Motion and Relativity

225 Pages·1960·6.547 MB·English

by  Leopold Infeld and Jerzy Plebanski (Auth.)

P O L S KA A K A D E M IA N A UK
MONOGRAFIE FIZYCZNE
Konntet redftkcyjny
L. INFELD,
L. NATANSON, W. RUBINOWICZ, L. SOSNOWSKl,
J. WEYSSENHOFF
WARSZAWA I960
P O L I SH ACADEMY OF S C I E N C ES
PHYSICAL MONOGRAPHS
MOTION AND RELATIVITY
by
Leopold Infeld and Jerzy Plebanski
PERGAMON PRESS · NEW YORK · OXFORD LONDON PARIS
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by Panstwowe Wydawnictwo Naokowe
1960
Warsaw, Poland
Library o! Congress Card No. 60-14864
Printed in Poland by Wroclawska Drukarnia Naukowa
mTBODUCTION
The problem of motion in gravitational theory was first solved
in a paper by Einstein, Infeld and Hoffmann in 1938. The calcula­
tions were so troublesome that we had to leave on reference at the
Institute for Advanced Study in Princeton a whole manuscript
of calculations for others to use. After that, Einstein and I made
some progress together on this problem. Twenty-two years have
elapsed since the first paper was published and I have again worked
on this problem with my students in Warsaw during the last few
years. This book presents the final results of all our work. Parallel,
independently of us and a little later, W. Pock and his school in
Leningrad tackled and solved the problem of motion in relativity
theory, too. His results are also piesented in his book entitled
"Theory of space, time and gravitation" Pergamon Press, London,
ITew York 1960. Though our approach is different from Fock's
and more in the spirit of Einstein, this work is not intended to
be polemic.
I have written this book with Dr Jerzy Plebanski. We discussed
the contents carefully over the four years it took to write it. Un­
fortunately we finished only the first chapter and appendix when
Dr Plebanski received a Rockefeller Fellowship to go to the
United States. Before leaving, Dr Plebanski prepared a sketch
in Polish of the rest of the manuscript with the exception of the
last chapter. This presentation was later very much changed by
me for which I take the full responsibility.
This book presupposes only a knowledge of the general principles
of relativity theory. Eeaders of greater mathematical inclination
are advised to read the appendix first and not merely the short
chapter on notation which is a summary of it.
8 INTRODUCTION
In writing the book, we were greatly helped by Dr Andrzej
Trautman who made many critical remarks, checked the formulas
and prepared the bibliography. Our thanks are also due to Dr
W. Tulczyjew who helped me greatly in preparing the last sections
of Chapters IV and V.
Leopold Infeld
Warsaw I960
NOTATION
A. NOTATION OF GENERAL RELATIVITY THEORY
We shall use throughout the tensor analysis of General Eelativ-
ity Theory (G· E. T. for short).
We shall denote by
x°j xl, x2, x* (0.1)
the time and space coordinates of a Biemannian manifold. If we
assume Special Belativity and a Cartesian coordinate system, then x°
corresponds to the time t from
o?° = at,
o being the velocity of light; for k = 1, 2, 3. the xk (or x) denote the
space coordinates.
All Greek indices run from 0 to 3, Latin indices from 1 to 3.
Bepetition of indices implies summation.
The geometry of the Biemannian space-time continuum is char­
acterized by a symmetrical metric tensor
9aßW)=9ßaW). (0.2)
To distinguish between time and space in all possible coordinate
systems we must assume that the metric tensor always satisfies
the condition:
</oo>0, L - ^ y V <0 (0.3)
for arbitrary ya Φ 0.
10 NOTATION
Instead-of these, we may assume the equivalent Hubert condi­
tions restricting the arbitrariness of spaoe-time transformations:
0oo? 0oi? 0021 (0.3a)
0oo? 0oi
0oo > 0, <o, 010, 011? ff» I > 0, 0 = detail < 0.
010? 011
020? 021? 022
The metric tensor g^ is a generalization of the Minkowski
metric tensor η^ of Special Belativity Theory, defined by
1 for a = ft,
*?oo = 1? Voa = 0, ~Vab = bob = (0.4)
0 for a Φ b.
To the covariant metric tensor g^ there corresponds a contia-
variant metric tensor tf* defined by
1 for a = £,
ft« = % = (0.5)
[0 for a φ β.
We shall denote the determinant of g^ by g and all quantities
that transform like
}/ — 0 x tensor
we shall call tensor densities and denote by
£g::£ = »^Ζί:::^· (°·β>
The ordinary derivative will usually be denoted by a stroke:
dS
(0.7)
8* ~ dx°
The Chrietoffel symbols, which do not have tensor character,
are:
(0.8)
(0.9)
NOTATION OF GENERAL RELATIVITY THEORY 11
These symbols allow us to differentiate tensors in a covariant
way. We shall denote such a covariant differentiation by a semi­
colon:
T:::: = τ^: +... + ^τ\ν.:+^ (ο.ιο)
iß φ
. T::ä. = T-V+... - [$ T-+.... (0.11)
;ß
The indices written after the semicolon have tensor character and
can be shifted up or down according to the ordinary rules.
Prom the Christoffel symbols we form the full Riemannian
tensor:
+
^={:}.-W,. täW-W(;}· ^
Prom it, by putting μ = σ we form the contracted Riemannian
tensor (Ricci tensor):
and the curvature scalar:
B = <r%^ (0.14)
The Einstein tensor is:
G^ = 22*- \g+R. (0.15)
AU the quantities g^, Ι^Λ, Βμ etc. are of a geometrical
κα1
character. With at least some of them one associates a definite
physical meaning. In a certain sense the Christoffel symbols repre­
sent the intensity of the gravitational field and the metric tensor
its potentials. We shall have much more to say about this later.
12 NOTATION
B. THE δ FUNCTIONS
We shall introduce "good" δ functions — that is, δ functions
which, besides having the properties of ordinary Dirac δ functions,
also satisfy the condition
- ^ dx = 0 for p=1.2,...,fc. (0.16)
/
Ixr
All calculations in which δ functions appear and no other assumption
is made, are performed with "good" δ functions. A more extensive
theory of these functions is given in Appendix 1.
C. THE FIELD VALUES UN THE WORLD-LINES
A
Let us call ξα(8 ) the world-line of the A'th singularity. Usually
Α
we take as the parameter on which the world-line depends, not the
"eigentime" 8 but the time #° = ct. Then the motion is character­
Aj
ized by |m(a?°) and f° = #°. Let us assume
Then the "tweedling process" belonging to the A'th singularity
and denoted by ~ written over the "tweedled" expression means
A
two things: first, the singular part at x = ξ of / is ignored; second,
A
it introduces the expression ξ instead of x into the regular part
of /. Or, using our "good" δ functions we define:
f =-Jdxo(x-tl*9))f(0, x\ ta). (0.17)
From this definition follows (if we omit the -l's):
Ψ)ο = 9> + 9Vf% = ^Γ,ο, (0.18)
dtp d<p ~-'
THE FIELD VALUES ON THE WORLD-LINES 13
We shall assume throughout that the functions with which we deal
obey the rule for "tweedling" the products, which is:
φψ == φψ . (0 19)
For more about this process, see Appendix 2.
D. THE COVARIANT CHARACTER OP THE (Τβ.
TENSORS ON WORLD-LINES
{
The four dimensional Dirac <3 function is a scalar density.
(4)
This follows from the relation:
fd (w)dx = 1. (0.20)
{4)
By "world-line tensors" we shall understand tensors defined
only on the world-lines ξβ. To such tensors we may apply the rules
of tensor algebra but not those of tensor analysis. To apply the
latter, we must have tensor fields. Wo may change a tensor defined
along a world-line, at least symbolically, into a tensor density field,
in the following way:
00
*?!:::&(·*)= / ^^-m)T^^(i). (0.21)
—00
The transformation properties of <5 follow from:
(s)
00
<5 = / \<d? (0.22)
(3) }
which, together with (0.20), gives:
Jo dx = l. (0 23)
(3)
This means that d dx can be treated as an invariant. If the d 's
{i) (3)
are our "good" (J's, we may regard (0.22) as the definition of the
"good" <5 .
(4)
For more about this, see Appendix 3.