Cosmic Perspective Fundamentals
3rd Edition
ISBN: 9780134988504
Author: Bennett, Jeffrey O., Donahue, M. (megan), SCHNEIDER, Nicholas, Voit, Mark
Publisher: Pearson,
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Textbook Question
Chapter 6, Problem 14SEQ
Explain all answers clearly, with complete sentences and proper essay structure if needed. An asterisk (*) designates a quantitative problem, for which you should show all your work.
The New View of titan. What other planet or moon in the solar system does Titan most resemble, in your opinion? Summarize the similarities and differences in a few sentences.
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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 6 Solutions
Cosmic Perspective Fundamentals
Ch. 6 - Choose the best answer to each of the following....Ch. 6 - Choose the best answer to each of the following....Ch. 6 - Choose the best answer to each of the following....Ch. 6 - Choose the best answer to each of the following....Ch. 6 - Choose the best answer to each of the following....Ch. 6 - Choose the best answer to each of the following....Ch. 6 - Choose the best answer to each of the following....Ch. 6 - Choose the best answer to each of the following....Ch. 6 - Choose the best answer to each of the following....Ch. 6 - Choose the best answer to each of the following....
Ch. 6 - Choose the best answer to each of the following....Ch. 6 - Prob. 12QQCh. 6 - Explain all answers clearly, with complete...Ch. 6 - Explain all answers clearly, with complete...Ch. 6 - Explain all answers clearly, with complete...Ch. 6 - Explain all answers clearly, with complete...Ch. 6 - Explain all answers clearly, with complete...Ch. 6 - Explain all answers clearly, with complete...Ch. 6 - Explain all answers clearly, with complete...Ch. 6 - Explain all answers clearly, with complete...Ch. 6 - Explain all answers clearly, with complete...
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