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The electromagnetic energy-momentum tensor in a material medium: A crucial thought experiment
Volume 24A, number 13
PHYSICS L E T T E R S
19 June 1967
THE ELECTROMAGNETIC ENERGY-MOMENTUM IN A MATERIAL MEDIUM: A CRUCIAL THOUGHT
TENSOR EXPERIMENT
J. AGUDIN Centre At6rnico Bariloche (C. N. E.A .) R~o Negro, Argentina Received 19May 1967
It is shown that Einstein's mass-energy relation makes Abraham's tensor the only choice for an electromagnetic energy-momentum tensor.
The question of the validity of A b r a h a m ' s or Minkowski's e l e c t r o m a g n e t i c e n e r g y - m o m e n t u m t e n s o r in m a t t e r has b e e n thoroughly d i s c u s s e d by s e v e r a l authors [1]. This letter r e p o r t s the a n a l y s i s of a s i m p l e thought e x p e r i m e n t , which shows that only A b r a h a m ' s s y m m e t r i c t e n s o r gives a g r e e m e n t with E i n s t e i n ' s w e l l - e s t a b l i s h e d m a s s - e n e r g y relation. The e x p e r i m e n t c o n s i s t s in the e m i s s i o n of a photon by an e l e c t r o n i n s i d e m a t t e r (~erenkov effect), and its s u b s e q u e n t a b s o r p t i o n by a d e tector fixed to the m e d i u m . This is a modified v e r s i o n of the e x p e r i m e n t proposed by E i n s t e i n [2] to e s t a b l i s h the m a s s - e n e r g y r e l a t i o n and, as in this e x p e r i m e n t , the c e n t e r - o f - m a s s t h e o r e m is the fundamental a s s u m p t i o n of the p r o o f * . The a r r a n g e m e n t of the thought e x p e r i m e n t is the following (see fig. 1):
-U . . . . . . .
X z. . . . . . . . . . .
At t i m e t = 0 an e l e c t r o n moving with velocity u - (Ux, O, O) e n t e r s a n o n - d i s p e r s i v e d i e l e c t r i c of r e f r a c t i o n index n. To keep the c e n t e r - o f m a s s of the whole s y s t e m fixed, t h e r e is a n other e l e c t r o n in the vacuum moving with v e l o city - u . At t = t 1 the e l e c t r o n in the medium e m i t s a photon of e n e r g y E in the d i r e c t i o n 0(see fig. 1), after what it t r a v e l s with velocity v at some angle ~. At t = t 2 the photon i s a b s o r b e d by the detector fixed to the d i e l e c t r i c in D. We use for the e l e c t r o n the o r d i n a r y r e l a t i v i s t i c equation for m o m e n t a in free space [4]. F o r the photon, we a l t e r n a t i v e l y use r e s u l t s obtained from both t e n s o r s . Both [1,2, 5] give the s a m e e x p r e s s i o n Ap e = -n E / c for the change of m o m e n t u m suffered by the electron. But while A b r a h a m ' s t e n s o r m a k e s a d i s t i n c t i o n between the m o m e n t u m t r a n s f e r r e d to the photon E / n c and the m o m e n t u m t r a n s f e r r e d to the medium ( n 2 - 1 ) E / n c , Minkowski's t e n s o r gives for the photon n E / c and no t r a n s f e r of m o m e n t u m to the medium at all. We look at the e x p e r i m e n t from the point of view of A b r a h a m ' s t e n s o r . To apply the c e n t e r o f - m a s s t h e o r e m , we equate the f i r s t o r d e r m o ment of the m a s s e s of the whole s y s t e m with r e s p e c t to the y - a x i s at t = 0 and at t = t 2, i.e.:
~ ¢ n i x i + m D x D + re(u) x u = ~ m ~ x
i + V x ( t 2 - t 1 )} +
+ (mD+m'){XD+Vx(t2-tl)} + m(u){xu-ut2} + .....
Xz . . . . . . . .
. ......
XE----
~ '
+ m ( v ) { u t 1 + (v.eos~) ( t 2 - t l ) }
F i g . 1. T h e a r r a n g e m e n t o f the thought e x p e r i m e n t . * T h i s t h e o r e m has been a p H i e d i n a d i f f e r e n t w a y t o e l u c i d a t e the v a l i d i t y o f one o r a n o t h e r t e n s o r b y
Balazs [3].
(1)
~X
where mD is the d e t e c t o r ' s m a s s , m' i s the u n known m a s s c o r r e s p o n d i n g to the e n e r g y E of the a b s o r b e d photon, re(u) and re(v) a r e the r e l a t i v i s tic m a s s e s of the e l e c t r o n s and Z m i x i is the m o m e n t of the r e s t of the s y s t e m . Now, 761
Volume 24A. number 13
PHYSICS LETTERS
(~Arni + m D ) V x = M V x is just the m o m e n t u m t r a n s f e r r e d to the m e d i u m during the e m i s s i o n p r o c e s s and t o g e t h e r with re(v) (v. c o s ~ ) can be r e p l a c e d by { rn(u).u - ( E / n c ) c o s 0} using c o n s e r vation of m o m e n t u m along the x - a x i s . F u r t h e r m o r e , it is V x ( t 2 - t 1) < xD(n2-1). E / M c 2 , - and s i n c e E << M c 2 , we can n e g l e c t Vx(t2-11) a g a i n s t x D in eq. (1), and a l s o justify d i s r e g a r d i n g the kinetic e n e r g y of the m e d i u m when c o n s i d e r i n g e n e r g y c o n s e r v a t i o n and n e g l e c t i n g a c o r r e c t i o n due to the d r a g g effect in eq. (1). On account of t h e s e , and i n t ro d u cin g c o n s e r v a t i o n of energy, eq. (1) leads to E i n s t e i n ' s r e l a t i o n , i.e.: rn ' = E / c 2
{ 1 + (n2-1)
x2
}
(3)
Xl+X2 This e x p r e s s i o n shows a r a t h e r involved g e o m e try d e p e n d e n c e which m u s t be r e g a r d e d as unphysi cal . On account of an e x p e r i m e n t by Goos and H~inchen [6], a modified v e r s i o n of our thought e x -
762
p e r i m e n t can be a n a l y s e d to obtain the e n e r g y m o m e n t u m r e l a t i o n of the photon in the total r e f l e c t i o n c a s e . This r e l a t i o n enables us to e l a b o e a t e a quantum t h e o r y of the e x t e r n a l ~ e r e n k o v effect [7], which l ead s to the r a d i a t i o n condition of the c l a s s i c a l t h eo r y . T h e s e r e s u l t s will be published e l s e w h e r e . The author w i s h e s to thank Lic. A r t u r o Ldpez for many valuable d i s c u s s i o n s .
References
(2)
If we r e w r i t e the above explanation using for the m o m e n t u m of the photon n E / c as given by M i n k o w s k i ' s t e n s o r , we a r r i v e at E rn' = ~
19 June 1967
1. G.Marx and G.Gy~rgi, Ann. Phys. 16 (1965) 241. 2. M. Born, Einstein's theory of relativity (Dover Publications, N. York, 1962) p.283. 3. N,L.Balazs, Phys. Rev. 91 {1953) 408, 4. These questions are discussed in J. V. Jelley, ~erenkov radiation and its applications (Pergamon Press, 1958) pp. 27-31. 5. G,GySrgi, Am. J, Phys. 28 (1960) 85; V.L.Ginzburg, Soviet Phys. Uspekhi 2 (1960) 876. 6. F.Goos and H. Httnchen, Ann. Ph. 1 (1947) 333. These authors measured the parallel displacement of the total-reflected ray. 7. M. Danos et al., Phys. Rev. 92 (1953) 828 (exp. work) J.G.Linhart, J. Appl. Phys. 26 (1955) 527 (theor. work).
PHYSICS L E T T E R S
19 June 1967
THE ELECTROMAGNETIC ENERGY-MOMENTUM IN A MATERIAL MEDIUM: A CRUCIAL THOUGHT
TENSOR EXPERIMENT
J. AGUDIN Centre At6rnico Bariloche (C. N. E.A .) R~o Negro, Argentina Received 19May 1967
It is shown that Einstein's mass-energy relation makes Abraham's tensor the only choice for an electromagnetic energy-momentum tensor.
The question of the validity of A b r a h a m ' s or Minkowski's e l e c t r o m a g n e t i c e n e r g y - m o m e n t u m t e n s o r in m a t t e r has b e e n thoroughly d i s c u s s e d by s e v e r a l authors [1]. This letter r e p o r t s the a n a l y s i s of a s i m p l e thought e x p e r i m e n t , which shows that only A b r a h a m ' s s y m m e t r i c t e n s o r gives a g r e e m e n t with E i n s t e i n ' s w e l l - e s t a b l i s h e d m a s s - e n e r g y relation. The e x p e r i m e n t c o n s i s t s in the e m i s s i o n of a photon by an e l e c t r o n i n s i d e m a t t e r (~erenkov effect), and its s u b s e q u e n t a b s o r p t i o n by a d e tector fixed to the m e d i u m . This is a modified v e r s i o n of the e x p e r i m e n t proposed by E i n s t e i n [2] to e s t a b l i s h the m a s s - e n e r g y r e l a t i o n and, as in this e x p e r i m e n t , the c e n t e r - o f - m a s s t h e o r e m is the fundamental a s s u m p t i o n of the p r o o f * . The a r r a n g e m e n t of the thought e x p e r i m e n t is the following (see fig. 1):
-U . . . . . . .
X z. . . . . . . . . . .
At t i m e t = 0 an e l e c t r o n moving with velocity u - (Ux, O, O) e n t e r s a n o n - d i s p e r s i v e d i e l e c t r i c of r e f r a c t i o n index n. To keep the c e n t e r - o f m a s s of the whole s y s t e m fixed, t h e r e is a n other e l e c t r o n in the vacuum moving with v e l o city - u . At t = t 1 the e l e c t r o n in the medium e m i t s a photon of e n e r g y E in the d i r e c t i o n 0(see fig. 1), after what it t r a v e l s with velocity v at some angle ~. At t = t 2 the photon i s a b s o r b e d by the detector fixed to the d i e l e c t r i c in D. We use for the e l e c t r o n the o r d i n a r y r e l a t i v i s t i c equation for m o m e n t a in free space [4]. F o r the photon, we a l t e r n a t i v e l y use r e s u l t s obtained from both t e n s o r s . Both [1,2, 5] give the s a m e e x p r e s s i o n Ap e = -n E / c for the change of m o m e n t u m suffered by the electron. But while A b r a h a m ' s t e n s o r m a k e s a d i s t i n c t i o n between the m o m e n t u m t r a n s f e r r e d to the photon E / n c and the m o m e n t u m t r a n s f e r r e d to the medium ( n 2 - 1 ) E / n c , Minkowski's t e n s o r gives for the photon n E / c and no t r a n s f e r of m o m e n t u m to the medium at all. We look at the e x p e r i m e n t from the point of view of A b r a h a m ' s t e n s o r . To apply the c e n t e r o f - m a s s t h e o r e m , we equate the f i r s t o r d e r m o ment of the m a s s e s of the whole s y s t e m with r e s p e c t to the y - a x i s at t = 0 and at t = t 2, i.e.:
~ ¢ n i x i + m D x D + re(u) x u = ~ m ~ x
i + V x ( t 2 - t 1 )} +
+ (mD+m'){XD+Vx(t2-tl)} + m(u){xu-ut2} + .....
Xz . . . . . . . .
. ......
XE----
~ '
+ m ( v ) { u t 1 + (v.eos~) ( t 2 - t l ) }
F i g . 1. T h e a r r a n g e m e n t o f the thought e x p e r i m e n t . * T h i s t h e o r e m has been a p H i e d i n a d i f f e r e n t w a y t o e l u c i d a t e the v a l i d i t y o f one o r a n o t h e r t e n s o r b y
Balazs [3].
(1)
~X
where mD is the d e t e c t o r ' s m a s s , m' i s the u n known m a s s c o r r e s p o n d i n g to the e n e r g y E of the a b s o r b e d photon, re(u) and re(v) a r e the r e l a t i v i s tic m a s s e s of the e l e c t r o n s and Z m i x i is the m o m e n t of the r e s t of the s y s t e m . Now, 761
Volume 24A. number 13
PHYSICS LETTERS
(~Arni + m D ) V x = M V x is just the m o m e n t u m t r a n s f e r r e d to the m e d i u m during the e m i s s i o n p r o c e s s and t o g e t h e r with re(v) (v. c o s ~ ) can be r e p l a c e d by { rn(u).u - ( E / n c ) c o s 0} using c o n s e r vation of m o m e n t u m along the x - a x i s . F u r t h e r m o r e , it is V x ( t 2 - t 1) < xD(n2-1). E / M c 2 , - and s i n c e E << M c 2 , we can n e g l e c t Vx(t2-11) a g a i n s t x D in eq. (1), and a l s o justify d i s r e g a r d i n g the kinetic e n e r g y of the m e d i u m when c o n s i d e r i n g e n e r g y c o n s e r v a t i o n and n e g l e c t i n g a c o r r e c t i o n due to the d r a g g effect in eq. (1). On account of t h e s e , and i n t ro d u cin g c o n s e r v a t i o n of energy, eq. (1) leads to E i n s t e i n ' s r e l a t i o n , i.e.: rn ' = E / c 2
{ 1 + (n2-1)
x2
}
(3)
Xl+X2 This e x p r e s s i o n shows a r a t h e r involved g e o m e try d e p e n d e n c e which m u s t be r e g a r d e d as unphysi cal . On account of an e x p e r i m e n t by Goos and H~inchen [6], a modified v e r s i o n of our thought e x -
762
p e r i m e n t can be a n a l y s e d to obtain the e n e r g y m o m e n t u m r e l a t i o n of the photon in the total r e f l e c t i o n c a s e . This r e l a t i o n enables us to e l a b o e a t e a quantum t h e o r y of the e x t e r n a l ~ e r e n k o v effect [7], which l ead s to the r a d i a t i o n condition of the c l a s s i c a l t h eo r y . T h e s e r e s u l t s will be published e l s e w h e r e . The author w i s h e s to thank Lic. A r t u r o Ldpez for many valuable d i s c u s s i o n s .
References
(2)
If we r e w r i t e the above explanation using for the m o m e n t u m of the photon n E / c as given by M i n k o w s k i ' s t e n s o r , we a r r i v e at E rn' = ~
19 June 1967
1. G.Marx and G.Gy~rgi, Ann. Phys. 16 (1965) 241. 2. M. Born, Einstein's theory of relativity (Dover Publications, N. York, 1962) p.283. 3. N,L.Balazs, Phys. Rev. 91 {1953) 408, 4. These questions are discussed in J. V. Jelley, ~erenkov radiation and its applications (Pergamon Press, 1958) pp. 27-31. 5. G,GySrgi, Am. J, Phys. 28 (1960) 85; V.L.Ginzburg, Soviet Phys. Uspekhi 2 (1960) 876. 6. F.Goos and H. Httnchen, Ann. Ph. 1 (1947) 333. These authors measured the parallel displacement of the total-reflected ray. 7. M. Danos et al., Phys. Rev. 92 (1953) 828 (exp. work) J.G.Linhart, J. Appl. Phys. 26 (1955) 527 (theor. work).