Physical Universe
16th Edition
ISBN: 9780077862619
Author: KRAUSKOPF, Konrad B. (konrad Bates), Beiser, Arthur
Publisher: Mcgraw-hill Education,
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Textbook Question
Chapter 1, Problem 23E
The earth takes almost exactly 24 h to make a complete turn on its axis, so we might expect each high tide to occur 12 h after the one before. However, the actual time between high tides is 12 h 25 min. Can you account for the difference?
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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 1 Solutions
Physical Universe
Ch. 1 - The scientific method is a. continuing process b....Ch. 1 - A scientific law or theory is valid a. forever b....Ch. 1 - A hypothesis is a. a new scientific idea b. a...Ch. 1 - The ongoing evolution of living things a. is one...Ch. 1 - The object in the sky that apparently moves least...Ch. 1 - A constellation is a. an especially bright star b....Ch. 1 - Which of the following is no longer considered...Ch. 1 - A planet not visible to the naked eye is a. Mars...Ch. 1 - The planet closest to the sun is a. earth b. Venus...Ch. 1 - Prob. 10MC
Ch. 1 - Kepler modified the copernican system by showing...Ch. 1 - The speed of a planet in its orbit around the sun...Ch. 1 - According to Keplers third law, the time needed...Ch. 1 - The law of gravity a. applies only to large bodies...Ch. 1 - The earth bulges slightly at the equator and is...Ch. 1 - The usual tidal pattern in most parts of the world...Ch. 1 - Tides are caused a. only by the sun b. only by the...Ch. 1 - High tide occurs at a given place a. only when the...Ch. 1 - The prefix micro stands for a. 1/10 b. 1/100 c....Ch. 1 - A centimeter is a. 0.001 m b. 0.01 m c. 0.1 m d....Ch. 1 - Of the following, the shortest is a. 1 mm b. 0.01...Ch. 1 - Of the following, the longest is a. 1000 ft b. 500...Ch. 1 - A person is 180 cm tall. This is equivalent to a....Ch. 1 - The measurements of a room are given as length =...Ch. 1 - Prob. 1ECh. 1 - Prob. 2ECh. 1 - What is the difference between a hypothesis and a...Ch. 1 - Prob. 4ECh. 1 - According to the physicist Richard Feynman,...Ch. 1 - What does a year correspond to in terms of...Ch. 1 - You are lost in the northern hemisphere in the...Ch. 1 - In terms of what you would actually observe, what...Ch. 1 - What must be your location if the stars move...Ch. 1 - How do leap years fit into the ptolemaic system?...Ch. 1 - From observations of the moon, why would you...Ch. 1 - The sun, moon, and planets all follow...Ch. 1 - What is the basic difference between the ptolemaic...Ch. 1 - Ancient astronomers were troubled by variations in...Ch. 1 - Compare the ptolemaic and copernican explanations...Ch. 1 - What do you think is the reason scientists use an...Ch. 1 - The average distance from the earth to the sun is...Ch. 1 - As the earth revolves around the sun, some stars...Ch. 1 - Why is gravity considered a fundamental force...Ch. 1 - What, if anything, would happen to the shape of...Ch. 1 - What is the difference between spring and neap...Ch. 1 - The length of the day has varied. When did the...Ch. 1 - The earth takes almost exactly 24 h to make a...Ch. 1 - Does the sun or the moon have the greater...Ch. 1 - In the following pairs of length units, which is...Ch. 1 - A European driving from Paris to Brussels finds...Ch. 1 - The worlds tallest tree is a sequoia in California...Ch. 1 - The diameter of an atom is roughly 104 times the...Ch. 1 - How many square feet are there in an area of 1.00...Ch. 1 - A swimming pool is 20.00 m long, 7.00 m wide, and...Ch. 1 - The speedometer of a European car gives its speed...Ch. 1 - A horse galloped a mile in 2 min 35 s. What was...Ch. 1 - How many microphones are there in a megaphone?Ch. 1 - Use the proper number of significant figures to...Ch. 1 - Prob. 35E
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