Bundle: Physics for Scientists and Engineers with Modern Physics, Loose-leaf Version, 9th + WebAssign Printed Access Card, Multi-Term
9th Edition
ISBN: 9781305932302
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Chapter 28, Problem 44P
To determine
The value of the integral
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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 28 Solutions
Bundle: Physics for Scientists and Engineers with Modern Physics, Loose-leaf Version, 9th + WebAssign Printed Access Card, Multi-Term
Ch. 28.1 - To maximize the percentage of the power from the...Ch. 28.2 - With the switch in the circuit of Figure 27.4a...Ch. 28.2 - With the switch in the circuit of Figure 27.6a...Ch. 28.2 - Prob. 28.4QQCh. 28.4 - Consider the circuit in Figure 27.17 and assume...Ch. 28 - Prob. 1OQCh. 28 - Prob. 2OQCh. 28 - Prob. 3OQCh. 28 - Prob. 4OQCh. 28 - Prob. 5OQ
Ch. 28 - Prob. 6OQCh. 28 - Prob. 7OQCh. 28 - Prob. 8OQCh. 28 - Prob. 9OQCh. 28 - Prob. 10OQCh. 28 - Prob. 11OQCh. 28 - Prob. 12OQCh. 28 - Prob. 13OQCh. 28 - Prob. 14OQCh. 28 - Prob. 15OQCh. 28 - Prob. 1CQCh. 28 - Prob. 2CQCh. 28 - Why is it possible for a bird to sit on a...Ch. 28 - Prob. 4CQCh. 28 - Prob. 5CQCh. 28 - Prob. 6CQCh. 28 - Prob. 7CQCh. 28 - Prob. 8CQCh. 28 - Is the direction of current in a battery always...Ch. 28 - Prob. 10CQCh. 28 - Prob. 1PCh. 28 - Two 1.50-V batterieswith their positive terminals...Ch. 28 - An automobile battery has an emf of 12.6 V and an...Ch. 28 - Prob. 4PCh. 28 - Prob. 5PCh. 28 - Prob. 6PCh. 28 - Prob. 7PCh. 28 - Prob. 8PCh. 28 - Prob. 9PCh. 28 - Prob. 10PCh. 28 - Prob. 11PCh. 28 - Prob. 12PCh. 28 - Prob. 13PCh. 28 - Prob. 14PCh. 28 - Prob. 15PCh. 28 - Prob. 16PCh. 28 - Prob. 17PCh. 28 - Prob. 18PCh. 28 - Prob. 19PCh. 28 - Why is the following situation impossible? A...Ch. 28 - Prob. 21PCh. 28 - Prob. 22PCh. 28 - Prob. 23PCh. 28 - Prob. 24PCh. 28 - Prob. 25PCh. 28 - The following equations describe an electric...Ch. 28 - Prob. 27PCh. 28 - Prob. 28PCh. 28 - Prob. 29PCh. 28 - Prob. 30PCh. 28 - Prob. 31PCh. 28 - Prob. 32PCh. 28 - Prob. 33PCh. 28 - Prob. 34PCh. 28 - Prob. 35PCh. 28 - Prob. 36PCh. 28 - An uncharged capacitor and a resistor are...Ch. 28 - Prob. 38PCh. 28 - Prob. 39PCh. 28 - A 10.0-F capacitor is charged by a 10.0-V battery...Ch. 28 - Prob. 41PCh. 28 - Prob. 42PCh. 28 - Prob. 43PCh. 28 - Prob. 44PCh. 28 - A charged capacitor is connected to a resistor and...Ch. 28 - Prob. 46PCh. 28 - Prob. 47PCh. 28 - Prob. 48PCh. 28 - Prob. 49APCh. 28 - Prob. 50APCh. 28 - Prob. 51APCh. 28 - Prob. 52APCh. 28 - Prob. 53APCh. 28 - Prob. 54APCh. 28 - Prob. 55APCh. 28 - Prob. 56APCh. 28 - Prob. 57APCh. 28 - Why is the following situation impossible? A...Ch. 28 - Prob. 59APCh. 28 - Prob. 60APCh. 28 - When two unknown resistors are connected in series...Ch. 28 - Prob. 62APCh. 28 - Prob. 63APCh. 28 - A power supply has an open-circuit voltage of 40.0...Ch. 28 - Prob. 65APCh. 28 - Prob. 66APCh. 28 - Prob. 67APCh. 28 - Prob. 68APCh. 28 - Prob. 69APCh. 28 - Prob. 70APCh. 28 - Prob. 71APCh. 28 - Prob. 72APCh. 28 - A regular tetrahedron is a pyramid with a...Ch. 28 - An ideal voltmeter connected across a certain...Ch. 28 - Prob. 75APCh. 28 - Prob. 76APCh. 28 - Prob. 77APCh. 28 - Prob. 78APCh. 28 - Prob. 79APCh. 28 - Prob. 80APCh. 28 - Prob. 81APCh. 28 - Prob. 82CPCh. 28 - Prob. 83CP
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