On the relativistic dynamics of polarized systems

Physica 37 165-181

Vlieger, J 1967

ON THE RELATIVISTIC DYNAMICS POLARIZED SYSTEMS*) by J. Instituut-Lorentz,

OF

VLIEGER Leiden,

Nederland

Synopsis Merller’s relativistic equations of motion for systems with an internal angular momentum in an arbitrary (non-gravitational) external field of force are applied to the special model of electric and magnetic dipole point atoms in an external electromagnetic field of force. The resulting equations of motion are used in order to derive the relativistic atomic energy-momentum tensor for a system, consisting of these dipole atoms. The tensor, found in this way, has exactly the same form as the energymomentum tensor, obtained recently by de Groot and Suttorp for dipole atoms, although these authors use a different definition for the centre of gravity of the atoms as the one used here, and also make several approximations in their calculations. The only difference is, that some of the quantities appearing in their final expression for the atomic energy-momentum tensor, are approximations of the corresponding quantities, found in the present paper. $ 1. Introduction. In a paper on the relativistic dynamics of systems with an internal angular momentum, Mprllerl) has derived the equations of motion for such systems in an arbitrary (non-gravitational) field of force. In connection herewith it was necessary to give a proper definition of the centre of gravity for these systems. In the present paper, after a short review of Moller’s theory (3 2), we

shall apply it to the special model of electric and magnetic dipole point atoms in an external electromagnetic field of force, and so derive the equations of motion for these particles (9 3). From these equations, we obtain in the next section (3 4) the relativistic atomic energy-momentum tensor for a system, consisting of N polarized dipole point atoms. This tensor has recently also been derived by de Groo t and S u t t orps) 3) for these atoms. Since these authors use a different definition for the centre of gravity of the atoms from the one used in the present paper, and also make several approximations in their calculations, (e.g. of the intraatomic fields), one would expect a different result for the energy-momentum *) Force,

This research was supported in part by the Office through its European Office.

-

165 -

of Aerospace

Research,

United

States

Air

166

J. VLIEGER

tensor in both cases. It will appear, however, that the tensor, found by de Groot and Su t t orp2) 3) has exactly the same form as theoneobtained by us, with the only difference, that some of the quantities, appearing in their expression,

are approximations

$ 2. Moller’s

of the corresponding

equations of motion.

quantities

Consider an arbitrary

in our case.

relativistic

(non-

gravitational) system, subjected to a given external field of force, with a density described by the four-vector fa(x), with CL= 0, 1, 2, 3 and x = (ct, x) the time-space co-ordinates. The energy-momentum law of this system can be written in the (covariant) form: +Tafl = fn,

(1)

where Tab(x) is the energy-momentum tensor of our system, which is supposed to be symmetric, and a, = (a/act, a/&). Using this symmetry, we also obtain the angular momentum balance: xflTay) = xeffl -

$,(x”T~~Y -

x6/f”.

(4

We now consider a four-dimensional cylindrical region 0, in which Tap # 0, around the world-line L of a representative point x&r) of the system, where r is the corresponding eigentime. Let 52 be bounded by two (three-dimensional) hyperplanes F’(T) and V(, + dT), perpendicular to L, in X(~)(T) and ~~~~(7+ d7). If one now integrates (1) and (2) over Q, one obtains the following equationsl) : dP” __ dr

= F”,

(3A)

dMa@ ___ = DaB, dr where the four-vectors

P”(T) and F~(T),

$

P”(T) = -

WV

defined by the surface

s

T@(x) u,&-) dV,

integrals:

(41

V(T) and F~(T) =

1+ &

(xy -

x&,(7)) ti,(t)’

(5:

V(T) are the energy-momentum of the system force, acting upon it, whereas the tensors the relations : M@(T)

= - -&

1 {x”Tfl”(x) V(T)

-

(as a whole) and the externa. M&~(T) and D@(T), defined bJ

xCP”(x))

U,,(T) dk’,

(6

ON THE

RELATIVISTIC

DYNAMICS

OF POLARIZED

167

SYSTEMS

and

represent respectively its angular momentum and the moment of the external force, both with respect to the origin of the Lorentz frame. In the above expressions dV is the magnitude of a surface-element of V(T). Furtherof the representative point, more U(T) = dx,,,(T)/d 7 is the four-velocity whereas ‘k(T) = dU(T)/dT is its acceleration. Introducing the internal angular momentum *) : Qap(T)

=

-

3 j [{

Xa -

$)(T))

Tfl”(X) -

{ZC”-

x$.,(T)} Tory(X)] %6?(T)dV

(8)

V(T) of the system with respect to Xc,,(T), (3B), with the help of (3A), as -

dT

one may alternatively

= da0 -

(@Pi3 /

where d‘@ =

P [{X” -

x$,(T)}

fP(X)

-

{xp

-

&T)}

write equation

uopq,

f+)]

J V(T) *

[

1+ &

{zG’-

x’~‘~,(T)}z&(T)

1 (‘0) dl’

is the moment of the external force with respect to Xcrj(T). The centre of gravity X~(T) is now defined by the conditioni): LPug

= 0,

(11)

which means that X~(T) is the centre of mass (or energy) in the momentary rest frame of inertia of the system, i.e. in the Lorentz frame, for which us(T) E dXar(T)/dT = 0. Introducing now the rest mass M*(T) of the system by means of: P”U,

= -

A!*$,

(‘2)

Molleri) obtains with (3A), (9) and (11) the following equations for the system as a whole: $

(M*U”)

d&d + U”nO *) This four-tensor e.g. in reference

aa@ should

4 (pg.

1716).

not be confused

+ 7ia = Fa,

(134

U8@ = d”B, with the three-dimensional

of motion

(W tensor a,

introduced

168

J. VLIEGER

where :

(14) and where the dots mean differentiation with respect to the eigentime These equations can be written more explicitly in the form:

&

(M*Un) +

$ & (f-2~~~~)=

where we have introduced

Fb - ;

&

(d”Wp),

7.

(15-4)

the tensor : 1

A”,g = Bap + c2 UWp,

(16)

and used the antisymmetry of da@; (& p are the elements of the unit-tensor). If we apply (11) to the left-hand side of (15B), this equation may alternatively be written as:

A”eA$&

= Aa,APcd”C,

(17)

using the antisymmetry of ~30. In the next section we shall evaluate the right-hand sides of (15A) and (15*) (or (17)) in the case, that the system is a (charged) point atom, with electric and magnetic dipole moments ,u and v, negligible higher order atomic moments, moving in an external electromagnetic field of force.

5 3. Equations of motion for electric and magrtetic dipole atoms. Let j&(x) be the charge-current density vector of the atom (as a whole) and W@(X) its polarization, then the density of the external electromagnetic force, acting upon this atom can be written as: f”(x) = f”%){j&)/c

+ Q%wl~

(18)

with f”“(x) the tensor of the external electromagnetic field and c the velocity of light. For a point atom with charge e, electric and magnetic dipole moments ,u and v, one has, if one neglects all higher order moments (cf. reference 4)) :

where ,u$(T)

ja(x)/c = e >mUa(~) 6(4)(X(~) -co

x> d7,

(19)

m,s(x)

x} dT,

(20)

is a tensor,

=

;-P&T) --M

depending

8(4)(X(~)

-

on ,u, v and Ua. The explicit

form of

ON THE

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DYNAMICS

OF POLARIZED

169

SYSTEMS

this tensor could be calculated with the help of the equations (83) and (84) of reference 4, but it would be of no importance for our further work. The essential point of the expression (20) for the afiproximate polarization tensor is the fact that it does not contain any derivatives of the S-function, in contrast with the general expression for this tensor, as given in reference 4. If we now substitute (18), with (19) and (20), into the right-hand sides of (5) and (lo), we obtain:

(x” -

1+ ;

xd(7))

&(T) up(+) d@(x(~‘) -x} dvdT’+ 1

V(r)

+ [

-cc

/j”“(x)

[ 1+ ;

(X6-Xd(T))

&(T)]@+‘)

&,s(4)(X(T1) - x> dV d+,

(21)

V(7) --

1s[{@ +m

&a(T)

=

e

xa(T))faY(X)

(x0 - X”(T))f’xy(X)]

-co

V(T)

* 1+

+

-

$

(X” - x'(T)}

t&(T)

ss

[{Xa - xa(T)}fPY(X)

--m

V(r)

* 1+

$

(X” - x'(T)}

Since the hyper-surface

v(T)

&(T)

1

uy(T') dc4){x(T')

- {x0 - x@(T))fay(X)]

1

,Uy"(T') &d(4)(x(T')

is characterized

{%” -

A@(T)}

- X) dl’ dT’ +

us(T)

- X) dV dT’.

by the orthogonality =

(22)

condition:

0,

(23)

we may rewrite (21) and (22) as volume integrals over the total time-space, by introducing the function 6[- (1 /c){xa - X&(T)) U,&)]. We then have:

1

1 + ,';-(X" - xd(T)) ad(T) &(T')8(4){x(T') - x}

FL"(T)= -co

- ;

+ j-

{Xc - x'(T)}

lpB(r) [

1+ ;

1

UC(T) d14)X dT’ $

(x” - x’(T)) ad(T)] ,U~y(T')[8yd(4)(x(T1) -X)]

--oo

- ?- {X8 - x'(T)} C

1

UC(T) d(4)% d+,

(24)

170

J. VLIEGER

L&@(T)= e

ss

[{x” - Xq-)yyX)

- (x0 - X0(T)} p(x)]

--M

* 1+ ; [

{x” -

X6(T)} U&)]

&’ -

-

U,(T’) 8(4)(X(7’)

-

x>

+j j [{x” -

x6(T)) u6(T)]d'4jxdT'

c

xa(~))fby(x)

-

--m

-(d

- xs(T)}fay(X)]

*,L+~(T')

1+ ;

[~&V~){X(T’)-

{X” - xd(T)) n,(T)]

Y,}] 6 - i

(~6 -

X&(T)}U6(7)

1

d(4) x dT’.

(25)

The first integrals at the right-hand sides of (24) and (25) are evaluated by first integrating over the variables x”, and subsequently over T’, using the formula (h(z’) monotonously increasing) : +m P

J

&‘(T’)d{h(T')} dT’ =

[ dh$;di’

-cc with

T

the value of

1+

*6

-

= e

j_

= ;

T’,

for which

& 1%”-

-i {X6 c

*/@{x(T’)}

X6(7)}

x'(T)}

[

h(~‘)

1+

/‘@{X(T)} u,(T),

=

],._,.

(26)

0. In this way we obtain

06(-r)] U~(T')S(~)(X(T') - X)

us(T)

1

d(4)% dT’ =

G {xd(T') -

x'(T)}

0,3(T)] ulj(T')

(27)

ON THE

RELATIVISTIC

(using the property

UaU,

DYNAMICS

OF POLARIZED

= -cs),

and:

-

{Xfi -

XqT)}fqX)]

{X” -

X6(7)} &(,)I

171

SYSTEMS

m += e ss

[{xa -

xqT)}foqX)

-Co * 1+ $

*6

!- {x”

-

-

x'(T)}

Uy(T’) W{X(T’)

-

X}

(28)

1

dc4)% dT’ = 0.

us

C

Formula (27) is the expression for the Lorentz force on the charged point atom, (28) expresses the fact, that the moment of this force with respect to the centre of gravity of the atom is zero. The second integrals of (24) and (25) are calculated by first integrating partially with respect to P, and then applying (26) : co

f

+-

jjao(X)

[

1 +

$

(xd

-

xd(T)}

lfd(T)]

,l#(~‘)

[&~(4)(X(T1)

-

%}I

--w

-6 -

L {Xc -

1

UC(T) d(4)% d+ = -

x'(T)}

C

-

f

i

/J&T)

-

fa’{X(T)) PS”tT)odT)+

-

;

(x6(?')

-

x&(T)) u&(T)

and :

1

dT’,

(29)

+m

ss c-2

[&,f‘@{X(T))] C

[{Xa - x&(T)) fPY(X) - {Cd - x0(T)} fny(X)]

--03

’

1+

-

= f

-${%” Lc {cc” -

xd(T))

x'(T)}

&(T)] &(T')

[i?,dc4){x(T)

1

UC(T) d(4)X dT’ =

[fay{X(T)} /%‘(T) - fPY{X(T)} /%J”(~)] +

-X}]

172

1.

VLIEGER

+m

s

[{X01(7’)

t-f

-

- {Xfl(T’) - X0(7)} f"~{X~~')}l

pJ(X(7’))

XcqT)}

--oo

-

+ {X&(T’)-

F or this

where 6’(y) = dd(y)/dy. formula (cf. (26)) :

1

X&(T)}UE(7) d+,

derivative

one derives

the

following

+oO g(~‘)

d7’ =

B’{~(T’)}

=

g(T’){d2h(T’)/dT’2) - {d&‘)/d+){dh(+)/d+} {dh(+)/d+}a

where T is again the value of h(T’)

we

=

-

T’,

for which

h(~‘)

‘-

{X”(d)

Xa(7))

-

17’=7 ’ (31)

0. If we take:

=

U,(T),

(32)

obtain in particular: +oo

.I 1

gb’) 6’ - ;

{X"(T')

-

U&(T)] dr’ = -

X~(T)}

;

.

[F]

T’S7

-co

With the help of this latter formula, the integrals of (29) and (30) can be calculated, and we find:

at the right-hand

(33)

sides

m +w J

JP%)[l

+ +x~(~)iU&)]

-0v *pp(7’)

[QW

(X(7’)

-

x}]

-

6

;

(XC

-

[

Z.=

-

X&(T)}

G(T).

1 ,

d(4) x d7’ =

71I--

& V(T)

= -

[a,p{x(T,}]

l-C [a,p{x(T)}]

/4?(T) UY(T)- f

#u&T)

- ;

fq=w/w(4

UYb)=

-$[f”“(X(T)) PLW UY(41

(34)

ON THE

RELATIVISTIC

DYNAMICS

OF POLARIZED

SYSTEMS

173

and

ss 00

+-

[{XIX- XqT)} p-y%) - {XP - X0(T)} p(X)]

--m

*

[ ’

-

1+ $

{X’ -

X0(T)} CL+)

{X” - x'(T)}

d - ;

&T’)[+v4)(X(T’)

&(T)] d(4)X dT’ =

fSy{X(T)}py’(T)] - ;

= ;

1

[u+)

[f”Y{X(T))P~d(‘d Ad’(T)-

=

--

+/@{x(d}

%}I

+ [/“‘{x(T))

,+0(T) -

f”Y{X(T))- u’(T) f”‘{X(T))lP~d(T) ud(T)= fsY{X(T)}/J~d(T)

using the definition (16). Substituting now (34) and (35), together right-hand sides of (24) and (25), we get:

Fatd

-

UP(T)

-;

(35)

AJa(T)lt

with

(27) and (28), into

[a~/@(xb,>l

the

PO'(T) -

:3& i?p{x(T)hy(T)UY(T)IJ

(36)

and :

das(T) = ; [faY{X(T))Py'(T)

Ad"(T)

- foY{X(T)),Uy'(T) d@(T)],

(37)

for the external electromagnetic force and its moment, acting on the dipole atom. With (36) and (37), the right-hand side of the equation of motion (15A) becomes :

--

c;&

and the right-hand

{(faypYp

- fPYpya) UP}+ f & (UaU/3fPy,bdUd),

side of (15B)

or (17) :

(38)

174

J.

VLIEGER

where we have used the properties: da/&P = 0,

Ll”&fl

= A”&

We therefore obtain the following equations of motion for an electric magnetic dipole point atom in an external electromagnetic field:

(40) and

(4lA) c AnBA@&~ = AapA”s(f”Ypy’ -

fQ’,+“).

(41B)

From these equations we shall derive in the next section the atomic energy-momentum tensor in a polarized medium, and compare it with the obtained recently by de Groot and Suttorps)s), for this expression, tensor. 3 4. The atomic energy-momentum tensor in a polarized medium. In recent papers, de Groot and Suttorpz) 3) have derived the relativistic energymomentum tensor for a system, consisting of N atoms, carrying both electric and magnetic dipole moments, in the presence of an electromagnetic field. Since these authors use a different definition for the centre of gravity of the atoms as the one used here, it is not evident that both theories lead to the same results. Moreover, several approximations are made in the calculations of de Groot and Suttorp, (e.g. of the intra-atomic fields). We shall see, however, that the energy-momentum tensor, which follows from the equations of motion (41), is formally the same as the one obtained by de Groot and Suttorp2)s). There are only small differences in the definitions of some of the quantities (as e.g. the atomic mass densities) in terms of the microscopic variables (e.g. particle co-ordinates). Within the approximatiorts made by de Groot and Suttorp, however, there is a complete agreement between the results of both theories. In order to compare the theories, we shall first derive the energy-momentum tensor, corresponding to the relativistic equations of motion (41). We consider a system, consisting of N point atoms, numbered by the index K. The rest mass of the kth atom will be denoted by m&, (instead of M*), its charge by eCkj (instead of e), whereas the tensor ,.u$ now becomes p(k)$, and O@ becomes Q$,. For the centre of gravity of the kt” atom we shall now write R&) (instead of Xa), for its velocity U$,, for the tensor A@,3 we get A&,,, and the eigentime of atom K becomes 7(k). If we suppose, that there are no external fields acting from outside the system, the (external) field acting on atom k is the sum of the partial electromagnetic fields f(z)@

ON THE

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DYNAMICS

OF POLARIZED

175

SYSTEMS

due to the other atoms 1 (# k) : f”Vw,) With these new notations,

(42)

= ~~~k~fewd.

the equations {~“;lc%,S)

of motion = e(k) 2

(41) become: f;;@(k))

U(k)8 -

l(ik)

-

z

Z(#k)

@(WY f$(&kd)Pu(k)(Jy -

+-rA

(

c

C4 dqk) Z(#k)

c d~I&d~~,~~~, = &&&,,

f$&kd /W’dUWd~

u~k)u(k,B

[ c (f;@(k)) Rfk)

,‘%)Y’ - fg@(k))

(43A) (43B)

P(k)?‘%

Let us now multiply both members of these equations with the fourdimensional &function C?(4){R(k)(T(k))- R), integrate over T(k), and sum the result over k. We then obtain, after partial integrations with respect to T(k), and with: d dT(k) the following

d(4){R(k)(T(k))

result

-

R)

=

-

U;;Cj(T(k))

acd(4) {R(k) (T(k))

-

(44)

R},

:

-co

+m

=

Is

k,Z,(kfZ)

-

Iz

k,Z, (kfl)

f;;@(k))

e(k:)

U(k)d(*)(&k)

-

R)

dT(k)

-

s

s

{aWh’f;Xlf;(RW,)~ ~(k)BY~(4)(hk)

-

R)

‘hk)

-

176

J. VLIEGER

’ f@(k))

~(k)O”U(k)d(4)(&k)

-

. J

+m d~k,,d~k,,~k,6’4’(R(k)

k

-

R)

dT(k)

=

x k, I, (k#l)

-cc

(45A)

,

d:k,sdfk,d;t;(R(k))

PU(kd-

--oo -

If we get e.g.:

1

t-

J

clz

dqk)

R)

f:$(&k))

change the integration

now

~(kh’$’

dc4)(&k)

variable

-

R)

dvk).

(45J3)

from -r(k) to t(k) = RFk,/c, we

+=J m;k,(T(kd

u:k)(T(kd

Ufk)(T(k))

~@{R(k)(T(k))

-

R)

dT(k)

=

s

--m

+oC P =

I

$k){T(k)(t(k)))

$k)@(k))

ufk,(t(k))

~(3){&k)(t(kd

J --oo

-

R}

6{C(t(k)

-

t)>

+m

ho (t(k)) * -dt(k)

dkc) =

$k,@)

J

@‘&,{T(k)(t(k)))

u$r,(t)

-co

’

=

d&k) =

8(3)‘&W(tW)) - R) %&4 - t)} 3(k) u:k,(t)

+$dT(k))

u:k,(t)

~(4){R(k)(W)

-

(46)

R2> dT(k)>

J

-cc where u;;c,(t(kc))

+w J~:,&hk))

=

and where we have changed finally back to

~(k)+(k))

-Co

u$&(k))

6(4){R(k)(7(k))

-

+W =

G:,&(k))

dc4){R(k)(T(k))

-

R}

hk)

uTk,(t)

8~.

L'S where we have introduced

In an analogous way we get

T(k).

RI

1

dT(k)

t&k)

=

zt(kdt))

“fk,(t),

(48)

the operator: D(k)

=

:

(47)

U;k){T(k)(t(k)))~

(49)

ON THE

In this manner

where jckja(R) whole) :

RELATIVISTIC

DYNAMICS

we can transform

density

vector

J Uw)a(w) ~(4){%(w)

qrc)

SYSTEMS

177

(45A) into:

is the charge-current

j(k)a(R)/c =

OF POLARIZED

of the kth atom

- RI dwc),

(as a

(51)

--03

(cf. eq. (19)), and m(k)2

the polarization

“WWP(R) z j ,~a~h,) -ca

(cf. eq. PO)). In equation

of this atom:

d(4)(R(k) (qzc))- R) dTtICj,

(52)

(50) :.

c j ‘$,,(w) ~(4){Rw)(w)- R} dw = --Do

z

$&)h

- tif&)/C2~(3){R(k)(t)- R) = p;;,(R)

is the rest mass density of atom k (i.e. the mass density rest frame of inertia of this atom), whereas: c jWQ$,(~crc,) ~(4){&&(k)) -cc. = Q$,@)J’

-

(53)

in the momentary

R} dr(k) = -

z&(t)/cs d@~{R~~~(t)-

R} = c@‘(R)

(54)

is a tensor, which in the atomic rest frame, represents the angular momentum density, (the “atomic angular momentum density tensor”). Introducing (53) and (54) in (50), we obtain:

178

J. VI_.IEGER

(55) If we finally

apply the Maxwell-equations: %f(k)n’ aaf(k)8y

= +

j(k)&

+

%f(k)w

to the first term at the right-hand result :

+

%m(k)aPJ a,f(k,M

=

(564 0

WV

side of (55), we can derive the following

c fwf$,P k,Z,(kfZ)

+ -

1

@

-

f

where

x

{f?Z?jm(k)N -

m~&fWyd}

+

uTk)“fk)

k,Z, (k#Z)

,,

k z z+Z)

h$

=

{~E,f(Z)ydm~k)EU.~k)}

f $,

-

m$,,

u:k,“:k,

1 )

-

(57) (58)

and where g@ has the elements go0 = - 1, g”” = 1 (for i = 1, 2, 3) and g@=O (for a#p). Now the right-hand side of (57) is equal to minus the divergence of the tensor: t;; =

2

f;;h:k,,

-

i{f(ZWd$,>

gas +

k,Z,(k#Z) C

(59)

ON THE RELATIVISTIC

DYNAMICS

OF POLARIZED

179

SYSTEMS

which is exactly the atomic energy-momentum tensor of the field, calculated by de Groot and Suttorps)a). The left-hand side of (57) is also the divergence of a tensor, which may be interpreted energy-momentum tensor tT;T :

as the atomic

material

Equation (57) can then be interpreted as the conservation law for energymomentum at the atomic level for a system, consisting of polarized atoms: ap{t;$

+ tg> = 0.

(61)

The tensor (60) differs, however, slightly from the expression for the atomic material energy-momentum tensor, found by de Groot and Sut t orps) 3). This is, of course, not very astonishing, since their definition of the centre of gravity of the atoms differs from the one used in the present paper. Moreover, de Gr o o t and Su t t or p give explicit calculations of the contributions of the intra-atomic fields to the material energy-momentum tensor, and this is only possible in an approximation *), whereas in the present paper these calculations could be avoided, so that all results are exact for the model of dipole point atoms. We shall now show that one can add a divergence-free tensor to t;$, and so obtain a new material energy-momentum tensor t&, which has exactly the same formas the final expression for this tensor, obtained by de Groot and Suttorps)s). The result will be that the total energy-momentum tensor t& + t GT is symmetric. We shall follow the same method below as de Groot and Suttorp, but again we will not have to make any approximations. First we derive from (45B) the atomic angular momentum balance: z O~~,,d~~,SaY{~~~,~SS> =

c

k

k,L(k#O

A”ucd~k,ccm~~,ftl,, - fp&)u 1,

(64

using (44), (52) and (54). Now the right-hand side of this equation is equal to twice the antisymmetric part of the tensor t$, eq. (59). For the left-hand side of (62) we find: x

d~k,,d~k,Sa,{zt~k,O;~i;}

=

-

@;it{

-

t;$)

k

using (60), the antisymmetry

of o;[f, $$N(k)B

which follows from (54), together therefore equal to minus twice *) They use the so-called

+

c a,(u;k,o;;~}> k

and the property = 0,

with (11). The left-hand the antisymmetric part

Darmin-approximation

(63)

for these fields.

(64) side of (62) is of the tensor

180

J. VLIEGER

t;$ - t Ck a~(u;k,a;;~ }. We still have the liberty of adding an arbitrary symmetric tensor to this, for which we take: 3 & ay{uTk,a$’ + u&)a;$$. We then define: (65) as the new atomic material energy-momentum tensor of the system, and this is allowed, since the tensor, added to t;AT in (65) is divergence-free, so that both t;$ and t$,., lead to the same physical results. The left-hand side of (62) is then equal to minus twice the antisymmetric part of the tensor cmj, and the atomic angular momentum balance therefore expresses the tmB fact that the total atomic energy-momentum tensor t:i, + t$ is symmetric. The conservation law (61) for energy-momentum can then be written as: %{t& If we now substitute

+ t(a;> = 0.

(66)

(60) into (65), we get:

and this has exactly the same form as the atomic material energy-momentum tensor, derived by de Groo t and Su t tarps) 3). However, as a consequence of the different definition of the centre of gravity of the atoms and the various approximations made by these authors, they obtain expressions for the rest mass-density (denoted by &,) and the atomic angular momenof the exact expressions tum density (agyP), which are approximations ‘taB used in our paper. (E.g. they obtain the contribution of the Pf,l, and O(k) intra-atomic field to pTLj only within the Coulomb-approximation). But taking into account the limits of the approximations, made by de Groo t and Sutt orp, there appears to be a complete agreement between their results and those obtained in the present paper, and this is, of course, very satisfactory. 3 5. Concluding remarks. We have derived in this paper exact expressions for both material and field part of the atomic energy-momentum tensor for a relativistic system, consisting of electric and magnetic dipole point atoms. The total atomic energy-momentum tensor is symmetric, and the conservation of total energy-momentum therefore implies the conservation of total angular momentum of the system plus the field; (see reference 3). For the explicit evaluation of the quantities, occurring in this angular momentum law, we refer to that paper.

ON THE

Finally energy

RELATIVISTIC

we want

to remark

and momentum

DYNAMICS

that

OF POLARIZED

the macroscopic

can be obtained

181

SYSTEMS

conservation

by averaging

laws of

the corresponding

atomic laws. This can be done in an analogous way as the macroscopic Maxwell-equations were obtained from the corresponding atomic field equations in reference 5. This programme will be carried out by de Gr oo t and Su t t orp,

as they have announced

at the end of reference

3.

TheauthorisindebtedtoProfessorS.R.deGrootandMr.L.G.Suttorp for sending him the manuscript of their papersa) 3) before publication journal Physica. Received

7-8-67

REFERENCES 1) Msller,

C., Ann.

Inst.

II. Poincarb

11 (1949-50)

251.

2)

de Groat,

S. R. and Suttorp,

L. G., Physica

37 (1967)

284.

3)

de Groat,

S. R. and Suttorp,

L. G., Physica

37 (1967)

297.

L. G., Physica

31 (1965)

1713.

4)

de Groot,

S. R. and Suttorp,

5)

de Groot,

S. R. and Vlieger,

J., Physica

31 (1965) 254.

in the