A NNALES DE L ’I. H. P., SECTION A
T Z . I. G UTZUNAYEV
On the problem of relativistically moving frames of reference
Annales de l’I. H. P., section A, tome 17, n
o1 (1972), p. 71-79
<http://www.numdam.org/item?id=AIHPA_1972__17_1_71_0>
© Gauthier-Villars, 1972, tous droits réservés.
L’accès aux archives de la revue « Annales de l’I. H. P., section A » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.
org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
http://www.numdam.org/
71
On the problem
of relativistically moving frames of reference
Tz. I. GUTZUNAYEV
Vol. XVII, no 1, 1972, Physique théorique.
ABSTRACT. - A method of derivation of relativistic formulae of trans- formation to accelerated
frames
of reference isdeveloped
in the caseof one-dimentional motion.
Special examples
of uni-accelerated frame and ofoscillating
frame are considered. Theproblem
of calculation ofelectromagnetic
field of uni-accelerated electron is solvedemploying
the non-inertial frame of reference.
1. INTRODUCTION
A
problem
offinding
the relativistic formulae which describe the transformations to accelerated frames ofreference
is not a new one.Now and
again
it becomes asubject
of discussions in the litera- ture([ 1]
a[6]).
The
employment
of relativistic non-inertial frames seems to be very useful in theattempts
to solve certainelectrodynamic problems.
Forexample,
when theparticle
motion indifferent types
of accelerators isstudied,
the introduction of a non-inertial frame ofreference
makes the calculations ofelectromagnetic
fields andparticle trajectories
substan-tially easier,
and also maysignificantly simplify
theproblem
ofstability
of
trajectories (c f. [7], [8]).
The
simplest
of the non-inertial frames arerectilinearly moving non-rotating
ones. Theuniformly
accelerated motion is oneparticular
case of such motions. In relativistic kinematics the
uniformly
acce-lerated motion is defined as one in which the
magnitude
of acceleration aremains constant with
respect
to proper(instantly co-moving inertial)
frame.
72 TZ. I. GUTZUNAYEV
The relativistic
formulae
of transformation touni-accelerated
frameswere first indicated
(without derivation) by
Kottler[1].
These are written as follows :
where
X, Y, Z,
T are defined withrespect
to inertial frame ofreferens.
Subject
to condition ~ 1 formulae(1~
take the form ofc
that if of non-relativistic formulae of transformation to uni-accelerated frame.
Moreover, if,
forexample,
the motion of theorigin
of non-inertial frame is
considered,
then it follows from(1)
thatwhich indicates that the
velocity
of thepoint
under consideration does not increaseindefinitely,
butapproaches asymptotically
that oflight.
Some time
later,
Miller[2]
made anattempt
to derive transformationformulae (1) from
theprinciple
ofequivalence,
his concernbeing
thequantitative analysis
of thetwin-paradox.
In the
present
paper the method ofderivation
offormulae
of trans- formation torelativistically
accelerated frames of reference is considered in the case of one-dimentional motion.2. ONE-DIMENTIONAL RELATIVISTIC MOTION OF FRAME OF REFERENCE
Consider
some inertial frame(X, Y, Z, T), designated
as K. Anotherframe
(x,
y, z,t) relativistically
accelerated withrespect
to K will bedesignated
as K’. Assume the initial moment T = 1 =0,
when theorigins
of K andK’, designated
as 0 and 0’coincide,
and let the frame K’move in the direction of Z-axis.
In this case
only
the co-ordinates Z and T areevidently subject
totransformation,
Under transformations
(2)
thespace-time
metricwill
acquire
theform
We may
further
transform the z and t so that two conditions will be f ullfilled :First the factor C
(z, t)
in the term with dz dt ofexpression (4)
willvanish,
and second - the factor A(z, t)
will beequal
tounity.
So,
when the one-dimentional motion of frame K’ isconsidered,
themetric
(4)
may without the loss ofgenerality
beanticipated
in theform :
The condition of
quadratic form (5) transformation
into theshape (3)
may be written as a
requirement,
that allcomponents
of the curvature tensor ofspace-time
vanish :which leads to the
following equation
for metric coefficient *(z, t) :
The solution of
(7)
is the functionwhere a
(t)
isarbitrary
function ofargument t
and has adimentionailty
of
acceleration.
As will be seenbelow,
to everyparticular
choice offunction a
(t) there corresponds
some definitenon-uniform
motion offrame
K’ withrespect
to K.Thus it is seen that the functions under
question
F andG,
which transform(3)
into(5)
mustsatisfy
theequations :
74 TZ. I. GUTZUNAYEV
Let us turn to somewhat more detailed consideration of the motion of the
origin
0’ of the frame K’.Assume,
that at thepoint
0’ theelectric
charge
e ispresent
and is acted uponby
static non-uniform electric fieldE(Z).
Let the Z-axis be directedalong
thefield
vector E
(Z).
In this case it follows from theequations
ofcharge
motion
that
If we introduce the proper time accelerated
particle
asthen the
equation
oftrajectory
of theparticle
foundby integration
ofthe
equation (11)
may be written in aparametric
formwhile the
equations (13)
may be viewed as conditions to which the functions F and G aresubject
at thepoint
z = 0.In this manner, the solution of the
problem posed by
the derivation of formulae oftransformation
to someparticular accelerated
frame ofreference
may becomprised
in thefollowing
sequenceof operations : (a)
Fromequations (11)
find the law of motion of theorigin
0’ of non-inertial frame.
(b)
Write thetrajectory
ofpoint
0’ inparametric
formwhere t is proper time of
point
0’.(c) Express
the acceleration of thepoint
0’ in terms of proper time(d)
Formulate theCauchy-problem
forequations (9)
and(10)
withinitial conditions
(14).
The functions Z
(z, t)
and T(z, t)
thenrepresent
the solution of theproblem
underquestion.
3. UNI-ACCELERATED FRAME OF REFERENCE The above advocated
procedure
forderivation
of the formulae oftransformation
to non-inertial frames may be illustratedby
someparticular examples.
The
integration
of theequation
of motion(11)
in the case of constantuniform field E
(Z)
=Eo
= Const.gives
Z aswhere
{The
motion of the materialpoint following
the law(16)
isusually
referred
to as "hiperbolic
motion".]
Write the law
(16)
in aparametric
formSince a
(t)
= ao =Const.,
theequations (9)
and(10)
areeasily integrated.
The
integral
surfaces of interestembedding correspondently given
curves
(17)
are definedby
that is we have
re-established (with
Vo =~o
=0)
thetransformation formulae
ofKottler-Mvller.
When ao = 0 theformulae (18)
become theLorentz
transformations.
’76 TZ. I. GUTZUNAYEV
4. OSCILLATING FRAME OF REFERENCE
Assume,
that the motion oforigin
0’ follows the lawimposed by equation (11)
with e E(Z)
= - x2Z, where
x2 is somepositive
constant.In this case the
first integral
ofequation (11)
iswhere
Wo
is energy constant. If theparticle
reaches the state of rest at Z =Ao,
thenWo
= mo c2 +T.
With the use of
equations (12)
and(19)
it is not difficult to find thetrajectory
ofpoint
0’ in aparametric
form :Here we denoted
sn
(u, k)
and dn(u, k) being
theelliptic
functions ofJacoby.
In the case considered
so, that after
integration
ofequations (9)
and(10)
with conditions(20)
we shall have
finally
If c ~ oo and
consequently
k ~0, 03C9
~A m0
there follow from equa- tion(21)
i. e. non-relativistic
formulae
of transformation to theoscillating
frameof
reference.
5. APPLICATION
OF NON-INERTIAL FRAME OF REFERENCE
TO CALCULATIONS OF THE ELECTROMAGNETIC FIELD OF ACCELERATED ELECTRON
Consider the
problem
of calculation of theelectromagnetic
field ofpoint
electron inhiperbolic
motionusing
the transformation to uni- accelerated frame of reference.As is well
known,
the Maxwellsequations
are written inarbitrary
curvi-linear co-ordinates as
whily
Lorentz conditions areUsing
uni-accelerated frame ofreference
describedby metric (5)
with @
(z)
=1 + a0 z c2)2
we may search for a solution ofequations (23)
and
(24)
in the form ofwhily A’o
satisfies theequation :
78 ... TZ. I. GUTZUNAYEV
its solution
being
theexpression
The
components
of covariant vectorAi
withrespect
to the inertial frame areequal
toEmploying
the transformation formulae(1)
weget
So that the relation
(27)
withexpressions (28)
insertedyelds
the finalresult :
The solution
(29)
was first obtainedby
Born[9]
andlater,
Schott[10]
who made use of retarded
potentials technique.
The author
acknowledges
hisdeep appreciation
of the valuablesuggestions by
Professor Ja. P.Terletzky.
[1] F. KOTTLER, Ann. Physik, vol. 56, 1918, p. 401.
[2] C. MØLLER, Kgl. Danske Videnskab. Selskab, Mat.-Fys. Medd., vol. 20, n° 19,
1943.
[3] E. L. HILL, Phys. Rev., vol. 84, 1951, p. 1165.
[4] E. P. NEWMAN et A. J. JANIS, Phys. Rev., vol. 116, 1959, p. 1610.
[5] J. E. ROMAIN, Rev. Mod. Phys., vol. 35, 1963, p. 376.
[6] L. M. MARSH, Amer. J. of Phys., vol. 33, 1966, p. 934.
[7] Tz. I. GUTZUNAYEV, Vestnik Mosk. Univ. (Fizika, Astron.), No. 5, 1960.
[8] Tz. I. GUTZUNAYEV, Zh. tehn. Fiz., vol. 31, No. 4, 1961, p. 389.
[9] M. BORN, Ann. Physik, vol. 30, 1909, p. 1.
[10] G. SCHOTT, Electromagnetic Radiation, Cambridge, 1912, p. 63-69.
(Manuserit reçu le 18 avril 1972.)
ANN. INST. POINCAR~~ A-XVII-1 6