- Email: [email protected]
Maxwell equations in nonuniformly moving media
ANNALS
OF
PHYSICS:
Maxwell
18, 264-273 (1962)
Equations
in Nonuniformly ALLAN
Lawrence
Radiation
Laboratory,
N.
Moving
M e d ia*
KAUFMAN
University
of California,
Livermore,
Calijornia
The covariant Maxwell equations for material media in arbitrary motion are derived by means of a statistical average of the microscopic equations. Special-relativistic considerations are used throughout, even though accelerated motion is treated. The Minkowski form, originally derived for uniform motion only, is found to be generally valid for nonuniform motion, if the macroscopic quantities are suitably defined. In particular, the eovariant polarization is defined in terms of the molecular electromagnetic dipole moments.
The Maxwell equations for moving material media, as derived by Minkowski from special-relativistic considerations, apply only to the case of uniform motion This point is made explicit in several textbooks (1-S). Nevertheless, these equations are frequently applied to problems involving nonuniform motion, and the question arises as to the validity of such application. A partial justification is the work of Dlillenbach (4)) who applied the Lorentz space-time averaging procedure to the microscopic Maxwell equations. However, Dallenbach’s work contains a restrictive assumption and an error, which we shall point out later. The corresponding nonrelativistic problem has been solved by Mazur and Nijboer (5)) who replaced the Lorentz averaging procedure by an average over a statistical ensemble, the latter method being mathematically simpler and more precise. They found that the conventional nonrelativistic equations for nonuniformly moving media (6) must be modified in general. (Their results are included in our relativistic ones below.) In this paper we shall derive the covariant Maxwell equations for nonuniformly moving media, using special relativity throughout. W e shall see that this does not eliminate the case of accelerated motion of the medium, for all observables are referred to inertial frames1 The technique used will be that of Mazur and Nijboer, generalized so as to apply to the covariant problem. In the statistical averaging method, we consider an ensemble of identical * Work performed under the auspices of the U. 8. Atomic Energy Commission. 1 Alternatively, it is possible (although not necessary) to use noninertial frames and general relativity. Such is the approach of Arzelies (7), who considers the macroscopic problem of a medium at rest in an accelerated frame. 264
NONUNIFORMLY
MOVING
265
MEDIA
systems in various states. The choice of states for the ensemble is the matter of statistical mechanics, and will not concern us here. We shall that the ensemble is given. Then we define a macroscopic quantity, F,,“(z) (the macroscopic electromagnetic field B(x, 2)) E(x, 1))) as the over the ensemble of the corresponding microscopic quantity f,,,,(x) :
subject assume such as average
FLAX) = (fLw>(x). Thus the inhomogeneous
Maxwell
equation’
u-w(x)
= (47rIc)$l(s)
yields, upon averaging, hF,“(X)
= (4nlc) W(x).
(1)
Our major concern in this paper will be the development molecular moments. The microscopic current density is
of (j,)(z)
in terms of
j(&(x) = iccz qis dz,i(Ti)84[x - zi(7i>],
(2)
where the sum is over all particles i, qi is the charge of particle i, ri its proper time, and z,,~(T’) its orbit. The integration is from 7i = - 00 to 7i = + m. Let us verify that (2) is correct by evaluating the integral in a given inertial frame. We set fJ4[x - x”(P)]
= S3[x - Zi(Ti)]8[X4 - x4yTi)],
and dzpi( TV) = (d~.,,~/‘dx4~)d.ai. Integrating
over 2ki (since 2di( TV) is monotonic), jJx,
2) = ic?
we find
pi(dxPi/d.r&S3[x - zi(t)],
(3a) = F qi(dz,i/‘dt)63[x where z;(l)
- zi(t)],
= zi[7i(zdi = t)]. Thus we have j(x, t) = F q”~“(t)6~[x - zi(t)], I
(3b) e(x, t) = 7
i as required by the definition 2 We use the Euclidean 1 to 3. [jr] = (j, ice).
metric,
qiS3[x - zi( t)],
of j, E. with 24 = ict. Greek indices
run from 1 to 4, Latin
from
266
KAUFMAN
Following Mazur and Nijboer’s method, we develop j,,(x) by supposing that the particles of the system are grouped into “molecules” (possibly charged and possibly consisting only of a single particle). Denoting the molecules by a, we write (2) as
jp(,> = ic c ‘2 qi 1 dz,i(Ti)8*[X: - Zi(Ti)]. a i
(4)
It is now necessary to specify the center of a molecule. This can be done in an arbitrary way, so long as it leads to a definite space-time orbit y,“( TV). For a monatomic molecule, for example, one could choose the nucleus. (Note, however, that the concept of center of mass is meaningless for a system in an external field (8)). The relative position of a particle in a molecule s,” = z/J 2) - Y,“(T”)
(5)
then is a function of two proper times. We can make it a function by introducing a definite relationship ri( 7a) ; then (5) reads
of only ra,
SNi(TO) 3 zpi[Ti(Ta)] - Y,“(TQ). Such a relationship
(6)
must be covariant, and we choose it to be u,a(Taa)spi(Ta) = 0,
(7)
where u,” = dy,“/dra is the molecular covariant velocity. rest-frame of the molecule (u$‘) = 0)) Eq. (7) reads i (0) 84
so that our condition
=
In the instantaneous
0,
(8)
says that $O)( p> =
y;(O){
T(1))
or t(O)(p) = t(O)(p)* In other words, 7i corresponds to ra at a given time in the instantaneous lar rest-frame. Now we may write Eq. (4) as
The 6” function 6*(s
molecu-
is expanded in powers of si: -
ya -
$1 =
6*(2
-
y”)
-
&$3,~*(x:
-
y")
+
. . . )
(10)
NONUNIFORMLY
MOVING
267
MEDIA
where & means d/ax, . We shall keep only the two terms written in (lo), for the reason given below (18). Expression (9) then consists of four terms: (a) The term in u,%~ is j,‘(x)
= ic C q” dT”U,“(Ta)6*[x - y*(Ta)] s a (11) = ic c
qa
s
dY,“b”)~4~x- Y”(T”~l,
’ t h e t o taa 1 c h arge of molecule a. This term, called the where pa = c: pi 1s “free” current density, is completely analogous in form to j,(x) of Eq. (2). (b) The term in (dsPi/d~‘) 64 is
ic F / dT”(gf) 67x-
yYTQ)],
(12a)
where (13) is the covariant -ic
C a
electric dipole moment. We integrate
s
(12a) by parts to obtain
dTaTa ds4[X- YO(TQ)l P
dra = -icC = -%a,
a s
dTarfla( -~u,“a,>6~[~ - y”(~“)l
c / d Ta7rTlla(Ta)Uy”(Ta)s”[x- y”(7”)l. a
(12b)
(c) The term in u,“s,~&~~ is -ic&
c dTau,‘ny”6”[x - y”(~“)l, a s
(141
and can be combined with (12b) to yield
which is the divergence of an antisymmetric tensor. (d) Finally, the term in (dspi/dT”) svidJ4 is --id,
c /” d Tap~JTa)64[~ - gyT”)J, a
(16)
268
KAUFMAN
where
is the covariant magnetic moment. Let us examine the first neglected term of (9), coming from the third term +>$s,~s~~&~S~(X - y”) in (lo), and from U, a in (9). It contains the quadrupole moment
thus we arc neglecting the effects of For media with gradual macroscopic take the statistical average (9). In but we shall not. We are thus entitled equal t,o zero:
Q% (and analogous higher order moments). density variation, this is justified when we any case, we could include it if we wished, to set the invariant time derivative of ( 18) d( &xi) ___ dra
= 0,
(19)
and write (17) equivalently as (20) which is manifestly antisymmetric. Now we shall combine (15) and ( 16)) to obtain j;(z)
= j/L(,) - jdw
(21a)
= -iiea, c d~“rn~,(~‘)6*[X: - y”(~~)l + . . . , a s
(21b)
where m;,(Ta)
+ ?r”e(Ta)UPa(Ta) - ?rpa(Ta)uva(.“)
(22)
is the covariant elect,romagnetic moment, and is antisymmetric. Let us investigate m”y, in the instantaneous molecular rest-frame. a:01 space components are (since uka.01= UC = 01, by Eq. PO),
The space-
mkl
= &(T=)
a(O)
=
&O’
=
$ I
2
qi
( sI(o,
dd:::
szl co) d@&))
,
(23)
2
which is c times the conventional magnetic components are, using (17) for /.L?,,
moment
tensor. The space-time
NONUNIFORMLY Q(O) m4k
=
MOVING
(since a4i (0) = 0)
NO)
--ic,&
269
MEDIA
a = -ic c q$d’O’, z
(24)
which is -ic times the conventional electric moment vector. We may now compare our result with Dallenbach’s (4). In the first place, he assumes explicitly that drp/dra vanishes. In the second place, he obtains (22) but with the last two terms multiplied by $6. As a result, he must make a compensating error to obtain (24). Let us introduce the conventional three-dimensional correspondence to noncovariant magnetic and electric dipole moments: (25) and note that only in the rest-frame are p$ = CejklplQand rja = pj” (see (22)). Corresponding to the Lorentz transformation of or& to the instantaneous molecular rest-frame : ??LZ”(
we have
Ta)
=
.&(
Ta)
d&x(
Ta) mZ:“‘(
TQ),
va =&(O ++),a $y’ -;)vaxpaC ) O > (26 Pa = p”;,‘ + -(a o’ &y’ + ; VQ )( g(O ) , r (26a)
where y” = [I - (TJ~)~/~~]-~‘~,the symbols (1and I refer to components parallel and transverse to vat and all the quantities may be time-dependent. Returning to our covariant treatment, we introduce the microscopic dipole density m,,(s)
= ic c
a
/ dT0m,“,(7a)84[J: - y”(~“)],
(27)
so that Eq. (21b) becomes jsb(z)
Upon averaging then obtain
(and denoting
=
dyT?Lpy(Z)
microscopic J,b(z)
+
. . .
quantities
(28)
.
by capital letters),
= a,hf~,(l:),
where we now may drop the (. . .) . Equation
we (29)
(1) then reads (30)
270
liAUFMAN
With t’he conventional
introduction
of
HPY(X:) = F&x) we finally obtain the Minkowski
- 2
(31)
equation
&H,,(s)
= +f J,‘(s),
valid for nonuniform motion of the medium. Analogously to Eqs. (25)) let us introduce zation P:
Mj4(2Z)
M,,(x),
s
iCPj(X,
(32)
the magnetization
M and polari-
t).
Then, from Eq. (27)) we have
(34a) (34b) We can express M and P in terms of yaco) and p”“’ by substituting (34) : M(x, t) = c / [+%)I-‘&“‘(t) a\ i
+ e(“:“‘(t> - ; v”(t)
(26b) into
X p”“‘(t) \ i
.63[x - Y”W,, ‘\ (35) P(x, t) = c [r”(t>]-‘p;‘“‘(t) a (i
+ p”:“‘(t)
+ i?(t)
x $-(O)(t)
1
. a3[x - y”(t)],, \ . To evaluate these statistical means, the choice of ensemble must be known. The Mazur-Nijboer results are obtained from (35) by keeping terms of order c-l (recalling that cL(I(‘)is of order c-‘) : M(x, t) = F \/
1
E?“‘“‘(t) - ; v”(t)
P(x, t) = c (p”(“)(t)63[X - y”(l)]). a
X p”“‘(t)
6% - y”(t)], \ , (36)
NONUNIFORMLY
The three-dimensional
MOVING
equivalents
271
MEDIA
of Eqs. (29) and (31) are
Jb(x, t) = cV X M + ~P/‘c%, (37)
Eb(X, t) = - V-P, and H = B - 47rM,
(38)
D E E + 47rP.
We may substitute either Eqs. (34) or (35) into these equations, to eliminate M and P. Still another form for M and P can be obtained, by introducing the concept of the mean velocity V(x, t) of the medium, and its covariant equivalent ( U,(x)) E y(x) (V, ic), with y(z) = [I - V2/cz]-“2. Let us Lorentz-transform U,(z) to a new inertial frame, denoted by X, U,(x)
= J&,(X> V(x),
such that the spacelike components
U:x’(x)
U!y(X)
(39)
vanish at the point X: = 0.
(49)
The new frame will be called the local rest-frame of X. Now we may settle on a suitable definition of V(z). For example, we may choose it to be such that the total momentum density vanishes in the local rest-frame of each point. Let us now transform MPy(z) to the local rest-frame of z: M Icp(x) = d:PC(x)~&)M~~‘(x),
(41)
i.e., of the same point at which MP,, is measured. We substitute
into Eq. (29) :
J,b(X) = &[&7(r) ~“&)Ms’(X)l,
(42)
and note that a, operates on the x-dependence of Z(X) as well as on the two-fold z-dependence of IMC2)(x) . This equation is no longer manifestly covariant, but it is still valid in any inertial frame. The three-dimensional equivalent of Eq. (41) is M(z) P(x)
= M\iz‘)(xCZ)) + r(z) = P;;)(z’~))
+ y(z)
[ [
M:‘(x:‘“‘)
- ; V(x)
x P”‘(.r”‘)]
,
x M’“‘(.?))]
,
(43) P~‘(.z’*‘)
+ ; V(s)
where )I and I refer to components parallel and transverse to V(x). The equivalent of (42) is obtained by substituting (43) into Eq. (37)) and noting the space-time-dependence of all the quantities in (43). The new quantities M’“‘,
272
KAUFMAN
PcZ’can be evaluated from ( 35) or (34)) with v’ measured in the local rest-frame. In the local rest-frame, the Mazur-Kijboer result (36) yields @(x,
t) = F \/
$(‘)(t)
- ;
#I)(t)
x
p”“‘(t)
t[X
-
Y”(t)],
\
,
:
P’“‘(x,t)
(44)
= c (p”‘“‘(1)63[x ,I
while the relations
y”(t)l),
(43) reduce (to the order c-‘) to M
= Mcz)
-
f v x PCZ), (45)
p = p(z). Combining these, we have the Mazur-Nijboer result (36) in another form, which is the one they explicitly present. The two forms are evidently equivalent, since v”(t) = V(Z) + vaiz)(t) to lowest order. Let us define an unmagnetized medium3 as one for which M’“‘(z) = 0 for all Z. Then, to order c-l, Eqs. (43) reduce for such a medium to M = -;
V x p(+, (46)
p = p(z). Upon substitution
into (37)) we obtain J” =
V X (Per’ x V) + aP’“‘/dt, (47)
2 = - V.P’“‘, in agreement with the macroscopic nonrelativistic theory motion of an unmagnetized medium. In the general case of relativistic velocities, one must fall expressions (43)) (35), (34)) (26). In summary, we have kowski formulation is indeed valid for nonuniform motion, is taken in the definition of macroscopic quantities.
(6) for nonuniform back on the general seen that the Minif only suitable care
REFERENCES 1. C. M$LI,ER, “The Theory of Relativity,” p. 200, Clarendon Press, Oxford, 1952. 2. R. C. TOLMAN, “Relativity, Thermodynamics, and Cosmology,” p. 101. Claran;lon Oxford, 1934. 3. W. PAULI, “Theory of Relativity,” p. 101. Pergamon Press, New York, 1958. 3 Mazur and Nijboer, on the other hand, would define an unmagnetized medium for which @(“’ = 0, so that M(r) # 0. Hence their result (45) differs from (46).
Press,
as one
NONUNIFORMLY
MOVING
MEDIA
273
4. W. DXLLENBACH, Ann. Physik 68, 523 (1919). Discussed in W. Pauli, op. cit., p. 104. 5. P. MAZUR AND B. R. A. NIJBOER, Physiea 19,971 (1953). 6. R. BECKER AND F. SAUTER, “Theorie der ElectrizitLt,” 16th ed., Vol. I, Sec. 70. Teuhner, Stuttgart, 1957. 7. H. ARZELI~S, “Milieux Conducteurs ou Polarisables en Mouvement.” Gauthier-Villars, Paris, 1959. 8. C. MILLER, op. cit., p. 190. 9. A. N. KAUFMAN, Aw. J. Phys. 29.626 (1961). RECEIVED:
October 27, 1961
OF
PHYSICS:
Maxwell
18, 264-273 (1962)
Equations
in Nonuniformly ALLAN
Lawrence
Radiation
Laboratory,
N.
Moving
M e d ia*
KAUFMAN
University
of California,
Livermore,
Calijornia
The covariant Maxwell equations for material media in arbitrary motion are derived by means of a statistical average of the microscopic equations. Special-relativistic considerations are used throughout, even though accelerated motion is treated. The Minkowski form, originally derived for uniform motion only, is found to be generally valid for nonuniform motion, if the macroscopic quantities are suitably defined. In particular, the eovariant polarization is defined in terms of the molecular electromagnetic dipole moments.
The Maxwell equations for moving material media, as derived by Minkowski from special-relativistic considerations, apply only to the case of uniform motion This point is made explicit in several textbooks (1-S). Nevertheless, these equations are frequently applied to problems involving nonuniform motion, and the question arises as to the validity of such application. A partial justification is the work of Dlillenbach (4)) who applied the Lorentz space-time averaging procedure to the microscopic Maxwell equations. However, Dallenbach’s work contains a restrictive assumption and an error, which we shall point out later. The corresponding nonrelativistic problem has been solved by Mazur and Nijboer (5)) who replaced the Lorentz averaging procedure by an average over a statistical ensemble, the latter method being mathematically simpler and more precise. They found that the conventional nonrelativistic equations for nonuniformly moving media (6) must be modified in general. (Their results are included in our relativistic ones below.) In this paper we shall derive the covariant Maxwell equations for nonuniformly moving media, using special relativity throughout. W e shall see that this does not eliminate the case of accelerated motion of the medium, for all observables are referred to inertial frames1 The technique used will be that of Mazur and Nijboer, generalized so as to apply to the covariant problem. In the statistical averaging method, we consider an ensemble of identical * Work performed under the auspices of the U. 8. Atomic Energy Commission. 1 Alternatively, it is possible (although not necessary) to use noninertial frames and general relativity. Such is the approach of Arzelies (7), who considers the macroscopic problem of a medium at rest in an accelerated frame. 264
NONUNIFORMLY
MOVING
265
MEDIA
systems in various states. The choice of states for the ensemble is the matter of statistical mechanics, and will not concern us here. We shall that the ensemble is given. Then we define a macroscopic quantity, F,,“(z) (the macroscopic electromagnetic field B(x, 2)) E(x, 1))) as the over the ensemble of the corresponding microscopic quantity f,,,,(x) :
subject assume such as average
FLAX) = (fLw>(x). Thus the inhomogeneous
Maxwell
equation’
u-w(x)
= (47rIc)$l(s)
yields, upon averaging, hF,“(X)
= (4nlc) W(x).
(1)
Our major concern in this paper will be the development molecular moments. The microscopic current density is
of (j,)(z)
in terms of
j(&(x) = iccz qis dz,i(Ti)84[x - zi(7i>],
(2)
where the sum is over all particles i, qi is the charge of particle i, ri its proper time, and z,,~(T’) its orbit. The integration is from 7i = - 00 to 7i = + m. Let us verify that (2) is correct by evaluating the integral in a given inertial frame. We set fJ4[x - x”(P)]
= S3[x - Zi(Ti)]8[X4 - x4yTi)],
and dzpi( TV) = (d~.,,~/‘dx4~)d.ai. Integrating
over 2ki (since 2di( TV) is monotonic), jJx,
2) = ic?
we find
pi(dxPi/d.r&S3[x - zi(t)],
(3a) = F qi(dz,i/‘dt)63[x where z;(l)
- zi(t)],
= zi[7i(zdi = t)]. Thus we have j(x, t) = F q”~“(t)6~[x - zi(t)], I
(3b) e(x, t) = 7
i as required by the definition 2 We use the Euclidean 1 to 3. [jr] = (j, ice).
metric,
qiS3[x - zi( t)],
of j, E. with 24 = ict. Greek indices
run from 1 to 4, Latin
from
266
KAUFMAN
Following Mazur and Nijboer’s method, we develop j,,(x) by supposing that the particles of the system are grouped into “molecules” (possibly charged and possibly consisting only of a single particle). Denoting the molecules by a, we write (2) as
jp(,> = ic c ‘2 qi 1 dz,i(Ti)8*[X: - Zi(Ti)]. a i
(4)
It is now necessary to specify the center of a molecule. This can be done in an arbitrary way, so long as it leads to a definite space-time orbit y,“( TV). For a monatomic molecule, for example, one could choose the nucleus. (Note, however, that the concept of center of mass is meaningless for a system in an external field (8)). The relative position of a particle in a molecule s,” = z/J 2) - Y,“(T”)
(5)
then is a function of two proper times. We can make it a function by introducing a definite relationship ri( 7a) ; then (5) reads
of only ra,
SNi(TO) 3 zpi[Ti(Ta)] - Y,“(TQ). Such a relationship
(6)
must be covariant, and we choose it to be u,a(Taa)spi(Ta) = 0,
(7)
where u,” = dy,“/dra is the molecular covariant velocity. rest-frame of the molecule (u$‘) = 0)) Eq. (7) reads i (0) 84
so that our condition
=
In the instantaneous
0,
(8)
says that $O)( p> =
y;(O){
T(1))
or t(O)(p) = t(O)(p)* In other words, 7i corresponds to ra at a given time in the instantaneous lar rest-frame. Now we may write Eq. (4) as
The 6” function 6*(s
molecu-
is expanded in powers of si: -
ya -
$1 =
6*(2
-
y”)
-
&$3,~*(x:
-
y")
+
. . . )
(10)
NONUNIFORMLY
MOVING
267
MEDIA
where & means d/ax, . We shall keep only the two terms written in (lo), for the reason given below (18). Expression (9) then consists of four terms: (a) The term in u,%~ is j,‘(x)
= ic C q” dT”U,“(Ta)6*[x - y*(Ta)] s a (11) = ic c
qa
s
dY,“b”)~4~x- Y”(T”~l,
’ t h e t o taa 1 c h arge of molecule a. This term, called the where pa = c: pi 1s “free” current density, is completely analogous in form to j,(x) of Eq. (2). (b) The term in (dsPi/d~‘) 64 is
ic F / dT”(gf) 67x-
yYTQ)],
(12a)
where (13) is the covariant -ic
C a
electric dipole moment. We integrate
s
(12a) by parts to obtain
dTaTa ds4[X- YO(TQ)l P
dra = -icC = -%a,
a s
dTarfla( -~u,“a,>6~[~ - y”(~“)l
c / d Ta7rTlla(Ta)Uy”(Ta)s”[x- y”(7”)l. a
(12b)
(c) The term in u,“s,~&~~ is -ic&
c dTau,‘ny”6”[x - y”(~“)l, a s
(141
and can be combined with (12b) to yield
which is the divergence of an antisymmetric tensor. (d) Finally, the term in (dspi/dT”) svidJ4 is --id,
c /” d Tap~JTa)64[~ - gyT”)J, a
(16)
268
KAUFMAN
where
is the covariant magnetic moment. Let us examine the first neglected term of (9), coming from the third term +>$s,~s~~&~S~(X - y”) in (lo), and from U, a in (9). It contains the quadrupole moment
thus we arc neglecting the effects of For media with gradual macroscopic take the statistical average (9). In but we shall not. We are thus entitled equal t,o zero:
Q% (and analogous higher order moments). density variation, this is justified when we any case, we could include it if we wished, to set the invariant time derivative of ( 18) d( &xi) ___ dra
= 0,
(19)
and write (17) equivalently as (20) which is manifestly antisymmetric. Now we shall combine (15) and ( 16)) to obtain j;(z)
= j/L(,) - jdw
(21a)
= -iiea, c d~“rn~,(~‘)6*[X: - y”(~~)l + . . . , a s
(21b)
where m;,(Ta)
+ ?r”e(Ta)UPa(Ta) - ?rpa(Ta)uva(.“)
(22)
is the covariant elect,romagnetic moment, and is antisymmetric. Let us investigate m”y, in the instantaneous molecular rest-frame. a:01 space components are (since uka.01= UC = 01, by Eq. PO),
The space-
mkl
= &(T=)
a(O)
=
&O’
=
$ I
2
qi
( sI(o,
dd:::
szl co) d@&))
,
(23)
2
which is c times the conventional magnetic components are, using (17) for /.L?,,
moment
tensor. The space-time
NONUNIFORMLY Q(O) m4k
=
MOVING
(since a4i (0) = 0)
NO)
--ic,&
269
MEDIA
a = -ic c q$d’O’, z
(24)
which is -ic times the conventional electric moment vector. We may now compare our result with Dallenbach’s (4). In the first place, he assumes explicitly that drp/dra vanishes. In the second place, he obtains (22) but with the last two terms multiplied by $6. As a result, he must make a compensating error to obtain (24). Let us introduce the conventional three-dimensional correspondence to noncovariant magnetic and electric dipole moments: (25) and note that only in the rest-frame are p$ = CejklplQand rja = pj” (see (22)). Corresponding to the Lorentz transformation of or& to the instantaneous molecular rest-frame : ??LZ”(
we have
Ta)
=
.&(
Ta)
d&x(
Ta) mZ:“‘(
TQ),
va =&(O ++),a $y’ -;)vaxpaC ) O > (26 Pa = p”;,‘ + -(a o’ &y’ + ; VQ )( g(O ) , r (26a)
where y” = [I - (TJ~)~/~~]-~‘~,the symbols (1and I refer to components parallel and transverse to vat and all the quantities may be time-dependent. Returning to our covariant treatment, we introduce the microscopic dipole density m,,(s)
= ic c
a
/ dT0m,“,(7a)84[J: - y”(~“)],
(27)
so that Eq. (21b) becomes jsb(z)
Upon averaging then obtain
(and denoting
=
dyT?Lpy(Z)
microscopic J,b(z)
+
. . .
quantities
(28)
.
by capital letters),
= a,hf~,(l:),
where we now may drop the (. . .) . Equation
we (29)
(1) then reads (30)
270
liAUFMAN
With t’he conventional
introduction
of
HPY(X:) = F&x) we finally obtain the Minkowski
- 2
(31)
equation
&H,,(s)
= +f J,‘(s),
valid for nonuniform motion of the medium. Analogously to Eqs. (25)) let us introduce zation P:
Mj4(2Z)
M,,(x),
s
iCPj(X,
(32)
the magnetization
M and polari-
t).
Then, from Eq. (27)) we have
(34a) (34b) We can express M and P in terms of yaco) and p”“’ by substituting (34) : M(x, t) = c / [+%)I-‘&“‘(t) a\ i
+ e(“:“‘(t> - ; v”(t)
(26b) into
X p”“‘(t) \ i
.63[x - Y”W,, ‘\ (35) P(x, t) = c [r”(t>]-‘p;‘“‘(t) a (i
+ p”:“‘(t)
+ i?(t)
x $-(O)(t)
1
. a3[x - y”(t)],, \ . To evaluate these statistical means, the choice of ensemble must be known. The Mazur-Nijboer results are obtained from (35) by keeping terms of order c-l (recalling that cL(I(‘)is of order c-‘) : M(x, t) = F \/
1
E?“‘“‘(t) - ; v”(t)
P(x, t) = c (p”(“)(t)63[X - y”(l)]). a
X p”“‘(t)
6% - y”(t)], \ , (36)
NONUNIFORMLY
The three-dimensional
MOVING
equivalents
271
MEDIA
of Eqs. (29) and (31) are
Jb(x, t) = cV X M + ~P/‘c%, (37)
Eb(X, t) = - V-P, and H = B - 47rM,
(38)
D E E + 47rP.
We may substitute either Eqs. (34) or (35) into these equations, to eliminate M and P. Still another form for M and P can be obtained, by introducing the concept of the mean velocity V(x, t) of the medium, and its covariant equivalent ( U,(x)) E y(x) (V, ic), with y(z) = [I - V2/cz]-“2. Let us Lorentz-transform U,(z) to a new inertial frame, denoted by X, U,(x)
= J&,(X> V(x),
such that the spacelike components
U:x’(x)
U!y(X)
(39)
vanish at the point X: = 0.
(49)
The new frame will be called the local rest-frame of X. Now we may settle on a suitable definition of V(z). For example, we may choose it to be such that the total momentum density vanishes in the local rest-frame of each point. Let us now transform MPy(z) to the local rest-frame of z: M Icp(x) = d:PC(x)~&)M~~‘(x),
(41)
i.e., of the same point at which MP,, is measured. We substitute
into Eq. (29) :
J,b(X) = &[&7(r) ~“&)Ms’(X)l,
(42)
and note that a, operates on the x-dependence of Z(X) as well as on the two-fold z-dependence of IMC2)(x) . This equation is no longer manifestly covariant, but it is still valid in any inertial frame. The three-dimensional equivalent of Eq. (41) is M(z) P(x)
= M\iz‘)(xCZ)) + r(z) = P;;)(z’~))
+ y(z)
[ [
M:‘(x:‘“‘)
- ; V(x)
x P”‘(.r”‘)]
,
x M’“‘(.?))]
,
(43) P~‘(.z’*‘)
+ ; V(s)
where )I and I refer to components parallel and transverse to V(x). The equivalent of (42) is obtained by substituting (43) into Eq. (37)) and noting the space-time-dependence of all the quantities in (43). The new quantities M’“‘,
272
KAUFMAN
PcZ’can be evaluated from ( 35) or (34)) with v’ measured in the local rest-frame. In the local rest-frame, the Mazur-Kijboer result (36) yields @(x,
t) = F \/
$(‘)(t)
- ;
#I)(t)
x
p”“‘(t)
t[X
-
Y”(t)],
\
,
:
P’“‘(x,t)
(44)
= c (p”‘“‘(1)63[x ,I
while the relations
y”(t)l),
(43) reduce (to the order c-‘) to M
= Mcz)
-
f v x PCZ), (45)
p = p(z). Combining these, we have the Mazur-Nijboer result (36) in another form, which is the one they explicitly present. The two forms are evidently equivalent, since v”(t) = V(Z) + vaiz)(t) to lowest order. Let us define an unmagnetized medium3 as one for which M’“‘(z) = 0 for all Z. Then, to order c-l, Eqs. (43) reduce for such a medium to M = -;
V x p(+, (46)
p = p(z). Upon substitution
into (37)) we obtain J” =
V X (Per’ x V) + aP’“‘/dt, (47)
2 = - V.P’“‘, in agreement with the macroscopic nonrelativistic theory motion of an unmagnetized medium. In the general case of relativistic velocities, one must fall expressions (43)) (35), (34)) (26). In summary, we have kowski formulation is indeed valid for nonuniform motion, is taken in the definition of macroscopic quantities.
(6) for nonuniform back on the general seen that the Minif only suitable care
REFERENCES 1. C. M$LI,ER, “The Theory of Relativity,” p. 200, Clarendon Press, Oxford, 1952. 2. R. C. TOLMAN, “Relativity, Thermodynamics, and Cosmology,” p. 101. Claran;lon Oxford, 1934. 3. W. PAULI, “Theory of Relativity,” p. 101. Pergamon Press, New York, 1958. 3 Mazur and Nijboer, on the other hand, would define an unmagnetized medium for which @(“’ = 0, so that M(r) # 0. Hence their result (45) differs from (46).
Press,
as one
NONUNIFORMLY
MOVING
MEDIA
273
4. W. DXLLENBACH, Ann. Physik 68, 523 (1919). Discussed in W. Pauli, op. cit., p. 104. 5. P. MAZUR AND B. R. A. NIJBOER, Physiea 19,971 (1953). 6. R. BECKER AND F. SAUTER, “Theorie der ElectrizitLt,” 16th ed., Vol. I, Sec. 70. Teuhner, Stuttgart, 1957. 7. H. ARZELI~S, “Milieux Conducteurs ou Polarisables en Mouvement.” Gauthier-Villars, Paris, 1959. 8. C. MILLER, op. cit., p. 190. 9. A. N. KAUFMAN, Aw. J. Phys. 29.626 (1961). RECEIVED:
October 27, 1961