Henri Poincare and Relativity Theory - LOGUNOV, A. A..pdf - Theoretical Physics - annam101

A.A. Logunov

HENRI POINCAR E

AND

RELATIVITY THEORY

Logunov A.A.

The book presents ideas by H. Poincare and H. Minkowski

according to those the essence and the main content of the rela-

tivity theory are the following: the space and time form a unique

four-dimensional continuum supplied by the pseudo-Euclidean ge-

ometry. All physical processes take place just in this four-dimen-

sional space. Comments to works and quotations related to this

subject by L. de Broglie, P.A.M. Dirac, A. Einstein, V.L. Ginzburg,

S. Goldberg, P. Langevin, H.A. Lorentz, L.I. Mandel’stam, H. Min-

kowski, A. Pais, W. Pauli, M. Planck, A. Sommerfeld and H. Weyl

are given in the book. It is also shown that the special theory of

relativity has been created not by A. Einstein only but even to a

greater extent by H. Poincare.

The book is designed for scientiﬁc workers, post-graduates

and upper-year students majoring in theoretical physics.

Devoted to 150th Birthday

of Henri Poincare – the greatest mathematician,

mechanist, theoretical physicist

Preface

The special theory of relativity “resulted from the joint efforts

of a group of great researchers – Lorentz, Poincare, Einstein,

Minkowski ” ( Max Born ).

“Both Einstein, and Poincare, relied on the preparatory works

of H.A. Lorentz, who came very close to the ﬁnal result, but was

not able to make the last decisive step. In the coincidence of re-

sults independently obtained by Einstein and Poincare I see the

profound sense of harmony of the mathematical method and the

analysis, performed with the aid of thought experiments based

on the entire set of data from physical experiments” . ( W. Pauli,

1955. ).

H. Poincare, being based upon the relativity principle formu-

lated by him for all physical phenomena and upon the Lorentz

work, has discovered and formulated everything that composes the

essence of the special theory of relativity. A. Einstein was coming

to the theory of relativity from the side of relativity principle for-

mulated earlier by H. Poincare. At that he relied upon ideas by

H. Poincare on deﬁnition of the simultaneity of events occurring

in different spatial points by means of the light signal. Just for this

reason he introduced an additional postulate – the constancy of the

velocity of light. This book presents a comparison of the article by

A. Einstein of 1905 with the articles by H. Poincare and clariﬁes

what is the new content contributed by each of them. Somewhat

later H. Minkowski further developed Poincare’s approach. Since

Poincare’s approach was more general and profound, our presen-

tation will precisely follow Poincare.

According to Poincare and Minkowski, the essence of relativ-

ity theory consists in the following: the special theory of relativ-

ity is the pseudo-Euclidean geometry of space-time. All phys-

ical processes take place just in such a space-time . The conse-

quences of this postulate are energy-momentum and angular mo-

mentum conservation laws, the existence of inertial reference sys-

tems, the relativity principle for all physical phenomena, Lorentz

transformations, the constancy of velocity of light in Galilean co-

ordinates of the inertial frame, the retardation of time, the Lorentz

contraction, the possibility to exploit non-inertial reference sys-

tems, the clock paradox, the Thomas precession, the Sagnac ef-

fect, and so on. Series of fundamental consequences have been

obtained on the base of this postulate and the quantum notions,

and the quantum ﬁeld theory has been constructed. The preser-

vation (form-invariance) of physical equations in all inertial ref-

erence systems mean that all physical processes taking place in

these systems under the same conditions are identical . Just for

this reason all natural etalons are the same in all inertial refer-

ence systems.

The author expresses profound gratitude to Academician of the

Russian Academy of Sciences Prof. S.S. Gershtein, Prof. V.A. Pet-

rov, Prof. N.E. Tyurin, Prof. Y.M. Ado, senior research associate

A.P. Samokhin who read the manuscript and made a number of va-

luable comments, and, also, to G.M. Aleksandrov for signiﬁcant

work in preparing the manuscript for publication and completing

Author and Subject Indexes.

A.A. Logunov

January 2004

1. Euclidean geometry

In the third century BC Euclid published a treatise on math-

ematics, the “Elements” , in which he summed up the preceding

development of mathematics in antique Greece . It was precisely

in this work that the geometry of our three-dimensional space –

Euclidean geometry – was formulated.

This happened to be a most important step in the development

of both mathematics and physics. The point is that geometry ori-

ginated from observational data and practical experience, i. e.

it arose via the study of Nature. But, since all natural phenom-

ena take place in space and time, the importance of geometry for

physics cannot be overestimated, and, moreover, geometry is ac-

tually a part of physics.

In the modern language of mathematics the essence of Eu-

clidean geometry is determined by the Pythagorean theorem .

In accordance with the Pythagorean theorem, the distance of a

point with Cartesian coordinates x,y,z from the origin of the re-

ference system is determined by the formula

ℓ 2

= x 2 + y 2 + z 2 ,

(1.1)

or in differential form, the distance between two inﬁnitesimally

close points is

(dℓ) 2

= (dx) 2 + (dy) 2 + (dz) 2 .

(1.2)

Here dx,dy,dz are differentials of the Cartesian coordinates. Usu-

ally, proof of the Pythagorean theorem is based on Euclid’s ax-

ioms, but it turns out to be that it can actually be considered a

deﬁnition of Euclidean geometry. Three-dimensional space, de-

termined by Euclidean geometry, possesses the properties of ho-

mogeneity and isotropy. This means that there exist no singular

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