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On some generalizations of the Lorentz transformation
Volume
80A, number
PHYSICS
2,3
ON SOME GENERALIZATIONS
6 August
rjolpdxadxfl =~~‘dx~‘dxfl’.
(1)
This condition is more restrictive than that of the invariance of the light velocity, the latter being quite satisfied also if golOdx”dxfl = -n(u,p’ dxCY’dxP’.
0)
Recently some approaches to a generalization of the LT based on relation (2) were discussed [4-71. Using (2) Parker [4], and later Mariwalla [5] and Antippa [6] have got the following generalization of the LT for superluminal frames in (1 + I)-dimensional ST (( 1 + 1) ST): t’ = y(t - ux) )
1)-lj2;
*’ We use common
(3)
u2 > 1 .
notes and agreements, particularly: light velocity C = 1, metric tensor nap = diag (+ - - -), summation over repeating indices etc.
102
the USSR, MOSCOK~ 117913, USSR
proposed by Recanii and Mignani, Antippa and Everett, and some are considered. It is shown that these approaches have a number of
The light velocity independence of the choice of a reference frame requires the invariance of the only ’ under the transzero interval dr2 = n Qpdxadxfl=O* formations of space-time (ST) coordinates. The transformations satisfying this condition form, in general, the 15parameter conformal group [ 1,2]. However, the uniform translation of a massive body is not invariant under the transformation of this group [3]. Because of this, its lo-parameter subgroup, the Lorentz group, is widely used. The Lorentz transformations (LT) satisfy the condition of the invariance of STinterval
y=sgnu(u2-
ofSciencesof
1980
The generalizations of the Lorentz transformation others based on a 6dimensional space-time formalism serious shortcomings making them inapplicable.
x’ = y(x ~ ut) )
1980
OF THE LORENTZ TRANSFORMATION
N.N. WEINBERG Zelinskv Institute of Organic Che1nistr.v of the Academy Received
24 November
LETTERS
The dimension of the ST in this generalization is of crucial importance, because of Gorini’s result [8] that the Lorentz group and the Galilean group are the only linear kinematical groups in (tz t I)-ST (n > 3) which are consistent with the space isotropy. Now there exist two approaches to the generalization of the transformation (3) onto (3 t I)-ST. These are the generalizations of Recami and Mignani [7] and Antippa and Everett [9]. A critical analysis of the Recam-Mignani approach has been given by Lee and Kalotas [lo]. The authors have pointed out the essential shortcoming of the Recami-Mignani generalization consisting in presence of complex quantities having no physical interpretation. Some attempts are made to refme its meaning in terms of (3 t 3)ST [ 1 l- 131, the formalism of the (3 t 3)-ST being available in at least three versions. In the first variant [ 14,l S] 6-ST decomposes on the direct product of three (1 + l)-ST. a motion in 6-ST decomposing on direct product of three one-dimensional motions in (1 t I)-ST. In this case the ST-coordinates transformation has the following form XI tf
= =
y.I =
yi(Xi Yj(ti
~~ Ui -
ti)
UiXi) .
) i=
1,2,3.
(4)
(1 - u?)~1/2 I
where ui are the components of the relative velocity of the reference frames. As indicated by Strnad [ 161 transformations (4) form a commutative group, and therefore, this formalism does not account for the Thomas precession although the latter is the well-known experi-
mental fact. One could also notice the following shortcoming of the transformation (4). If the time, t, observed in 4-ST is expressed in terms of time-coordinates of 6-ST as t = (t: t t$ + ti)li2, then passing from 6-ST to 4-ST and suitably transformating (4) one will get the linear transformation which differs from the LT contrary to the Gorini theorem [8]. The same result can be obtained on the assumption that t equals to one Of ti. In the second variant of the 6-ST formalism [ 131 the ST-coordinates transformation in suitable frames have the form
t; = Y($-
t;= t2; t; = t3 )
t; =y;
$)-l/2;
y” = (6 z” = z;
x’=‘Y(x-ut),
y’=y;
z’=z, (7)
u2<1,
LJX))
(5)
t; =z,
T=sgnu(u2-l)-W;$>l.
1ft=(tftt2tt3) 2 2 1/2, then just as in the preceeding case the transformation (5) does not transform into the LT. Should t = tl, then if u2 < 1, (5) coincides with the LT, but the question arises how to interpret t2 and t3 which not only are unobservable but even have no effect on any observable ones. One could substitute tl in (5) by t = (t: + titt$u2, thus removing the difficulties in transforming 6-ST to 4-ST, so far as (5) will transform into LT. But first, this new transformation is nonlinear (though, it is not evident, of course, that this transformation should be linear). And second, being applied twice: initially for the transformation from the frame S to the frame S’ moving with the relative velocity II = (u, 0, 0), and then for the transformation S’ + S” with the relative velocity v = (0, u, 0), this new transformation for S’-+S”, withu>landu>l,gives X”Z [(&-
t’=Y(t-ux),
y = sgnu(u2 - 1)-lj2
y’ = t.2; z’ = tg )
z’=z,
t; = Y(Q - ux) )
y = (I-
on introducing the matrix velocity ~ii = ax,/&,. However, this formalism seems to be developed insufficiently: the matrix of the ST-coordinates transformation is not obtained (only matrix equations for its 3-dimensional blocks are available), and some used quantities are not defined correctly enough. Particularly, this concerns the matrix velocity ~ii playing a leading role in the formalism. Antippa and Everett [9] have proposed the following generalization of transformation (3), alternative to that of Recami and Mignani:
x’=y(x-q),
x’ = y(x - I$), y’=y;
24 November 1980
PHYSICS LETTERS
Volume 80A, number 2,3
l)-l(t-u@
-
y2 _ $]1/2
1)-l/2 [t2 - u(u2 - I)-1qt
-ux)]
Their approach is based on the breakdown of the Lorenz invariance under the transformations to superluminal frames. The breakdown of the Iorentz invariance seems to be a too categorical assumption. First, (7) is the continuation of (3) onto (3 t I)-ST, and (3) logically follows from the invariance of the light velocity. But at the same time, the light velocity is not invariant under the transformation (7), i.e. generalizing (3) the transformation (7) destroys its logical background. Secondly, if the breakdown of the Lorentz invariance is accepted, the transformation (7) is not, by any means, the single generalization of (3) ** , Such a freedom can hardly be regarded as the merit of a physical theory. And thirdly, as shown in the Appendix, equations of the type B”,=O,
)
(6)
etc.
As seen from (6), for some u > 1 and u > 1 (e.g. u = u = 1.6) the relative velocity of the frame S” with respect to the frame S is w < 1. For such u and u the transformation (6) should coincide with LT, but this condition is not satisfied. The third variant of the 6-ST formalism [ 171 is based I
(8)
are not covariant under the transformation (7). From this it follows that such conservation laws as the conservation of charge principle J(y+=O, and the conservation T+#
)
; u2 > 1 .
of energy-momentum
principle
=0 ,
are not covariant under this transformation. This is the most serious shortcoming of the transformation (7). To summarize, one could see that all the considered generalizations of the LT have a number of serious short. comings making them inapplicable, at least in the present form. ** Some alternative variants have been discussed even by one of the authors of the transformation (7) [ 181. 103
Volume
80A, number
PHYSICS
2,3
LETTERS
Appendix. Let the equation (8) hold in some unprimed frame, and matrix AP (y transforming contrava riant components of tensors satisfy the condition
nfiu’V,AU’,AGO = B,p , Bap = 0 for (Y# fl .
(9)
References
If eq. (8) holds in the primed frame, then
$X&+-p
frame in mind, it gives
+(K~+K~)B~=O,
from which, in virtue of linear independency %Y+p and Bi’li=i, we get K;=
p 6;;
of
p = const.
With (10) in mind, it gives the necessary condition of covariance of the equation (8) under the transformation 8ap = P%p
’
(1’1)
the case of the transformation (7) 8,ap = diag(-t--J, and hence, (8) is not covariant under this transformation. In
104
1980
It can be easily seen, that the equation (8) is covariant under those linear transformations which satisfy either (1) or (2). This follows from (11) with an accuracy of scaling.
Then respective matrix transforming the covariant components can be presented as follows
With (8) for the unprimed
24 November
\
[ 11 H. Weyl, Space, time, matter (Dover, New York, 1950). ]2] T. F&on, F. Rohrlich and L. Witten, Rev. Mod. Phys. 34 (1962) 442. [ 31 S. Weinberg, Gravitation and cosmology (Wiley, New York, 1972). [4] L. Parker, Phys. Rev. 188 (1969) 2287. [5] K.H. Mariwalla, Am. J. Phys. 37 (1969) 1281. [61 A.1:. Antippa, Nuovo Cim. 10A (1972) 389. [71 E. Recami and R. Mignani, Rivista Nuovo Cim. 4 (1974) 209. [81 V. Gorini, Commun. Math. Phys. 21 (1971) 150. [91 A.F. Antippa and A.E. Everett, Phys. Rev. D8 (1973) 2352. A.R. Lee, TM. Kalotas, Nuovo Cim. 41B (1977) 365. H.C. Corben, Lett. Nuovo Cim. 11 (1974) 533. R. Mignani and E. Recami, Lett. Nuovo Cim. 16 (1976) 449. [I31 E.A.B. Cole, Nuovo Cim. 40A (1977) 171. [I41 P.T. Pappas, Lett. Nuovo Cim. 22 (1978) 601; 25 (1979) 429. [I51 G. Ziino, Lett. Nuovo Cim. 24 (1979) 171; Phys. Lett. 70A (1979) 87. [I61 J. Strnad. Lett. Nuovo Cim. 26 (1979) 535. [I71 E.A.B. Cole, Nuovo Cim. 44B (1978) 157. ,18l A.E. Everett, Phys. Rev. D13 (1976) 785.
80A, number
PHYSICS
2,3
ON SOME GENERALIZATIONS
6 August
rjolpdxadxfl =~~‘dx~‘dxfl’.
(1)
This condition is more restrictive than that of the invariance of the light velocity, the latter being quite satisfied also if golOdx”dxfl = -n(u,p’ dxCY’dxP’.
0)
Recently some approaches to a generalization of the LT based on relation (2) were discussed [4-71. Using (2) Parker [4], and later Mariwalla [5] and Antippa [6] have got the following generalization of the LT for superluminal frames in (1 + I)-dimensional ST (( 1 + 1) ST): t’ = y(t - ux) )
1)-lj2;
*’ We use common
(3)
u2 > 1 .
notes and agreements, particularly: light velocity C = 1, metric tensor nap = diag (+ - - -), summation over repeating indices etc.
102
the USSR, MOSCOK~ 117913, USSR
proposed by Recanii and Mignani, Antippa and Everett, and some are considered. It is shown that these approaches have a number of
The light velocity independence of the choice of a reference frame requires the invariance of the only ’ under the transzero interval dr2 = n Qpdxadxfl=O* formations of space-time (ST) coordinates. The transformations satisfying this condition form, in general, the 15parameter conformal group [ 1,2]. However, the uniform translation of a massive body is not invariant under the transformation of this group [3]. Because of this, its lo-parameter subgroup, the Lorentz group, is widely used. The Lorentz transformations (LT) satisfy the condition of the invariance of STinterval
y=sgnu(u2-
ofSciencesof
1980
The generalizations of the Lorentz transformation others based on a 6dimensional space-time formalism serious shortcomings making them inapplicable.
x’ = y(x ~ ut) )
1980
OF THE LORENTZ TRANSFORMATION
N.N. WEINBERG Zelinskv Institute of Organic Che1nistr.v of the Academy Received
24 November
LETTERS
The dimension of the ST in this generalization is of crucial importance, because of Gorini’s result [8] that the Lorentz group and the Galilean group are the only linear kinematical groups in (tz t I)-ST (n > 3) which are consistent with the space isotropy. Now there exist two approaches to the generalization of the transformation (3) onto (3 t I)-ST. These are the generalizations of Recami and Mignani [7] and Antippa and Everett [9]. A critical analysis of the Recam-Mignani approach has been given by Lee and Kalotas [lo]. The authors have pointed out the essential shortcoming of the Recami-Mignani generalization consisting in presence of complex quantities having no physical interpretation. Some attempts are made to refme its meaning in terms of (3 t 3)ST [ 1 l- 131, the formalism of the (3 t 3)-ST being available in at least three versions. In the first variant [ 14,l S] 6-ST decomposes on the direct product of three (1 + l)-ST. a motion in 6-ST decomposing on direct product of three one-dimensional motions in (1 t I)-ST. In this case the ST-coordinates transformation has the following form XI tf
= =
y.I =
yi(Xi Yj(ti
~~ Ui -
ti)
UiXi) .
) i=
1,2,3.
(4)
(1 - u?)~1/2 I
where ui are the components of the relative velocity of the reference frames. As indicated by Strnad [ 161 transformations (4) form a commutative group, and therefore, this formalism does not account for the Thomas precession although the latter is the well-known experi-
mental fact. One could also notice the following shortcoming of the transformation (4). If the time, t, observed in 4-ST is expressed in terms of time-coordinates of 6-ST as t = (t: t t$ + ti)li2, then passing from 6-ST to 4-ST and suitably transformating (4) one will get the linear transformation which differs from the LT contrary to the Gorini theorem [8]. The same result can be obtained on the assumption that t equals to one Of ti. In the second variant of the 6-ST formalism [ 131 the ST-coordinates transformation in suitable frames have the form
t; = Y($-
t;= t2; t; = t3 )
t; =y;
$)-l/2;
y” = (6 z” = z;
x’=‘Y(x-ut),
y’=y;
z’=z, (7)
u2<1,
LJX))
(5)
t; =z,
T=sgnu(u2-l)-W;$>l.
1ft=(tftt2tt3) 2 2 1/2, then just as in the preceeding case the transformation (5) does not transform into the LT. Should t = tl, then if u2 < 1, (5) coincides with the LT, but the question arises how to interpret t2 and t3 which not only are unobservable but even have no effect on any observable ones. One could substitute tl in (5) by t = (t: + titt$u2, thus removing the difficulties in transforming 6-ST to 4-ST, so far as (5) will transform into LT. But first, this new transformation is nonlinear (though, it is not evident, of course, that this transformation should be linear). And second, being applied twice: initially for the transformation from the frame S to the frame S’ moving with the relative velocity II = (u, 0, 0), and then for the transformation S’ + S” with the relative velocity v = (0, u, 0), this new transformation for S’-+S”, withu>landu>l,gives X”Z [(&-
t’=Y(t-ux),
y = sgnu(u2 - 1)-lj2
y’ = t.2; z’ = tg )
z’=z,
t; = Y(Q - ux) )
y = (I-
on introducing the matrix velocity ~ii = ax,/&,. However, this formalism seems to be developed insufficiently: the matrix of the ST-coordinates transformation is not obtained (only matrix equations for its 3-dimensional blocks are available), and some used quantities are not defined correctly enough. Particularly, this concerns the matrix velocity ~ii playing a leading role in the formalism. Antippa and Everett [9] have proposed the following generalization of transformation (3), alternative to that of Recami and Mignani:
x’=y(x-q),
x’ = y(x - I$), y’=y;
24 November 1980
PHYSICS LETTERS
Volume 80A, number 2,3
l)-l(t-u@
-
y2 _ $]1/2
1)-l/2 [t2 - u(u2 - I)-1qt
-ux)]
Their approach is based on the breakdown of the Lorenz invariance under the transformations to superluminal frames. The breakdown of the Iorentz invariance seems to be a too categorical assumption. First, (7) is the continuation of (3) onto (3 t I)-ST, and (3) logically follows from the invariance of the light velocity. But at the same time, the light velocity is not invariant under the transformation (7), i.e. generalizing (3) the transformation (7) destroys its logical background. Secondly, if the breakdown of the Lorentz invariance is accepted, the transformation (7) is not, by any means, the single generalization of (3) ** , Such a freedom can hardly be regarded as the merit of a physical theory. And thirdly, as shown in the Appendix, equations of the type B”,=O,
)
(6)
etc.
As seen from (6), for some u > 1 and u > 1 (e.g. u = u = 1.6) the relative velocity of the frame S” with respect to the frame S is w < 1. For such u and u the transformation (6) should coincide with LT, but this condition is not satisfied. The third variant of the 6-ST formalism [ 171 is based I
(8)
are not covariant under the transformation (7). From this it follows that such conservation laws as the conservation of charge principle J(y+=O, and the conservation T+#
)
; u2 > 1 .
of energy-momentum
principle
=0 ,
are not covariant under this transformation. This is the most serious shortcoming of the transformation (7). To summarize, one could see that all the considered generalizations of the LT have a number of serious short. comings making them inapplicable, at least in the present form. ** Some alternative variants have been discussed even by one of the authors of the transformation (7) [ 181. 103
Volume
80A, number
PHYSICS
2,3
LETTERS
Appendix. Let the equation (8) hold in some unprimed frame, and matrix AP (y transforming contrava riant components of tensors satisfy the condition
nfiu’V,AU’,AGO = B,p , Bap = 0 for (Y# fl .
(9)
References
If eq. (8) holds in the primed frame, then
$X&+-p
frame in mind, it gives
+(K~+K~)B~=O,
from which, in virtue of linear independency %Y+p and Bi’li=i, we get K;=
p 6;;
of
p = const.
With (10) in mind, it gives the necessary condition of covariance of the equation (8) under the transformation 8ap = P%p
’
(1’1)
the case of the transformation (7) 8,ap = diag(-t--J, and hence, (8) is not covariant under this transformation. In
104
1980
It can be easily seen, that the equation (8) is covariant under those linear transformations which satisfy either (1) or (2). This follows from (11) with an accuracy of scaling.
Then respective matrix transforming the covariant components can be presented as follows
With (8) for the unprimed
24 November
\
[ 11 H. Weyl, Space, time, matter (Dover, New York, 1950). ]2] T. F&on, F. Rohrlich and L. Witten, Rev. Mod. Phys. 34 (1962) 442. [ 31 S. Weinberg, Gravitation and cosmology (Wiley, New York, 1972). [4] L. Parker, Phys. Rev. 188 (1969) 2287. [5] K.H. Mariwalla, Am. J. Phys. 37 (1969) 1281. [61 A.1:. Antippa, Nuovo Cim. 10A (1972) 389. [71 E. Recami and R. Mignani, Rivista Nuovo Cim. 4 (1974) 209. [81 V. Gorini, Commun. Math. Phys. 21 (1971) 150. [91 A.F. Antippa and A.E. Everett, Phys. Rev. D8 (1973) 2352. A.R. Lee, TM. Kalotas, Nuovo Cim. 41B (1977) 365. H.C. Corben, Lett. Nuovo Cim. 11 (1974) 533. R. Mignani and E. Recami, Lett. Nuovo Cim. 16 (1976) 449. [I31 E.A.B. Cole, Nuovo Cim. 40A (1977) 171. [I41 P.T. Pappas, Lett. Nuovo Cim. 22 (1978) 601; 25 (1979) 429. [I51 G. Ziino, Lett. Nuovo Cim. 24 (1979) 171; Phys. Lett. 70A (1979) 87. [I61 J. Strnad. Lett. Nuovo Cim. 26 (1979) 535. [I71 E.A.B. Cole, Nuovo Cim. 44B (1978) 157. ,18l A.E. Everett, Phys. Rev. D13 (1976) 785.