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A Student’s Manual for A First Course in General Relativity
This comprehensive student manual has been designed to accompany the leading textbook
by Bernard Schutz, A First Course in General Relativity, and uses detailed solutions, cross-
referenced to several introductory and more advanced textbooks, to enable self-learners,
undergraduates, and postgraduates to master general relativity through problem solving.
The perfect accompaniment to Schutz’s textbook, this manual guides the reader step-by-
step through over 200 exercises, with clear easy-to-follow derivations. It provides detailed
solutions to almost half of Schutz’s exercises, and includes 125 brand-new supplementary
problems that address the subtle points of each chapter. It includes a comprehensive index
and collects useful mathematical results, such as transformation matrices and Christoffel
symbols for commonly studied spacetimes, in an appendix. Supported by an online table
categorizing exercises, a Maple worksheet, and an instructors’ manual, this text provides
an invaluable resource for all students and instructors using Schutz’s textbook.
Robert B. Scott is a Senior Lecturer with CNRS Chaire d’Excellence at the Université de
Bretagne Occidentale, France where he specializes in relativity and geophysical fluid
dynamics and turbulence.
A Student’s Manual for A First
Course in General Relativity
Robert B. Scott
Université de Bretagne Occidentale
University Printing House, Cambridge CB2 8BS, United Kingdom
Cambridge University Press is part of the University of Cambridge.

It furthers the University’s mission by disseminating knowledge in the pursuit of
education, learning and research at the highest international levels of excellence.
www.cambridge.org
Information on this title: www.cambridge.org/9781107638570
© Cambridge University Press 2016
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2016
Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall
A catalog record for this publication is available from the British Library
Library of Congress Cataloging in Publication data
Scott, Robert B., 1965– author.
A student’s manual for A first course in general relativity / Robert B. Scott,
Université de Bretagne Occidentale.
pages cm
Includes bibliographical references and index.
ISBN 978-1-107-63857-0 (Paperback)
1. General relativity (Physics)–Problems, exercises, etc. 2. Astrophysics–Problems, exercises, etc.
I. Schutz, Bernard F. First course in general relativity. II. Title.
III. Title: First course in general relativity.
QC173.6.S37 2015
530.11076–dc23 2015020004
ISBN 978-1-107-63857-0 Paperback
Additional resources for this publication at www.cambridge.org/scott
Cambridge University Press has no responsibility for the persistence or accuracy
of URLs for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
Contents
Preface page vii
1 Special relativity 1
1.1 Exercises 1
1.2 Supplementary problems 15
2 Vector analysis in special relativity 23

2.1 Exercises 23
3 Tensor analysis in special relativity 47

3.1 Exercises 47
4 Perfect fluids in special relativity 76

4.1 Exercises 76
5 Preface to curvature 98
5.1 Exercises 98
6 Curved manifolds 121

6.1 Exercises 121
7 Physics in curved spacetime 160

7.1 Exercises 160
8 The Einstein field equations 186

8.1 Exercises 186
v
vi Contents
9 Gravitational radiation 210

9.1 Exercises 210
10 Spherical solutions for stars 227

10.1 Exercises 227
11 Schwarzschild geometry and black holes 247

11.1 Exercises 247
12 Cosmology 268
12.1 Exercises 268
Appendix A Acronyms and definitions 293

A.1 Acronyms 293
A.2 Mathematical and physical symbols 294
Appendix B Useful results 295

B.1 Linear algebra 295
B.2 Series approximations 295
B.3 Transformations between spherical polar and Cartesian coordinates 296
B.4 Selection of spacetimes 296
References 303
Index 305
Preface
General relativity is a beautiful theory, our standard theory of gravity, and an essential
component of the working knowledge of the theoretical physicist, cosmologist, and
astrophysicist. It has the reputation of being difficult but Bernard Schutz, with his
groundbreaking textbook, A First Course in General Relativity (first edition published
in 1984, current edition in 2009), demonstrated that GR is actually quite accessible to
the undergraduate physics student. With this solution manual I hope that GR, using
Schutz’s textbook as a main resource and perhaps one or two complementary texts (see
recommendations at the end of this preface), is accessible to all “technically minded
self-learners” e.g. the retired engineer with some time to devote to a dormant interest, a
philosopher of physics with a serious interest in deep understanding of the subject, the
mathematics undergraduate who wants to become comfortable with the language of the
physicist, etc.
You can do it too!

I’m speaking with some experience when I say that an engineer can learn GR and in
particular starting with Schutz’s textbook. My bachelor’s and master’s degrees are in
engineering and I started learning GR on my own when my academic career had gained
enough momentum that I could afford a bit of time to study a new area in my free time.
I must admit it wasn’t always easy. I personally found the explanations of mathematics
in the excellent textbook by Misner, Thorne and Wheeler (1973) more confusing then
helpful. (In retrospect I’m at a loss to explain why; in no way do I blame the authors.)
Soon two children arrived miraculously in our household, free time became an oxymoron,
but with the constant reward I found from beavering away at Schutz’s exercises I continued
to learn GR, albeit slowly and with screaming (not always my own) interruptions. In his
autobiography John A. Wheeler explains that he started learning GR in the 1940s when
he finally got the chance to teach the subject. Similarly the real breakthrough for me came
when I was offered the possibility to teach the subject to third-year undergraduate students
at the Université de Bretagné Occidentale in Brest, France. Suddenly my hobby became my
day job, fear of humiliation became my motivation, and most significantly I was forced to
view the subject from the student’s point of view. I also had to learn French. I can honestly
state with no exaggeration that, even with Canadian high-school French instruction and
years living in Montreal, it was much harder to learn the local language than to learn GR!
Vraiment!
vii
viii Preface
Is it better to start with a popular level book?

Popular level books are for people who want a superficial overview. There’s nothing
wrong with that if that’s your ultimate goal. But if you want to really understand GR,
skip the popular books for now. By the way, the French word for popular level books
is vulgarisation. If they’re good then they’ll be at least as difficult, probably much more
difficult, to understand then the real thing. Why? Because the author is obliged to explain
mathematical concepts in an artificial language, divorced from the logical and precise
language in which the ideas were developed. Popularizers are forced to come up with
creative analogies that are insightful if you already understand the idea but are always
somewhat misleading. If you made it through your college calculus class and basic linear
algebra course then you’ll find learning the tools of GR a natural extension of these ideas.
Once you understand what the metric on a Riemann manifold is (say after working through
Chapter 6 of Schutz), then you can easily and completely understand the Robertson–
Walker metric that explains the expansion of the Universe. You won’t need bread pudding
analogies but if you want to create them yourself you’ll be free to do so; just be sure to
explain to your listener all the caveats.
Using this book

Suppose you’d like to learn some to play chess, or guitar, or ice hockey. You go to the
library, or amazon.co.uk, or a bookstore and find an instruction manual. Reasonable start,
but you wouldn’t expect that after reading the manual you would be a chess master,
awesome guitar player, or hockey star. You would have to practice first, a lot, and learn
from your experience. Learning physics takes practice too and the initial practice comes
from doing exercises. So don’t make the mistake of reading this solution manual like a
novel or recipe book thinking you’ve bypassed the practice sessions. Schutz’s A First
Course in General Relativity has a lot of exercises (338 in fact), many more than most
of the other textbooks at that level. Do them!
Some advice to GR students and self-learners in particular:
1 Don’t give up. It’s normal to not understand something the first time you read it. Make
note of what you don’t understand, try to articulate your question to yourself as clearly
as possible, then press on.
2 Read several textbooks. See additional resources for suggestions.
3 Work the exercises yourself, using this solution manual as a guide to get you over the
hurdles and to verify your answers.
You’ll probably find that sometimes you can find a simpler solution to the exercise then
the solution I have offered. Your solution is probably fine. I have followed the idea that the
exercises are designed to teach you to use the new mathematical techniques of differential
geometry and tensor calculus (the “big machinery”) in a simple setting where you might
guess the answer or find it easily with simpler tools. For example you certainly don’t need
the Minkowski space metric tensor to know that the unit basis vector ex is orthogonal to
ey . But when you learn that in fact the component of the metric ηxy = ex · ey , it’s nice to
notice that of course ηxy = 0.
ix Preface
You’ll also find that I explain the solution steps in more detail than you really need. I
certainly don’t mean to insult you! My aim was to be complete, to spell it all out. I endeavor
to explain the steps with brief comments to the right of most equation lines that anticipate
and answer your question: “what did he do to get this line from the previous line?” If you
find it too easy, read it quickly!
To distinguish between references to my equations and those in Schutz’s book I use the
form Schutz Eq. (n.m) for his equations and eqn.(n.m) for equations in this book. If you
see something like
−1/2
t¯ = (t − vx) 1 − v 2 used Schutz Eq. (1.12)

1
= (t − vx) 1 + v 2 + O(v 4 ) , used eqn.(B.2) (0.1)
2
this means that the first line follows from Eq. (1.12) in Schutz’s textbook, while the RHS of
the second line used the equation eqn.(B.2), which in this case is found in Appendix B of
the book you’re holding. Some of you might not have seen O(x 2 ) before; look in Table A.2
because it’s an equation symbol. For abbreviations and acronyms in the text, like “RHS”
in the sentence before the previous one, look in Table A.1.
From time to time I make reference to an accompanying MapleTM worksheet. This is
available for free download from the Cambridge University Press website. Please also
visit the authors website for this book at http://stockage.univ-brest.fr/~scott/Books/Schutz/
index_schutz.html.
Additional resources
There are many good introductory resources for learning GR and throughout this manual
you’ll find references to them. Eric Poisson (Poisson, 2004, preface) recommends you
read Schutz’s textbook to get started, then Misner, Thorne, and Wheeler’s mammoth tome
(Misner et al., 1973) for breadth, and finally Robert Wald’s monograph (Wald, 1984) for
rigor. It would be hard to improve upon that advice. I suggest if you have time and find you
can read Misner, Thorne, and Wheeler (1973) straight off you could even skip Schutz
and this solution manual. Otherwise I agree with Eric, start here. But to complement
Schutz’s book I recommend books at a similar level, for example either Hobson et al.
(2006) or Rindler (2006). The first is similar to Schutz’s book but at times may be a bit
more challenging to the reader. Rindler is a bit weaker on tensor analysis, but great for
geometrical and physical insight. Sean Carroll (2004) has a flair for clear explanation and
has covered a lot of the material in Wald (1984) in a more concrete fashion.
If you find you are struggling with Schutz’s book you are probably missing some basic
background. The most important background is a working knowledge of basic differential
calculus, for which there are countless good begining university level books. If you have
this but your math skills need polishing, you could work through the first six chapters
of Felder and Felder (2014) concurrently with Schutz and this solution manual. After
completing a good number of Schutz’s exercises you’ll be ready for advanced books
(Misner et al., 1973; Hawking and Ellis, 1973; Wald, 1984; Poisson, 2004) and can even
x Preface
read some of the literature, especially American Journal of Physics, European Journal of
Physics, and Foundations of Physics articles.
Thanks
I would especially like to thank Gary Felder who read carefully the first six chapters of
this textbook and offered valuable suggestions for improvement. Jean-Philippe Nicolas,
Jose Luis Jaramillo, Richard Tweed, and Fred Taylor also had helpful input. I dedicate
this book to Dr. Donald Taylor, who was my first instructor in relativity, my first physics
supervisor, and the first to encourage me in a career in physics.
If you find any errors, or have suggestions for learning GR, you can first check this
book’s website: http://stockage.univ-brest.fr/~scott/Books/Schutz/index_schutz.html and,
if it is not already there, contact the author via email: robert.scott@univ-brest.fr.
1 Special relativity
The essence of a physical theory expressed in mathematical form is the identification

of mathematical concepts with certain physically measurable quantities. This must be
our first concern . . .
Bernard Schutz, §7.1
Minkowski pointed out that it is very helpful to regard (t, x, y, z) as simply four
coordinates in a four-dimensional space [that] we now call spacetime. This was the
beginning of the geometrical point of view, which led directly to general relativity in
1914–16.
Bernard Schutz, §1.1
1.1 Exercises
1.1 Convert the following to [natural] units in which c = 1, expressing everything in

terms of m and kg:
(a) Worked example: 10 J.
Solution:
10 kg m2 s−2
10 J = 10 N m = 10 kg m2 s−2 = = 1.11 × 10−16 kg.
(3 × 108 m s−1 )2
(c) Planck’s reduced constant, h̄ = 1.05 × 10−34 J s. (Note the definition of h̄ in

terms of Planck’s constant h: h̄ ≡ h/2π .)
Solution:
1.05 × 10−34 kg m2 s−1
h̄ = 1.05 × 10−34 J s = = 3.52 × 10−43 kg m.
3 × 108 m s−1
(e) Momentum of a car.
1
Solution:
30 ms−1
p= × 1000 kg = 10−4 kg.
3 × 108 ms−1
(g) Water density, 103 kg m−3 .
Solution:
103 kg m−3 .
We will learn in Chapter 8 how to express mass in terms of meters, see in particular
eqn. (8.8).
1.2 Convert from natural units (c = 1) to SI units

(a) Velocity, v = 10−2 :
Solution:
v = 10−2 × c[m s−1 ] = 3 × 106 [m s−1 ].
(c) Time, 1018 [m]:
Solution:
1018 [m]
= 3.3 × 109 [s].
c[m s−1 ]
(e) Acceleration, 10 [m−1 ]:
Solution:
10 [m−1 ] × c2 [m2 s−2 ] = 9 × 1017 [m s−2 ].
1.3 Draw the t and x axes of the spacetime coordinates of an observer O and then draw:
(c) The t¯ and x̄ axes of an observer Ō who moves with velocity v = 0.5 in the
positive x-direction relative to O and whose origin (t¯ = x̄ = 0) coincides with
that of O.
3 Exercises
t t
Figure 1.1 The x̄ and t¯ axes are the solution to Exercise 1.3(c). The dotted line is the invariant hyperbola with s 2 = −4.
The solution to 1.3(h) is the horizontal line. The solution to 1.3(i) is the sloping line, parallel to the x̄-axis. It is
tangent to the invariant hyperbola at the t¯-axis. These plots were made using the Mapletm worksheet that
accompanies this book.
Solution: Recall from Schutz §1.5 that the t¯-axis follows from simple kinematics; it
is just the line t = x/v, so here t = 2x. Recall also from §1.5 (see Schutz Fig. 1.5)
that the x̄-axis was a straight line with slope equal to the inverse of that of the t¯-axis,
x = t/v. (In SP1.3 you will prove this.) Here t = x/2. The solution was plotted in
fig. 1.1.
(h) The locus of events, all of which occur at the time t = 2 m (simultaneous as seen
by O).
Solution: See fig. 1.1.
(i) The locus of events, all of which occur at the time t¯ = 2 m (simultaneous as seen
by O).
Solution: The locus of events, all of which occur at the time t¯ = 2 m, have arbitrary
x̄, and so the solution is a straight line parallel to the x̄-axis. The coordinates in the
O frame are easily found with the Lorentz transformation. (See SP1.13 for a different
approach.) From Schutz Eq. (1.12) we have
t − vx √
t¯ = 2 = √ ⇒ t = vx + 2 1 − v 2 = x/2 + 3.
1 − v2
The solution was plotted in fig. 1.1.
1.5 (c) A second observer O moves with speed v = 0.75 in the negative x-direction
relative to O. Draw the spacetime diagram of O and in it depict the experiment
performed by O. Does O conclude that the particle detectors sent out their signals
simultaneously? If not, which signal was sent first?
Hint: See Schutz Fig. 1.5(b) for how the time and space axes look for a reference frame
moving in the negative x-direction. Think carefully about what the t¯ and x̄ mean.
(d) Compute the interval s 2 between the events at which the detectors emitted their
signals, using both the coordinates of O and those of O.
Hint: Use the Lorentz transformation for a velocity boost to obtain the coordinates of
the events in O.
1.6 Show that the interval

3
3
s =
2
Mαβ (x α )(x β ), Schutz Eq. (1.2) (1.1)
α=0 β=0
contains only Mαβ + Mβα when α β, not Mαβ and Mβα independently. Argue that
this allows us to set Mαβ = Mβα without loss of generality.
Solution: Pick a pair of indices, α = α∗ and β = β∗ say, with α∗ β∗, and where
α∗ and β∗ are fixed integers in the set {0, 1, 2, 3}. So s 2 contains a term like,
Mα∗β∗ (x α∗ )(x β∗ ).
But s 2 also contains a term like,
Mβ∗α∗ (x β∗ )(x α∗ ) = Mβ∗α∗ (x α∗ )(x β∗ ).
5 Exercises
The equality follows because of course the product does not depend upon the order
of the factors. So we can group these two terms and factor out the (x α∗ )(x β∗ )
leaving,
(x α∗ )(x β∗ )(Mα∗β∗ + Mβ∗α∗ ).
Because the off-diagonal terms always appear in pairs as above, we could without
changing the interval (and therefore without loss of generality) replace them with
their mean value
M̃αβ ≡ (Mαβ + Mβα )/2.
Thus the new tensor M̃αβ is by construction symmetric. The RHS of eqn. (1.1) is
called a quadratic form, and thus the interval of SR can be written as a symmetric
quadratic form.
1.8 (a) Derive,
s 2 = M00 (r)2 + 2M0i x i r + Mij x i x j , Schutz Eq. (1.3) (1.2)

where r = (x)2 + (y)2 + (z)2 , from eqn. (1.1) for general Mαβ . [You
can assume s 2 = 0 and t > 0.]
Solution: Start with eqn. (1.1), and partially expand the summations

3
3
3
3
s 2 = M00 (t)2 + M0i tx i + Mi0 x i t + Mij x i x j
i=1 i=1 i=1 j =1

3
3
3
= M00 (t)2 + 2 M0i tx i + Mij x i x j . used Mi0 = M0i
i=1 i=1 j =1
Consider
the case s 2 = 0, so from Schutz Eq. (1.1), t = ±r =
± (x) + (y)2 + (z)2 . Then, when t > 0,
2

3
3
3
s 2 = M00 (r)2 + 2r M0i x i + Mij x i x j ,
i=1 i=1 j =1
which is eqn. (1.2).
(b) Since s 2 = 0 in eqn. (1.2) for any {x i }, replace x i by −x i in eqn. (1.2)
and subtract the resulting equations from eqn. (1.2) to establish that M0i = 0 for
i = 1, 2, 3.
Solution: Let us first recall why s 2 = 0 in eqn. (1.2) for any {x i }. We have set
s 2 = 0 (because we were considering the path of a light ray) and it followed, based
upon the universality of the speed of light, that we required also s 2 = 0. Now why
does s 2 = 0 for any x i ? Because we have imposed that we are considering the
path of a light ray, and regardless of the spatial point x i on the light ray path we
choose, it always has (t)2 = (r)2 , so s 2 = −(t)2 + (r)2 = 0.
Now note that changing x i to −x i does not change

r = (x)2 + (y)2 + (z)2 .
Thus the only term in eqn. (1.2) to change sign when changing x i to −x i is the
middle term, the sum over 2M0i x i r. The final term does not because changing
x i to −x i also changes x j to −x j ; the i and j are just dummy indices. So
when we subtract s̄ 2 (t, x i ) − s̄ 2 (t, −x i ) as instructed, using eqn. (1.2), we
find:
0 = 0 − 0 = s̄ 2 (t, x i ) − s̄ 2 (t, −x i )

3
3
3
= M00 (r) + 2r
2
M0i x + i
Mij x i x j
i=1 i=1 j =1

3
3
3
− M00 (r)2 + 2r M0i (−x i ) + Mij (−x i )(−x j )
i=1 i=1 j =1

3
= 4r M0i x i . (1.3)
i=1
This must be true for arbitrary x i so M0i = 0.
(c) Derive
Mij = −M00 δij , (i, j = 1, 2, 3) Schutz Eq. (1.4b) (1.4)
using eqn. (1.2) with s̄ 2 = 0. Hint: x, y, and z are arbitrary.
Solution: Recall from Exercise 1.8(b) that adding to eqn. (1.2) the following

3
3
3
0 = s 2 = M00 (r)2 − 2r M0i x i + Mij x i x j
i=1 i=1 j =1
gives

3
3
0 = M00 (r)2 + Mij x i x j . (1.5)
i=1 j =1
7 Exercises
Suppose, x = r, y = z = 0. Substituting into eqn. (1.5) then gives M00 =

−M11 . Or, when y = r, x = z = 0, we see that M00 = −M22 . Similarly,
M00 = −M33 . To see that
√ the off-diagonal terms are zero, note that it is also possible
that x = y = r/ 2 and z = 0. Substitution into eqn. (1.5) gives that
0 = (M12 + M21 )(r)2 /2 + M11 (r)2 /2 + M22 (r)2 /2 + (r)2 M00
= (M12 + M21 )(r)2 /2 − M00 (r)2 /2 − M00 (r)2 /2 + (r)2 M00
= (M12 + M21 )(r)2 /2 = M21 (r)2 . (1.6)
The final step used Mαβ = Mβα , as proved in Exercise 1.6. And since (r)2 was
arbitrary, we have M21 = 0 = M12 . Similarly, M13 = M31 = 0 = M23 = M32 . In
summary,
Mij = −M00 δij , (i, j = 1, 2, 3),
which is eqn. (1.4).
1.9 Explain why the line PQ in Schutz Fig. 1.7 is drawn in the manner described in the
text. [Note that in Schutz Fig. 1.7 the F should be a Q to be consistent with the text
and with the corresponding figure in the first edition (Schutz, 1985, Fig. 1.7).]
Solution: The line PQ is described in the paragraph after Schutz Eq. (1.5) as
perpendicular to the y-axis, parallel to the t–x plane, and parallel to the t¯-axis in
Schutz Fig. 1.5(a). The line PQ represents the path of a clock that is stationary in the
O frame. Because the O frame moves in the x-direction its path must be orthogonal
to the y-axis. And furthermore it must be parallel to the t–x plane, as argued for
a clock at the origin of the O frame in Schutz §1.5. In fact the clock is simply
displaced a fixed distance from y = 0 along the y- or ȳ-axis and moves parallel to the
t-axis.
1.11 Show that the hyperbolae −t 2 + x 2 = a 2 and −t 2 + x 2 = −b2 are asymptotic to the
lines t = ±x, regardless of a and b.
Hint: Regardless of how large a and b are, consider the approximate behavior when
|x| and |t| are much greater than |a| and |b|.
1.12 (a) Use the fact that the tangent to the hyperbola DB in Schutz Fig. 1.14 is the line
of simultaneity for O to show that the time interval AE is shorter than the time
recorded on O’s clock as it moved from A to B.
Figure 1.2 Similar to Schutz Fig. 1.14. The dotted line is the path of a second clock at rest in O needed to infer that the moving
clock along the t-axis runs slowly.
Solution: This example shows that time dilation is self-consistent. From the perspec-
tive of an observer in O, the time interval AE = τ corresponds to the proper time
of a moving clock, whose world line in Schutz Fig. 1.14 is the t-axis, see fig. 1.2.
An observer at rest in O needs two clocks to record the time interval t¯ = t¯E − t¯A
corresponding to the proper time interval τ . The clock moving from A to B is one of
those two clocks, recording t¯A . The other is drawn as a dotted line (fig. 1.2) that passes
through E, recording t¯E . The fact that the line of simultaneity in O passes through B
and E means that t¯E = t¯B , and hence t¯ = t¯B − t¯A . Recall the time dilation formula,

τ = t 1 − v 2 . Schutz Eq. (1.10) (1.7)
where t was the so-called improper time, an interval measured by two clocks. Here
t¯ plays the role of t (improper time measured by two clocks):

τ = t¯ 1 − v 2 , (1.8)
implying τ < t¯ for |v| > 0.

Don’t be thrown off by the t in eqn. (1.7) not having a bar above it, while it does
in eqn. (1.8) above. It is not the symbol that is important but the role played by the
thing it depicts. The roles of the O and O frame have been reversed in this exercise,
which was the point of discussion around Schutz Fig. 1.14.
9 Exercises
1.12 (b) Calculate that1

(s 2 )AE = (1 − v 2 ) (s 2 )AB . (1.9)
Solution: Start with the LHS of eqn. (1.9):

(s 2 )AE ≡ −(tE − tA )2 + (xE − xA )2 definition of the interval
= −tE + xE
2 2
A is the origin
= −tE2 . E on t-axis (1.10)
From fig. 1.2 herein it is clear that
tE = tB − xB v dashed line parallel to x̄-axis, slope is v
= tB − (tB v)v
= tB (1 − v 2 ). (1.11)
Now consider the RHS of eqn. (1.9),
(s 2 )AB = −tB2 + xB2 = −tB2 + (vtB )2 = −tB2 (1 − v 2 ). (1.12)
Combining eqns. (1.10, 1.11, 1.12) one finds,
(s 2 )AE = −tB2 (1 − v 2 )2 = (1 − v 2 ) (s 2 )AB . (1.13)
1.12 (c) Use (b) to show that O regards O’s clocks to be running slowly, at just the right
rate.
Solution: This corresponds to verfying eqn. (1.8) above; recall τ = tE and t¯ = t¯B .
To find t¯B use the fact that the interval is invariant between Lorentz frames,
(s 2 )AB = −tB2 + xB2 = −t¯B2 + x̄B2
= −t¯B2 . B on t¯-axis (1.14)
Combining eqns. (1.10, 1.13, 1.14)
−tE2 = (s 2 )AE = (1 − v 2 ) (s 2 )AB = −(1 − v 2 ) t¯B2

tE = t¯B 1 − v 2 . took square root (1.15)
1.13 The half-life of the elementary particle called the pi meson (or pion) is 2.5 × 10−8 s
when the pion is at rest relative to the observer measuring its decay time. Show, by
1 We had corrected a typo in the original question, replacing AC with AE. SP1.15 explores the other possible
interpretation.
the principle of relativity, that pions moving at speed v = 0.999 must have a half-life
of 5.6 × 10−7 s, as measured by an observer at rest.
Hint: Study the solution to Exercise 1.12, and make the analogy with the situation
here. Think of the pion as a clock of sorts; its birth is say at time zero and its decay
is another tick of the clock. In making the analogy with Exercise 1.12, pay attention
to which time intervals are measured by one clock (proper time intervals) and which
involve two physically separated clocks.
1.14 Suppose that the velocity v of O relative to O is small, v = |v| 1. Show

that the time dilation, Lorentz contraction, and velocity-addition formulae can be
approximated by, respectively:

1
(a) t ≈ 1 + v 2 t¯, (1.16)
2

1
(b) x ≈ 1 − v 2 x̄, (1.17)
2
(c) w ≈ w + v − wv(w + v), (with |w| 1 as well). (1.18)
What are the relative errors in these approximations when v = w = 0.1.
(a) Solution: Recall the time dilation formula was given in eqn. (1.7), with here
τ = t¯. Solving for t, and expanding the RHS in a Taylor series in the
small parameter v we obtain
1
t = t¯ √ = t¯ (1 − v 2 )−1/2
1 − v2

1 2 3 4
¯
= t 1 + v + v + · · · used eqn. (B.2)
2 8

1
t¯ 1 + v 2 . (1.19)
2
For the Taylor series we have used the binomial series, eqn. (B.2) of Appendix
B, a result well worth remembering! The largest term we ignored was 38 v 4 . You
will often see this written as O(v 4 ), read “of order v to the fourth.” This means
that we are focusing attention on the important part, i.e. v 4 , and ignoring the
irrelevant numerical factor 3/8 that is close to unity. The higher order terms
in the series were O(v 6 ) and these are clearly much smaller since v 1. The
relative error is then
3 4
8v 3 4
≈ v = 3.75 × 10−5 .
(1 − v 2 )−1/2 8
In fact the relative error can be calculated exactly to be 3.76 × 10−5 , see
accompanying MapleTM worksheet.
11 Exercises
(b) Solution: Recall the Lorentz contraction formula was given in Schutz
Eq. (1.11), which we can write as

x = l 1 − v 2 = x̄ 1 − v 2 , cf. Schutz Eq. (1.11) (1.20)
where x̄ is the so-called proper length of the rod, i.e. as measured in a
frame in which the rod is at rest, x is the length of the rod measured in a
frame in which the rod has speed v. Using again the binomial series we have
immediately

1
x = x̄ (1 − v 2 )1/2 x̄ 1 − v 2 , used eqn. (B.2) (1.21)
2
where we have dropped the terms O(v 4 ) and higher order terms in the
binomial series because v 1. The largest error term here is 18 v 4 , which
gives a relative error of about 1.25 × 10−5 . The exact calculation of relative
error gives −1.26 × 10−5 , see accompanying MapleTM worksheet.
(c) Solution: Finally the Einstein law of composition of velocities was
w+v
w = , Schutz Eq. (1.13) (1.22)
1 + wv
where w is the speed of a particle measured in some inertial frame O, v is the
speed of an observer A measured in O, and w is the speed of the particle, in
the direction as v, measured by observer A. Using again the binomial series
we have immediately
w = (w + v)(1 + wv)−1 (w + v)(1 − wv), used eqn. (B.2) (1.23)
where we have dropped terms O(w 2 v 2 ) and higher order terms in the binomial
series because wv 1. More precisely, in fact the largest term we dropped
was w 2 v 2 leading to a relative error of
w2 v2
≈ w 2 v 2 = 1 × 10−4 ,
1/(1 + wv)
which is a very good estimate, agreeing with the exact relative error to one
part in 10−10 ; see accompanying MapleTM worksheet.
1.16 Use the Lorentz transformations,

1
t¯ = γ t − vγ x, with γ ≡ √
1 − v2
x̄ = −vγ t + γ x,
ȳ = y,
z̄ = z, Schutz Eq. (1.12) (1.24)
to derive (a) the time dilation, and (b) the Lorentz contraction formulae. Do this by
identifying the pairs of events where the separations (in time or space) are to be
compared, and then using the Lorentz transformation to accomplish the algebra that
the invariant hyperbolae had been used for in the text.
Solution: It is helpful to have short catchphrases to orient you, e.g. “Moving clocks
run slowly.” More precisely, in SR time dilation occurs when a clock is moving at
constant velocity as observed from an inertial reference frame O. This situation was
depicted in Schutz Fig. 1.14, (see fig. 1.2) with the “moving clock” following the
t¯-axis, passing through events A and B during proper time
τ = t¯B − t¯A = t¯B . (1.25)
Note we chose the origins to coincide in the two frames so that the algebra is
simplified, i.e. tA = t¯A = 0. We want to relate τ to the time between these same
events observed in O, wherein:
t = tB − tA = tB . (1.26)
With eqns. (1.25, 1.26) we have the two time intervals we want to relate expressed
in terms of the same event. The Lorentz transformations eqn. (1.24) give the O
coordinates in terms of the O coordinates when O is moving at speed v along the
x axis. Substituting v → −v then gives us the transformation back to O:
t¯B (−v)x̄B 1
tB = − = √ t¯B .
1 − (−v)2 1 − (−v)2 1 − v2
Using eqn. (1.25) and eqn. (1.26) we obtain

τ = t¯B = 1 − v 2 tB = 1 − v 2 t,
in agreement with the time dilation formula, cf. eqn. (1.7).
“Moving rods contract.” More precisely, the Lorentz contraction (or Lorentz–
Fitzgerald contraction) occurred when the clock was replaced by a rod. The geometry
and algebra are simplified when the rod lies along the x̄-axis as depicted in Schutz
Fig. (1.13). The proper length of the rod is the length observed in the frame wherein
the rod is stationary,
l = x̄C − x̄A = x̄C = x̄B , (1.27)
where the final equality holds because the trajectory of the tip of the rod through x̄B
and x̄C is parallel to the t¯-axis. We want to relate the proper length l to that observed
in O. The length in O is the distance between the ends of the rod measured at a given
instant in O, e.g. at t = 0 the length is
x = xB − xA = xB . (1.28)
With (1.27) and (1.28) we have the two lengths we want to relate expressed in terms
of the same event. We can use the Lorentz transformations eqn. (1.24) to transform
the O coordinates of event B to that of O
−vtB xB xB
x̄B = √ + √ = √ .
1−v 2 1−v 2 1 − v2
13 Exercises
Using (1.27, 1.28) we find

x = l 1 − v2,
in agreement with the Lorentz contraction formula eqn. (Schutz Eq 1.11).
Time dilation and Lorentz contraction were derived using the Lorentz transforma-
tion by Hobson et al. (2006, §1.7).
1.18 (a) The Einstein velocity-addition law, eqn. (1.22), has a simpler form if we
introduce the concept of the velocity parameter V , defined by the equation
v = tanh V . Notice that for −∞ < V < ∞, the velocity is confined to the
acceptable limits −1 < v < 1. Show that if u = tanh U and w = tanh W , then
eqn. (1.22) implies
w = tanh(W + U ).
This means that velocity parameters add linearly.
Solution: Simply substitute the definition of velocity parameter into eqn. (1.22):
tanh(U ) + tanh(W )
w = (1.29)
1 + tanh(U ) tanh(W )
(tanh(U ) + tanh(W )) cosh(W ) cosh(U )
= . (1.30)
cosh(W ) cosh(U ) + sinh(U ) sinh(W )
The numerator can be written as,
N = sinh(W ) cosh(U ) + cosh(W ) sinh(U ),
so that
sinh(W ) cosh(U ) + cosh(W ) sinh(U )
w = .
cosh(W ) cosh(U ) + sinh(U ) sinh(W )
The following identities are useful:

exp(a) + exp(−a) exp(b) + exp(−b)
cosh(a) cosh(b) =
2 2
exp(a + b) + exp(−(a + b)) exp(a − b) + exp(−(a − b))
= +
4 4
cosh(a + b) cosh(a − b)
= + , (1.31)
2 2

exp(a) − exp(−a) exp(b) − exp(−b)
sinh(a) sinh(b) =
2 2
exp(a + b) + exp(−(a + b)) exp(a − b) + exp(−(a − b))
= −
4 4
cosh(a + b) cosh(a − b)
= − , (1.32)
2 2
and

exp(a) − exp(−a) exp(b) + exp(−b)
sinh(a) cosh(b) =
2 2
exp(a + b) − exp(−(a + b)) exp(a − b) − exp(−(a − b))
= +
4 4
sinh(a + b) sinh(a − b)
= + . (1.33)
2 2
Using eqns. (1.31) and (1.32) the denominator above simplifies to D =
cosh(U + W ). Using (1.33) the numerator simplifies to N = sinh(U + W ). So,
w = tanh(U + W ),
which reveals that we can linearly add velocity parameters, then apply tanh to reduce
the final parameter to the final velocity.
1.18 (b) Use this to solve the following problem. A star measures a second star to be
moving away at speed v = 0.9c. The second star measures a third to be receding
in the same direction at 0.9c. Similarly, the third measures a fourth, and so on,
up to some large number N of stars. What is the velocity of the N th star relative
to the first? Give an exact answer and an approximation useful for large N .
Solution: The velocity of second star relative to first is u2 = 0.9. The velocity of N th
star relative to (N − 1)th, uN − uN−1 = 0.9. So the velocity of the N th star relative
to the first is,
uN = tanh[(N − 1)U ],
where 0.9 = tanh(U ), so U ≈ 1.47222. For large N the argument of the tanh becomes
large, so
ez − e−z 1 − e−2z
tanh(z) = = [1 − e−2z ]2 1 − 2e−2z .
ez + e−z 1 + e−2z
The first approximation used the binomial series eqn. (B.2) to first order, and the
second ignored the square of the small number, i.e. e−4z . So uN 1 − 2e[−2(N −1)U ] .
1.20 Write the Lorentz transformation equations in matrix form.
Solution: The Lorentz transformation equations were given in eqn. (1.24). These can
be written in matrix for as:
x = Ax,
15 Supplementary problems
where
⎛ ⎞ ⎛ ⎞
t t
⎜x ⎟ ⎜x ⎟
x=⎜ ⎟
⎝y ⎠ , x=⎜
⎝y ⎠
⎟
z z
and
⎛ ⎞ ⎛ ⎞
γ −vγ 0 0 cosh(V ) − sinh(V ) 0 0
⎜−vγ 0 0⎟ ⎜ ⎟
A=⎜
γ ⎟ = ⎜− sinh(V ) cosh(V ) 0 0⎟ , (1.34)
⎝ 0 0 1 0⎠ ⎝ 0 0 1 0⎠
0 0 0 1 0 0 0 1
√
where γ = 1/ 1 − v 2 . The second matrix above used the velocity parameter defined
as v = tanh(V ) in Exercise 1.18.
1.2 Supplementary problems
SP 1.1 Construct an experiment to show that one is in an inertial reference frame.

Assume that there are no gravitational forces on your test object. (See next problem for
complications arising from gravity!)
Solution
When gravity is not present one can use Newton’s laws to argue that test particles
with no external forces on them should have zero acceleration and therefore move at
constant velocity relative to each other in an inertial frame. So any experiment that
tests for these conditions suffices. For example, if you let go of an object at rest with
respect to you, it should appear to hover unmoving in front of you. A more elaborate
experimental test of an inertial frame, that works also in presence of gravity, was
presented by Misner et al. (1973, Fig. 1.7).
SP 1.2 Answer the question posed in footnote 4 on page 3 of Schutz: An astronaut is in

orbit about Earth, holding a bowl of soup. Does the soup climb up the side of the bowl?
You can make some idealizations: assume that the astronaut has a lid on his soup so that
the surface of the soup is initially flat, and he removes the lid without accelerating the
bowl relative to him; the bowl is small relative to the radius of the orbit and the experiment
short enough that the astronaut and all of his soup can be treated as being essentially in
the same orbit. (Calculating the effects associated with nonuniform gravitational fields
will be an important consideration much later when we get into general relativity.) The
footnote alludes to problems gravity poses for SR revealed by considering two astronauts
in different orbits. Explain these problems, and in particular, address these questions: (i)
Does the astronaut in orbit pass the test for a local inertial reference frame? (ii) Can
two inertial reference frames accelerate relative to each other? (iii) Argue that two non-
coincident orbits have non-zero relative acceleration.
Philosophers and historians of physics will recognize this footnote as bearing on the
famous bucket experiment of Isaac Newton (e.g. Maudlin, 2012).
SP 1.3 Referring to Schutz Fig. 1.5, explain why the angle of the x-axis to the x-axis is
φ = arctan(v), where v = |v| is the magnitude of the velocity of O along the x-axis. The
result follows from the construction of the x-axis, but the steps involved are not trivial.
Following the spirit of Schutz §1.5, try to find a geometric argument rather than using the
Lorentz transformation.
Solution
Refer to Schutz Fig. 1.4. We will find the coordinates of the point P, an arbitrary point
on the x-axis. This can be found from the intersection of the line passing through
events E and P and the line passing through R and P as follows. Let the equation for
the t-axis be t = mx. (Any non-vertical line through the origin can be written in this
form.) The events E and R then have coordinates (−x0 , −mx0 ) and (x0 , mx0 ), where
x0 is a parameter related to a. The line through EP has slope unity and contains the
point (−x0 , −mx0 ), so its equation is t = x + x0 − mx0 . Similarly, the line through
RP has equation t = −x + x0 + mx0 . By setting those equal to each other you can
easily find that the point of intersection P is (mx0 , x0 ), which shows that the x-axis is
t = x/m.
Using the Lorentz transformation one can obtain the same result more easily.
SP 1.4 A particle that follows the curve t = x in the S coordinate system has the speed
of light in the x-direction of that frame. Based upon that fact, what do you anticipate for
the equation of that curve under the Lorentz transformations of a velocity boost in the
x-direction? Verify your prediction using the Lorentz transformations eqn. (1.24).
SP 1.5 The special theory of relativity has led to a revision in our notion of space and time
from that of Euclidean space and absolute time used for Newtonian mechanics to that of
Minkowski spacetime. Is Newton’s first law of motion consistent with Special Relativity?
Solution
Newton’s first law of motion is consistent with Special Relativity. For a free particle
moves at uniform speed along a straight line in SR as well as in Newtonian mechanics.
Newton’s second law can also be made to be consistent with the two hypotheses of
17 Supplementary problems
Special Relativity, but that requires a reinterpretation of mass, and a transformation

law of forces that would have surprised Newton.
SP 1.6 Prove that the Lorentz transformation for a boost of velocity, which is linear
by construction, transforms straight lines to straight lines so that a particle of constant
velocity in one frame also has a constant velocity in the other frame. Thus the Lorentz
transformation for a boost of velocity respects Newton’s first law of motion.
SP 1.7 In Schutz §1.6 when deriving the transformation from the O frame to the O frame
it was assumed that the transformation must be linear.
(a) If we exclude the transformation that reduces all curves to a point, prove that only a
linear transformation, possibly followed by a translation, (i.e. an affine transformation)
is consistent with Newton’s first law that a particle subjected to zero net external force
must travel in a straight line at constant speed in all inertial reference frames.
Solution
Let x μ (t) be the world line of a free particle in inertial frame O, parameterized by the
time coordinate t. For a general transformation F ᾱ to another inertial frame O, the
world line of the particle in O will be
x ᾱ (t¯) = F ᾱ (x μ ) ◦ t (t¯), (1.35)
where initially we entertain the possibility that F ᾱ is a nonlinear function of x μ . To
help keep track of dependencies, we explicitly parameterize the world line coordinates
in O with the time coordinate t¯. Because the particle is free of external forces
Newton’s first law requires zero acceleration,
d2 x μ
= 0. (1.36)
dt 2
To be consistent with Newton’s first law this must hold true in all inertial reference
frames. Consider the inertial frame O moving at constant velocity relative to O.
Applying transformation (1.35) and the chain rule the first derivative is
dx ᾱ ∂F ᾱ dx μ dt
= , (1.37)
dt¯ ∂x μ dt dt¯
so Newton’s first law requires
2
d2 x ᾱ ∂ 2 F ᾱ dx μ dx ν dt ∂F ᾱ dx μ d2 t
= + = 0, (1.38)
dt¯2 ∂x μ ∂x ν dt dt dt¯ ∂x μ dt dt¯2
where we have used the chain rule and eqn. (1.36). Arbitrary free particles satisfying
eqn. (1.36) have constant but otherwise arbitrary velocities dx μ /dt. Furthermore we
exclude the possibility that dt/dt¯ = 0, otherwise, from eqn. (1.37), we would have
stationary particles in O for arbitrary free particles in O. (This requirement has been
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that very night. She felt that she could not meet the eyes of the
baronet, his fiancée, or Jack Rotherfield again.
The evening seemed a long one; she had to go to bed, to avoid
exciting suspicion as to her intention, which was to steal out of the
house when everybody else was asleep. But before retiring she
witnessed a sight that set her thinking. For after dinner Sir Robert
walked with Lady Sarah up and down the terrace close under
Rhoda’s window, and the girl fancied, both by the affectionate
manner in which they smiled at each other, and by the defiant half-
glances which the baronet cast stealthily up towards her window,
that he had told his fiancée of the doubts expressed as to her
sincerity, and that Lady Sarah had set him quite at rest upon that
score.
Rhoda did not sleep. At one o’clock, when all was silent in the
house, she rose, dressed herself hastily, and glided softly out of her
room and down the stairs. She had written a letter, directed to Sir
Robert, and left it in her room. She had said in it that, having had the
misfortune to offend him, she could not meet him again, but that she
begged his pardon with all her heart, and hoped that he would
forgive her, as she felt sure he would do, if he could only understand
the pain she felt at having given a moment’s displeasure to one to
whom she owed so much. She added that she would never forget
his goodness to her as long as she lived.
She had reached the hall, with the intention of leaving the house
by the front-door, and had withdrawn the bolts, when she was
startled by the sound of some one rapidly descending the stairs. She
thought she was discovered, and hastily hid herself in the dark
corner beside the tall grandfather’s clock that stood near the door.
But she had scarcely done so when she caught sight of something
which she could dimly discern to be a man, disappearing into the
drawing-room, and the next moment she heard sounds within the
room as of a scuffle and stifled cries.
Trembling and horror-struck, Rhoda was unable to decide whether
she ought to go upstairs and call for help, when, panting and drawing
deep breaths the figure stole out of the room again, shutting the door
softly.
The man was in such deep darkness and Rhoda was so far
entrenched in her corner that she could see but little of him, and that
little very dimly, until he was half-way up the stairs, when, dragging
his way up by the stair-rails, he laid his hand for a moment upon that
spot of the banisters where a single ray of moonlight fell upon them
from between the heavy velvet curtains that draped the staircase
window.
And Rhoda saw, with a shudder, that across the hand was the red
line of a cut which was still bleeding.
Before she could even be sure whether the figure was that of Sir
Robert, as she believed, it had disappeared.
Confused, trembling, wondering what it was that had happened,
Rhoda opened the front-door and slipped out, closing it softly behind
her.
She thought that she must have made enough noise for the
shutting of the door to have attracted attention, and she hoped, as
she went slowly down the narrow slip of garden which was all that
lay between the front of the house and the road, that the baronet
would come out after her, waylay her, and perhaps insist upon her
return.
But nobody came out, nobody followed her; and so, mystified, sick
with terror, and asking herself as she went whether she ought to
have come out without an effort to find out what had happened, she
went down the road towards the harbour.
She put up, for the rest of the night, at an hotel where she had
stayed before with her parents, and where travellers from off the
boats came at all times of the night, so that her late arrival attracted
no particular attention.
On the following morning she took the first train to Deal, and
reached the lodgings where her parents were in such a condition of
exhaustion that she was promptly put to bed. She insisted, however,
upon being allowed to tell her mother the singular circumstances that
had occurred at the moment of her departure from Mill-house, and
begged that they would let her know at once if it should come to her
parents’ ears that anything serious had happened that night at Sir
Robert’s residence.
For four days she was kept in bed, and assured that nothing had
happened as far as any one knew.
But when she was well enough to get up again, the truth was
gradually broken to her. The dead body of the butler, Langton, had
been found in the drawing-room, where it was evident that some sort
of a scuffle had taken place. The drawing-room window had been
found open, and it was supposed that a burglar had got in, and that
the butler, hearing a noise, had gone down and had been murdered
by the intruder.
The inquest had been held, and the verdict brought in: “Wilful
murder by some person or persons unknown.”
But the rumour about the neighbourhood was that there had been
a serious quarrel between Langford and his master, that he was
known to have been under notice to leave his situation, and that it
was in a scuffle between master and man that Langford came by his
death.
Rhoda sprang up with a cry.
“It’s not true!” she cried. “Sir Robert is incapable of such a thing!
Besides, I know! I can prove—Oh let me go and tell what I know!”
But the next moment the light faded out of her eyes and she sank
back, trembling.
What did she know? What could she prove? Nothing, nothing.
CHAPTER III.
TEN YEARS AFTER
Ten years passed before Rhoda Pembury saw Sir Robert Hadlow or
the old Mill-house again, and during those ten years all that she
heard of him or of his doings was through an announcement in the
newspapers, some six months after her stay there, of his marriage
with Sarah, third daughter of the Marquis of Eridge.
After that, although Rhoda did, from time to time, see brief
paragraphs in the papers concerning the doings of Lady Sarah
Hadlow, and incidental mention in connection with her, of her
husband, Sir Robert, she held no communication with them, or with
any of the household at the Dourville Mill-house, and she believed,
during the whole of that period, that the baronet who had saved her
life and who had been kind to her, had passed out of her life for ever.
In the meantime, having developed into a beautiful and
accomplished woman from the half-fledged girl she had been then,
Rhoda received a good deal of attention and more than one offer of
marriage.
But she cared little for admiration, and her heart was never
touched. Greatly to the annoyance of her parents, who had a large
family, and who were both eager to settle their handsome daughter
in marriage and a home of her own, Rhoda made light of all the
attentions paid to her, refused her lovers without compunction, and
announced, when reproached with her coldness and obstinacy, that
she intended to remain single through life, and that, as her parents
would never be able to get her off their hands in the way they
desired, she would meet their wishes by earning her own living.
This was not at all what they wanted, and her mother prevailed
upon Rhoda to give way on this point for a time. But the thought was
ever in the girl’s mind, and Mrs. Pembury was not surprised when,
ten years after the episode at the Mill-house, Rhoda came to her
with a newspaper in her hand, and, pointing to an advertisement in
one of the columns, said briefly:
“Mother, I’m going to answer this.”
The announcement to which she pointed ran like this:
“A lady wanted, as nurse-companion to an invalid boy.

Apply personally, if possible, at the Old Mill-house, Dourville.”
Mrs. Pembury having put on her glasses, read the advertisement,

and laid down the paper with an exclamation of something like
dismay.
“Why, it’s Sir Robert Hadlow’s, where the murder was committed!
Surely you wouldn’t go back there!”
Rhoda, who was very pale, asked briefly:
“Why not?”
But Mrs. Pembury was too much disturbed to reply. Hastily leaving
the room with some excuse, she went straight to her husband, who
was in his surgery, and laid the paper before him.
“What do you think of that?” she asked in consternation. “I’d
always had an idea it was something that happened while she was
there that prevented Rhoda’s marrying, and now I’m sure of it. I
believe she fell in love with Sir Robert’s ward, Mr. Rotherfield.”
But Dr. Pembury thought this idea high-flown and far-fetched, and
said he could not see any likelihood of such a thing. Rhoda would
have betrayed herself before this if she had nourished a passion for
so long, and in any case, it did not matter, as it was more than likely
that Sir Robert had left the Mill-house by this time, since his wife
appeared to be always in London, or, if not, that Mr. Rotherfield had
settled down somewhere else with a wife of his own.
Mrs. Pembury was troubled, but she always submitted, even
against her better judgment, to her husband’s wishes, and in this
case it was not even her judgment, but only a sort of feminine
instinct, which told her that Rhoda had strong sentimental reasons
for wishing to take this step.
On the following day, therefore, Rhoda, who was now twenty-
seven, and better capable of looking after herself than she had been
at seventeen, started alone for Dourville, and presented herself,
early in the afternoon, at the house she remembered so well.
Emotion made her eyes fill and her limbs tremble as she
approached the house, and recognised that it had undergone such
changes, since she first knew it, that she could scarcely be sure she
had not made a mistake and come to the wrong gate, when she
found herself standing before a long, white house, with wings
extending far in each direction, and with the modern big, wide
windows replacing the little narrow old ones.
Many of the trees that had surrounded the old house so closely
had been cut down, with considerable advantage as regards light,
but at a great loss of picturesqueness. Gone was the cosy, old-
fashioned look, together with the dark red curtains, the heavy square
portico, the homely look that she had loved.
It was into a stately, handsome hall that she was shown, a room
having been sacrificed to enlarge the entrance; and when she found
herself following a footman across a wide expanse of parquetted
floor to a new part of the building, and ushered into a lofty, light
library, Rhoda was almost ready to believe that she had made a
mistake, and that the Sir Robert, who would, so the footman said,
see her at once, could not be the man who had saved her life, the
memory of whose kindness she had treasured in secret for so long,
the man whose image had, almost unknown to herself, so effectually
shut out that of every other man from her heart during these ten
years.
But there he was, not indeed the Sir Robert she remembered, but
perfectly recognisable to her, although the dark hair had become
thickly streaked with grey, and the face more deeply lined than that
of a man of forty-five ought to be.
She had felt her heart beating very fast as she was led towards
the library, but she was totally unprepared for the reception she met.
She was duly announced as “Miss Pembury,” but she perceived at
once that the name awoke no memories in Sir Robert.
He had forgotten her.
In part, perhaps, the fact that he was very short-sighted, and that
he was not wearing his glasses, was answerable for his lack of
recognition, but however that might be, Rhoda felt cut to the heart
when he rose, bowed formally, and offered her a chair.
“I am sorry that my wife is away, as she would have been better
capable than I of arranging these matters with you, Miss Pembury,”
he said. “Ladies can get to the point over such things more quickly
than a mere man can hope to do. But I’ll tell you all I can, and you
will perhaps be able to judge whether you would care to come to
take charge of my boy. But perhaps you would like to see him first?
He is an invalid, as I suppose you know.”
“I should like to see him,” said Rhoda, in a low and gentle voice.
But the voice recalled no memories, and Sir Robert, ringing the
bell, told the servant to have Master Caryl’s carriage brought along
the terrace.
Rhoda was still so much under the influence of the strong
emotions called up by this meeting that she was glad there was no
need for her to say much.
On learning that she lived in London, and that she had come all
the way from the great city that morning, Sir Robert seemed
surprised, and ringing the bell again, told the footman to have some
luncheon prepared for Miss Pembury.
Rhoda protested, and thanked the baronet, who seemed already
to take it for granted that she would stay. The fact being that her
refined manner, sweet voice and sympathetic appearance had at
once predisposed the rather absent-minded gentleman in the
visitor’s favour.
A few minutes later there was a slight sound of footsteps upon the
terrace, and Rhoda, looking out, saw, with tears in her eyes, a boy
lying on a long, spinal chair, looking in at her through the window
with big, soft, dark eyes that, while they recalled in colour and
brilliancy those of his mother, had something of the far-away
expression of his father in them too.
She hastened across the floor, and bending over the boy kissed
him on the forehead.
He flushed a little, put out his hand and laid it upon hers.
“Are you coming to stay with me?” he asked simply.
“It is for you to say, dear,” said Rhoda.
He moved his head slowly and looked at her with great intentness.
“I should like it very much,” he said. “What am I to call you?”
Rhoda threw a hasty glance at Sir Robert, who was standing by
them, so intent in watching his son’s face that he took but little heed
of the visitor. So she thought she might venture to give her name
without fear of discovery. Since he had begun by non-recognition, it
was better to go on without undeceiving him, she thought.
“Call me Rhoda,” she said softly.
He smiled at her. Though he was scarcely more than eight years
old, his condition had made him older in many ways than his age,
and his manner was almost that of a grown person as he said:
“Rhoda. Yes, I like that.”
“You will stay with him then?” asked Sir Robert, evidently pleased
at the fancy the child had taken to the lady.
Rhoda hesitated. There were details to be settled, of which the
man took no cognizance. Perceiving her hesitation, he smiled, and
waving his hand, said:
“You will want to know a great many things, about hours and
holidays and—and other things. We can leave all those until Lady
Sarah comes back next week, can’t we? My housekeeper will assign
you rooms, any you care for, and you can do just as you please, as
long as you make my boy happy. He is left too much alone. His
mother doesn’t like Dourville; it doesn’t agree with her very well. I
hope it will please you, however.”
“Thank you. I shall like it, I know,” said Rhoda.
“Come and talk to me,” said Caryl, “and let me show you my
monkey, and my rabbits. I’ve got three, and some budgerigars. I
hope you like birds. And I’ve got a dog. Would you like to see him? I
want you to see him do his tricks.”
Off they went together, the lady and the child, and Sir Robert,
standing blinking in the sunshine, seemed to have, Rhoda thought, a
vague impression of having seen or heard something which the
lady’s presence recalled.
But to her regret, it was evident that the recollection was fraught
with pain.
She and the boy made the tour of the garden together, for he had
dismissed the servant, asking Rhoda if she would draw him along.
By the time they had been an hour together, moving slowly along
the shady walks, and visiting the boy’s numerous pets in rabbit-hutch
and aviary, they were already firm friends; and when they returned to
the terrace, Rhoda had the satisfaction of seeing Sir Robert standing
at the library window, with a faint smile upon his face. He was
pleased by the pleasure of his boy.
“It’s good of you to humour him by walking about so long when
you must be tired, Miss Pembury,” he said. “I have sent for Mrs.
Hawkes, the housekeeper, and directed her to have some tea for
you, and to show you your rooms.”
“Thank you very much,” said Rhoda.
She had a transient fancy that Sir Robert recalled something in her
name or in her person, for he looked at her suddenly with a slight
frown and with vague curiosity. He did not, however, ask her any
questions, and a few minutes later Rhoda was going upstairs,
escorted by the footman, towards the rooms where the housekeeper
was busy preparing for her reception.
The man threw open a door and announced:
“Miss Pembury, Mrs. Hawkes,” and Rhoda entered.
The man went away, and Rhoda heard an exclamation from the
grey-haired woman in spectacles who was drawing a cover over a
little table in the pretty sitting-room.
“Why, it’s Miss Rhoda!” cried the grey-haired woman.
The visitor exclaimed in her turn.
“Bessie!” cried she.
Neither woman could restrain her tears.
“I’m housekeeper here now,” said Bessie, wiping her eyes. “But,
oh, Miss Pembury, to think Sir Robert shouldn’t know you! And to
think of your turning up here, after all this time, and us wanting you
so bad ten years ago!”
“What do you mean?” asked Rhoda, trembling.
“Let me get your rooms ready, and get rid of the maid, who is in
the next room and who will be in here in a minute, and then I’ll tell
you everything,” said Mrs. Hawkes.
And within five minutes, the two rooms having been got ready, and
the maid dismissed in search of tea and sandwiches for Miss
Pembury, the two old friends sat down in the window-seat together,
and the housekeeper began her story.
CHAPTER IV.
RHODA RETURNS TO MILL-HOUSE
“Ah, Miss Pembury, there’s been a many changes since the night
when you ran away from here!” she said, as she sighed and folded
her hands in her lap. “But why did you go so quick and so quiet? And
why didn’t you come forward when the inquest was held?”
“I—I went away because I’d displeased Sir Robert,” said Rhoda.
“So that I couldn’t bear to meet him again. And as for the inquest, if
you mean that on the poor butler, I never heard anything about it till
long after it was over. I fell ill, you know, and they wouldn’t let me
know anything.”
Mrs. Hawkes nodded.
“I know that was what they said, but we all thought that it was only
an excuse, and that the truth was you didn’t want to come forward,
because you knew too much.”
“Too much!” faltered Rhoda.
“Yes. By the time you were missed, and by what we heard of your
arriving at the hotel where you stayed the night, we thought as how
you couldn’t but have heard or seen something of the murderer of
poor Langton.”
Rhoda trembled at the recollection.
“Who was the murderer?” she asked in a whisper.
The housekeeper shook her head.
“Nobody knows from that day to this,” she answered. “The inquest
was held, after being put off, and they brought it in ‘by some person
unknown.’ But people talked, and it was very unpleasant for us all.”
“What did they say?” asked Rhoda hoarsely.
The housekeeper closed the window, and went to the door, looked
out and came back again.
“These aren’t things one likes to talk about, even now,” she said.
“Of course the thing was really clear enough. It was a thief tried to
rob the house, did get in a little way, and poor Langford went down
and struggled with him and got killed.”
“How was he killed?” asked Rhoda.
“He must have been flung down into the fireplace with so much
force that it killed him, they said. He was found with his head in the
stone fireplace, covered with blood and dead. Fractured skull, the
doctors said he died of. But his hands were gashed as if he’d been
struggling with some one for a knife.”
Rhoda was listening, in a state of stupefaction with horror. But she
would not betray herself. Sitting very still, with her head bent, she
listened.
The housekeeper went on:
“No knife was found, and though they saw some footsteps coming
to the house, they found none going away again. That was odd and
mysterious. Especially,” the housekeeper looked round her again,
and dropped her voice, “as Sir Robert had been out in the grounds
very late.”
“Sir Robert!” echoed Rhoda, appalled.
Mrs. Hawkes nodded.
“That was the part of it that made us all uncomfortable,” she said,
below her breath. “And that was why they wanted you to come
forward. And you would have had to come, only your father said you
knew nothing about it at all, and that it would have endangered your
life to have had to come.”
“Oh!” gasped Rhoda.
“For everybody thought even more than they said. Everybody
wanted to know if you had seen anybody.”
She paused, and tried to look into Rhoda’s face. But the girl kept
her head obstinately bent. Not for the world would she have had the
nurse see the look of horror which she felt there must be in her own
eyes.
It was not that she thought that Sir Robert had killed his servant:
not for one moment would she have admitted such a possibility. But
she could herself have borne witness to the fact that some one did
go upstairs after the struggle in the drawing room.
Who could it have been?
“There was lots of talk and idle gossip,” went on Mrs. Hawkes.
“And even after the verdict was given, the talk went on just the same.
You see it was known that nobody had any quarrel with Langford
except the master, and it was known that Langford had had his
notice, though why he got it was not rightly known.”
There was a pause, but still Rhoda refrained from asking any
questions.
“And it never has been known,” added the housekeeper solemnly,
“from that day to this.”
“I couldn’t have said anything to help,” said Rhoda at last in a
stifled voice.
“Didn’t you see anything, or hear anything then?”
“Yes. I heard a noise in the drawing-room,” admitted Rhoda, “and I
went out by the front door.”
“Yes, we knew that, for some one heard it shut. And that was one
reason why we thought you must have known something.”
Rhoda suddenly sat up.
“Surely,” she said sharply, “nobody was so foolish and wicked as
to think that Sir Robert, the best man in the world, had anything to do
with it?”
The housekeeper answered quickly:—
“Of course we, who knew him, didn’t think so. But there were
plenty of unkind things said outside, you may be sure, miss.”
“How shocking!”
“And folks thought as the marriage would be broken off, for the
Marquis was a good deal cut up about the gossip. But then Lady
Sarah she stood up like a high-minded lady, and she said as how
she didn’t allow such foolishness to disturb her for one moment. And
she married him, and even married him the sooner for the talk.
Which was handsome of her, and which Sir Robert he thought the
world of in her, you may be sure.”
Rhoda nodded. From what she had seen of the flippant and
vivacious flirt she wondered whether high-mindedness was really the
quality to which Sir Robert owed her steadfastness.
There was a pause, and Mrs. Hawkes gave a deep sigh, which
made Rhoda look at her, and perceive that an expression of the
deepest disappointment was on the good woman’s features.
“I was in hopes as you would be able to tell something, something
that would have cleared things up, miss,” she said.
Rhoda’s eyes filled with tears, while a hot blush rose to her
cheeks. It was quite true that she did know something, just a little
more than anybody else appeared to know, about the doings of that
fatal night. But as it was nothing definite enough to absolve anybody
or to convict anybody, she felt that wisdom lay in keeping that little to
herself, for the present, at any rate.
“And so Lady Sarah was staunch, and earned Sir Robert’s
gratitude?” she said, her constraint making her words sound rather
stiff.
Mrs. Hawkes looked enigmatic for a moment, and then came a
little closer.
“Seeing you know so much about them, I may tell you, in
confidence, that it’s not been as happy a marriage as, from such a
beginning, one might have hoped,” she said. “You see it was a
disappointment there being only the one child, this poor boy that
never was strong. And then, well, Lady Sarah’s tastes and Sir
Robert’s they don’t seem to go well together. So my lady’s most
often away, either in town or abroad for her health, and Sir Robert,
he don’t seem to care to leave his house and his boy that he loves
so much.”
“And doesn’t Lady Sarah care for her boy too?”
The housekeeper’s face altered a little in expression.
“Of course she does,” she replied diplomatically. “But there’s
different ways of caring, and the sight of him with his little couch and
his spinal chair, well, it hurts my lady, who would have liked to have a
boy handsome and tall and strong.”
Rhoda felt chilled.
“It’s a pity she ever married Sir Robert,” she cried impulsively.
The housekeeper looked rather shocked.
“Well, miss, he wouldn’t let her be till she’d promised him, he was
so much in love,” she said quickly. “And anyhow, he’s pleased his
fancy. He married the lady he liked best.”
“Yes.”
Another question was on Rhoda’s tongue, but it was one she was
shy of uttering.
It took a different form from the one at first in her mind when at last
she said, timidly:
“Is Mr. Rotherfield married?”
Mrs. Hawkes looked at her quickly.
“No, he’s not married,” she said slowly. “I think he’s in love too
often to fix upon any one lady.”
There was something in her face that prevented Rhoda from
asking any more questions on that subject. Indeed, Mrs. Hawkes
was not prepared to answer any more, for she changed the subject
and said: “Do you remember the two children who were here at the
time of your accident, miss?”
“Why, yes, of course I do. George and Minnie. What has become
of them? The Terrors you used to call them.”
“And the Terrors they are still,” said Mrs. Hawkes emphatically.
“They’re away now; Master George he’s at Sandhurst, and Miss
Minnie she’s staying in Normandy with friends for the summer
holidays. But they live here still, and I don’t say I’d be without them,
though their battles with my lady don’t give one much peace.”
“Battles?”
“Yes, they’re just what they always were, and the plague of all our
lives.”
“I wonder whether they’ll recognise me!” said Rhoda.
“Trust them for it!”
“But Sir Robert doesn’t.”
Mrs. Hawkes looked at her.
“Well, there’s no need to be astonished, for he’s so short-sighted,
and he lives so much shut up with his books and his collections, that
he hasn’t much memory for anything else. He’s taken to collecting
since you were here, miss, and he’s got a gallery of pictures that
people come for miles to see. That’s what the north wing was built
for, to put them in. And the south wing, that was for my lady’s
dances. Not that she gives many of them now.”
There was a little constraint on both sides now that Rhoda had
confessed that Sir Robert had failed to recognise her. Mrs. Hawkes
looked disturbed. At last she said:
“I was wondering, if I may make so bold as say so, miss, whether
Sir Robert would let you stay here again, if he was to remember
you.”
Rhoda looked startled and uneasy.
“Why should he mind?” she asked quickly.
“Oh, only that he doesn’t in general like to be reminded of that
time. And if he had recognised you, he couldn’t but have thought of
it, could he?”
“N-no,” said Rhoda, beginning to feel nervous.
There was another silence, and then Mrs. Hawkes ventured:
“Would it be taking too great a liberty, miss, to ask how you came
to want to come back here, after all these years? For you must have
remembered too, what happened, and have felt uncomfortable about
it, I should think?”
Rhoda blushed hotly.
“Of course I knew what happened, through the newspapers and
what I was told,” she said. “But I didn’t think it could matter. How
should it? I didn’t know anything.”
“No, miss.”
Mrs. Hawkes looked down again.
“May I venture to ask whether you found the master altered,
miss?”
Rhoda’s lips trembled a little as she replied:
“Yes, I did. He doesn’t look so young, of course, as he did then.”
“Nor so happy,” suggested the housekeeper almost under her
breath. “And do you still think him as handsome as you did?”
Rhoda tried to laugh.
“You want to know, I suppose, whether I still feel the infatuation I
felt then about him?” she said. “Of course I don’t. It was a young
girl’s childish fancy. But I do think he is a most sympathetic, kindly
natured man, and I should be very glad, considering what my
obligations are to him, if I could be of any use in taking care of his
child.”
She was wondering, as she spoke, what Lady Sarah would say
when she found her installed at the Mill-house. Until that moment,
strange to tell, she had felt no curiosity on this point; it was only now,
when she saw the view the housekeeper took of her coming, that
this question suggested itself to her. However, there were some days
to pass before Lady Sarah would return from abroad, and in the
meantime Rhoda might pass her time very happily with the child, she
thought.
And so it fell out. Within a few minutes her tête-à-tête with the
housekeeper was interrupted by a message to the effect that Master
Caryl wanted to see her, wanted to know whether she would have
tea with him, and Rhoda, hastily divesting herself of her hat, went
downstairs to the boy’s room, where she found him, flushed and
eager, awaiting her coming and welcoming her with a cry of delight.
The next few days were among the happiest she had ever passed.
Caryl was a charming companion, affectionate, docile on the whole,
though somewhat spoilt. He had taken a great fancy to Rhoda, and
would not leave her much time to herself, while Sir Robert, delighted
at his son’s finding an interest in life, overwhelmed her with signs of
his appreciation.
Rhoda wondered sometimes whether he did not begin to
remember her; for she would find him regarding her as it were by
stealth, with a frown of pain upon his face, and although he asked no
questions, she felt sure that he must already be wondering whether
he had not met her before.
To Rhoda the sadness in his quiet face was infinitely touching, and
little by little she found ways of making herself useful to him, by
copying the notes he had made concerning his curios, as well as by
letting him talk to her concerning them.
“It’s very good of you to let yourself be bored, Miss Pembury,” he
would say to her with a shy laugh when he had been expatiating
upon the beauties of his enamels or of his old Sèvres china. “When
Lady Sarah comes back, she will say that you have spoilt me. I’m not
used to having my dull dissertations listened to with so much
appearance of interest. And I’m quite sure,” he added archly, “that it
can’t be more than an appearance.”
“Indeed I wouldn’t pretend to be interested if I were not, Sir
Robert,” Rhoda assured him humbly and earnestly.
And she told the truth. She would not, indeed, have found the
pictures and curios so intensely interesting as she did, if they had not
belonged to the man who had once saved her life. But for his sake
she liked them, and her sympathy delighted the grave and rather
lonely gentleman.
He was profusely grateful to her for the pains she took in collecting
and copying his notes, and in sorting his papers for him. And he said
to her with intense appreciation, one day when she had succeeded
in deciphering some of his notes which he himself could not read:
“Miss Pembury, if you hadn’t come here as companion to my boy, I
should have had to keep you here as my secretary.”
He could not guess the pleasure the simple words gave to the
sensitive and grateful Rhoda. She had to pause a moment before
she could reply with calmness:
“I wonder you have never before thought of having a secretary, Sir
Robert.”
He shook his head.
“I wouldn’t have one for worlds,” he answered with decision,
“unless I could get one to undertake the duties of free will. What! To
have a professional secretary fingering my papers, and handling my
treasures coldly, because it was his or her duty to do it!” And with a
little playful assumption of horror, he added: “Do you know, I really
think it would injure the pictures and the china too, to be subjected to
the perfunctory care of some one specially engaged to look after
them? No. I’m fanciful about my treasures. Whatever work is done in
connection with them, must be done for love.”
The ingenuous words struck a responsive chord within the breast
of Rhoda, and she did not say a word.
But the implied compliment to her thoughtful help was treasured
up in her heart, and it made her happy for the day.
Lady Sarah’s return was delayed for a week, so that, when at last
Mrs. Hawkes received word that she was to prepare her rooms,
Rhoda had been a fortnight at the Mill-house, and was already
feeling quite at home.
She spent the day between Caryl and Sir Robert; very often now,
indeed, Caryl would insist upon her taking him into his father’s study,
where he would lie in a corner watching Rhoda while she deciphered
notes and copied inscriptions.
Sir Robert began to entrust more and more of his work to her,
always prefacing any request with a humble apology for taking up so
much of her time, and always receiving the quiet assurance that
what he asked her to do was just what she had been wishing herself
that she might do.
Caryl, his father said, was happier than he had ever been before.
“You fill just that place to him,” said Sir Robert enthusiastically, one
evening, “that I had always hoped would be filled by my niece
Minnie. But of course you don’t know her, so you don’t understand.”
Rhoda remained silent. She did know Minnie, and she knew, too,
how hopeless it would have been to expect quiet sympathy from that
young lady, if she had fulfilled her childish promise and grown up the
mischievous torment she seemed to be inclined to develop into.
It seemed almost tragic to Rhoda that, while speaking thus of his
niece, he left out all mention of his wife, who would have seemed to
be the boy’s natural companion.
“You’ll be very, very glad to see mama again, won’t you, Caryl?”
Rhoda asked that evening, when he had been put to bed and she
was bending over him to bid him good-night.
“It doesn’t make so much difference to me whether she’s here or
not,” replied the child, in the quaint, old-fashioned way children have
who see few playfellows or companions of their own age.
Perhaps Rhoda looked rather shocked. So the boy added:
“Mama is not like you. She likes to be out in her motor-car all day,
or playing tennis or dancing. She isn’t quiet, like you.”
“She will have brought you something pretty, I expect,” suggested
Rhoda.
“Oh, yes, but she never brings the things that I like,” complained
Caryl. “What I want is a book full of pictures of hunting. I know she
won’t bring me that.”
Rhoda was struck with the pathos of this wish. For poor little Caryl,
condemned to lie on his back and unable to run about and play like
other children, had a passion for sport of all kinds, and was never
happier than when watching a cricket or a football match; and even
now, in early September, he was talking eagerly about the fox-
hunting season, and asking Rhoda if she would take him to a meet
of foxhounds when cub-hunting began.
She had begun by this time to dread Lady Sarah’s return, to
wonder whether her presence at the Mill-house would be resented
by the flighty beauty, who would certainly remember her, and who
might perhaps look upon her as an interloper, and be jealous of the
help she gave to Sir Robert and of the love which little Caryl had
already bestowed upon her.
It was the next day that the mistress of the house was to arrive
and Rhoda was now on thorns. In the old days, indeed, Lady Sarah
had scarcely spoken to her, but she might not look upon her with the
same indifference now.
For Rhoda was conscious that there were whispers abroad
concerning herself; and she guessed that, although the whole of the
household, with the single exception of Mrs. Hawkes, was changed
since she was there last, the housekeeper must have told some of
the servants about the bicycle accident and the flight of Miss
Pembury on the night of the tragedy at the Mill-house, and that there
was a certain curiosity abroad concerning her.
It was late in the day when Lady Sarah arrived, and coming up to
the bedroom of her little son when he had retired for the night, found
Rhoda in the room.
Rhoda, however, regretting that she should have been found
there, and fearing that Lady Sarah would think she was trying to take
the mother’s place already with the boy, kept in the background, and
witnessed, unremarked by Lady Sarah, the meeting between mother
and son.
“Well, Caryl, and how are you?” cried she, as she bent over him
and gave him a light kiss on the forehead. “They tell me you’ve been
getting on famously and that you’ve got an awfully nice companion
now.”
“Yes. I love Rhoda, and so will you, mama. Rhoda, come here.
You shall see her, mama,” cried the boy in excitement.
Lady Sarah stood up and Rhoda had a good view of her. She saw
that the ten years which had passed since she met her first had only
served to ripen her beauty. Lady Sarah, though not quite so slim and
slender, so like a fairy as she had been in the days of her girlhood,
was lovelier than ever. Her dark eyes were just as bright, her
complexion was as brilliant, while a little dignity of manner now
added to her charms.
She held out her hand graciously, and Rhoda came forward.
But the moment she came within the range of light thrown by the
shaded electric lamp on the table at the foot of the bed, Lady Sarah’s
face changed. A look of intense horror appeared in her face, and her
hand dropped, as she met Rhoda’s eyes with a startled look, and,
recognising her at once, said hoarsely, under her breath:
“Miss Pembury!”

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A Student s Manual for A First

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Introduction to General Relativity A Course for

General relativity a first examination Second Edition

Renewable Energy, Second Edition: A First Course Robert

The Psychology Student Writer s Manual and Reader s

The General Theory of Relativity A Mathematical

A Mathematical Journey to Relativity Deriving Special

A First Course in Statistics for Signal Analysis Wojbor

Asymptotic Analysis in General Relativity 1st Edition

Cambridge University Press is part of the University of Cambridge.

Preface page vii

2 Vector analysis in special relativity 23

3 Tensor analysis in special relativity 47

4 Perfect fluids in special relativity 76

6 Curved manifolds 121

7 Physics in curved spacetime 160

8 The Einstein field equations 186

9 Gravitational radiation 210

10 Spherical solutions for stars 227

11 Schwarzschild geometry and black holes 247

Appendix A Acronyms and definitions 293

Appendix B Useful results 295

You can do it too!

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Using this book

The essence of a physical theory expressed in mathematical form is the identiﬁcation

1.1 Convert the following to [natural] units in which c = 1, expressing everything in

(c) Planck’s reduced constant, h̄ = 1.05 × 10−34 J s. (Note the deﬁnition of h̄ in

(e) Momentum of a car.

(g) Water density, 103 kg m−3 .

1.2 Convert from natural units (c = 1) to SI units

(c) Time, 1018 [m]:

(e) Acceleration, 10 [m−1 ]:

Solution: See ﬁg. 1.1.

1.6 Show that the interval

1.8 (a) Derive,

s 2 = M00 (r)2 + 2M0i x i r + Mij x i x j , Schutz Eq. (1.3) (1.2)

which is eqn. (1.2).

This must be true for arbitrary x i so M0i = 0.

Mij = −M00 δij , (i, j = 1, 2, 3) Schutz Eq. (1.4b) (1.4)

Suppose, x = r, y = z = 0. Substituting into eqn. (1.5) then gives M00 =

implying τ < t¯ for |v| > 0.

1.12 (b) Calculate that1

Solution: Start with the LHS of eqn. (1.9):

1.14 Suppose that the velocity v of O relative to O is small, v = |v|  1. Show

1.16 Use the Lorentz transformations,

Using (1.27, 1.28) we ﬁnd

1.20 Write the Lorentz transformation equations in matrix form.

1.2 Supplementary problems

SP 1.1 Construct an experiment to show that one is in an inertial reference frame.

SP 1.2 Answer the question posed in footnote 4 on page 3 of Schutz: An astronaut is in

Special Relativity, but that requires a reinterpretation of mass, and a transformation

“A lady wanted, as nurse-companion to an invalid boy.

Mrs. Pembury having put on her glasses, read the advertisement,

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1.14 Suppose that the velocity v of O relative to O is small, v = |v| 1. Show