Elementary General Relativity - Luther College

111 Pages·2015·3.88 MB·English

by  Alan Macdonald

General Relativity
in a
Nutshell
October 2, 2017
Alan Macdonald
Luther College, Decorah, IA USA
mailto:[email protected]
http://faculty.luther.edu/~macdonal
(cid:13)c
To Ellen
“Themagicofthistheorywillhardlyfailtoimposeitselfonanybody
who has truly understood it.”
Albert Einstein, 1915
“The foundation of general relativity appeared to me then [1915],
and it still does, the greatest feat of human thinking about Nature,
themostamazingcombinationofphilosophicalpenetration,physical
intuition, and mathematical skill.”
Max Born, 1955
“One of the principal objects of theoretical research in any depart-
mentofknowledgeistofindthepointofviewfromwhichthesubject
appears in its greatest simplicity.”
Josiah Willard Gibbs
“Thereisawidespreadindifferencetoattemptstoputacceptedthe-
ories on better logical foundations and to clarify their experimental
basis, an indifference occasionally amounting to hostility. I am con-
cerned with the effects of our neglect of foundations on the educa-
tion of scientists. It is plain that the clearer the teacher, the more
transparent his logic, the fewer and more decisive the number of ex-
periments to be examined in detail, the faster will the pupil learn
and the surer and sounder will be his grasp of the subject.”
Sir Hermann Bondi
“Things should be made as simple as possible, but not simpler.”
Albert Einstein

Contents
Contents v
Preface vii
1 Flat Spacetimes 1
1.1 Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Inertial Frame Postulate . . . . . . . . . . . . . . . . . . . . 4
1.3 The Metric Postulate . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 The Geodesic Postulate . . . . . . . . . . . . . . . . . . . . . . . 15
2 Curved Spacetimes 17
2.1 Newton’s theory of gravity. . . . . . . . . . . . . . . . . . . . . . 17
2.2 The Key to General Relativity . . . . . . . . . . . . . . . . . . . 19
2.3 The Local Inertial Frame Postulate . . . . . . . . . . . . . . . . . 23
2.4 The Metric Postulate . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 The Geodesic Postulate . . . . . . . . . . . . . . . . . . . . . . . 30
2.6 The Field Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Spherically Symmetric Spacetimes 39
3.1 Stellar Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Schwarzschild Metric . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Solar System Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4 Kerr Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5 Extreme Binary Systems . . . . . . . . . . . . . . . . . . . . . . . 55
3.6 Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4 Cosmological Spacetimes 61
4.1 Our Universe I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 The Robertson-Walker Metric . . . . . . . . . . . . . . . . . . . . 65
4.3 Expansion Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4 Our Universe II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5 General Relativity Today . . . . . . . . . . . . . . . . . . . . . . 77
A Appendix 79
Index 101
v
vi
Preface
Thepurposeofthislittlebookistoprovideaclearandcarefulaccountofgeneral
relativity with a minimum of mathematics. The book has fewer prerequisites
thanothertexts,andlessmathematicsisdeveloped. Theprerequisitesaresingle
variable calculus, a few basic facts about partial derivatives and line integrals,
and a little matrix algebra. A little knowledge of physics is useful but not
essential. The algebra of tensors plays only a minor role.1 (Similarly, many
elementary differential geometry texts develop the intrinsic geometry of curved
surfaces without using tensors.)
Despitethebook’sbrevityandmodestprerequisites,itisaseriousintroduc-
tion to the theory and applications of general relativity which demands careful
study. It can be used as a textbook for general relativity or as an adjunct to
standard texts. It is also suitable for self-study by more advanced students.
All this should make general relativity available to a wider audience than
before. Awellpreparedsophomorecanlearnaboutexcitingcurrenttopicssuch
ascurvedspacetime,blackholes,thebigbang,darkenergy,andtheaccelerating
universe.
Chapter 1 is a self-contained introduction to those parts of special relativity
werequireforgeneralrelativity. Wetakeanonstandardapproachtothemetric,
analogous to the standard approach to the metric in Euclidean geometry. In
geometry, distance is first understood geometrically, independently of any coor-
dinatesystem. Ifcoordinatesareintroduced,thendistancescanbeexpressedin
terms of coordinate differences: ∆s2 = ∆x2+∆y2. The formula is important,
but the geometric meaning of the distance is fundamental.
Analogously,wedefinethespacetimeintervalofspecialrelativityphysically,
independently of any coordinate system. If inertial frame coordinates are in-
troduced, then the interval can be expressed in terms of coordinate differences:
∆s2 = ∆t2 −∆x2 −∆y2 −∆z2. The formula is important, but the physical
meaning of the interval is fundamental. This approach to the metric provides
easier access to and deeper understanding of special and general relativity, and
facilitates the transition from special to general relativity.
1 The notion of covariant and contravariant indices is developed and used in one page of
thetext.
vii
Chapter 2 introduces the four fundamental principles of general relativity
as postulates. The purpose of the postulates is not to achieve rigor – which
is neither desirable nor possible in a book at this level – but to state clearly
the principles, and to exhibit clearly the relationship to special relativity and
the analogy with surfaces. The first three principles are expressed in terms
of local inertial frames, which tells us their physical meaning. They are then
translatedtoglobalcoordinates,whichisnecessaryforcalculations. Thefourth
principle,thefieldequation,isthenmotivatedandstated. Finally,aninteresting
consequence of the field equation relating curvatures and density is obtained.
Thefirsttwochapterssystematicallyexploitthemathematicalanalogywhich
led to general relativity: a curved spacetime is to a flat spacetime as a curved
surface is to a flat surface. Before introducing a spacetime concept, its analog
forsurfacesispresented. Thisisnotanewidea,butitisusedheremoresystem-
atically than elsewhere. For example, when the metric ds of general relativity
is introduced, the reader has already seen a metric in three other contexts.
Chapter 3 solves the field equation for a spherically symmetric spacetime to
obtain the Schwarzschild metric. The geodesic equations are then solved and
appliedtotheclassicalsolarsystemtestsofgeneralrelativity. Thereisasection
on the Kerr metric, which includes gravitomagnetism and the Gravity Probe B
experiment. The chapter closes with sections on the double pulsar and black
holes. Inthischapter,aselsewhere,Ihavetriedtoprovidethecleanestpossible
calculations.
Chapter4appliesgeneralrelativitytocosmology. WeobtaintheRobertson-
Walker metric in an elementary manner without the field equation. We review
the evidence for a spatially flat universe with a cosmological constant. We
then apply the field equation with a cosmological constant to a spatially flat
Robertson-Walker spacetime. The solution is given in closed form. Recent
astronomical data allow us to specify all parameters in the solution, giving the
new “standard model” of the universe with dark matter, dark energy, and an
accelerating expansion.
Many recent spectacular astronomical discoveries and observations are rel-
evant to general relativity. They are described at appropriate places in the
book.
Some tedious (but always straightforward) calculations have been omitted.
They are best carried out with a computer algebra system. Some material has
beenplacedinabout20pagesofappendicestokeepthemainlineofdevelopment
visible. They may be omitted without loss of anything essential. Appendix 1
gives the values of various physical constants. Appendix 2 contains several ap-
proximation formulas used in the text.
viii
Chapter 1
Flat Spacetimes
1.1 Spacetimes
The general theory of relativity is our best the-
ory of space, time, and gravity. Albert Einstein
created the theory during the decade following
the publication of his special theory of relativ-
ity in 1905. The special theory is a theory of
space and time which does not take gravity into
account. The general theory, published in 1915,
generalizes the special theory to include gravity.
It is commonly felt to be the most beautiful of
all physical theories.
We will explore many fascinating aspects of
relativity, such as the behavior of moving clocks,
curved spacetimes, Einstein’s field equation, the
Fig. 1.1: Albert Einstein,
perihelion advance of Mercury, black holes, the
1879-1955.
big bang, the accelerating expansion of the uni-
verse, and dark energy. Let us begin!
1
1 Flat Spacetimes 1.1 Spacetimes
Definitions. Wegivedefinitionsofthreefundamentalconceptsusedthrough-
out this book: event, spacetime, and worldline.
Event. In geometry the fundamental entities are points. A point is a
specific place. In relativity the fundamental entities are events. An event is
a specific time and place. It has neither temporal nor spatial extension. For
example, the collision of two particles is an event. To be present at the event,
you must be at the right place at the right time.
Spacetime. A flat or curved surface is a set of points. (We shall prefer
the term “flat surface” to “plane”.) Similarly, a spacetime is a set of events.
Chapter 3 examines the spacetime around the Sun. We will study the motion
of planets and light in the spacetime. Chapter 4 examines a spacetime for the
entire universe! We will study the origin and evolution of the universe.
Aflat spacetime isaspacetimewithoutsignificantgravity. Specialrelativity
describes flat spacetimes. A curved spacetime is a spacetime with significant
gravity. General relativity describes curved spacetimes.
There is nothing mysterious about the words “flat” or “curved” attached to
a spacetime. They are chosen because of a remarkable analogy, already hinted
at, concerningthe mathematical descriptionof aspacetime: a curved spacetime
is to a flat spacetime as a curved surface is to a flat surface. The analogy will
be a major theme of this book: we will use our intuitive understanding of flat
and curved surfaces to guide our understanding of flat and curved spacetimes.
Worldline. Acurveinasurfaceisacontinuoussuccessionofpointsinthe
surface. A worldline in a spacetime is a continuous succession of events in the
spacetime. A moving particle or a pulse of light emitted in a single direction is
present at a continuous succession of events, its worldline. Even if a particle is
at rest, time passes, and the particle has a worldline. A planet has a worldline
in the curved spacetime around the Sun. Note that if the planet returns to a
specificpointinspaceduringitsorbit,thenitdoesnotreturntothesameevent
in the spacetime, as it returns at a later time.
Time. Relativity theory contradicts everyday views of time. The most
direct illustration of this is the Hafele-Keating experiment, which we now de-
scribe.
The length of a curve between two given points depends on the curve. Sim-
ilarly, the time between two given events measured by a clock moving between
theeventsdependsontheclock’sworldline! In1971J.C.HafeleandR.Keating
brought two atomic clocks together, placed one of them in an airplane which
circled the Earth, and then brought the clocks together again. Thus the clocks
had different worldlines between the event of their separation and the event
of their reunion. The clocks measured different times between the two events.
The difference was small, about 10−7 sec, but was well within the ability of the
clocks to measure. There is no doubt that the effect is real.
We shall see that relativity predicts the measured difference. We shall also
seethatrelativitypredictslargedifferencesbetweenclockswhoserelativespeed
is close to that of light.
2