- Email: [email protected]
The relativistic ether hypothesis
Volume 78A, number 1
PHYSICS LETFERS
7 July 1980
THE RELATiVISTIC ETHER HYPOTHESIS J. REMBIELII~SKI The Institute of Physics, University of Ldd~90-l36Lddzç Poland Received 27 March 1980
It is found that the “generalised Gahlean transformations” introduced by Chang form a subclass of the nonlinear (with respect to three-velocity) transformations of the Lorentz group. The concept of the relativistic covariance in a modified form leads to the necessity of existence of the absolute reference frame. The (formal) connection with usual formalism of special relativity is given. The form of the field equations and equations of motion is analysed. Some possible implications of this hypothesis are discussed.
Recently Chang [1,21 has proposed the anisotropic space-time variant of the Maxwell equations. This description of the electromagnetic phenomena seems
Firstly we note that the transformations (1). do not have the group property. However we can relate two inertial frames L,7 and La’ by the composed transfor-
not to conflict with present experimental status. Moreover it can be a basis for the interpretation of the cosnuc background radiation anisotropy [3,8] and some data from quasars and strong compact radio-galaxies radiation In his paper [1] Chang the existence of [4]. the absolute reference frameassumes E 0 (ether) and defines a class of the inertial frames La by the socalled generaliseci Galilean transformations
mationL ~ —~L0
(1) =
R
—
(e/c)~R0 + (e/o2)(y
—
—
— —).
-+
2 —1 0
(li-ow/c)
W
r
=
r ~W
7w’ ÷~ ~
ow
r
LT + br ,-
2 + ~/
w
iwr
where the vector-parameter w is given by the transformation formula for a’ 2)7~(1 +7w)~ e’= w+o~iç’+w(ew/c li-a ‘C2 (3)
1)eR.
Here y 2/c2)112 r°= Ct and r are the space0 = (1 c in time coordinates while R0 cT and R in Lo; a is the velocity of E 0 with respect2to= Lo.~dR~dR~’ The metric in Lo has the =Minkowskian form ds where diag(+ In this article we show that the inertial frames introduced by Chang are related by the transformations belonging to the nonlinear (with respect to three-velocity a) realization of the Lorentz group which linearizes for rotations from the subgroup 0(3). Consequently the variant of the Maxwell equations proposed in ref. [1] a) is Lorentz covariant, b) it can be obtained from the usual one by the a-dependent transformation. However the physical consequences of the Chang equations are different from the usual ones. —
rO=’y
—i
Note that eq. (3) is just the transformation law for velocity in special relativity. However, as will be evident later, the parameter can be identified withinerthe relative velocity for a verywrestricted class of the tial frames only. Nevertheless eq. (3) suggests that there is a connection between the transformation laws (2), (3) and the Lorentz group ones. Let us consider a class of the Lorentz frames related by the Lorentz group transformations A’~ ~ A. x v~C under which the metric form ds2 ~dx~dx~’ dxTn dx remains unaffected. Furthermore, let us introduce the four-vector uM satisfying u2 = c2. As is —
14
33
Volume 78A, number I
PHYSICS LETTERS
well known [5] the one-sheet hyperboloid defined by the invariant constraints u2 = c2, u0 >0 is the underlying manifold for the nonlinear realization of the @rthochronous) Lorentz group Lt which becomes linear with respect to the 0(3) subgroup. If we introduce the new coordinates a —uc/u°then the Lorentz boosts generate the transformations (3). Next, we define the “Galilean” coordinates r’2 by the formula 4: T’2~(e)xi’
rIL
T(u) =
ri_a/cl
Lo
I
j
=
D(A, u)r(u)
where
4(u)
~—1
=
dt
.~,
1Iuu/c2)2
—
u2/c2.
(9)
Here u 4: dr/dt is three-velocity and r0 = Ct. Note that the relative velocity V of La’ with respect to La has the form = I~0I~~ = w/(l + 0w/c2)
(5)
(6)
defines the absolute timec is(measured inL0). The velocity of light direction-dependent
Note that T~(a)= T(—a) and det T = 1. It is easy to see that r’2 transforms under the action of V as follows ~‘ r’(u’)
dT
i.e. w coincides with V only for a perpendicular to V. Here r 0 denotes the position of E0. with respect to E0. Furthermore, the invariant differential 2u~dx’2= c~2uT?iT(_a)dr = ‘y dT c_ 0dt (10)
where the matrix T(a) has the following form T(a)
by
7 July 1980
(7a)
c = cn/(1 + an/c) (11) where n = c/Id. Nevertheless result of the Michelson— Morley experiment is the same as in the special relativity: the measurement of the velocity of a light signal
7b
running around any closed path gives the average value c always [6]. Because of the relation
D(A, u) = T(Au)AT For rotations from 0(3), r splits on the singlet (r0) and triplet (r). For the Lorentz boosts *2 we obtain the formula (2) and consequently for a 0, w =a’ the Chang formula (1). In conclusion, the Chang transformations belong to the nonlinear realization of the Lorentz group which is induced by the nonlinear action of V on the symmetric homogenous space Lt/0(3). 2(x) reads in the Chang form The [1] invariant length ds ds2(r)=drTg(a)dr
u= ~/(1 +aVc2) (12) where ~ c dx/dx0 is the lorentzian three-velocity, we have dr(u) = dr(~)and thus we can derive the following formula connecting the “Galilean” and lorentzian (p and ir respectively) four-momentum dr
[p’2] ~P ~m
0 dr
—=
m0T(u)
=
T(e)ir
(13a)
or ~
mp=g(a)p=TT(_a)~.
(13b)
Explicitly (dr0)2 _2!drdr° +(!~dr)2—(dr)2
(8)
where [g~~] ~g(a)TT(_a),~T(_,)i.e. [g’2’~] = T( 0)~TT(q).Thus the invariant proper time is given *1
*2
The transformation law (7a) should be interpretedas passing from 2~,to E 0’; for this reason we wrote explicitly r = r(u). We adopt the following parametrization of A12~,(w): A(w)
34
1 7w[~
7-i5k
+
_w/c
wkwl
+
~
(l4a)
pm
2/c2. (14b) 0u/\/~i~i~u/c) 2 —v = invariant and the (conNote that p~p’2= m~ c served) translation generators are associated with p~ rather than p12.
For the electromagnetic waves which are radiated by a source approaching or receding from the observer (L0), the modified Doppler formula reads
Volume 78A, number 1
PHYSICS LETTERS 1/2
(I±~J-~)1hI2/(l uu~n) (15) \ ~ cc c ~ where n = u/u is the direction of the source velocity while c~and w denote initial and observed frequen-
-~-=
cies respectively. The above described formalism admits the following geometrical interpretation. With every inertial reference frame L0 we associate the set of the linear coordinate frames {r’2(e);a fixed} related by the transformations of the semidirect product 43 S0 = 0(3) ~ T~(rotations and space-time translations). Then the manifold of all linear frames at all inertial frames can be treated as the With we each point of 2 =fiber c2, u0bundle: >0 (basis) associate the set hyperboloid u the of linear frames {rM(a); a fixed} (fiber). S 0 is the structural group of this fiber i.e. TM. The transformations ofthe the stability Lorentz group group ofu relate the different fibres (points of the hyperboloid). Now let us consider the field concept in this formalism. Because of the relation (5) there is no difficulty for defining of the tensor fields. If 3~MiM2 ‘2n spans a standard tensor representation space of L then the quantity ...
F’21’22
‘2fl(r)
T’2’
(a)T’22
a
7 July 1980
a
1
~V
~j~2 ~‘
—~
ax”
~V =
0
—
‘ax”
a
T(a)
—
ar” i.e. —
—
!j’2, _~_F’2”= 0 C arv
x B +~x (! X E)l
a
I
F12z~=
I
=
V E C
C
VB = 0, VXI(B
J
C
a.o
V X E= —
~
XE)
=
~
(17)
—
-
+ ~~
at L
c
c
In the classical we must complement these equations with physics the equation of motion for a test charge 4:4 e dir dx” T(a) _!-~eF dr” (18) = e ~V dr dr 12” ~ A question arises how to construct the field equations for spinor fields. The correspondence (16) suggests that the bilinear forms ~(x)7M Li(x) and ~(r)’y’2(a)~(r), where x(r) denote the bi-spinors matrices in and the 712(0) presented formalism, shouldand be connected as can follows: ~(r)7M(a)X(r) = TM~(a)~(x)7”~J(x). Thus we identify ~-
1)2
(a)
...
T’2’~ (a)9~’1’-’2 “7(x) (16) Vfl -~
transforms covariantly with respect to the transformation law (7). The mixed tensors can be obtained with help of the metric tensorg 12~(a).Thus the tensor field equations are related to the usual (lorentzian) ones by the formula (16) (the derivatives a/arM and a/ax’2 are related by TT(_0)). If, for example, we identify the physical electric field E(r) and the magnetic induction B(r) with the components ~F0k and —eoe,kiFkJ of the antisymmetric tensor F,~(r)then the usual Maxwell equations for 9~12~(x) must be rewritten in the Chang form
712(0) = T12~(a)’yV
(19a)
and ~(r) = ~t,[x(r)]
=
i~i(T(—a)r).
(19b)
In the framework of the Rarita—Schwinger formalism of spinor—tensor fields we replace (19b) by x’2i’22...’2n(r) T’2fl
=
TMi~1(a)TM21)2(a)
(19c) 121122 Mn behaves i.e. under the (bi-) spinorspinor indices x Thus the field equaas standard (lorentzian) a)~”iV2
“7(x)
Un
~.
The eq. (18) is in fact an operational definition of the electromagnetic field. Note, that loosing of this equation causes (via the nondiagonality of g 121)(a)) an ambiguity for the definition of the physical (measured) quantities E and B. ~ See following page. 44
43
These transformations do not affect the u-velocity of ~a with respect to ~. The rotations belonging to S0 axe given by D(Ru, u) where RuU = U.
35
Volume 78A, number 1
PHYSICS LETTERS
tions for spinors can be obtained analogously to the tensor ones with the additional replacement 712 -+712(0). For example, the Dirac equation takes the form (~12
the reference frame in which the cosmic background radiation is isotropic [8]. Therefore it can be reasonable to start from the beginning with such assumption. It is interesting that under the conditions: a) space-
—+
~
12
a
(i ~ (a) ‘2
i.e. r
a/a~’2)
=
—
moc)x(r)
0
20 (
a)
,,
L~ (,~O÷! ~) ~
7 July 1980
time coordinates (r) form the singlet (rO) ÷triplet (r) of 0(3), b) rM’ and r’2 are related by the transformations linear with respect to r which form a realization of the Lorentz group, c) the physical three-space is
i~7v— mocj ~(r) = U (20b) Let us summarize the results. We have shown that the modified Lorentz covariance postulate leads to the necessity of existence of the absolute reference frame, The Chang [1]transformations form a subclass of the modifieda-dependent Lorentz transformations. In
physically euclidean one acceptable and r° possibilities: time, there theexist special only relativtwo ity and the presented hypothesis [91. Finally, we note that on the classical level the presented formalism can be equivalent to the special relativity one if dynamics does not mark out the absolute reference frame (via terms like U121/17127P in the lagrangian) [9].In fact, it
particular we have obtained in this framework the Chang variant of the Maxwell equations. Note that from the relations (5, 6) we have equality of the “Galilean” and lorentzian space coordinates: r = x. Therefore, as follows from the above described relation between the field equations (and equations of motion) in both formalisms (see eq. (16) and below), the trajectories of the charged particle moving in the electromagnetic field in the presented case and the special relativity one are identical. The difference lies in0the time development of the kinematical variables ‘=xO + (a/c)x). Thus the Trouton—Noble experi(r ment [71 gives in our case the negative result also. Furthermore, as follows from the eqs. (8—15) all kinematical effects are significant only for a/c 1. In the relativistic region for velocities (u/c 1), the specia! relativity effects dominate for a/c ~ 1. In conclusion, if we treat seriously a version of the hot big bang cosmological model then we should take mto account such possibility, despite of the fact that the coexistence of the Lorentz covariance and the absolute reference frame is surprising from the usual point of view. In fact, the big bang model marks out
may be possible to synchronize the clocks in the standard (lorentzian) fashion with the help of light signals. However on the quantum level both formalisms can be unitary nonequivalent even without dynamical requirements. Moreover, the eventual noninvariance of the vacuum state, caused by the big bang initial conditions for example, can be equivalent to the existence of the absolute reference frame [9].
C
5 *
36
+
Such situation is admissible because the spinor fields are not simply measured quantities.
-~
—
References [1] T. Chang, Phys. Lett. 70A (1979) 1. [21 T. Chang, J. Phys. A: Math. Gen. 12 (1979) L203. [31G.F. Smoot et aL, Phys. Rev. Lett. 39 (1977) 898, R.B. Partridge and D.T. Wilkinson, Phys. Rev. Lett. 18 (1967) 557. [4] M.H. Cohen et a!., Nature 268 (1977) 405. [5] S. Coleman et a!., Phys. Rev. 177 (1969) 2239; A. Salam and J. Strathdee, Phys. Rev. 184 (1969) 1750;
J.D. Hind, I! Nuovo Cim. 4A (1971) 71.
[6] F.R. Tangherlini, Nuovo Cim. Suppl. 20 (1961) no. 1. [71F.T. Trouton, Trans. Roy. Soc. 202 (1903) 165; Proc. Roy. Soc. 72(1903)132. [8] For a review of topics related to this subject see for example Physics of the expanding universe, ed. M. Demianski, (Springer-Verlag, Berlin 1979). [9] J. Rembielii!ski, in preparation.
PHYSICS LETFERS
7 July 1980
THE RELATiVISTIC ETHER HYPOTHESIS J. REMBIELII~SKI The Institute of Physics, University of Ldd~90-l36Lddzç Poland Received 27 March 1980
It is found that the “generalised Gahlean transformations” introduced by Chang form a subclass of the nonlinear (with respect to three-velocity) transformations of the Lorentz group. The concept of the relativistic covariance in a modified form leads to the necessity of existence of the absolute reference frame. The (formal) connection with usual formalism of special relativity is given. The form of the field equations and equations of motion is analysed. Some possible implications of this hypothesis are discussed.
Recently Chang [1,21 has proposed the anisotropic space-time variant of the Maxwell equations. This description of the electromagnetic phenomena seems
Firstly we note that the transformations (1). do not have the group property. However we can relate two inertial frames L,7 and La’ by the composed transfor-
not to conflict with present experimental status. Moreover it can be a basis for the interpretation of the cosnuc background radiation anisotropy [3,8] and some data from quasars and strong compact radio-galaxies radiation In his paper [1] Chang the existence of [4]. the absolute reference frameassumes E 0 (ether) and defines a class of the inertial frames La by the socalled generaliseci Galilean transformations
mationL ~ —~L0
(1) =
R
—
(e/c)~R0 + (e/o2)(y
—
—
— —).
-+
2 —1 0
(li-ow/c)
W
r
=
r ~W
7w’ ÷~ ~
ow
r
LT + br ,-
2 + ~/
w
iwr
where the vector-parameter w is given by the transformation formula for a’ 2)7~(1 +7w)~ e’= w+o~iç’+w(ew/c li-a ‘C2 (3)
1)eR.
Here y 2/c2)112 r°= Ct and r are the space0 = (1 c in time coordinates while R0 cT and R in Lo; a is the velocity of E 0 with respect2to= Lo.~dR~dR~’ The metric in Lo has the =Minkowskian form ds where diag(+ In this article we show that the inertial frames introduced by Chang are related by the transformations belonging to the nonlinear (with respect to three-velocity a) realization of the Lorentz group which linearizes for rotations from the subgroup 0(3). Consequently the variant of the Maxwell equations proposed in ref. [1] a) is Lorentz covariant, b) it can be obtained from the usual one by the a-dependent transformation. However the physical consequences of the Chang equations are different from the usual ones. —
rO=’y
—i
Note that eq. (3) is just the transformation law for velocity in special relativity. However, as will be evident later, the parameter can be identified withinerthe relative velocity for a verywrestricted class of the tial frames only. Nevertheless eq. (3) suggests that there is a connection between the transformation laws (2), (3) and the Lorentz group ones. Let us consider a class of the Lorentz frames related by the Lorentz group transformations A’~ ~ A. x v~C under which the metric form ds2 ~dx~dx~’ dxTn dx remains unaffected. Furthermore, let us introduce the four-vector uM satisfying u2 = c2. As is —
14
33
Volume 78A, number I
PHYSICS LETTERS
well known [5] the one-sheet hyperboloid defined by the invariant constraints u2 = c2, u0 >0 is the underlying manifold for the nonlinear realization of the @rthochronous) Lorentz group Lt which becomes linear with respect to the 0(3) subgroup. If we introduce the new coordinates a —uc/u°then the Lorentz boosts generate the transformations (3). Next, we define the “Galilean” coordinates r’2 by the formula 4: T’2~(e)xi’
rIL
T(u) =
ri_a/cl
Lo
I
j
=
D(A, u)r(u)
where
4(u)
~—1
=
dt
.~,
1Iuu/c2)2
—
u2/c2.
(9)
Here u 4: dr/dt is three-velocity and r0 = Ct. Note that the relative velocity V of La’ with respect to La has the form = I~0I~~ = w/(l + 0w/c2)
(5)
(6)
defines the absolute timec is(measured inL0). The velocity of light direction-dependent
Note that T~(a)= T(—a) and det T = 1. It is easy to see that r’2 transforms under the action of V as follows ~‘ r’(u’)
dT
i.e. w coincides with V only for a perpendicular to V. Here r 0 denotes the position of E0. with respect to E0. Furthermore, the invariant differential 2u~dx’2= c~2uT?iT(_a)dr = ‘y dT c_ 0dt (10)
where the matrix T(a) has the following form T(a)
by
7 July 1980
(7a)
c = cn/(1 + an/c) (11) where n = c/Id. Nevertheless result of the Michelson— Morley experiment is the same as in the special relativity: the measurement of the velocity of a light signal
7b
running around any closed path gives the average value c always [6]. Because of the relation
D(A, u) = T(Au)AT For rotations from 0(3), r splits on the singlet (r0) and triplet (r). For the Lorentz boosts *2 we obtain the formula (2) and consequently for a 0, w =a’ the Chang formula (1). In conclusion, the Chang transformations belong to the nonlinear realization of the Lorentz group which is induced by the nonlinear action of V on the symmetric homogenous space Lt/0(3). 2(x) reads in the Chang form The [1] invariant length ds ds2(r)=drTg(a)dr
u= ~/(1 +aVc2) (12) where ~ c dx/dx0 is the lorentzian three-velocity, we have dr(u) = dr(~)and thus we can derive the following formula connecting the “Galilean” and lorentzian (p and ir respectively) four-momentum dr
[p’2] ~P ~m
0 dr
—=
m0T(u)
=
T(e)ir
(13a)
or ~
mp=g(a)p=TT(_a)~.
(13b)
Explicitly (dr0)2 _2!drdr° +(!~dr)2—(dr)2
(8)
where [g~~] ~g(a)TT(_a),~T(_,)i.e. [g’2’~] = T( 0)~TT(q).Thus the invariant proper time is given *1
*2
The transformation law (7a) should be interpretedas passing from 2~,to E 0’; for this reason we wrote explicitly r = r(u). We adopt the following parametrization of A12~,(w): A(w)
34
1 7w[~
7-i5k
+
_w/c
wkwl
+
~
(l4a)
pm
2/c2. (14b) 0u/\/~i~i~u/c) 2 —v = invariant and the (conNote that p~p’2= m~ c served) translation generators are associated with p~ rather than p12.
For the electromagnetic waves which are radiated by a source approaching or receding from the observer (L0), the modified Doppler formula reads
Volume 78A, number 1
PHYSICS LETTERS 1/2
(I±~J-~)1hI2/(l uu~n) (15) \ ~ cc c ~ where n = u/u is the direction of the source velocity while c~and w denote initial and observed frequen-
-~-=
cies respectively. The above described formalism admits the following geometrical interpretation. With every inertial reference frame L0 we associate the set of the linear coordinate frames {r’2(e);a fixed} related by the transformations of the semidirect product 43 S0 = 0(3) ~ T~(rotations and space-time translations). Then the manifold of all linear frames at all inertial frames can be treated as the With we each point of 2 =fiber c2, u0bundle: >0 (basis) associate the set hyperboloid u the of linear frames {rM(a); a fixed} (fiber). S 0 is the structural group of this fiber i.e. TM. The transformations ofthe the stability Lorentz group group ofu relate the different fibres (points of the hyperboloid). Now let us consider the field concept in this formalism. Because of the relation (5) there is no difficulty for defining of the tensor fields. If 3~MiM2 ‘2n spans a standard tensor representation space of L then the quantity ...
F’21’22
‘2fl(r)
T’2’
(a)T’22
a
7 July 1980
a
1
~V
~j~2 ~‘
—~
ax”
~V =
0
—
‘ax”
a
T(a)
—
ar” i.e. —
—
!j’2, _~_F’2”= 0 C arv
x B +~x (! X E)l
a
I
F12z~=
I
=
V E C
C
VB = 0, VXI(B
J
C
a.o
V X E= —
~
XE)
=
~
(17)
—
-
+ ~~
at L
c
c
In the classical we must complement these equations with physics the equation of motion for a test charge 4:4 e dir dx” T(a) _!-~eF dr” (18) = e ~V dr dr 12” ~ A question arises how to construct the field equations for spinor fields. The correspondence (16) suggests that the bilinear forms ~(x)7M Li(x) and ~(r)’y’2(a)~(r), where x(r) denote the bi-spinors matrices in and the 712(0) presented formalism, shouldand be connected as can follows: ~(r)7M(a)X(r) = TM~(a)~(x)7”~J(x). Thus we identify ~-
1)2
(a)
...
T’2’~ (a)9~’1’-’2 “7(x) (16) Vfl -~
transforms covariantly with respect to the transformation law (7). The mixed tensors can be obtained with help of the metric tensorg 12~(a).Thus the tensor field equations are related to the usual (lorentzian) ones by the formula (16) (the derivatives a/arM and a/ax’2 are related by TT(_0)). If, for example, we identify the physical electric field E(r) and the magnetic induction B(r) with the components ~F0k and —eoe,kiFkJ of the antisymmetric tensor F,~(r)then the usual Maxwell equations for 9~12~(x) must be rewritten in the Chang form
712(0) = T12~(a)’yV
(19a)
and ~(r) = ~t,[x(r)]
=
i~i(T(—a)r).
(19b)
In the framework of the Rarita—Schwinger formalism of spinor—tensor fields we replace (19b) by x’2i’22...’2n(r) T’2fl
=
TMi~1(a)TM21)2(a)
(19c) 121122 Mn behaves i.e. under the (bi-) spinorspinor indices x Thus the field equaas standard (lorentzian) a)~”iV2
“7(x)
Un
~.
The eq. (18) is in fact an operational definition of the electromagnetic field. Note, that loosing of this equation causes (via the nondiagonality of g 121)(a)) an ambiguity for the definition of the physical (measured) quantities E and B. ~ See following page. 44
43
These transformations do not affect the u-velocity of ~a with respect to ~. The rotations belonging to S0 axe given by D(Ru, u) where RuU = U.
35
Volume 78A, number 1
PHYSICS LETTERS
tions for spinors can be obtained analogously to the tensor ones with the additional replacement 712 -+712(0). For example, the Dirac equation takes the form (~12
the reference frame in which the cosmic background radiation is isotropic [8]. Therefore it can be reasonable to start from the beginning with such assumption. It is interesting that under the conditions: a) space-
—+
~
12
a
(i ~ (a) ‘2
i.e. r
a/a~’2)
=
—
moc)x(r)
0
20 (
a)
,,
L~ (,~O÷! ~) ~
7 July 1980
time coordinates (r) form the singlet (rO) ÷triplet (r) of 0(3), b) rM’ and r’2 are related by the transformations linear with respect to r which form a realization of the Lorentz group, c) the physical three-space is
i~7v— mocj ~(r) = U (20b) Let us summarize the results. We have shown that the modified Lorentz covariance postulate leads to the necessity of existence of the absolute reference frame, The Chang [1]transformations form a subclass of the modifieda-dependent Lorentz transformations. In
physically euclidean one acceptable and r° possibilities: time, there theexist special only relativtwo ity and the presented hypothesis [91. Finally, we note that on the classical level the presented formalism can be equivalent to the special relativity one if dynamics does not mark out the absolute reference frame (via terms like U121/17127P in the lagrangian) [9].In fact, it
particular we have obtained in this framework the Chang variant of the Maxwell equations. Note that from the relations (5, 6) we have equality of the “Galilean” and lorentzian space coordinates: r = x. Therefore, as follows from the above described relation between the field equations (and equations of motion) in both formalisms (see eq. (16) and below), the trajectories of the charged particle moving in the electromagnetic field in the presented case and the special relativity one are identical. The difference lies in0the time development of the kinematical variables ‘=xO + (a/c)x). Thus the Trouton—Noble experi(r ment [71 gives in our case the negative result also. Furthermore, as follows from the eqs. (8—15) all kinematical effects are significant only for a/c 1. In the relativistic region for velocities (u/c 1), the specia! relativity effects dominate for a/c ~ 1. In conclusion, if we treat seriously a version of the hot big bang cosmological model then we should take mto account such possibility, despite of the fact that the coexistence of the Lorentz covariance and the absolute reference frame is surprising from the usual point of view. In fact, the big bang model marks out
may be possible to synchronize the clocks in the standard (lorentzian) fashion with the help of light signals. However on the quantum level both formalisms can be unitary nonequivalent even without dynamical requirements. Moreover, the eventual noninvariance of the vacuum state, caused by the big bang initial conditions for example, can be equivalent to the existence of the absolute reference frame [9].
C
5 *
36
+
Such situation is admissible because the spinor fields are not simply measured quantities.
-~
—
References [1] T. Chang, Phys. Lett. 70A (1979) 1. [21 T. Chang, J. Phys. A: Math. Gen. 12 (1979) L203. [31G.F. Smoot et aL, Phys. Rev. Lett. 39 (1977) 898, R.B. Partridge and D.T. Wilkinson, Phys. Rev. Lett. 18 (1967) 557. [4] M.H. Cohen et a!., Nature 268 (1977) 405. [5] S. Coleman et a!., Phys. Rev. 177 (1969) 2239; A. Salam and J. Strathdee, Phys. Rev. 184 (1969) 1750;
J.D. Hind, I! Nuovo Cim. 4A (1971) 71.
[6] F.R. Tangherlini, Nuovo Cim. Suppl. 20 (1961) no. 1. [71F.T. Trouton, Trans. Roy. Soc. 202 (1903) 165; Proc. Roy. Soc. 72(1903)132. [8] For a review of topics related to this subject see for example Physics of the expanding universe, ed. M. Demianski, (Springer-Verlag, Berlin 1979). [9] J. Rembielii!ski, in preparation.