Table Of ContentASTRONOMY AND(cid:13)
ASTROPHYSICS LIBRARY
SeriesEditors: G.Bo(cid:127)rner,Garching,Germany
A.Burkert,Mu(cid:127)nchen,Germany
W.B.Burton,Charlottesville,VA,USAand
Leiden,TheNetherlands
M.A.Dopita,Canberra,Australia
A.Eckart,Ko(cid:127)ln,Germany
T.Encrenaz,Meudon,France
B.Leibundgut,Garching,Germany
J.Lequeux,Paris,France
A.Maeder,Sauverny,Switzerland
V.Trimble,CollegePark,MD,andIrvine,CA,USA
Peter Hoyng
Relativistic Astrophysics
and Cosmology
A Primer
With 114 Figures, 16 in color and 12 Tables
123
PeterHoyng
SRONNetherlands Institute forSpaceResearch
Sorbonnelaan2
3584CAUtrecht,TheNetherlands
[email protected]
Cover picture: 'ZwarteKom op Geel Vlak' 2002 by Olav Cleofas van Overbeek, picture Galerie Lieve Hemel,
Amsterdam.
LibraryofCongressControlNumber: 2006920111
ISSN0941-7834
ISBN-10 1-4020-4521-2SpringerBerlinHeidelbergNewYork
ISBN-13 978-1-4020-4521-9SpringerBerlinHeidelbergNewYork
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Preface
This textbook offers a succinct and self-contained introduction into general
relativity and its main areas of application: compact objects, gravitational
wavesandcosmology.IthasevolvedfromlecturecoursesIhavetaughtatthe
University of Utrecht since 1990. The book is intended for advanced under-
graduate and beginning graduate students in physics and astrophysics.
The past decades have seen spectacular new developments in our knowl-
edgeofcosmology,thephysicsofcompactobjectsandinhighprecisiongravity
experiments.Asaresult,relativisticastrophysicsandcosmologyhavebecome
a very attractive element in the (astro)physics curriculum, and there is a va-
riety of excellent textbooks. But most of these are either too advanced, too
elementary, or too voluminous for my purpose, or they do not cover all top-
ics. My object in writing this book has been to provide a concise text that
addresses general relativity and its applications homogeneously, at an inter-
mediate level, conveying a maximal physical insight with a minimal amount
of formalism. It is often a revelation for students to see that it is possible, at
leastfortherangeofsubjectsaddressedhere,tocutdowntheusualtangle of
math to manageable proportions without watering down the discussion. My
guiding principle has been to keep only what is really useful, but that does
notmeanthatitisalwaysthemoredifficulttopicsthathavebeeneliminated.
Forexample,Ikeptverylittleformaltensorcalculusasitisnotreallyneeded
– only the basics are indispensable. But variational calculus is used exten-
sively because it is by far the simplest way to compute Christoffel symbols
and therefore very useful.
The approach is theoretical, but the text is interlaced with discussions
of observational, instrumental and historical aspects where appropriate. The
book is divided into (1) preparatory material: special relativity, geometry of
Riemann spaces, and general relativity, (2) Schwarzschild metric and appli-
cations: classical tests, binary pulsars, gravitational lenses, neutron stars and
black holes, (3) experimental gravity: gravitational waves and their detec-
tors, Gravity Probe B, and finally (4) cosmology: Robertson-Walker metric,
evolution of the universe, observational cosmology, and inflation. Due to the
self-imposedrestrictionsseveraltopicshadtobeskipped.Butinviewoftheir
current interest, extra attention has been given to the operation of interfer-
VI Preface
ometerdetectorsforgravitationalwaves,totheGravityProbeBmission,and
to structure formation in relation to the results of the Wilkinson Microwave
Anisotropy Probe (WMAP).
The reader is supposed to be familiar with linear algebra and calculus,
ordinary differential equations, and with elementary thermal physics, electro-
dynamics,specialrelativityandquantummechanics-inotherwords,thebasic
educationofadvancedphysicsundergraduates.Priorknowledgeofdifferential
geometry, general relativity and astrophysics is helpful but not required. The
necessary mathematical techniques are introduced informally, following geo-
metrical intuition as much as possible. The admirable texts of Dirac (1975),
Price (1982) and Schutz (1985) have been a source of inspiration for me in
this regard. And the astrophysical concepts are likewise briefly introduced to
a level where they should beintelligible for physics students. There are about
145 exercises with hints for their solution. These exercises are an indispens-
able element in helping students to come to grasp with the subject matter,
and to train them to solve elementary problems independently. In my expe-
rience 40−45 lectures (45 min.) of oral instruction would suffice to expound
all material, excluding tutorials for exercises.
References to the literature are eclectic rather that complete, and appear
as footnotes in the text. General references (mostly textbooks) are given in
Appendix A. The finiteness of the alphabet did cause some problems of nota-
tion.Thereaderisalertedtomypropensityforthesymbola.Therearemany
different constants a in the text, but confusion is unlikely as they have only a
localmeaning.Likewisehhasthreedifferentmeanings,(H /100,theconstant
0
of the motion r2ϕ˙ in the Schwarzschild metric, and Planck’s constant).
The cover picture is a still life by the Dutch artist Olav Cleofas van
Overbeek entitled Black bowl on yellow plane (2002). Its simplicity and well-
balanced design epitomize the rotational and translational symmetries that
are so ubiquitous in physics, and in this book embodied in the Schwarzschild
metric and the Robertson-Walker metric, respectively. The cartoons oppo-
site to the chapter headings have been drawn by Roeland van Oss, and I am
gratefulforhispermissiontoreproducethemhere.Thedraftingofthefigures
reflects the technical developments of the period, and began on rice paper, to
proceed entirely by electronic means in the end. I wish to thank Hans Braun,
Arjan Bik, and in particular Artur Pfeifer for their assistance in this area.
There are instances where we have been unable to trace or contact the copy-
right holder of some of the reproduced figures. If notified the publisher will
be pleased to rectify any errors or omissions at the earliest opportunity.
I want to express my gratitude to Jan van der Kuur for his help in solv-
ing my Latex problems, and to Constance Jansen who generously provided
Preface VII
library assistance. Lucas van der Wiel has helped me with the first English
translation. In the course of the years that led up to this book I have ben-
efitted from discussions and correspondence with many colleagues. I cannot
name them all, but I do wish to thank Bram Achterberg and Ed van den
Heuvel and several unknown referees who read sections of the manuscript. I
am in particular indebted to my friend and colleague John Heise who since
many years is my discussion partner on matters relativistic and other. His
influence is pervasive throughout the book. And last but not least, I should
thank all the students who continually forced me to improve the presentation
of the material, from my first notes in 1990 (to which I think in slight em-
barrassment), to the present text which is, I hope, of some use to the reader.
Utrecht, Peter Hoyng
February 2005
Contents
1 Introduction............................................... 1
1.1 Special relativity (SR) ................................... 3
1.2 General relativity (GR) .................................. 10
1.3 The need for GR in astrophysics........................... 14
2 Geometry of Riemann Spaces.............................. 19
2.1 Definition .............................................. 19
2.2 The tangent space ....................................... 21
2.3 Tensors ................................................ 23
2.4 Parallel transport and Christoffel symbols .................. 26
2.5 Geodesics .............................................. 30
2.6 The covariant derivative.................................. 33
2.7 Riemann tensor and curvature ............................ 35
3 General Relativity......................................... 43
3.1 Co-ordinates, metric and motion .......................... 43
3.2 Weak fields (1).......................................... 47
3.3 Conservation of mass .................................... 50
3.4 The field equations ...................................... 52
3.5 Weak fields (2).......................................... 56
3.6 Discussion .............................................. 59
4 The Schwarzschild Metric ................................. 65
4.1 Preliminary calculations.................................. 65
4.2 The Schwarzschild metric................................. 69
4.3 Geodesics of the Schwarzschild metric...................... 72
4.4 The classical tests of GR ................................. 77
4.5 Gravitational lenses...................................... 82
X Contents
5 Compact Stars............................................. 89
5.1 End products of stellar evolution .......................... 89
5.2 The maximum mass M .................................. 95
c
5.3 The Tolman-Oppenheimer-Volkoff equation ................. 97
5.4 A simple neutron star model..............................101
5.5 Realistic neutron star models .............................103
6 Black Holes................................................109
6.1 Introduction ............................................109
6.2 Observations............................................110
6.3 Elementary properties....................................113
6.4 Kruskal-Szekeres co-ordinates .............................121
6.5 Rotating black holes: the Kerr metric ......................125
6.6 Hawking radiation.......................................128
7 Gravitational waves........................................133
7.1 Small amplitude waves ...................................133
7.2 The effect of a gravitational wave on test masses ............136
7.3 Generation of gravitational radiation.......................138
7.4 Bar detectors ...........................................143
7.5 Interferometer detectors ..................................145
8 Fermi-Walker Transport ...................................155
8.1 Transport of accelerated vectors ...........................155
8.2 Thomas precession.......................................159
8.3 Geodesic precession......................................161
8.4 Gravity Probe B ........................................164
9 The Robertson-Walker Metric .............................169
9.1 Observations............................................169
9.2 Definition of co-ordinates.................................174
9.3 Metric and spatial structure ..............................177
9.4 Equations of motion .....................................181
9.5 The cosmological constant ................................183
9.6 Geodesics ..............................................185
10 The Evolution of the Universe .............................189
10.1 Equation of state........................................189
10.2 The matter era..........................................191
10.3 The radiation era........................................197
10.4 The formation of structure................................203
Contents XI
11 Observational Cosmology ..................................213
11.1 Redshift and distance ....................................213
11.2 The visible universe and the horizon .......................218
11.3 Luminosity distance and Hubble relation ...................224
11.4 The microwave background ...............................228
11.5 Light-cone integrals......................................231
12 The Big Bang .............................................237
12.1 Nuclear reactions........................................237
12.2 The first 100 seconds.....................................240
12.3 The synthesis of light elements ............................246
13 Inflation ...................................................253
13.1 The horizon problem.....................................254
13.2 Evolution of a universe with a scalar field...................258
13.3 Chaotic inflation ........................................261
13.4 Discussion ..............................................266
