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Comment on “the behavior of maxwell's equations under real superluminal lorentz transformations”
Volume 113B, number 2
PHYSICS LETTERS
10 June 1982
COMMENT ON "THE BEHAVIOR OF MAXWELL'S EQUATIONS UNDER REAL SUPERLUMINAL LORENTZ TRANSFORMATIONS" L. MARCHILDON, A.F. ANTIPPA De'partement de Physique, Universit~ du Quebec ~ Trois-Rivi~res, Trois-Rivi~res~ Quebec, G9A 5H7, Canada and
A.E. EVERETT Department of Physics, Tufts University, Medford, MA 02155, USA Received 2 March 1982
Negi et al. have recently obtained field equations for the superluminal electromagnetic field, in theories based on real supeduminal transformations along a "tachyon corridor". Their results differ from equations obtained some time ago by the present authors. We trace the source of the discrepancy to the failure of Negi et al. to consistently transform all relevant quantities from the old to the new Lorentz frame.
In this note we intend to comment on a recent paper by Negi et al. bearing the same title [1]. These authors investigated the transformation properties of Maxwell's equations and the Lorentz force under a given type of supeduminal transformations. Results on this problem were obtained some time ago by the present authors [2]. Since some of the results in ref. [1] disagree with those obtained by us, and since somewhat subtle questions of interpretation arise in this context, we wish to examine the paper by Negi et al. in the light of our earlier resul{s. We will refer to equations from ref. [1] as eq. (I-1), etc. Refs. [1,2] both begin with the following superluminal transformations, assumed to hold between the coordinates of a frame K and a frame K' [3] : t = 7 p ( t ' + oz'),
x = x',
z = 7t~(ot' + z'),
y = y',
7 = ( °2 -- i) -1/2,
P = o/Iv[,
Iv[ ~> 1.
(1)
These transformations are interpreted as holding between a class o f preferred inertial frames, moving with respect to each other along a preferred direction called the tachyon Corridor. Other inertial frames are re180
lated to the preferred ones through usual (subluminal) Lorentz transformations. We note that eqs. (1) are the same as eqs. (1) in ref. [1]. Assume there is in K' a world of bradyons. In K they will be seen as tachyons. Two very different questions can now be asked: (i) what is the behavior of these tachyons and their fields, interacting among themselves, as seen in K? (ii) If there are bradyons in K, what are their interactions with the tachyons? The first question is rather easy to answer, but the second is not. The second question, however, is the one that is really relevant to tachyon detection. We know how to describe the behavior of bradyons in K'. As the coordinate transformations between K' and K are given, it is straightforward to infer from these the behavior of tachyons in K. For electromagnetic interactions this is accomplished as follows: in K', the electromagnetic field is characterized by a second-rank antisymmetric tensor F~x , related to the potential A'K through F '~ . = ~'~A'~ - ~'x A'~"
(2)
t
In terms of FKx., its dual (G')Uu = ~-g~"~uV~XF' ~~ 2~ ca, and the current four-vector (j')K, Maxwell's equations read as
0 031-9163/82]0000-0000/$02.75
© 1982 North-Holland
Volume 113B, number 2 =
(a') ",
PHYSICS LETTERS = 0.
(3)
The Lorentz force acting on a particle of charge e, mass m, proper time ~-and momentum four-vector (p')~ can be obtained from the following equation: (f,)K = d(p,)K/dr = (e/m)(F')KXp' x.
(4)
In the frame K, one can define quantities F Kx, G ~x, f~, etc., which are related to (/7') Kx, (G')~x, (f~)K by tensor transformations. That is, each contravariant index transforms like (x') K, i.e. with the matrix that brings (t',x',y',z') to (t, x,y, z) according to eqs. (1). These quantities then form representation spaces for the group of all subluminal and superluminal transformations along the tachyon corridor. One can also define covariant quantities obtained from the contravariant ones with the help of the metric tensor. From well-known results of tensor analysis, the same tensor equations then hold in K as hold in K'. They describe the behavior of the particles and fields seen in K, that is, of tachyons and the electromagnetic field. Notice that the metric g'Kx gets changed in going over to K, that is, g'~x CgKx- This is a consequence of the transformation law ofg'Kx, and is related to the fact ¢ g ! that under eqs. (1), the world interval ( x ) x~ is transformed in the following way: ,' , ')K (X) ~'Tt = --(t') 2 + (X') 2 + (.V')2 + (Z') 2 gKh(X
= t 2 + x 2 + y 2 _ Z2 =gKxxKx x.
(5)
Because of the non-invariance of the world interval under superluminal transformations, the velocity of light is not invariant. That is, particles (or waves) propagating with a speed e in K' will not in general propagate with the same speed in K. This in turn implies that Maxwell's equations will not hold in K for superluminal fields. This can be seen explicitly as follows: define in K an electric field E and a magnetic field B so that they enter the tensor G Kx in the same was as E' and B' enter (G')Kx. The transformation laws of (G') ~x into G Kx then imply that
E z = E'z,
#
Bz = - B z '
-%), Ey=Tlx(Ei-VB'x),
By =7/X(By +rE'x).
Eqs. (6) are the field transformation equations ob-
(6)
10 June 1982
tained in ref. [2] except that we have made a cyclic permutation x -+ z -+y -+ x in order to follow the notation of ref. [ 1] in taking the preferred direction along the z axis rather than the x axis. By making use of eqs. (3), written in K, one finds that E and B obey the same homogeneous equations as E' and B', but that the equations relating E and B to the sources (that is, the inhomogeneous equations) are given by + OEylOy -
= p,
aByl x - OBxlay.+ a e zl t : ]z, Bzlay + ~Byl~z + ~Exl~t = -Jx,
aex/ Z + aBzlax -
aE y / a t = Jy.
(7)
The equations of motion of a charge moving in these fields with a superluminal velocity u are obtained from the analogue in K of eq. (4), and one finds
dpJdt = e(E z + UxBy -
UyBx),
@xidt = - e ( E x + UyB z - UzBy), @yldt = -e(Ey + UzB x - UxBz).
(8)
As,expected, the differential laws that the fields E and B obey in Kare different from Maxwell's equations. Furthermore, the equations of motion o f a superluminal charge in these fields are not given by the Lorentz force. There is a certain arbitrariness in the definition of E and B. For instance, matching components of (F') ~x and F ~x instead of (G') K~'~and G Kx would have resulted in a few sign differences. But the point is that no definition of E and B as linear combinations of E' and B' will yield Maxwell's equations for E and B. That is, these fields are fundamentally different from the usual (subluminal) electromagnetic fields. Eqs. (7) differ from the corresponding equations, eqs. (I-10), in ref. [1]. The origin of the discrepancy lies in the procedure apparently used by Negi et al. in obtaining these equations. They begin, as we do, by assuming that the electric and magnetic fields satisfy Maxwell's equations in the superluminal reference frame K'. They then make the replacements t'=z,
z'=t
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Volume 113B, number 2
PHYSICS LETTERS
in Maxwell's equations. Eq. (9) correspond to (1-3). They are the correct form of the transformation eqs. (1) for the case that the relative velocity of the two reference frames is infinite, though not in general. (Note that we use the notation t', x', etc. for the coordinates in K' rather than the notation T, X, etc. adopted in ref. [ 1] .) If one now begins, e.g., with the divergence equation for the electric field E' in K', ' '+ BE'y]Oj+ a e ; / 0 z ' = p ' aE'xlaX
(10)
and makes use of (9), one arrives at aE'x/aX + aE'y/ay + aE'z/at = p'.
(11)
Eq. (11) is just the first of e qs. (I-10), except that in ref. [1] the primes on p and E are omitted. The remainder of eqs. (I-10) are obtained in the same way, i.e., by beginning with MaxweU's equations in K', substituting eqs. (9) and defining: E = E',
B = B',
J " = (J')".
(12a,b,c)
Thus eqs. (1-10) represent a kind of hybrid in which the coordinates are transformed in going from K' to K but the field components and those of the current four-vector are not, so that one has equations involving derivatives with respect to unprimed coordinates of primed components of the fields and primed components of the current. If one replaces eq. (12c) by the transformation equations (1) for a four.vector in the limit v = 0% so that in particular one has p' = Jz, and also replaces eqs. (12a) and (12b) by the field transformation equations (6) with v = co, one finds that eq. (11) reduces to the second of our field equations (7). In the same way after one transforms the fields and current, as well as the coordinates, from K' to K, the entire set of eqs. (I-10) reduces to the set of eqs. (7). Thus the discrepancy between the superluminal field equations of ref. [1] and those derived previously by us in ref. [2] results entirely from the failure of Negi et al. to transform the fields and current as well as the coordinates in going from K' to K. A similar transformation of the fields, together with a careful consideration of the signs involved, would also bring their expressions (I-14) for the Lorentz force in agreement with the right-hand side of our eqs. (8) for the corresponding case, namely E z = B z = O. We also note that, in view of the procedures by which eqs. (I-10) were derived, the fact that eqs. (1-10) differ from Maxwell's equations does not, by itself, establish that the field equations are not invariant un182
10 June 1982
der supefluminal Lorentz transformations. Maxwell's equations, in fact, are not invariant under the analogs of the procedure used to derive eqs. (I-10), in particular the use of eqs. (12), even for transformations between ordinary Lorentz frames with subluminal relative velocities. Electrodynamics is invariant under ordinary Lorentz transformations only when all physical quantities are properly transformed from one reference frame to another. As we have noted above, in the superluminal case the transformation equations for the fields are ambiguous, depending on how the superluminal fields are derived in terms of the welldefined components of the field tensor in subluminal frames. The crucial point is, again, that no definition o f E i and B i as linear combinations of the components E~ and B~ yields Maxwell's equations for the components E i and B i. In conclusion we emphasize that the problem of real interest, for predicting the observable behavior of tachyons, is how do they interact with subluminal electromagnetic fields, or how do their fields interact with bradyons. It is very important to stress that unless such interactions are specified, the supefluminal fields E and B have no observational meaning. This is partly reflected in the arbitrariness inherent in their definition, which we have already alluded to. For instance, in contrast with many claims made in the literature, one cannot conclude anything about the monopole character of tachyons, or their ~erenkov radiation, unless these specifications are made. In the absence of experimental data on tachyon-bradyon interactions (other than upper bounds), there is considerable freedom for one to postulate what these interactions could be. Since E and B do not obey Maxwell's equations, there is no a priori reason to believe that they should superpose with fields produced by bradyons, or that they should act on bradyons through the Lorentz force. A proposal for a possible ~teraction mechanism was given by us in ref. [2]. Two of us (L.M. and A.F.A.) are grateful to the Natural Sciences and Engineering Research Council of Canada for financial support. [1] O.P.S. Negi, H.C. Chandola, K.D. Purohit and B.S. Rajput, Phys. Lett. 105B (1981) 281. [2] L. Marchildon, A.E. Everett and A.F. Antippa, Nuovo Cimento 53B (1979) 253;and referencestherein. [3] A.F. Antippa and A.E. Everett, Phys. Rev. D8 (1973) 2352.
PHYSICS LETTERS
10 June 1982
COMMENT ON "THE BEHAVIOR OF MAXWELL'S EQUATIONS UNDER REAL SUPERLUMINAL LORENTZ TRANSFORMATIONS" L. MARCHILDON, A.F. ANTIPPA De'partement de Physique, Universit~ du Quebec ~ Trois-Rivi~res, Trois-Rivi~res~ Quebec, G9A 5H7, Canada and
A.E. EVERETT Department of Physics, Tufts University, Medford, MA 02155, USA Received 2 March 1982
Negi et al. have recently obtained field equations for the superluminal electromagnetic field, in theories based on real supeduminal transformations along a "tachyon corridor". Their results differ from equations obtained some time ago by the present authors. We trace the source of the discrepancy to the failure of Negi et al. to consistently transform all relevant quantities from the old to the new Lorentz frame.
In this note we intend to comment on a recent paper by Negi et al. bearing the same title [1]. These authors investigated the transformation properties of Maxwell's equations and the Lorentz force under a given type of supeduminal transformations. Results on this problem were obtained some time ago by the present authors [2]. Since some of the results in ref. [1] disagree with those obtained by us, and since somewhat subtle questions of interpretation arise in this context, we wish to examine the paper by Negi et al. in the light of our earlier resul{s. We will refer to equations from ref. [1] as eq. (I-1), etc. Refs. [1,2] both begin with the following superluminal transformations, assumed to hold between the coordinates of a frame K and a frame K' [3] : t = 7 p ( t ' + oz'),
x = x',
z = 7t~(ot' + z'),
y = y',
7 = ( °2 -- i) -1/2,
P = o/Iv[,
Iv[ ~> 1.
(1)
These transformations are interpreted as holding between a class o f preferred inertial frames, moving with respect to each other along a preferred direction called the tachyon Corridor. Other inertial frames are re180
lated to the preferred ones through usual (subluminal) Lorentz transformations. We note that eqs. (1) are the same as eqs. (1) in ref. [1]. Assume there is in K' a world of bradyons. In K they will be seen as tachyons. Two very different questions can now be asked: (i) what is the behavior of these tachyons and their fields, interacting among themselves, as seen in K? (ii) If there are bradyons in K, what are their interactions with the tachyons? The first question is rather easy to answer, but the second is not. The second question, however, is the one that is really relevant to tachyon detection. We know how to describe the behavior of bradyons in K'. As the coordinate transformations between K' and K are given, it is straightforward to infer from these the behavior of tachyons in K. For electromagnetic interactions this is accomplished as follows: in K', the electromagnetic field is characterized by a second-rank antisymmetric tensor F~x , related to the potential A'K through F '~ . = ~'~A'~ - ~'x A'~"
(2)
t
In terms of FKx., its dual (G')Uu = ~-g~"~uV~XF' ~~ 2~ ca, and the current four-vector (j')K, Maxwell's equations read as
0 031-9163/82]0000-0000/$02.75
© 1982 North-Holland
Volume 113B, number 2 =
(a') ",
PHYSICS LETTERS = 0.
(3)
The Lorentz force acting on a particle of charge e, mass m, proper time ~-and momentum four-vector (p')~ can be obtained from the following equation: (f,)K = d(p,)K/dr = (e/m)(F')KXp' x.
(4)
In the frame K, one can define quantities F Kx, G ~x, f~, etc., which are related to (/7') Kx, (G')~x, (f~)K by tensor transformations. That is, each contravariant index transforms like (x') K, i.e. with the matrix that brings (t',x',y',z') to (t, x,y, z) according to eqs. (1). These quantities then form representation spaces for the group of all subluminal and superluminal transformations along the tachyon corridor. One can also define covariant quantities obtained from the contravariant ones with the help of the metric tensor. From well-known results of tensor analysis, the same tensor equations then hold in K as hold in K'. They describe the behavior of the particles and fields seen in K, that is, of tachyons and the electromagnetic field. Notice that the metric g'Kx gets changed in going over to K, that is, g'~x CgKx- This is a consequence of the transformation law ofg'Kx, and is related to the fact ¢ g ! that under eqs. (1), the world interval ( x ) x~ is transformed in the following way: ,' , ')K (X) ~'Tt = --(t') 2 + (X') 2 + (.V')2 + (Z') 2 gKh(X
= t 2 + x 2 + y 2 _ Z2 =gKxxKx x.
(5)
Because of the non-invariance of the world interval under superluminal transformations, the velocity of light is not invariant. That is, particles (or waves) propagating with a speed e in K' will not in general propagate with the same speed in K. This in turn implies that Maxwell's equations will not hold in K for superluminal fields. This can be seen explicitly as follows: define in K an electric field E and a magnetic field B so that they enter the tensor G Kx in the same was as E' and B' enter (G')Kx. The transformation laws of (G') ~x into G Kx then imply that
E z = E'z,
#
Bz = - B z '
-%), Ey=Tlx(Ei-VB'x),
By =7/X(By +rE'x).
Eqs. (6) are the field transformation equations ob-
(6)
10 June 1982
tained in ref. [2] except that we have made a cyclic permutation x -+ z -+y -+ x in order to follow the notation of ref. [ 1] in taking the preferred direction along the z axis rather than the x axis. By making use of eqs. (3), written in K, one finds that E and B obey the same homogeneous equations as E' and B', but that the equations relating E and B to the sources (that is, the inhomogeneous equations) are given by + OEylOy -
= p,
aByl x - OBxlay.+ a e zl t : ]z, Bzlay + ~Byl~z + ~Exl~t = -Jx,
aex/ Z + aBzlax -
aE y / a t = Jy.
(7)
The equations of motion of a charge moving in these fields with a superluminal velocity u are obtained from the analogue in K of eq. (4), and one finds
dpJdt = e(E z + UxBy -
UyBx),
@xidt = - e ( E x + UyB z - UzBy), @yldt = -e(Ey + UzB x - UxBz).
(8)
As,expected, the differential laws that the fields E and B obey in Kare different from Maxwell's equations. Furthermore, the equations of motion o f a superluminal charge in these fields are not given by the Lorentz force. There is a certain arbitrariness in the definition of E and B. For instance, matching components of (F') ~x and F ~x instead of (G') K~'~and G Kx would have resulted in a few sign differences. But the point is that no definition of E and B as linear combinations of E' and B' will yield Maxwell's equations for E and B. That is, these fields are fundamentally different from the usual (subluminal) electromagnetic fields. Eqs. (7) differ from the corresponding equations, eqs. (I-10), in ref. [1]. The origin of the discrepancy lies in the procedure apparently used by Negi et al. in obtaining these equations. They begin, as we do, by assuming that the electric and magnetic fields satisfy Maxwell's equations in the superluminal reference frame K'. They then make the replacements t'=z,
z'=t
(9) 181
Volume 113B, number 2
PHYSICS LETTERS
in Maxwell's equations. Eq. (9) correspond to (1-3). They are the correct form of the transformation eqs. (1) for the case that the relative velocity of the two reference frames is infinite, though not in general. (Note that we use the notation t', x', etc. for the coordinates in K' rather than the notation T, X, etc. adopted in ref. [ 1] .) If one now begins, e.g., with the divergence equation for the electric field E' in K', ' '+ BE'y]Oj+ a e ; / 0 z ' = p ' aE'xlaX
(10)
and makes use of (9), one arrives at aE'x/aX + aE'y/ay + aE'z/at = p'.
(11)
Eq. (11) is just the first of e qs. (I-10), except that in ref. [1] the primes on p and E are omitted. The remainder of eqs. (I-10) are obtained in the same way, i.e., by beginning with MaxweU's equations in K', substituting eqs. (9) and defining: E = E',
B = B',
J " = (J')".
(12a,b,c)
Thus eqs. (1-10) represent a kind of hybrid in which the coordinates are transformed in going from K' to K but the field components and those of the current four-vector are not, so that one has equations involving derivatives with respect to unprimed coordinates of primed components of the fields and primed components of the current. If one replaces eq. (12c) by the transformation equations (1) for a four.vector in the limit v = 0% so that in particular one has p' = Jz, and also replaces eqs. (12a) and (12b) by the field transformation equations (6) with v = co, one finds that eq. (11) reduces to the second of our field equations (7). In the same way after one transforms the fields and current, as well as the coordinates, from K' to K, the entire set of eqs. (I-10) reduces to the set of eqs. (7). Thus the discrepancy between the superluminal field equations of ref. [1] and those derived previously by us in ref. [2] results entirely from the failure of Negi et al. to transform the fields and current as well as the coordinates in going from K' to K. A similar transformation of the fields, together with a careful consideration of the signs involved, would also bring their expressions (I-14) for the Lorentz force in agreement with the right-hand side of our eqs. (8) for the corresponding case, namely E z = B z = O. We also note that, in view of the procedures by which eqs. (I-10) were derived, the fact that eqs. (1-10) differ from Maxwell's equations does not, by itself, establish that the field equations are not invariant un182
10 June 1982
der supefluminal Lorentz transformations. Maxwell's equations, in fact, are not invariant under the analogs of the procedure used to derive eqs. (I-10), in particular the use of eqs. (12), even for transformations between ordinary Lorentz frames with subluminal relative velocities. Electrodynamics is invariant under ordinary Lorentz transformations only when all physical quantities are properly transformed from one reference frame to another. As we have noted above, in the superluminal case the transformation equations for the fields are ambiguous, depending on how the superluminal fields are derived in terms of the welldefined components of the field tensor in subluminal frames. The crucial point is, again, that no definition o f E i and B i as linear combinations of the components E~ and B~ yields Maxwell's equations for the components E i and B i. In conclusion we emphasize that the problem of real interest, for predicting the observable behavior of tachyons, is how do they interact with subluminal electromagnetic fields, or how do their fields interact with bradyons. It is very important to stress that unless such interactions are specified, the supefluminal fields E and B have no observational meaning. This is partly reflected in the arbitrariness inherent in their definition, which we have already alluded to. For instance, in contrast with many claims made in the literature, one cannot conclude anything about the monopole character of tachyons, or their ~erenkov radiation, unless these specifications are made. In the absence of experimental data on tachyon-bradyon interactions (other than upper bounds), there is considerable freedom for one to postulate what these interactions could be. Since E and B do not obey Maxwell's equations, there is no a priori reason to believe that they should superpose with fields produced by bradyons, or that they should act on bradyons through the Lorentz force. A proposal for a possible ~teraction mechanism was given by us in ref. [2]. Two of us (L.M. and A.F.A.) are grateful to the Natural Sciences and Engineering Research Council of Canada for financial support. [1] O.P.S. Negi, H.C. Chandola, K.D. Purohit and B.S. Rajput, Phys. Lett. 105B (1981) 281. [2] L. Marchildon, A.E. Everett and A.F. Antippa, Nuovo Cimento 53B (1979) 253;and referencestherein. [3] A.F. Antippa and A.E. Everett, Phys. Rev. D8 (1973) 2352.