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On the uniqueness of the relativistic energy-momentum tensor in polarized media
Volume25A. number 2
ON THE
PHYSICS LETTERS
31 July 1967
UNIQUENESS OF THE RELATIVISTIC ENERGY-MOMENTUM TENSOR IN POLARIZED MEDIA S. R. de GROOT and L. G. SUTTORP
Institute of Theoretical Physics. University of A msterdam, Amsterdam Received 17 June 1967 The energy-momentum tensor in a polarized medium, as derived from microscopic theory, can be looked upon as the sum of a material part. defined at zero field, and a field part. It can then be s h o ~ that both Minkowski's and Abraham's tensora are unjustifiable.
The m a c r o s c o p i c relativistic e n e r g y - m o m e n t u m conservation laws in dipole substances have been derived [1] f r o m the c o r r e s p o n d i n g m i c r o s c o p i c laws. In this way a statistical expression for the icons e r v e d and s y m m e t r i c a l ) e n e r g y - m o m e n t u m tensor TC~fl in t e r m s of atomic p a r a m e t e r s has been found. In this expression a part appears, which reads in the rest f r a m e of the medium (i,j = 1,2,3; g/i = 1; gij= = 0, i ~ j):
\(ExH) i
-EiDJ-tliBJ+
(1)
(½E2 + ½B2 _ B . M ) g i j
it depends on the Maxwell fields only. ( T(0?) and c T (0i f ) a r e the energy density and flow; c - l T i0 (f) and T "~) a/3 _ a~ a r e the momentum density and flow.) The remaining part T (m) = Taft T i f ) ' which contains bulk material, velocity fluctuation and interatomic electromagnetic interaction correlation t e r m s , is d i r e c t ly related to the statistical t h e r m o d y n a m i c a l expressions. For the electric dipole case the second line of (1) was p r o p o s e ~ already by Lorentz [2] and Einstein-Laub [3]. If instead o~T/mP~ one defines a s i m i l a r tensor T~m~,, for the s y s t e m with same t e m p e r a t u r e T and d2~.ttloPns but: ? t / a / : q f i i l l i d : ~ u a C:th:sp:ndXfrngafi:l:~dten~::SrTl~aftfl)visT:flvelTc~y,~;:ctbtt?::::; # ? d e : e t ? relations P = KEand ,44 = x B it reads in the rest f r a m e 00
,
T (f,) = -~E.D + ½B.tl + -~E2T OK/~T + ½B 2 TOx/~T, Oi
iO
T(f,) = r(f,) = (ExH) i,
(2)
(3)
ij = - E i D J - I - I i B J (~E.D + ½ B , I t - -~E2 ag/ap - ~B2ax/ap)g ij T(f,) + + _ ~ ( p i p j +A4iMJ) + ~ (p2+/|42)gij"
(4)
F o r induced dipole media the last two t e r m s of eq. (2) a r e negligible; for permanent dipole media, which obey the Langevin-Debye laws (with K(K + 3) -1 and ×(3- 2×) -1 inversely proportional to T), the field energy density (2) b e c o m e s T oo ( I , ) = ½(E 2 +h,2) _ ~ ( p 2 + M 2 )
(5)
103
Volume25A. number 2
PHYSICS LETTERS
31 July 1967
Using the C l a u s i u s - M o s o t t i f o r m u l a for K(p) and X(P) one gets for the m o m e n t u m flow
T~f,) = -EiDJ - HiBJ + (½E2
+ -~/./2 _ _~p2
_ }M2) gij _ ts_(pipj+MiMj ) + ~ (1)2 +M2) giJ.
(6)
Consequently for induced dipole media with K << 1 and X << 1 the t e n s o r T~f~) r e d u c e s to
a~ ~ E ' D + ½B'H T (f,) =\(E x/'l) i
(E×IT) i 1 _EiD j _HiBJ + (½E2 + ½1.12)gij] ,
(7)
where the f i r s t line o c c u r s also in the t e n s o r s proposed by Minkowski [4] and A b r a h a m [5]; the m o m e n tum density c-lD ×B of Minkowski i s not found, it r e m a i n s c - l E ×/'/, as in Ab.ra.ham's t e n s o r ; the m o m e n t u m flow (field p r e s s u r e ) is different from both M i n k o w s k i ' s e x p r e s s i o n -E~D3 - HzBJ+ + ( 1~ E . D + ½ B ./-/)g~J and A b r a h a m ' s e x p r e s s i o n which is the s y m m e t r i c a l p a r t thereof. In the past the d i s c u s s i o n focused m a i n l y on the r e l a t i v e m e r i t s of M i n k o w s k i ' s and A b r a h a m ' s t e n s o r s , while L o r e n t z ' s and E i n s t e i n - L a u b ' s e x p r e s s i o n s were hardly taken into account, even in P a u l i ' s review [6]. The widely adopted a r g u m e n t of yon Laue [7] in favour of Minkowski was invalidated by Tang and M e i x n e r [8]. Thus it has been shown h e r e that s t a r t i n g f r o m the field t e n s o r (1), b a s e d on m i c r o s c o p i c theory, one can define - u n d e r two l i m i t i n g conditions - a field t e n s o r (2)-(4) which c o r r e s p o n d s to a m a t e r i a l t e n s o r at zero fields. In a m o r e s p e c i a l case it takes the form (7), which shows some s i m i l a r i t y with the t e n s o r s of A b r a h a m and Minkowski. However even in this case d i f f e r e n c e s between the r e s u l t s obtained from m i c r o s c o p i c theory and those of Minkowski and A b r a h a m r e m a i n , such that the l a t t e r cannot be justified. This i n v e s t i g a t i o n is p a r t of the r e s e a r c h p r o g r a m m e of the ~Stichting voor F u n d a m e n t e e l Onderzoek der M a t e r i e (F.O.M.)", which is financially supported by the " O r g a n i s a t i e voor Z u i v e r Wetenschappelijk Onderzoek (Z.W.O.)". A detailed account will a p p e a r in the j o u r n a l " P h y s i c a " .
References 1. 2. 3. 4. 5. 6. 7. 8.
S.R.de Groot and L.G.Suttorp. Phys. Letters 24A (1967) 385. H.A.Lorentz, Eric. math. Wiss. V2, fasc. 1. Leipzig (1904). A.Einstein and J.Laub, Ann. Physik (4}26 {1908} 541. H.Minkowski. Nachr. Ges. Wiss. G6ttingen {1908} 63. M.Abraham. R.C. Circ. mat. Palermo 28 (1909) 1. W. Pauli, Ene. math. Wiss. V2, fasc. 4; Relativit/~tstheorie, Leipzig (1921}; Theory of Relativity, London {1958}. M.von Laue, Z. Physik 128 (1950) 387. C.L.Tang and J.Meixner, Phys. Fluids 4 (1961} 148.
104
ON THE
PHYSICS LETTERS
31 July 1967
UNIQUENESS OF THE RELATIVISTIC ENERGY-MOMENTUM TENSOR IN POLARIZED MEDIA S. R. de GROOT and L. G. SUTTORP
Institute of Theoretical Physics. University of A msterdam, Amsterdam Received 17 June 1967 The energy-momentum tensor in a polarized medium, as derived from microscopic theory, can be looked upon as the sum of a material part. defined at zero field, and a field part. It can then be s h o ~ that both Minkowski's and Abraham's tensora are unjustifiable.
The m a c r o s c o p i c relativistic e n e r g y - m o m e n t u m conservation laws in dipole substances have been derived [1] f r o m the c o r r e s p o n d i n g m i c r o s c o p i c laws. In this way a statistical expression for the icons e r v e d and s y m m e t r i c a l ) e n e r g y - m o m e n t u m tensor TC~fl in t e r m s of atomic p a r a m e t e r s has been found. In this expression a part appears, which reads in the rest f r a m e of the medium (i,j = 1,2,3; g/i = 1; gij= = 0, i ~ j):
\(ExH) i
-EiDJ-tliBJ+
(1)
(½E2 + ½B2 _ B . M ) g i j
it depends on the Maxwell fields only. ( T(0?) and c T (0i f ) a r e the energy density and flow; c - l T i0 (f) and T "~) a/3 _ a~ a r e the momentum density and flow.) The remaining part T (m) = Taft T i f ) ' which contains bulk material, velocity fluctuation and interatomic electromagnetic interaction correlation t e r m s , is d i r e c t ly related to the statistical t h e r m o d y n a m i c a l expressions. For the electric dipole case the second line of (1) was p r o p o s e ~ already by Lorentz [2] and Einstein-Laub [3]. If instead o~T/mP~ one defines a s i m i l a r tensor T~m~,, for the s y s t e m with same t e m p e r a t u r e T and d2~.ttloPns but: ? t / a / : q f i i l l i d : ~ u a C:th:sp:ndXfrngafi:l:~dten~::SrTl~aftfl)visT:flvelTc~y,~;:ctbtt?::::; # ? d e : e t ? relations P = KEand ,44 = x B it reads in the rest f r a m e 00
,
T (f,) = -~E.D + ½B.tl + -~E2T OK/~T + ½B 2 TOx/~T, Oi
iO
T(f,) = r(f,) = (ExH) i,
(2)
(3)
ij = - E i D J - I - I i B J (~E.D + ½ B , I t - -~E2 ag/ap - ~B2ax/ap)g ij T(f,) + + _ ~ ( p i p j +A4iMJ) + ~ (p2+/|42)gij"
(4)
F o r induced dipole media the last two t e r m s of eq. (2) a r e negligible; for permanent dipole media, which obey the Langevin-Debye laws (with K(K + 3) -1 and ×(3- 2×) -1 inversely proportional to T), the field energy density (2) b e c o m e s T oo ( I , ) = ½(E 2 +h,2) _ ~ ( p 2 + M 2 )
(5)
103
Volume25A. number 2
PHYSICS LETTERS
31 July 1967
Using the C l a u s i u s - M o s o t t i f o r m u l a for K(p) and X(P) one gets for the m o m e n t u m flow
T~f,) = -EiDJ - HiBJ + (½E2
+ -~/./2 _ _~p2
_ }M2) gij _ ts_(pipj+MiMj ) + ~ (1)2 +M2) giJ.
(6)
Consequently for induced dipole media with K << 1 and X << 1 the t e n s o r T~f~) r e d u c e s to
a~ ~ E ' D + ½B'H T (f,) =\(E x/'l) i
(E×IT) i 1 _EiD j _HiBJ + (½E2 + ½1.12)gij] ,
(7)
where the f i r s t line o c c u r s also in the t e n s o r s proposed by Minkowski [4] and A b r a h a m [5]; the m o m e n tum density c-lD ×B of Minkowski i s not found, it r e m a i n s c - l E ×/'/, as in Ab.ra.ham's t e n s o r ; the m o m e n t u m flow (field p r e s s u r e ) is different from both M i n k o w s k i ' s e x p r e s s i o n -E~D3 - HzBJ+ + ( 1~ E . D + ½ B ./-/)g~J and A b r a h a m ' s e x p r e s s i o n which is the s y m m e t r i c a l p a r t thereof. In the past the d i s c u s s i o n focused m a i n l y on the r e l a t i v e m e r i t s of M i n k o w s k i ' s and A b r a h a m ' s t e n s o r s , while L o r e n t z ' s and E i n s t e i n - L a u b ' s e x p r e s s i o n s were hardly taken into account, even in P a u l i ' s review [6]. The widely adopted a r g u m e n t of yon Laue [7] in favour of Minkowski was invalidated by Tang and M e i x n e r [8]. Thus it has been shown h e r e that s t a r t i n g f r o m the field t e n s o r (1), b a s e d on m i c r o s c o p i c theory, one can define - u n d e r two l i m i t i n g conditions - a field t e n s o r (2)-(4) which c o r r e s p o n d s to a m a t e r i a l t e n s o r at zero fields. In a m o r e s p e c i a l case it takes the form (7), which shows some s i m i l a r i t y with the t e n s o r s of A b r a h a m and Minkowski. However even in this case d i f f e r e n c e s between the r e s u l t s obtained from m i c r o s c o p i c theory and those of Minkowski and A b r a h a m r e m a i n , such that the l a t t e r cannot be justified. This i n v e s t i g a t i o n is p a r t of the r e s e a r c h p r o g r a m m e of the ~Stichting voor F u n d a m e n t e e l Onderzoek der M a t e r i e (F.O.M.)", which is financially supported by the " O r g a n i s a t i e voor Z u i v e r Wetenschappelijk Onderzoek (Z.W.O.)". A detailed account will a p p e a r in the j o u r n a l " P h y s i c a " .
References 1. 2. 3. 4. 5. 6. 7. 8.
S.R.de Groot and L.G.Suttorp. Phys. Letters 24A (1967) 385. H.A.Lorentz, Eric. math. Wiss. V2, fasc. 1. Leipzig (1904). A.Einstein and J.Laub, Ann. Physik (4}26 {1908} 541. H.Minkowski. Nachr. Ges. Wiss. G6ttingen {1908} 63. M.Abraham. R.C. Circ. mat. Palermo 28 (1909) 1. W. Pauli, Ene. math. Wiss. V2, fasc. 4; Relativit/~tstheorie, Leipzig (1921}; Theory of Relativity, London {1958}. M.von Laue, Z. Physik 128 (1950) 387. C.L.Tang and J.Meixner, Phys. Fluids 4 (1961} 148.
104