show that e2wt it is a constant of geodesic motion.
[4 marks]
e)
Solve the geodesic equations for null geodesics.
[4 marks]
Page 2
SECTION A
Answer ALL questions in Section A
[Expect to use one single-sided A4 page for each Section-A sub question.]
Question A1
SPA6308 (2024)
Consider Minkowski spacetime in Cartesian coordinates th
=
(t, x, y, z), such that
ds² = dt² + dx² + dy² + dz².
(a) Consider the vector with components V" = (1,-1,0,0). Determine V and V. V.
(b) Consider now the coordinate system x' (u, v, y, z) such that
u =t-x,
v=t+x.
[2 marks]
Write down the line element, the metric, the Christoffel symbols and the Riemann curvature
tensor in the new coordinates. [See the Appendix of this document.]
[5 marks]
(c) Determine V", that is, write the object in question A1.a in the coordinate system x'. Verify
explicitly that V. V is invariant under the coordinate transformation.
Question A2
[5 marks]
Suppose that A, is a covector field, and consider the object
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Chapter 38 Solutions
Physics for Scientists and Engineers with Modern Physics
Ch. 38.1 - Which observer in Figure 38.1 sees the balls...Ch. 38.1 - Prob. 38.2QQCh. 38.4 - Suppose the observer O on the train in Figure 38.6...Ch. 38.4 - Prob. 38.4QQCh. 38.4 - Prob. 38.5QQCh. 38.4 - You are observing a spacecraft moving away from...Ch. 38.6 - You are driving on a freeway at a relativistic...Ch. 38.8 - Prob. 38.8QQCh. 38 - In a laboratory frame of reference, an observer...Ch. 38 - Prob. 2P
Ch. 38 - Prob. 3PCh. 38 - Prob. 4PCh. 38 - Prob. 5PCh. 38 - An astronaut is traveling in a space vehicle...Ch. 38 - Prob. 7PCh. 38 - You have been hired as an expert witness for an...Ch. 38 - Prob. 9PCh. 38 - Prob. 10PCh. 38 - Prob. 11PCh. 38 - A cube of steel has a volume of 1.00 cm3 and mass...Ch. 38 - Review. In 1963, astronaut Gordon Cooper orbited...Ch. 38 - You have an assistantship with a math professor in...Ch. 38 - Prob. 15PCh. 38 - Prob. 16PCh. 38 - A moving rod is observed to have a length of =...Ch. 38 - Prob. 18PCh. 38 - Prob. 19PCh. 38 - You have been hired as an expert witness in the...Ch. 38 - Figure P38.21 shows a jet of material (at the...Ch. 38 - Prob. 22PCh. 38 - Prob. 23PCh. 38 - Prob. 24PCh. 38 - Prob. 25PCh. 38 - Prob. 26PCh. 38 - Prob. 27PCh. 38 - (a) Find the kinetic energy of a 78.0-kg...Ch. 38 - Prob. 29PCh. 38 - Prob. 30PCh. 38 - Prob. 31PCh. 38 - Prob. 32PCh. 38 - Prob. 33PCh. 38 - Prob. 34PCh. 38 - Prob. 35PCh. 38 - Prob. 36PCh. 38 - Prob. 37PCh. 38 - Prob. 38PCh. 38 - Prob. 39PCh. 38 - An unstable particle with mass m = 3.34 1027 kg...Ch. 38 - Prob. 41PCh. 38 - Prob. 42APCh. 38 - Prob. 43APCh. 38 - Prob. 44APCh. 38 - Prob. 45APCh. 38 - Prob. 46APCh. 38 - Prob. 47APCh. 38 - Prob. 48APCh. 38 - Prob. 49APCh. 38 - Prob. 50APCh. 38 - Prob. 51APCh. 38 - Prob. 52APCh. 38 - The creation and study of new and very massive...Ch. 38 - Prob. 54CPCh. 38 - Prob. 55CP
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