The Cosmic Perspective (8th Edition)
8th Edition
ISBN: 9780134059068
Author: Jeffrey O. Bennett, Megan O. Donahue, Nicholas Schneider, Mark Voit
Publisher: PEARSON
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Chapter S1, Problem 7EAP
What are declination and right ascension? How are they similar to latitude and longitude? How are the different?
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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 S1 Solutions
The Cosmic Perspective (8th Edition)
Ch. S1 - Prob. 1VSCCh. S1 - Use the following questions to check your...Ch. S1 - Use the following questions to check your...Ch. S1 - Use the following questions to check your...Ch. S1 - Use the following questions to check your...Ch. S1 - Prob. 6VSCCh. S1 - Prob. 7VSCCh. S1 - Explain the differences between a (a) sidereal day...Ch. S1 - Prob. 2EAPCh. S1 - Prob. 3EAP
Ch. S1 - Prob. 4EAPCh. S1 - Prob. 5EAPCh. S1 - Prob. 6EAPCh. S1 - What are declination and right ascension? How are...Ch. S1 - 8. How and why do the Sun’s celestial coordinates...Ch. S1 - 9. Suppose you ire at the North Pole. Where is the...Ch. S1 - 10. Describe the Sun’s path through the local sky...Ch. S1 - 11. What is special about the tropics of Cancer...Ch. S1 - 2. Briefly describe how you can use the Sun or...Ch. S1 - Does It Make Sense? Decide whether the statement...Ch. S1 - Does It Make Sense?
Decide whether the statement...Ch. S1 - Does It Make Sense?
Decide whether the statement...Ch. S1 - Does It Make Sense?
Decide whether the statement...Ch. S1 - Does It Make Sense?
Decide whether the statement...Ch. S1 - Does It Make Sense?
Decide whether the statement...Ch. S1 - Does It Make Sense?
Decide whether the statement...Ch. S1 - Does It Make Sense?
Decide whether the statement...Ch. S1 - Does It Make Sense?
Decide whether the statement...Ch. S1 - Does It Make Sense? Decide whether the statement...Ch. S1 - Choose the best answer to each of the following...Ch. S1 - Choose the best answer to each of the following...Ch. S1 - Choose the best answer to each of the following...Ch. S1 - Choose the best answer to each of the following...Ch. S1 - Choose the best answer to each of the following...Ch. S1 - Choose the best answer to each of the following...Ch. S1 - Choose the best answer to each of the following...Ch. S1 - Choose the best answer to each of the following...Ch. S1 - Choose the best answer to each of the following...Ch. S1 - Choose the best answer to each of the following...Ch. S1 - Transits and the Geocentric Universe. Ancient...Ch. S1 - Geometry and Science. As discussed in Mathematical...Ch. S1 - Find Your Way Home. Roles: Scribe (takes notes on...Ch. S1 - Prob. 36EAPCh. S1 - Prob. 37EAPCh. S1 - Prob. 38EAPCh. S1 - Prob. 39EAPCh. S1 - Sydney Sky. Repeat Problem 39 for the local sky in...Ch. S1 - Prob. 41EAPCh. S1 - Prob. 42EAPCh. S1 - Be sure to show all calculations clearly and state...Ch. S1 - Be sure to show all calculations clearly and state...Ch. S1 - Be sure to show all calculations clearly and state...Ch. S1 - Prob. 46EAPCh. S1 - Be sure to show all calculations clearly and state...Ch. S1 - Be sure to show all calculations clearly and state...Ch. S1 - Be sure to show all calculations clearly and state...Ch. S1 - Be sure to show all calculations clearly and state...Ch. S1 - Be sure to show all calculations clearly and state...Ch. S1 - Northern Chauvinism. Why is the writing on maps...Ch. S1 - Celestial Navigation. Briefly discuss how you...Ch. S1 - Prob. 54EAPCh. S1 - Prob. 55EAPCh. S1 - Global Positioning System. Learn more about the...
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