Principles of Physics: A Calculus-Based Text
5th Edition
ISBN: 9781133104261
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Chapter 9, Problem 5P
At what speed does a clock move if it is measured to run at a rate one-half the rate of a clock at rest with respect to an observer?
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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
Fv=AAμ.
(a) Show explicitly that F is a tensor, that is, show that it transforms appropriately under a
coordinate transformation.
[5 marks]
(b)…
Chapter 9 Solutions
Principles of Physics: A Calculus-Based Text
Ch. 9.4 - Suppose the observer O on the train in Active...Ch. 9.4 - Prob. 9.2QQCh. 9.4 - Prob. 9.3QQCh. 9.4 - Prob. 9.4QQCh. 9.5 - You are driving on a freeway at a relativistic...Ch. 9.7 - The following pairs of energiesparticle 1: E, 2E;...Ch. 9 - An astronaut is traveling in a spacecraft in outer...Ch. 9 - A distant astronomical object (a quasar) is moving...Ch. 9 - Prob. 3OQCh. 9 - A spacecraft zooms past the Earth with a constant...
Ch. 9 - Which of the following statements are fundamental...Ch. 9 - Prob. 6OQCh. 9 - Prob. 7OQCh. 9 - Prob. 8OQCh. 9 - Two identical clocks are set side by side and...Ch. 9 - You measure the volume of a cube at rest to be V0....Ch. 9 - A train is approaching you at very high speed as...Ch. 9 - Explain why, when defining the length of a rod, it...Ch. 9 - A particle is moving at a speed less than c/2. If...Ch. 9 - Prob. 5CQCh. 9 - Prob. 6CQCh. 9 - Prob. 7CQCh. 9 - (a) “Newtonian mechanics correctly describes...Ch. 9 - Prob. 9CQCh. 9 - (i) An object is placed at a position p > f from a...Ch. 9 - With regard to reference frames, how does general...Ch. 9 - In a laboratory frame of reference, an observer...Ch. 9 - Prob. 2PCh. 9 - Prob. 3PCh. 9 - An astronaut is traveling in a space vehicle...Ch. 9 - At what speed does a clock move if it is measured...Ch. 9 - Prob. 6PCh. 9 - Prob. 7PCh. 9 - Prob. 8PCh. 9 - Prob. 9PCh. 9 - Prob. 10PCh. 9 - Prob. 11PCh. 9 - Prob. 12PCh. 9 - A friend passes by you in a spacecraft traveling...Ch. 9 - Prob. 14PCh. 9 - Prob. 15PCh. 9 - Prob. 16PCh. 9 - Prob. 17PCh. 9 - Prob. 18PCh. 9 - An enemy spacecraft moves away from the Earth at a...Ch. 9 - Prob. 20PCh. 9 - Figure P9.21 shows a jet of material (at the upper...Ch. 9 - Prob. 22PCh. 9 - Prob. 23PCh. 9 - Prob. 24PCh. 9 - Prob. 25PCh. 9 - Prob. 26PCh. 9 - Prob. 27PCh. 9 - Prob. 28PCh. 9 - Prob. 29PCh. 9 - Prob. 30PCh. 9 - Prob. 31PCh. 9 - Prob. 32PCh. 9 - Prob. 33PCh. 9 - Prob. 34PCh. 9 - Prob. 35PCh. 9 - Prob. 36PCh. 9 - Prob. 37PCh. 9 - Prob. 38PCh. 9 - Prob. 39PCh. 9 - Prob. 40PCh. 9 - Prob. 41PCh. 9 - Prob. 42PCh. 9 - Prob. 43PCh. 9 - Prob. 44PCh. 9 - Prob. 45PCh. 9 - Prob. 46PCh. 9 - Prob. 47PCh. 9 - Prob. 48PCh. 9 - Prob. 49PCh. 9 - Prob. 50PCh. 9 - Prob. 51PCh. 9 - Prob. 52PCh. 9 - An alien spaceship traveling at 0.600c toward the...Ch. 9 - Prob. 54PCh. 9 - Prob. 55PCh. 9 - Prob. 56PCh. 9 - Prob. 57PCh. 9 - Prob. 58PCh. 9 - Spacecraft I, containing students taking a physics...Ch. 9 - Prob. 60PCh. 9 - Prob. 61PCh. 9 - Prob. 62PCh. 9 - Owen and Dina are at rest in frame S, which is...Ch. 9 - A rod of length L0 moving with a speed v along the...Ch. 9 - Prob. 65P
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Length contraction: the real explanation; Author: Fermilab;https://www.youtube.com/watch?v=-Poz_95_0RA;License: Standard YouTube License, CC-BY