Introduction To General Relativity - G. T.Hooft.pdf

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INTRODUCTION TO GENERAL RELATIVITY
G. ’t Hooft
CAPUTCOLLEGE 1998
Institute for Theoretical Physics
Utrecht University,
Princetonplein 5, 3584 CC Utrecht, the Netherlands
version 30/1/98
PROLOGUE
General relativity is a beautiful scheme for describing the gravitational fieldandthe
equations it obeys. Nowadays this theory is often used as a prototype for other, more
intricate constructions to describe forces between elementary particles or other branches of
fundamental physics. This is why in an introduction to general relativity it is of importance
to separate as clearly as possible the various ingredients that together give shape to this
paradigm.
After explaining the physical motivations we first introduce curved coordinates, then
addto this the notion of an a1ne connection fieldandonly as a later step addto that the
metric field. One then sees clearly how space and time get more and more structure, until
finally all we have to do is deduce Einstein’s field equations.
As for applications of the theory, the usual ones such as the gravitational redshift,
the Schwarzschild metric, the perihelion shift and light deflection are pretty standard.
They can be found in the cited literature if one wants any further details. I do pay some
extra attention to an application that may well become important in the near future:
gravitational radiation. The derivations given are often tedious, but they can be produced
rather elegantly using standard Lagrangian methods from field theory, which is what will
be demonstrated in these notes.
LITERATURE
C.W. Misner, K.S. Thorne andJ.A. Wheeler, “Gravitation”, W.H. Freeman andComp.,
San Francisco 1973, ISBN 0-7167-0344-0.
R. Adler, M. Bazin, M. Schiffer, “Introduction to General Relativity”, Mc.Graw-Hill 1965.
R. M. Wald, “General Relativity”, Univ. of Chicago Press 1984.
P.A.M. Dirac, “General Theory of Relativity”, Wiley Interscience 1975.
S. Weinberg, “Gravitation andCosmology: Principles andApplications of the General
Theory of Relativity”, J. Wiley & Sons. year ???
S.W. Hawking, G.F.R. Ellis, “The large scale structure of space-time”, Cambridge Univ.
Press 1973.
S. Chandrasekhar, “The Mathematical Theory of Black Holes”, Clarendon Press, Oxford
Univ. Press, 1983
Dr. A.D. Fokker, “Relativiteitstheorie”, P. Noordhoff, Groningen, 1929.
1
J.A. Wheeler, “A Journey into Gravity andSpacetime, Scientific American Library, New
York, 1990, distr. by W.H. Freeman & Co, New York.
CONTENTS
Prologue
1
literature
1
1. Summary of the theory of Special Relativity. Notations.
3
2. The Eotvos experiments andthe equaivalence principle.
7
3. The constantly acceleratedelevator. Rindler space.
9
4. Curvedcoordinates.
13
5. The a1ne connection. Riemann curvature.
19
6. The metric tensor.
25
7. The perturbative expansion andEinstein’s law of gravity.
30
8. The action principle.
35
9. Spacial coordinates.
39
10. Electromagnetism.
43
11. The Schwarzschildsolution.
45
12. Mercury andlight rays in the Schwarzschildmetric.
50
13. Generalizations of the Schwarzschildsolution.
55
14. The Robertson-Walker metric.
58
15. Gravitational radiation.
62
2
1. SUMMARY OF THE THEORY OF SPECIAL RELATIVITY. NOTATIONS.
Special Relativity is the theory claiming that space andtime exhibit a particular
symmetry pattern. This statement contains two ingredients which we further explain:
(i) There is a transformation law, andthese transformations form a group.
(ii) Consider a system in which a set of physical variables is described as being a correct
solution to the laws of physics. Then if all these physical variables are transformed
appropriately according to the given transformation law, one obtains a new solution
to the laws of physics.
A “point-event” is a point in space, given by its three coordinates x =( x,y,z ), at a given
instant t in time. For short, we will call this a “point” in space-time, andit is a four
component vector,
x 0
x 1
x 2
x 3
ct
x
y
z
x =
=
.
(1 . 1)
Here c is the velocity of light. Clearly, space-time is a four dimensional space. These
vectors are often written as x µ ,where µ is an index running from 0 to 3. It will however
be convenient to use a slightly different notation, x µ ,µ =1 ,..., 4, where x 4 = ict and
i = √ −
{} µ ) andsubscript indices (
{} µ )is
of no significance in this section, but will become important later.
In Special Relativity, the transformation group is what one couldcall the “velocity
transformations”, or Lorentz transformations . It is the set of linear transformations,
4
( x µ ) =
L µ ν x ν
(1 . 2)
ν =1
subject to the extra condition that the quantity σ defined by
4
σ 2 =
( x µ ) 2 =
|
x
|
2
−
c 2 t 2
( σ
≥
0)
(1 . 3)
µ =1
remains invariant. This condition implies that the coe1cients L µ ν form an orthogonal
matrix:
4
L µ ν L α ν
= δ µα ;
ν =1
(1 . 4)
4
L α µ L α ν
= δ µν .
α =1
3
1. The intermittent use of superscript indices (
266000034.001.png
Because of the i in the definition of x 4 , the coe1cients L i 4 and L 4
i
must be purely
imaginary. The quantities δ µα and δ µν are Kronecker delta symbols:
δ µν = δ µν =1 if µ = ν, and0 otherwise .
(1 . 5)
One can enlarge the invariance group with the translations:
4
( x µ ) =
L µ ν x ν + a µ ,
(1 . 6)
ν =1
in which case it is referredto as the Poincar´egroup .
We introduce summation convention :
If an index occurs exactly twice in a multiplication (at one side of the = sign) it will auto-
matically be summed over from 1 to 4 even if we do not indicate explicitly the summation
symbol Σ. Thus, Eqs (1.2)–(1.4) can be written as:
( x µ ) = L µ ν x ν , 2 = x µ x µ =( x µ ) 2 ,
L µ ν L α ν
= δ µα , α µ L α ν
= δ µν .
(1 . 7)
If we do not want to sum over an index that occurs twice, or if we want to sum over an
index occuring three times, we put one of the indices between brackets so as to indicate
that it does not participate in the summation convention. Greek indices µ,ν,... run from
1 to 4; latin indices i,j,... indicate spacelike components only and hence run from 1 to 3.
A special element of the Lorentz group is
L µ ν
=
→ ν
10 0 0
01 0 0
,
(1 . 8)
↓
00 osh χi sinh χ
µ
00 −i sinh χ cosh χ
where χ is a parameter. Or
x = x ; y = y ;
z = z cosh χ
−
ct sinh χ ;
(1 . 9)
z
c sinh χ + t cosh χ.
t = −
This is a transformation from one coordinate frame to another with velocity
v/c =tanh χ
(1 . 10)
4
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