Four-dimensional symmetry of taiji relativity and
coordinate transformations based on a weaker postulate
for the speed of light. – II
JONG PING HSU(1) and LEONARDO HSU(2)
(1) Physics Department, University of Massachusetts Dartmouth
North Dartmouth, MA 02747, USA
(2) Physics Department, University of California at Berkeley - Berkeley, CA 94720, USA
(ricevuto il 15 Maggio 1996; approvato il 9 Luglio 1996)
Summary. — Extended relativity is a theory of four-dimensional symmetry with Reichenbach’s time, in which only the 2-way speed of light is a universal constant. It includes special relativity as a special case. The theory is shown to be consistent with experiments such as Fizeau’s experiment, aberration of light and precision Doppler shifts. The formulations of classical and quantum electrodynamics are discussed. They are shown to be dependent on the four-dimensional symmetry rather than on the usual constant one-way speed of light. The four-dimensional symmetry also dictates a new coordinate transformation, called the “Wu transformation”, for constant-linear-acceleration frames.
PACS 03.30 – Special relativity.
PACS 11.30.Cp – Lorentz and Poincaré invariance.
We continue to demonstrate that the four-dimensional symmetry is necessary and essential [1] for discussing physical laws from Reichenbach’s viewpoint of time [2] or Edwards’ weaker postulate for the speed of light [3]. Edwards attempted in 1963 to formulate a relativity theory based on a weaker postulate that the 2-way speed of light in a vacuum is a universal constant. He derived space and time transformations which involve Reichenbach’s time but which do not form a four-dimensional Lorentz group in general. As a result, it leads to an incorrect expression for the relativistic energy-momentum of a particle in the Lagrangian formalism of mechanics and electrodynamics, as shown in paper I [1]. Furthermore, it appears to be impossible to obtain invariant forms of Maxwell equations and the Dirac equation if Reichenbach’s time is used as an evolution variable, as we shall see later. The reason is that Reichenbach’s time does not transform covariantly as the zeroth component of the coordinate 4-vector in general. Therefore, the lack of four-dimensional symmetry makes Edwards’ original transformations untenable.
Recently, we have formulated and discussed taiji relativity [4] based solely on the
first postulate of relativity, i.e. the invariance of physical laws. A four-dimensional transformation between any two inertial frames, F(w , x , y , z) and F 8(w 8, x 8, y 8, z8), is derived. Since taiji relativity does not make any assumption regarding the speed of light, the zeroth components w and w8 cannot be factored into a well-defined speed of light and time. In fact, the speed of light and usual time (measured in seconds) are unspecified and undefined in the theory. However, the theory of taiji relativity possesses the four-dimensional symmetry which is shown to be the only essential ingredient for the theory to be consistent with previous experiments. This sheds light on the difficulty encountered by Edwards’ transformations.
In paper I, we show that, guided by the four-dimensional symmetry of taiji relativity, Reichenbach’s general convention of time (or, equivalently, the universal 2-way speed of light) can be used as the “second postulate” for the construction of a new four-dimensional formalism of coordinate transformation which is termed extended relativity. The second postulate is necessary to factorize, say, w8 into a well-defined velocity function b8 (called “ligh”) and Reichenbach’s time t8 in the F8 frame, i.e. w 84b 8 t 8 which is called “lightime”.
(
See eqs. (2.1)-(2.3) in sect. 2.)
It turns out that the lightime w8, rather than Reichenbach’s time, plays the role of evolution variable in physical laws and makes extended relativity consistent with established energy-momentum of a particle, the Lorentz group, etc. Furthermore, the covariant lightime embedded in the four-dimensional symmetry is also crucial for the formulation of a covariant quantum electrodynamics (QED) based on extended relativity, as we shall see in sect. 6.2. – Extended relativity—A theory with universal 2-way speed of light
Reichenbach’s synchronization procedure amounts to imposing a second postu-late [1] upon taiji relativity [4], so that we have a well-defined “extended” time, which includes Einstein’s time as a special case. For simplicity and without loss of generality, we choose q 40 in the synchronization of clocks in the F frame, so that we have Einstein’s time t 4tE; while in the F8 frame, clocks are synchronized to read
Reichenbach’s time t8. An event is, as usual, denoted by (ct, x, y, z) in F. Following taiji relativity [4], the same event must be denoted by (b 8 t 8, x 8, y 8, z8) in F8 which is moving along the 1x axis with a constant velocity b4VOc, as measured in F. We stress that it is necessary to introduce the “ligh function” b8 so that (b 8 t 8, x 8, y 8, z8) 4 (w 8, r8) transforms like a four vector and laws of physics can display four-dimensional symmetry. It was shown by Edwards on the basis of the universal 2-way speed of light that times t and t8 were related by [3]
t 84g[ (12bq 8) t2 (b2q 8) xOc] , q 842e821 , b 4VOc , (2.1)
which was the basic property of Reichenbach’s time. Assuming relation (2.1) is effectively the same as assuming the universality of the 2-way speed of light over a closed path in any inertial frame within four-dimensional framework. The extended four-dimensional coordinate transformation was derived [1],
.
/
´
w 8fb 8 t 84g(ct2bx) , x 84g(x2bct) , y 84y , z 84z ; b 4VOc , g 41O(12b2)1 O2, b 84 (ct2bx)O[t(12bq 8)2 (b2q 8) xOc] 4c2q 8 x 8 Ot 8 , (2.2)where t8 is given by (2.1) and b8 is determined by (2.1) and (2.2). Although t8 and b8 in (2.1) and (2.2) separately have complicated non-covariant transformation property, their product w 84b 8 t 8 has a simple transformation property and is covariant.
The extended transformation of the velocities v and acceleration a can be derived from (2.2): we obtain
. / ´
c 8fd(b 8 t 8)Odt 8fvw8 4 [ dtOdt 8 ] g(c 2 bvx) ,
vx8 4 ( dtOdt 8 ) g(vx2 bc) ; vy8 4 ( dtOdt 8 ) vy, vz8 4 ( dtOdt 8 ) vz
(2.3) and
.
`
/
`
´
aw8 f dc 8Odt 84 [ d2tOdt 82] g(c 2bvx) 2 [ dtOdt 8]2gbax,
ax8 4 ( d2tOdt 82) g(vx2 bc) 1 [ dtOdt 8 ]2gax,
ay8 4 ( d2tOdt 82) vy1 [ dtOdt 8 ]2ay,
az8 4 ( d2tOdt 82) vz1 [ dtOdt 8 ]2az;
(2.4)
where v 8x4 dx 8 Odt 8, vx4 dxOdt , etc. and dtOdt 8 may be obtained from (2.1),
. / ´
1 O[ dtOdt 8] 4dt 8Odt4g][12bq 8]2 [b2q 8] vxOc( ,
d2tOdt 824 ( dtOdt 8 ) Z , Z f [ (b 2q 8) axOc] O[ ( 1 2 bq 8 ) 2 (b 2 q 8 ) vxOc] .
(2.5)
We stress that the definition c 84d(b 8 t8)Odt 8 is quite natural because when ds2
4 0 , i.e. [ d(b 8 t8) ]22 d r824 0 is the law of light propagation, c 82dt 822 d r824 0 . Note that we still have the four-dimensional law for the propagation of light in the F8 frame, although the speed of light is not isotropic and Reichenbach’s time is not covariant. Also, the transformations of the velocity ratios v8 Oc 8 turn out to be precisely the same as those in special relativity,
vx8 Oc 84 (vxOc 2 b) O( 1 2 vxbOc) ; vy8 Oc 84 (vyOc) O[g( 1 2 bvxOc) ] , etc .
(2.6)
This is crucial for the extended coordinate transformations (2.2) to form the Lorentz group. For the ratios of accelerations, we have
. / ´
ax8 Oaw8 4 [Z(vx2 cb) 1 ( dtOdt 8 ) gax]
O
[Z(vx2 cb) 2 ( dtOdt 8 ) gbax] ,ay8 Oaw8 4 [Zvy1 ( dtOdt 8 ) ay]
O
[Z(vx2 cb) 2 ( dtOdt 8 ) gbax] , etc . ,(2.7)
which are more complicated than those in special relativity because q 8c0 in Z given by (2.5).
The momentum pm
4 ( p0, px, py, pz) and wave vector km4 (k0, kx, ky, kz) transform
like a coordinate four-vector (2.2). For example, we have
k08 4 g(k02 bkx) , kx8 4 g(kx2 bk0) , ky8 4 ky, kz8 4 kz.
(2.8)
For a light wave, k 4 (k cosu, k sinu, 0) and k84 (k 8 cosu8, k 8 sinu8, 0), eq. (2.8) leads to the formula for the aberration of light,
cos u 84 ( cos u2b)O(12b cos u) , (2.9)
sin u 84sin uO[g(12b cos u) ] , (2.10)
where we have used k 4NkN4k0for a light wave. These results are the same as those
in special relativity because they both have the four-dimensional symmetry. 3. – Some experimental implications of extended relativity
In the Fizeau experiment, the observed drag coefficient can be explained by the addition law (2.6) for velocity ratios with vxOc 4 1 On :
vx8 Oc 84 ( 1 On 2 b) O( 1 2 bOn) , b 4VOc .
(3.1)
The speed of light relative to the medium (at rest in the F frame) is vx4 cOn in F, where
n is the refractive index of the medium. Now, if the medium is moving with speed V 4 bc parallel to the direction of light, the ratio v 8xOc 8 observed by a person at rest in F8 is
given by (3.1) as
vx8 Oc 84 1 On28 4 ( 1 On 2 b) O( 1 2 bOn) ` 1 On 2 b( 1 2 1 On 2) ,
(3.2)
where n 82is the “effective refractive index” of the moving medium with velocity 1V.
Assuming each tube in the Fizeau experiment has length L8 and the speed of water is 6V 4 6 bc , the optical path difference D L 8 of the two beams of light is
D L 842L 8 n28 2 2 L 8 n18 4 4 L 8 n 2 (VOc)(121On2) , (3.3) 1 On68 4 1 On 6 (VOc)( 1 2 1 On 2) (3.4)
as expected, since the optical path is just the distance in vacuum equivalent to the actual path length traveled by each beam [7].
The formula for the aberration of light can be obtained from the inverse transformations of (2.9) and (2.10):
tg u 4sin u8O[g( cos u81b) ] . (3.5)
This shows the deviation of light when transforming to a new reference frame. Note that the invariant law for the propagation of light
c 82
dt 82
2 d r824 0 (3.6)
does not refer to any specific source and, hence, it holds for light emitted from any source. From a microscopic viewpoint, the state of motion of a macroscopic source of light is actually irrelevant because photons are emitted from atoms in violent motion which may not even be uniform.
4. – Doppler shifts of frequency and atomic energy levels
The extended transformation of wave four-vectors k 4 (vOc, k) and k 84 (v 8 Oc 8, k8) is given by (2.8):
v 8Oc 84g(vOc2bkx) , kx8 4 g(kx2 bvOc) , ky8 4 ky, kz8 4 kz.
(4.1)
This implies that the Doppler wavelength shift in extended relativity is
1 Ol84g(1Ol2bOl) 4 (1Ol)[ (12b)O(11b) ]1 O2,
where kx4 2 pOl , ky4 kz4 0 , k 8x4 2 pOl 8 . This is the same as that in special
relativity.
However, the consistency of the Doppler frequency shift in (4.1) with the result of laser experiments is more subtle because c8 is not a constant in extended relativity. Since experiments which measure the frequency shift involve the absorption of photons by atoms, we must first re-examine the nature of atomic levels from the point of view of extended relativity.
In extended relativity, Dirac’s Hamiltonian HD for a hydrogen atom is
. / ´ icˇ ¯cOˇw4HDc , HD4 2aQ Pc 2bmc22 e2O( 4 pr) , P 42iˇ˜ , (4.3)
which is given in the Dirac equation (6.6) below. Using the usual method, it can be shown that (4.3) leads to atomic energy levels
. / ´ En4 mc 2 O] 1 1 a2eO[n 2 h01 (h022 ae 2 )1 O2]2(1 O2, h04 j 1 1 O2 , ae4 e2O( 4 pcˇ) . (4.4)
Thus when an electron jumps from a state n1 to another state n2 , it will emit or absorb an energy quantum cˇk0:
En22 En14 cˇk04 ˇv , in F ,
(4.5)
E 8n22 E 8n14 cˇk 804 cˇv 8Oc 8 , in F 8 ,
(4.6)
where c8 and c in (4.6) do not cancel in F8, except for the special case q 840. If two photons with “energy” (v0Oc) cˇ and (v 80Oc8) cˇ are emitted from two hydrogen atoms
at rest in F and F8, respectively, then by the equivalence of F and F8 frames, (v0Oc) cˇ 4 (v08 Oc 8 ) cˇ .
(4.7)
Here, the ratio v 80Oc 8 is isotropic. However, if (vOc) cˇ and (v 8 Oc8) cˇ are energies of
the same photons measured from F and F8, respectively, then they are related by (4.1)
v 8Oc 84g(vOc)(12b cos u) , kx4 k cos u 4 (vOc) cos u .
(4.8)
Evidently, only when c 84c do we have the usual relation for the Doppler frequency shift. In general, we can only talk about an “energy shift” or a “k0 shift” in extended
relativity because the energy of a quantum particle is not always proportional to the corresponding frequency in all frames. In other words, the energy [1] of a particle or k0
is the zeroth component of a 4-vector, but the frequency (defined on the basis of Reichenbach’s time) is in general not covariant.
Experimentally, one never measures the frequency directly unless constant one-way speed of light is presupposed (so that the frequency becomes well-defined in all frames); instead, what one really measured is the shift of atomic levels in interaction with radiation or a laser that is measured in a general frame. Thus, (4.4)-(4.8) are consistent with precision measuring experiments of Doppler shifts.
Since Reichenbach’s time is used in F8, the speed c8 of a photon and its frequency v8 in F8 is anisotropic in general; however, the ratio v8 Oc8 is isotropic. Four-dimensional symmetry dictates the mixture of two waves in F8 in terms of (w 8, r8) and (v8 Oc8, k8)
rather than (t 8, r8) and (v8, k8), so their superposition is in general given by A0sin (k 801w 82k81Q r8)1B0sin (k 802w 82k28 Q r8 ) , w 84b 8 t 8 .
(4.9)
Only in the F frame, where c is, by definition, isotropic and constant, does one have the usual expression A0sin (v1t 2k1Q r) 1B0sin (v2t 2k2Q r).
5. – Classical electrodynamics in extended relativity
Although the one-way speed of light in extended relativity is not a universal constant, one still has the universal 2-way speed of light c. Thus, the usual action for a free particle, 2
s
mc ds , is an invariant in extended relativity. If a charged particle with mass m and charge e is moving in an electromagnetic field in a general frame, the invariant action is assumed to be [1]S 42
mc ds 2 (eOc)Am dxm2 ( 1 O4 c)FmnFmnd3r dw , (5.1) Fmn4 ¯mAn2 ¯nAm, (5.2) ds 4 (gmndxm dxn)1 O24 dt (C22 v2)1 O2, C 4dwOdt , v 4drOdt , (5.3) eOc421.6021891Q10220( 4 p)1 O2( g Q cm )1 O2, (5.4)where gmn4 ( 1 , 21 , 21 , 21 ), xm4 (w , x , y , z), eOc is in Heaviside-Lorentz units and
Am is the same as the usual electromagnetic vector potential in special relativity. Note
that C 4dwOdt is in general not a constant in extended relativity.
For a charged particle moving in an electromagnetic potential field Am, the
invariant action is given by Scp,
Scp4
Lcpdt , Lcp4 mc( 1 2 v2OC)1 O22 (eOc) Am dxmOdt ,(5.5)
which consists of the first two terms in (5.1), dxm
Odt 4 (C , v) and Am4 (A0, 2A). The
canonical momentum of a charged particle is now given by
. / ´ P 4¯LcpO¯ v 4 p 1 eAOc , p 4 (mcvOC)
O
( 1 2v2 OC2)1 O2. (5.6)Note that we have used the universal 2-way speed of light c to make p having the usual dimension of mass times velocity. In contrast, there is no universal speed of light in taiji relativity, so that the covariant momentum must have the dimension of mass [4]. Following taiji relativity, we define the covariant Hamiltonian H as
. / ´ H f [ (¯LO¯v)Qv2L](cOC) 4cp01 eA0, p04 mc
O
( 1 2v 2 OC2)1 O2. (5.7)Note that the factor cOC in definition (5.7) is necessary for the Hamiltonian HOc to
transform as the zeroth component of the momentum 4-vector, HOc4P0. Otherwise,
the Hamiltonian will not be meaningful. From (5.6), (5.8) and HOc4P0 we have
(P02 eA0Oc)22 (P 2 eAOc)24 m2c2.
(5.8)
The Lagrange equation of motion for a charged particle in the electromagnetic field has the usual form
dpmds 4 (eOc) FmndxnOds ,
(5.9) where pm
4 ( p0, p) and xm4 (w , 2r).
Making the substitutions P K2iˇ˜ and P0K iˇ ¯O¯w , we obtain the extended
relativistic Klein-Gordon equation
[ (iˇ¯O¯w2eA0Oc)22 (2iˇ˜ 2 eAOc)2F(w , r) 4m2c2F(w , r) .
(5.10)
For a continuous charge distribution in space, the second term in (5.1) should be replaced by 2
s
AmJmd3r dw . In this case, the variation of (5.1) leads to the invariantMaxwell equations in a general frame
. / ´ ¯nFmn4 Jm, ¯lFmn1 ¯mFnl¯nFlm4 0 , Fmn4 ¯mAn2 ¯nAm, ¯m4 ¯O¯xm, xl4 x(w , r) , (5.11)
Thus, Maxwell’s equations have the invariant form, even if the one-way speed of light is not universal. This is consistent with the result in taiji relativity.
6. – Quantum electrodynamics based on extended relativity
We demonstrate that an extended quantum electrodynamics (QED) can be formulated because extended relativity has the four-dimensional symmetry with the lightime w8 as the evolution parameter in the F8 frame. This would be impossible if one attempts to use Reichenbach’s time t8 in Edwards’ transformation [3] as the evolution parameter because t8 does not transform as the zeroth component of a 4-vector. In
extended QED, the invariant action Sq involves c and Am is assumed to be
Sq4
L d4x , L 4c[gm(iˇ¯m1 eAmOc 2 mc] c 2 ( 1 O4 c) FmnFmn,(6.1)
where e E0 and d4
x 4dw d3r . For quantization of the fermion fields, for example, the
“canonical momentum” pb conjugate to cb is defined as usual to be
pb4 ¯LcO¯(¯0cb) , Lc4 c[gmiˇ¯m2 mc] c ,
(6.2)
and the Hamiltonian density for a free electron is H 4p¯0c 2Lc. For free-photon (Am)
and electron (c) fields, we have (6.3) Am(w , r) 4 (1OV1 O2)
!
p ; a(ˇO2p0)
1 O2Q
(6.4) c(w , r) 4 (1OV1 O2)
!
p ; s(mOp0)
1 O2Q
Q
[
b( p , s) u( p , s) exp [2ipQxOˇ]1d†( p , s) v( p , s) exp [ip Q xOˇ]
]
, p Q x 4pmxm,where
. / ´
[A( p , a), A†( p8, a8) ] 4dpp8daa 8, [b( p , s), b†
( p8, s 8) ] 4dpp8dss 8, [d( p , s), d†( p8, s 8) 4dpp8dss 8;
(6.5)
and all other commutators vanish.
The Dirac equation in extended relativity can be derived from (6.1). We obtain iˇ ¯cO¯w4 [aQ (iˇ˜ 1eAOc)2bmc1eA0Oc] c .
(6.6)
In view of the equations of motion (5.10) and (6.6), we must use the lightime w in a
general frame as the evolution variable for a state F( S )(w) in the Schroedinger
representation: iˇ¯F ( S ) (w) ¯w 4 H ( S )(w) F( S )(w) , H( S ) 4 H0( S )1 HI( S ), (6.7)
because the evolution of a physical system is assumed to be described by a Hamiltonian operator which has the same transformation property as that of w.
The usual covariant formalism of perturbation theory [5] can also be applied to quantum field theory based on extended relativity. To illustrate this point, let us briefly consider the interaction representation and the S-matrix based on extended relativity. The transformations of the state vector F(w) and operator O from the Schroedinger representation to the interaction representation are
F(w) f F( I )
(w) 4exp [iH0( S )wOˇ] F( S )(w) ,
(6.8)
O(w) 4O( I )
(w) 4exp [iH0( S )wOˇ] O( S )exp [2iH0( S )wOˇ]
(6.9) Since O( S )
and O(w) are the same for w 40, we have
(6.10) iˇ¯F(w)
¯w 4 HI(w) F(w) , HI4 exp [iH0
( S )
wOˇ] HI( S ) exp [2iH0( S )wOˇ] ,
(6.11) O(w) 4exp [iH0( S )wOˇ] O(0) exp [2iH0( S )wˇ] .
The U-matrix can be defined in terms of the lightime w: F(w) 4U(w, w0) F(w0),
U(w0, w0) 41. It follows from (6.10) and (6.11) that iˇ ¯U(w , w0)
¯w 4 HI(w) U(w , w0) . (6.12)
If a physical system is in the initial state Fiat lightime w0, the probability of finding it
in the final state Ff at a later lightime w is
N
aFfNU(w , w0) FibN
24NUfi(w , w0) N2.Evidently, the average transition probability per unit lightime for FfK Fi is
NUfi(w , w0) 2dfiN2O(w 2 w0) .
(6.14)
As usual, we can express the S-matrix in terms of the U-matrix, i.e. S 4U(Q, 2Q) and obtain the following form:
(6.15) S 412 (iOˇ)
2Q Q HI(w) dw 1 (2iOˇ)2 2Q Q HI(w) dw 2Q w HI(w 8) dw 81R .For w-dependent operators, one can introduce a w-product W (corresponding to the usual chronological product), so that one can write (6.15) in an exponential form:
S 4W
{
expy
2(iOˇ) 2Q Q HI(xm) dw d3rz
}
, (6.16) 2Q Q HI(xm) d3r 4HI(w) . (6.17)For simplicity, one may set ˇ 4c41, where c is the 2-way speed of light. (These are the “natural units” in extended relativity.)
For the QED Lagrangian (6.1), we can derive Feynman rules (based on extended relativity). Let us summarize the Feynman rules for QED with the Lagrangian L in (6.1) with a gauge-fixing term (¯mAm)2O( 2 a8):
LQED4 L 2 (¯mAm)2O( 2 a 8 ) , ˇ 4 c 4 1 , e E0 .
(6.18)
The covariant photon and electron propagators are
2i[gmn2 ( 1 2 a 8 ) kmknO(kl21 ie) ,
(6.19)
iO(gmpm2 m 1 ie) .
(6.20)
The vertex factor is
iegm.
(6.21)
There is a factor emfor each external photon line and a factor u( p , s) for each absorbed
electron, a factor 21 for each closed fermion loop, etc. These rules are identical to those in the conventional theory, if the natural unit is used.
Thus, if one calculates scattering cross-sections and decay rates (with respect to the lightime w) of a physical process in a general frame, one will get the same result as that in special relativity [5]. For example, let us consider the decay rate G( 1 K21 3 1R1N) for a physical process 1 K2131R1N. It is given by
G( 1 K2131R1N) 4 lim w KQ
N
a fNSNibN
2 w d3x 2d3p2 ( 2 pˇ)3 R d3x Nd3pN ( 2 pˇ)3 , (6.22)which has the dimension of inverse length. The decay length D is given by D 4 1 OG(1 K2131R1N). Thus in extended relativity, we have the dilatation of the decay length for a particle decay in flight. Since we have the universal 2-way speed of
light c, we can use it to define the lifetime t of a particle by t 4DOc .
(6.23)
We stress that such a lifetime has little to do with Reichenbach’s time in general. The result (6.23) is also consistent with experiments because the decay length dilatation is what one actually measured in the high-energy laboratory. This is not surprising because we use the same four-dimensional symmetric Lagrangian as that in special relativity.
7. – Limiting four-dimensional symmetry and accelerated Wu transformation One may wonder whether the power of four-dimensional symmetry is strong enough to say something about constant-linear-acceleration (CLA) frames. The attempt to generalize the coordinate transformation for inertial frames to that for CLA frames through a symmetry consideration is very natural because the transformation for a CLA frame must reduce to that for an inertial frame in the limit of zero acceleration. So far, no satisfactory transformation for such non-inertial frames has been obtained in the literature, even though one has general relativity and the correspondence principle [6]. All those accelerated transformations discussed previously are not based on a symmetry principle and do not naturally reduce to the four-dimensional transformation for an intertial frame when the acceleration approaches zero.
By a stroke of luck, we have found a transformation for CLA frames which does reduce to the correct four-dimensional transformation in the limit of zero acceleration and reduces to the Galilean accelerated transformation when the velocity is small. For simplicity of notation and calculations in an accelerated frame, let us denote a CLA frame by F(w , x , y , z) and an inertial frame by FI(wI, xI, yI, zI). Suppose a CLA
frame F(w , x , y , z) is moving with a constant acceleration a , so that its velocity is b 4 aw 1b0, along the 1x axis. We find that the accelerated transformation between FI
and F is given by
.
/
´
wI4 gb(x 1 1 Oag0 2 ) 2b0Oag0, xI4 g(x 1 1 Oag0 2 ) 21Oag0, yI4 y , zI4 z ; b 4aw1b0, g 41O(12b2)1 O2, g04 1 O( 1 2 b20)1 O2, (7.1)which will be called the “Wu transformation”. If one wishes, one may define wI4 ctI,
where tI is the usual Einstein time, in (7.1) for easy comparson with special relativity.
(But this definition is not necessary for deriving experimental results.) The inverse Wu transformation of (7.1) is
.
`
/
`
´
w 4 wI1 b0Oag0 a(xI1 1 Oag0) 2 b0 a , x 4 [ (xI1 1 Oag0) 2 2 (wI1 b0Oag0) 2 ]1 O22 1 Oag20, y 4yI, z 4zI. (7.2)One can verify that (7.1) and (7.2) reduce to four-dimensional transformations of the form (2.2) in the limit of zero acceleration a. (See appendix.) We may remark that the
coordinate transformation between two CLA frames can be derived on the basis of (7.1) or (7.2).
From the viewpoint of limiting four-dimensional symmetry, the CLA transforma-tion must be expressed in terms of the Cartesian coordinates rather than other coordinates, just like the Lorentz transformation. Furthermore, the coordinates of CLA frames should play the same role and have a similar physical meaning as those of inertial frames. This appears to be different from the usual viewpoint that coordinates for accelerated frames have no physical meaning. The Wu transformation (7.1), based on the four-dimensional symmetry, differs from that obtained by Møller [6] based on the approximate principle of equivalence in general relativity because they give different spatial measurements by meter sticks or the Bohr radius of hydrogen atoms. We believe that such a difference should be tested by, say, measuring a Doppler shift of wavelength emitted from a source with a constant linear acceleration. We may remark that the constant acceleration a in (7.1) can be shown to be related to constant change of “energy” (or “moving mass”) per unit length measured in an inertial frame. This differs from the usual definition of acceleration in (2.4). It is interesting to note that such a constant acceleration a dictated by the limiting four-dimensional symmetry is precisely what has been actually realized in linear accelerators in laboratories. Physical implications of the Wu transformation and their experimental tests will be discussed in a separate paper.
8. – Remarks and discussions
In the formulation of QED in sect. 6, the electron is, as usual, assumed to be a point particle. However, if the physical electron is really a fuzzy point (in the sense of fuzzy
set theory with a bell-shape membership function having a width L0) rather than a
geometric point, then there will be a departure from the four-dimensional symmetry at short distances or large momentum [7]. A fuzzy-point model of a particle has been interpretated as follows: a particle by itself is a structureless-point particle, but it can simultaneously exist at different places with a different probabilities. As a result, the
position uncertainty of such a quantum particle has a minimum width Dx AL0. The
Coulomb potential will be modified when r EL0, and the photon propagator in (6.19)
will be modified when momentum becomes larger than ˇOL0. For a detailed discussion
of the fuzzy-point model of particles, we refer to ref. [7].
Let us compare and summarize basic differences in various relativity theories: a) Taiji relativity: it is based solely on the first postulate of relativity, namely, the invariance of physical laws. A four-dimensional transformation between two inertial frames, F(w , x , y , z) and F 8(w 8, x 8, y 8, z8), can be derived. The usual concept of time and speed of light are undefined and completely unknown; nevertheless, the theory agrees with all experiments. Taiji-times w and w8 play the roles of evolution variables, and the dimensionless “taiji-velocities” dxOdw, dx 8 Odw 8, etc. are well-defined. It is interesting that there are only two universal and fundamental constants in QED based on taiji relativity.
b) Extended relativity: it is based on two postulates. In addition to the invariance of physical laws, its second postulate is the universality of the 2-way speed of light. Reichenbach’s time and one-way speeds of light (non-isotropic in general) are well-defined. However, lightime w plays the role of the evolution variable in
four-dimensional physical laws. There are three universal and fundamental constants in QED based on extended relativity. Special relativity is a special case
(
q4q 840 in (2.2))
.c) Common relativity: it is based on two postulates. The additional second postulate is a common time t 84t for all observers [8]. The speed of light is, roughly speaking, relative. Lightimes w and w8 are evolution variables in four-dimensional laws. There are only two universal and fundamental constants in QED based on common relativity, precisely the same as those in taiji relativity. Common relativity has the unique advantage for dealing with many-particle systems where canonical evolution of the system is essential and for obtaining covariant thermodynamics and invariant Planck’s law of black-body radiations [9].
One may ask: how can one realize the evolution variable w8 in the extended coordinate transformation (2.2) by physical means? Since the invariant phase of an electromagnetic wave in the F8 frame is given by k 80w 82k8Qr8, where k 804 Nk8N , we can
define the lightime w8 in terms of k 80, just as the length can be defined by the
wavelength l8 or Nk8N. We note that the “clocks”, which show lightime in this theory, are the same as those in taiji relativity [4] because they have exactly the same four-dimensional transformation property. However, the taiji-time w 8 in F 8 cannot be factored into two well-defined b 8 and t 8 because of the absence of a second postulate while the lightime w 8 in extended relativity and common relativity can be factored into two well-defined functions b 8 and t 8, as shown in (2.1), (2.2) and ref. [8].
Our discussions show that it is extremely important to be aware of what quantities are actually measured in the experiments and what effects the assumption of a universal speed of light may have had on the interpretation of the results. For example, we have seen in paper I that the lifetime dilatation of unstable particle decay in flight has little to do with the property of Reichenbach’s time with a general parameter q or q8, because the lifetime t is basically defined as the decay length divided by the universal 2-way speed of light c. The basic reason is that the four-dimensional symmetry dictates that the decay rates in, say, QED based on extended relativity can only be defined in terms of the covariant lightime w or w8 which has the dimension of length.
The constant 2-way speed of light in extended relativity is in general not the maximum speed of physical objects in the universe. Rather, it is the one-way speed of light in a given direction, that is the maximum speed of any object in that direction, as shown in (2.3). This holds for any inertial frame. It is worthwhile to note that this property of light, being the “maximum speed” of all physical objects in any given direction, is a logical consequence of the first postulate of relativity, as shown in taiji relativity [4].
We have examined a number of experimental tests of special relativity and the formulations of classical electrodynamics and QED. All of them are consistent with extended relativity. These discussions can be generalized to other field theories such as unified electroweak theory and quantum chromodynamics. As we have seen, only the four-dimensional symmetry of physical laws is absolutely essential for explanations of experimental results and for the formulation of classical electrodynamics and QED; the universality of the one-way speed of light is irrelevant. In this connection, we stress that according to the theory of taiji relativity [4], the universality of the one-way or the two-way speed of light is a convention rather than an inherent part of the physical world [10].
Note added in proofs
Suppose one writes dwI4 g(T dw 1 Ub dx), dxI4 g(V dx 1 Wb dw), dyI4 dy , dzI4 dz ;
g 41/(12b2)1 O2, where T , U , V and W are four unknown functions of x and w . The new Wu
transformation (7.1) for a constant-linear-acceleration (CLA) frame can be derived from the postulate of the limiting four-dimensional symmetry of taiji relativity and the initial condition that a CLA transformation reduces to the spatial identity rI4 r when the taiji-time w 4 0 and the
initial “velocity” b04 0 . This initial condition holds also for the Lorentz transformation. Thus,
once the principle (or the first postulate) of relativity is rigorously stated to include the limiting cases, the concept of acceleration is determined in the physical theory based on “extended four-dimensional framework”.
Within the present conceptual framework, the taiji-time w in the Wu transormation (7.1) or (A.2) is a primary concept and has the dimension of length. The motion of physical objects, including light signals, is a derived concept and described by dimensionless “taiji-velocities” dr/dw . The taiji-time w can be realized by computerized “Leonardo clocks” [4]: We could program any Leonardo clock in a CLA frame F to obtain a reading wIfrom the nearest clock in an inertial
frame FI and, based on its FIframe position xI and given parameters a and b0, compute the
taiji-time w it should display, w 4 (wI1 b0/ag0) /[a(xI1 1 /ag0) ] 2b0/a .(See (7.2).)In the limit
of zero acceleration w shown on a Leonardo clock will automatically reduce to the taiji-time in the four-dimensional transformation, w 4y0(wI1 b0xI). It will not reduce to relativistic time, unless
the second postulate of universal constant for the speed of light (w f ct , wIfctI) is made in this
limit [4].
* * *
This paper is dedicated to Prof. TA-YOUWUfor his wonderful and tireless teaching of physics and his ninetieth birthday. The work was supported in part by The Jing Shin Research Fund of the UMass Dartmouth and by a grant from the Potz Science Fund.
AP P E N D I X
Limiting four-dimensional symmetry and constant-linear-acceleration frames For simplicity, let us denote a CLA frame by F(w , x , y , z) and an inertial frame by FI(wI, xI, yI, zI). Suppose a CLA frame F(w , x , y , z) is moving with a constant
acceleration a, so that its velocity is
b 4aw1b0,
(A.1)
along the 1x axis. Guided by the limiting four-dimensional symmetry, we find that the
linearly accelerated transformation between FI and F should be
.
/
´
wI4 gb(x 1 1 Oag20) 2b0Oag0, xI4 g(x 1 1 Oag20) 21Oag0, yI4 y , zI4 z , b 4aw1b0, g 41O(12b2)1 O2, g04 1 O( 1 2 b20)1 O2. (A.2)This is a generalization of the accelerated transformation obtained by Wu and Lee [7] based on a kinematic approach to satisfy limiting four-dimensional symmetry. It will be
called the “Wu transformation”. When b0approaches zero, the accelerated
transforma-tion (A.2) can reduce to the well-known transformatransforma-tion obtained in ref. [7], provided
one uses a new time t : b 4aw1b04 tgh (at). Furthermore, one can verify that the
Wu transformation (A.2) indeed reduces to four-dimensional transformations in the limit of zero acceleration a K0: wI4 g0(w 1b0x), xI4 g0(x 1b0w), yI4 y , zI4 z ;
where g04 1 O( 1 2 b20)1 O2.
With the definitions wI4 ctIand w 4ct, the “extented relativistic time” t in the CLA
frame F is completely determined by (A.2). In other words, if the time tI and the
position xI of an event as observed in the inertial frame FI is known, then the
corresponding time t in the CLA frame can be calculated, provided the constants c, a
and b0 are given. Such a time t in a CLA frame can be physically realized by
computerized “Leonardo clocks” [4]: evidently, the time t in the CLA frame is not the relativistic time in general and, hence, the constant c by itself is not physically meaningful. Only when the acceleration a vanishes, the time t in (A.2) for the frame F reduces to the relativistic time and c becomes physically meaningful. We may remark that the definitions wI4 ctIand w 4ct are not necessary for deriving observable results
because we may directly use w as evolution variables.
When b0K 0 , the inverse of the Wu transformation (A.2) leads to
. / ´ w 4ctIO( 1 1 axI) BctI( 1 2axI) , ctI4 wI, x 4 (1Oa)[112axI1 a2(xI22 c2tI2) ]1 O22 1 Oa B xI2 c2atI2O2 . (A.3)
Thus, c2a is related to a constant acceleration g in Newtonian mechanics by the relation a4gOc2
(A.4)
when velocities are small. In this sense, the Wu transformation (A.2) is a four-dimensional generalization of the Galilean transformation for accelerated frames in classical mechanics.
From (A.1) we obtain ds2
4 c2dt2
I2 drI24 g00dw22 dr2, g004 g4(g220 1 ax)2,
(A.5)
where x in a CLA frame F is restricted to the region x Dxsf 21 O(ag20) which may be
pictured as a “wall singularity” at xs. We may remark that finite Wu transformation
(A.2) implies that the space-time of the CLA frame F(W , x , y , z) is flat, i.e. the Riemann curvature tensor vanishes, Rijkm4 0 , which can also be directly calculated by
using the metric tensors in (A.5).
The velocity of a fixed point xI in FI as measured by F-observers using evolution
variable [4] w is dxOdw with xI fixed. From (A.2), we find
( dxOdw)xI4 b(g00)
1 O2; x D21O(ag2
0) , 1 Db2.
(A.6)
We see that only in the approximation g00B 1 do we have (dxOdw)xIB b and
( d2xOdw2)XIB a 4 constant .
(A.7)
We note that the Wu transformation (A.2) holds for general wIand w. In the limit of
zero acceleration, it reduces to the four-dimensional taiji transformation [4]. If one wishes, one may define
wI4 ctI and w 4bt , b f (ctI2 bxI) O[tI( 1 2bq 8)2 (b2q 8) xIOc] ,
where tI and t are, respectively, Einstein’s time and “extended” Reichenbach’s time
(and b is the corresponding “ligh function”), then the limit of zero acceleration of (A.2) is the extended transformation (2.2)
(
where the inertial frame F 8 corresponds to the CLA frame F of (A.2) in the limit of zero acceleration)
. One can formulate, say, classical electrodynamics in a CLA frame. According to taiji relativity, physical results in the CLA frame F should be independent of the definition in (A.8).R E F E R E N C E S
[1] HSUL., HSUJ. P. and SCHNEBLED., Nuovo Cimento B, 111 (1996) 1299. This is referred as paper I in the text.
[2] REICHENBACH H., The Philosophy of Space and Time (Dover, New York) 1958. [3] EDWARDS W.F., Am. J. Phys., 31 (1963) 482.
[4] HSU J. P. and HSU L., Phys. Lett. A, 196 (1994) 1; 217 (1996) 359; HSUL. and HSUJ. P.,
Nuovo Cimento B, 111 (1996) 1283.
[5] See, for example, BJORKEN J. D. and DRELL S. D., Relativistic Quantum Mechanics (McGraw-Hill, New York) 1964, pp. 261-268 and pp. 285-286; SAKURAI J. J., Advanced
Quantum Mechanics (Addison-Wesley, Reading, Mass.), 1967, pp. 171-172 and pp.181-188;
WEINBERG S., The Quantum Theory of Fields (Cambridge University Press, New York) 1995, pp. 134-147.
[6] MØLLER C., Danske Vid, Sel. Mat.-Fyz., xx, No. 19 (1943); FOCK V., The Theory of Space
Time and Gravitation (Pergamon, New York) 1958, pp. 206-211; WUT. Y. and LEEY. C., Int.
J. Theor. Phys., 5 (1972) 307; TA-YOUWU, Theoretical Physics, Vol. 4, Theory of Relativity (Lian Jing Publishing Co., Taipei) 1978, pp. 172-175.
[7] HSUJ. P., Nuovo Cimento B, 80 (1984) 183; 88 (1985) 140; HSUJ. P. and PEIS. Y., Phys. Rev.
A, 37 (1988) 1406.
[8] For a detailed discussion of common time in four-dimensional framework and its implica-tions, see HSU J. P., Nuovo Cimento B, 74 (1983) 67; 88 (1985) 140; 89 (1985) 30; Phys.
Lett. A, 97 (1983) 137; HSU J. P. and WHAN C., Phys. Rev. A, 38 (1988) 2248, appendix. [9] HSU J. P., Nuovo Cimento B, 93 (1986) 178.
[10] In other words, all physical results in taiji relativity or extended relativity can be derived by simply using the quantities (w , x , y , z) and (w 8, x 8, y 8, z8) without ever mention time t or t8 (measured in seconds) and speeds of light or other physical objects.
