uniformly with respect to the frame X, then the guy are reduced to rlrav and. I"~v = O. ..... H. Nordenson, Relativity, Time and Reality (George Allen and 11nwin,.
Physics Essays
volume 4, number 1, 1991
Relativistic Kinematics I: A Theory of Relativistic Kinematics Based on Physical Reality ~u~-~aHu~g
Abstract
This is the first in a series of papers on a new theory of relativistic kinematics ("Relativistic Kinematics It, III, and IV,," submitted to Phys. Essays). A new theory of relativity is deduced from assumptions whose validity is closely based on generally accepted experiments. The formalism of this theory provides a unified theoretical scheme that relativistically modifies the classical force laws: the electromagnetic force law and the gravitational force law ("Relativistic Kinematics II and III"). We emphasize that the relativistic transformation, the Lorentz transformation, shouM be interpreted as a transformation in velocity space only, not a transformation in space-time coordinates space as is done in Einstein's theory of relativity [A. Einstein, The Meaning of Relativity, 5th ed. (Princeton University Press, 1956)]. The concept of space and time entailed by this theory is different than Einstein's space-time concept.
Key words: relativity, concepts of space and time, global Lorentz transformation in spacetime coordinates space, instantaneous Lorentz transformation in velocity space, relativistic equations of motion.
1. INTRODUCTION Since Einstein presented his theory of relativity, many experimental tests have been claimed as corroboration of the theory. However, his theory is not a mere attempt to describe the physical phenomena. His theory is a far-reaching attempt to give a theory of space and time/~) The new spacetime concept introduced in Einstein's theory of relativity is not completely convincing. (2)-(5) If one tries to redefine the space and time structure of a coordinate reference frame by mathematical terms and formulas which are formulated in terms of this coordinate reference frame, one will fall into a tautology from the logical point of view. For instance, in Einstein's general relativity, the metric and the imaginary coordinate-independent invariants such as curvature have definitional problems when one tries to specify the units used in space and time that are dependent on the matter present. The presence of matter disturbs the structure of the space-time, whereas units for space and time are needed to specify the distribution of the matter. A new theory of relativity, which is closely based on the experimentally measured physical reality, is formulated from a purely kinematic point of view. Some mathematical formulas obtained by this theory are the same as in Einstein's special relativity, for example, the formulas for the relativistic addition of velocities and the relativistic energy-momentum forms. Nevertheless, the conceptual foundations of this theory are different than Einstein's special relativity. For example, the time concept of this theory is 68
still the same as the Newtonian concept of universal time, and the relativistic transformation of this theory is an instantaneous Lorentz transformation in velocity space, not a transformation in the whole space-time coordinates space. In contrast, the relativistic transformation of Einstein's special relativity is a global Lorentz transformation in the space-time coordinates space, and Einstein's space-time concept is then introduced accordingly. The paper is organized as follows. In Sec. 2 the assumptions of this new theory of relativity based on experimental results are first presented. In Sec. 3 the invariant differential length of the infinitesimal displacement 4-vector of a uniformly moving particle with respect to reference frames is formulated from those assumptions. In Sec. 4 the differential Lorentz transformation, which is an instantaneous transformation in velocity space only, is derived. In Sec. 5 the essential idea of this theory of relativity for describing a nonuniformly moving material particle is presented, and the relativistic equations of motion are then derived. In Sec. 6 the differences between this theory and Einstein's special relativity are discussed.
2. PRELIMINARIES 2.1 Coordinate Reference Frames When one tries to describe the motion of particles, a coordinate reference frame should be presupposed. A concrete method for describing the
Young-Sea Huang motion of particles is first to introduce a reference frame that has stipulated measurements for the space and time units. That is, unit standards of measurements should be used to build up the three-dimensional space coordinates and one-dimensional time coordinate which can then be employed for describing the motion of particles. The relativistic transformations in relativity relate the values of physical quantities between reference frames. In order to ensure that the relativistic transformations of physical quantities among reference frames are physically meaningful, these reference frames should be presumed to have the same stipulation of the space and time units. Hereafter, all the coordinate reference frames under consideration are supposed to have the same stipulation of the space and time measurements. It is an empirical fact that if a particle moves uniformly with respect to any one of the reference frames, which have relative constant velocities among them, then this particle moves uniformly with respect to the others. If one of them is an inertial frame, then the others are also inertial reference frames. In order to avoid lengthy (perhaps endless) arguments, 16) we will not discuss how to construct physical reference frames and whether available physical reference frames are inertial frames. 2.2 The Constancy of the Speed of Light Suppose two reference frames X and 5( move uniformly relative to one another. Then the values of light speed emitted by a light source measured with respect to the X frame and 5( frame, respectively, are the same. The assumption that the speed of light emitted by a light source is independent of the state of motion of its source is an assumption in Einstein's special relativity and is generally accepted as confirmed by experiment/7} Mthough this generally accepted interpretation of the constancy of light speed may' be debatable, ~8) we accept it as an assumption of this new theory of relativity. 2.3 Transverse Doppler Shift Suppose that the frequency of light emitted by a light source is Vo with respect to a reference frame Xs in which this light source is at rest, that is, this light source has zero velocity, zero acceleration, zero derivative of acceleration with respect to time, and so on, with respect to that frame. The reference frame Xs is called the rest frame of that light source. Suppose, also, that this light source moves with a constant velocity v with respect to another reference frame X. Then the frequency of light emitted by this light source measured transversely with respect to that reference frame X is experimentally confirmed as v = [1 - ( v / c ) 2 ] l/2vo, where c is the speed of light. 19y,{l~ This relativistic frequency shift, inertially framed, is a prediction of Einstein's special relativity. We take it as another assumption of this new theory of relativity, holding even for noninertial reference frames with constant velocities between them. 3. THE INVARIANT LENGTH OF THE INFINITESIMAL FOUR-DIMENSIONAL DISPLACEMENT VECTOR OF A UNIFORMLY MOVING MATERIAL PARTICLE Suppose two reference frmnes, X frame and 5( frame, move with constant velocity with respect to one another. Consider a light-source particle moving with a constant velocity v with respect to the X frame. Then this light source moves with a constant velocity v with respect to the 5( frame. Suppose, also, that the frequency of light emitted by this light source is v0 measured with respect to its rest frame. By the assumptions in Sec. 2, the frequency of light emitted by this light source measured with respect to the X frame is V = [1 -- (F/C)2]I/2VO .
(t)
Similarly, the frequency of that light measured with respect to the 5( frame is 9 = [ 1 -- (~/C) 2] t/2Vo" (2) Rewrite Eq. (1) as (3)
A~ = [1 - ( v / c ) 2 ] l / 2 A t , where Ato --= 1/Vo, and At -- l/v. Similarly, from Eq. (2) we have Ato = [1 - (~/c) 2 ] 1/2A~7,
(4)
where Ai - 1/v. As usual in describing the motion of particles in classical mechanics, during the time interval At with respect to the X frame, the particle will move from x i to x i +Ax i (i = 1, 2, 3), where Ax 1 = VxAt, A x 2 = v y A t and Ax 3 = vzAt. Then we define Ax ~ - c A t and define the length As, which is associated with the 4-vector of displacement Ax c~ = (Ax o ' Ax 1 ' Ax 2 , Ax 3) of the moving particle, as As ~ [ (z~gO)2 __ (Z~L~cl)2 __ ( ~ 2 ) 2
__ (Z~k,%.3)2] 1/2.
(5)
By the definition of zXxC~(ot = 0, 1, 2, 3), we have
(6)
As = [1 -- (V/C) 2 ] 1/2~k,~C0.
Similarly, during the time interval Ai with respect to the 5( frame, this particle will move from s to a7i + k.f i. Consequently, we have Ag ~ [ (,~L~O)2, ( ~ 1 ) 2 ,
(Z!L~2)2, (Z%:~3)2] 1/2,
(7)
and = [1 - (/,7/c) 2 ] 1/2Z~)~~
(8)
From Eqs. (3), (4), (6), and (8) we obtain As = ~
= czXto.
(9)
Therefore, we have for the uniformly moving light-source particle with respect to the X frame and 5( frame k'c - CAto = (rk,~Ax~Ax I~) 1/2 = ( r l c ~ k . ~ )
1/2
(o~, [3 = 0, 1, 2, 3),
where rlcq3 are defined as
13~13 ~
(0o o) -1
0
0
0
-1
0
0
0
-1
.
(10)
(11)
In our notation any index, like cz and [3 in Eq. (10), that appears twice, once as a subscript and once as a superscript, is understood to be summed over. 69
Relativistic Kinematics I: A Theory of Relativistic KinematicsBased on Physical Reality
In general, suppose a material particle moves uniformly with respect to the X frame, and thus also moves uniformly with respect to the 5~ frame. Consider a reference frame Xs moving with this particle. That is, this particle is at rest with respect to this rest frame Xs. During an infinitesimal time interval dlo with respect to the X~ frame, the particle will move from x i to x i + du~ with respect to the X frame, or from ~ to :?* + ds "* with respect to the X frame. From Eq. (10) and taking the limits of d t o = Ato/n, du~ = A x W n and d~ ~ = A~C~/n as n --+ oo, we obtain the invariant length of infinitesimal displacement, d'c = cdt0 = (qc~13dxC~du13) l/2 : (~q~d~d~3) 1/2.
(12)
It should be emphasized that although the formula Eq. (12) is the same as Minkowski's differential space-time distance in the usual theory of relativity, the physical interpretations of this formula are different than the theory presented here and the usual theory of relativity. The dx ~ is a four-dimensional infinitesimal displacement vector which characterizes the state of motion. The d't is the invariant infinitesimal length of the fourdimensional vector du~. Equation (12) gives a relationship of the physical quantities of motion, rather than a relationship of the space-time structure of coordinate systems, between the two reference frames X and 5:. In classical mechanics the dxC~(x) and d z ( u ) are functions of the space and time coordinates x of the particle with respect to the frame X. However, according to quantum mechanics, those physical quantities should be considered as functions of time only, due to the impossibility of simultaneously determining the definite values of positions and velocities of particles/n) Hereafter, for convenience, we do not explicitly express the dependent variables of these physical quantities of motion.
the direction of a chosen axis, for example X~-axis in the derivation below. For convenience, we also simplify the notation of a~(X, X) as a~. (1) If a particle is at rest in the X frame, then for this particle d , ~ 1 = d:? 2 = d:? 3 = 0 with respect to the X frame and dx 2 = du3 = 0 with respect to the X frame. Therefore, from the assumed linear equations of transformation Eq. (13), we have ao2 = ao~ = 0 and al /a o : d u l / d u 0
__
V/c ~ [3.
(2) If a particle is at rest in the X frame, then dul = du2 = du3 = 0 a n d d x 2 = d:? 3 = 0. Therefore, we have a [ = a~ = 0 and - a 1/a~ = d s 1 6 3 ~ = - V / c -- -[3.
(3) If a particle moves in the X1-X2 plane, then du3 = 0 and d:~ 3 = 0. Therefore, we have a23 = 0. (4) If a particle moves in the X1-X 3 plane, then du2 = 0 and d:~ 2 = 0. Therefore, we have a32 = 0. (5) If a particle moves in the X2-X3 plane, then d:c 1 = 0 and d ~ I / d:? ~ = -[3. Therefore, we have a 1 ( d~a / d s ~ + a~ (ds / d s ~ = 0. Because d s 1 6 3 ~ and d ~ 3 / d s ~ are arbitrary, a I = a~ = 0. (6) If a particle moves in the 512-5~3 plane, then d:~ 1 = 0 and d x l / d x ~ = [3. Therefore, we have d x l / d u ~ = a l / [ a ~ + a ~ 1 6 3 1 6 3 ~ + a ~ ( d s 1 6 3 1 7 6 = [3. From (1) and (2) we obtain a~ = ao~ Since d ~ 2 / d s ~ and d s 1 6 3 ~ are arbitrary, we have a ~ = a 3o = 0 Now, we have reduced the universal transformation to duo = aOd~O +aOd~l dul = [3aOd~O +aOd~l du2 = a2ds
(14)
du 3 = a~ d,~ 3 .
4. DIFFERENTIAL LORENTZ TRANSFORMATIONS 4.1 The Relativistic Transformation for the 4-vector of Infinitesimal Displacement Consider a reference frame X moving with a constant velocity V with respect to a reference frame X. As usual, it must be postulated that given a velocity V of the .~ frame relative to the X frame, then when measured relative to the 5( franle, the X frame has velocity -V. Suppose that at an arbitrary instant a material particle has the state of motion dx c~ with respect to the frame X; at that instant the material particle has the state of motion d:?U with respect to the frame 5~. Assume that the general functional relationship between the four-dimensional vectors of displacements du c~ and d.~ ~t is linear, that is, duc~ = a ~ ( X , X)daP t
(or, l a : O ,
1, 2, 3).
(13)
In general, the coefficients a~(X, X) depend on the motion of the particle, as well as the relationship between the frames X and X, for example, the orientation of coordinate systems and the relative velocity V between the reference frames. Suppose there exists a universal transformation that could be applied to particles with arbitrary motion, that is, the coefficients a~(X, X) of the universal transformation depend on the relationship between the two frames, not on the motion of particles. This supposed existing universal transformation of the four-dimensional infinitesimal vector will be derived below. We simplify the mathematical problem by keeping the corresponding coordinate axes of the frames X and X parallel and the relative velocity V in
70
(7) Suppose a particle moves uniformly with respect to the frames X and 5~. Then the reduced transformation Eq. (14) should also satisfy Eq. (12). Consequently, we obtain ao~177
-122,
a~
a~
a22==klanda33=+l.
According to the chosen orientation of coordinate systems between the frames X and ~,, we finally obtain the relativistic transformation of the infinitesimal displacement vector duc~: dx0 = 7(d.~ ~ + [3d:? 1) dul = 7(d:71 + [3doT0) du2 = d.~2
(15)
du3 = d:?3, where Y - ( 1 - [32 ) -1/2. The mathematical form of the differential Lorentz transformation Eq. (15) is close to that of the usual Lorentz transformation in Einstein's special relativity, x o = 7(2 0 + I~ 1) X1
,y(.~ 1 + [ ~ 0 )
x2
s
x3
.f3.
(16)
Young-Sea Huang
However, the physical interpretation of the differential Lorentz transformation by this theory is entirely different from that by Einstein's special relativity. The differential Lorentz transformation is considered as a physical rule to instantaneously transform the infinitesimal displacement 4-vector dac~ with respect to one reference frame to another. The infinitesimal 4-vector da c~ characterizes the state of motion of a particle with respect to the frame X; the components are not the units grid of space-time measurements of the frame X. The da'z are physical quantities of the motion of the particle with respect to the frame X; the units grid of space and time of the frame X should be presumed in the beginning, in order to measure the physical quantity of motion dac,. The differential Lorentz transformation derived is not a transformation of the space-time structure from one reference to another as it is so interpreted in Einstein's special relativity. 4.2 The Relativistic Addition of Velocities and the Formulas of Relativistic Energy-Momentum The equations of uniform motion of a material particle with respect to a reference frame X are d2xC~/d'~ 2 = 0,
(17)
d'c = (qct~dac~da ~) 1/2
(18)
Now, we define the energy and momentum for a uniformly moving material particle with respect to the frame X, respectively, as E = Yv mc 2,
(24)
P -- "/vm v,
(25)
and
where m is the rest mass of the particle. Therefore, we have the relativistic relationship of the energy and momentum for a uniformly moving material particle, E 2 - p2C2 = m 2 c 4 .
(26)
The formulas of relativistic energy-momentum and the relativistic addition of velocities obtained by the theory presented here are the same as that in Einstein's special relativity. Therefore, experiments that attempt to corroborate these formulas cannot distinguish this theory from Einstein's special relativity.
where The equations of uniform motion are covariant under this differential Lorentz transformation Eq. (15). The general equations of motion of a nonuniformly moving particle, which will be introduced in the next section, are also covariant under this differential Lorentz transformation. Now consider a material particle moving with constant velocity v with respect to the reference frame X. From Eq. (17) we have daa/ d'~ = V c~,
(19)
where d'c = [ 1 - (v/c) 2 ] l/2dao and U c~ are constants. Consequently, we have U~ =[1-(v/c)2]
-v2 = T , . , a n d U i = Y v v * / c
5. THE RELATIVISTIC EQUATIONS OF MOTION OF A NONUNIFORMLY MOVING PARTICLE Consider a material particle moving with respect to the reference frames X and X, which move uniformly relative to each other. Assume that at any arbitrary instant, there exists an instantaneous reference frame Xs with respect to which this particle is at rest (zero velocity, zero acceleration, zero time derivative of acceleration, etc.). It should be noted that the rest frame Xs is not necessarily an inertial frame and is associated with the particle under consideration. The generalization of this assumption and comparisons of it with the equivalence principle of Einstein's general relativity are presented in the Appendix. Therefore, at any instant we have the equations of motion of this particle with respect to the chosen rest frame Xs,
( i = 1, 2, 3). (20) da d = 0
With respect to the reference frame 5( this particle moves with constant velocity 9. By similar argumentation as the above, we have ds
= ~c~,
(7~ = {t
-- ( V / C ) 2 ] -1/2
~'~,
and[3 i =g,' UIc (i =1, 2, 3). (22)
By applying the differential Lorentz transformation, we obtain the formulas for the relativistic addition of velocities,
(27)
d2xff/dx ~ = 0.
(28)
Or, we have the equations of motion with respect to the rest frame Xs, dZxff/d'c 2 = 0,
(29)
where a " [ = da's0 = (i]o.~d~s(Zdg~)1/2
(30)
We assume that there exists a linear transformation of the infinitesimal displacement dau of the moving particle with respect to the frame X to the das~ with respect to the rest frame Ks., that is,
~,, = g/,~(1 + }~*/c) vi
2, 3),
and
(21)
where d'~ = [1 - ( H c ) 2 ] V2ds176and/~c~ are constants. Hence we have also
(i=l,
~+zTt/C
7- = ]-7-+}717 t,2
~21c
v3
~3/c
(23)
daft =A~(Xs, X)da la
(~, hi. =0, 1, 2, 3).
(31)
The coefficients A~(Xs, X) depend on the relationship between the reference frames X and Xs, that is, the instantaneous motion of the rest frame Xs with respect to the frame X. If the particle moves uniformly with respect to
71
Relativistic Kinematics I: A Theory of Relativistic Kinematics Based on Physical Reality
the frame X, then the associated rest frame Xs moves with constant velocity with respect to the frame X. The coefficients A~'(Xs, X) are then reduced to the coefficients a~ of the differential Lorentz transformation which depend on the relative constant velocity between the frames X and Xs. In general, the coefficients A~(Xs, X) depend not only on the velocity, but also on the acceleration, the time derivative of acceleration, etc., of the instantaneous rest frame Xs with respect to the frame X. In other words, the coefficients A~(Xs, X) depend on the velocity, the acceleration, the time derivative of acceleration, etc., of the particle at that instant with respect to the frame X. From Eqs. (30) and (31) we obtain d'c= ('qap/l~(Xs, X)Av~(Xs, X)dxgd~V) 1/2
(32)
d't = (guvdx~tdx v) 1/2,
(33)
Or, we have
where gvv are defined as
d'r, -- (goo)1/2 ds176
g.~ - n=~(x~, X)A~(X~, x).
(34)
From Eqs. (29), (30), (31), (33), and (34) we obtain the relativistic equations of motion of this material particle with respect to the reference frame X, d2x la
dx 7v dx v
d't 2 +F~v d't
d't - 0,
(35)
1,
if ~L= v;
0,
if X-iv.
(37)
(i=1,
2, 3).
(38)
By similar arguments to those above, we obtain relativistic equations of motion with respect to the frame X, d2s t
ds
d.t2 + ['~v dx
d't: - O,
d't: = (,~tavd2ltd2 v) 1/2.
(43)
(44)
The function f(v, a, it, ...) is, in general, a function of velocity v, acceleration a, and time derivative of acceleration it, ..., of the particle with respect to the frame X at that instant. When the particle moves uniformly with respect to the frame X, Eq. (43) must be reduced to d'c = [1 (v/c)2]l/2dx ~ That is, f(v, a = 0, it = 0 . . . . ) = 1, for any value of v. Hence the function f(v, a, fit, ...) must not contain terms that depend solely on velocity. When the particle moves with zero velocity, but not necessarilv zero acceleration, Eq. (43) is reduced to d't = f ( v = 0, a, it . . . . )dx ~ The invariant length d't depends not only on velocity, but also on acceleration, time derivative of acceleration, and so on. As usual in classical mechanics, v, a, it, and so on are considered as functions of the location and time of the particle with respect to the frame X, that is, v(x, t), a(x, t), it(x, t), and so on. Therefore, Eq. (43) can be written as d'c = f ( x ) {1 - [ v ( x ) / c ] 2 }l/2dx~
ds v
(45)
(39)
where (40)
It must be emphasized that g~tv and l~v are not the geometrical elements of the curved space-time as interpreted in Einstein's general relativity. The
72
Therefore, from Eqs. (41) and (42) we obtain
f(v, a, it. . . . ) ~ (,~00) 1/2
In our notation ~,, g, v, ~ = 0, 1, 2, 3. Rewriting the relativistic equations of motion explicitly in terms of the coordinates of the reference frame X, we obtain ) dxX dx v dx o dr 0
(42)
(36)
and G ~'v are defined by
d2x' ( abe" dxO.. - F~.v ~ - F ~ v
ds ~ = [1 - (v/c)2]V2dx ~
where the functionf(v, a, it . . . . ) is defined as
{a&,. agv,, a&~} ~+,gxX ax~ '
G~agow - 8~ -
(41)
In addition, from the DLT between the frames X and 5~, as well as ~i/c = d2i/d~ ~ = 0 of this particle with respect to the frame X, we have
d'l: : f ( v , a, it. . . . ) [1 - (v/c) 2 ] 1/2d~7~
where F~v are defined as
r~=5-c;~"
g~v and I~v are the physical quantities associated with the state of motion of the particle, that is, velocity, acceleration, and so on. If the particle moves uniformly with respect to the frame X, then the guy are reduced to rlrav and I"~v = O. The general equations of motion Eq. (35) are then reduced to the equations of uniform motion Eq. (17). Furthermore, from the definitions of g~v and 1-~v, we can show that guy and I~v are covariant under the differential Lorentz transformation (DLT) between the frames X and X. Consequently, the general equations of motion, Eq. (35), are covariant under the DLT. We can further reduce the equation of the invariant length of infinitesimal displacement 4-vector, Eq. (33). Suppose that with respect to the reference frame X, at an arbitrary instant t, the velocity of the particle is v. At that instant consider a reference frame 5( moving with constant velocity v with respect to the reference frame X. Then, with respect to the frame X, at that instant ,7, this particle moves with zero velocity, but not necessarily zero acceleration. Therefore, from Eq. (40) we have
where x represents the space and time coordinates of the moving particle with respect to the frame X. The quantities f(v, a, it. . . . ) and [1 - ( v / c ) 2 ] m are physical quantities related to the state of motion. In order to measure these physical quantities of motion, v, a, it, and so on, the units of space and time coordinates should be presumed in the beginning. Equation (45) should not be thought of as a metric equation of the curved space-time coordinates
Young-Sea Huang
in Einstein's general relativity such as Schwarzschild solution of Einstein's gravitational field equations. /he theory presented here is not a metric
theory as Einstein's theory of relativity: special relativily and general relativily. The formalism of this new theory of relativity provides a unified theoretical scheme that rdativistically modifies the classical force laws: the electromagnetic force law and the gravitational force law. The unified scheme uniquely fixes the relativistic modification to given force laws. In contrast, relativistic modifications by the usual theory of relativity, via the Lorentz covariance restriction of mathematical forms, often are not unique. For the detailed applications of this theory to modifying the classical force laws, see this author's coming papers. (12),(13)
6. COMPARISONS BETWEEN THE THEORY PRESENTED HERE AND EINSTEIN'S THEORY OF RELATIVITY 6.1 The Principle of Relativity Historically the principle of relativity has various meanings. What is the prevailing meaning of the principle of relativity? The principle of relativity: there exists a triply infinite set of equivalent Euclidean reference frames moving with constant velocities in rectilinear paths relative to one another in which all physical phenomena occur in an identical manner. (7) Inertial reference frames are presumed. It is then asserted that the results of all experiments performed in a given inertial frame are independent of the uniform translational motion of the frame as a whole. Consequently, a physical law of nature should be formulated in the form that is covariant under the relativistic transformation between the inertial frames, if that law can be expressed in a mathematical form. Einstein's theory of relativity has the global Lorentz transformation (GET), for example, Eq. (16), as the relativistic transformation. The initial conditions or boundary conditions of a physical system are not required to be covariant under the GLT. The theory of relativity presented here is formulated from a purely kinematic point of view. This theory entails the principle of relativity as follows: all physical phenomena of the motion of particles occur in an identical manner in sets of Euclideml reference frames (not necessarily inertial reference frames) which move with constant velocities relative to each other. The laws of motion are independent on the reference frame to which they are referred, as long as these reference frames move, as a whole for each reference frame, with uniform translational velocities relative to each other. This theory has the differential Lorentz transformation, for example, Eq. (15), as the relativistic transformation. The laws of motion should be formulated in the mathematical forms which are covariant under the DLT. It has been mentioned in previous sections that the equations of uniform motion, Eq. (17), as well as the equations of nonuniform motion, Eq. (35), are covariant under the DLT. The initial conditions and the boundary conditions of a physical system are not required to be covariant under the relativistic transformation. 6.2 The Conditions Between the Reference Frames Utilized in Deriving the Relativistic Transformations Since the relativistic transformations relate the values of physical quantities, for example, velocity, in one frame to another, in order to ensure that the transformations are physically meaningful, the reference frames should satisfy certain allowable conditions between them. The conditions between the reference frames usually employed in deriving the relativistic transformations are as follows: (1) The coordinate systems of the reference frames have the same stipulation for the units of space and time measurement between
them. (2) The coordinate systems of the two reference flames are chosen such that the corresponding coordinate axes between them are parallel and the relative constant velocity between them is in the direction of a chosen axis. (3) Set the same reading of the clocks at the origins of coordinate systems of the reference flames at the moment when these origins coincide. Meanwhile, one is supposed to be synchronizing all the clocks at every point in the same reference frame, respectively, for each reference flame. Without condition (1), the relativistic transformations are physically meaningless. The adoption of condition (2) is merely for the convenience of deriving, or describing, the relativistic transformations. The main difference between Einstein's special relativity and this new theory of relativity is in condition (3). Einstein's special relativity entails the GLT which is a transformation of space-time coordinates. The GLT requires an initial condition of space-time coordinate systems between reference frames. That is, the GLT needs condition (3), or a condition that is equivalent to condition (3), in order to make a transformation of the space-time coordinates. However, this new theory entails the DLT, which is an instantaneous transformation of the infinitesimal displacement of motion. Hence to make relativistic transformations, the DLT does not require any initial condition which is equivalent to condition (3). Furthermore, it should be noted that Einstein's special relativity does not explicitly state by what laws or rules the set-up of the initial condition between reference frames can be achieved and verified; this is especially relevant to a quantum object, such as a photon.
6.3 The Physical Interpretations of the Relativistic Transformations The most controversial problem raised in Einstein's theory of relativity is the physical interpretation of the mathematical symbols and formulas used in the theory. (2)-(4) The physical interpretation of the GLT in Einstein's special relativity is also beyond the comprehension of this author. Hence we do not present the exact physical meaning of the GET here. However, we want to emphasize that the GLT in Einstein's special relativity is not equivalent to the DLT in the theory presented here. The GLT in Einstein's special relativity is a flame-to-flame transformation assumed beyond the restrictions of the events along the so-called world line. The GLT transforms the space-time coordinate labels assigned to events from one reference flame to another. However, in the theory herein, the conceptual bedrock is different. The DLT is considered as a physical rule that relates the infinitesimal displacements from one reference flame to another. The DLT does not transform the space-time labels of events from one frame to another. That is, the DLT is a transformation of the physical quantities dx", which characterize the state of motion, not a transformation of the spacetime coordinates xU. According to the principle of relativity, we believe that the function of the relativistic transformations is a transformation of physical quantities, instead of the space-time coordinates, to make the physical laws covariant. Further distinctions between the GLT and the DLT from the point of view of quantum mechanics are presented in this author's coming paper/u) 6.4 The Concepts of Space and Tune The revolutionary concepts of space and time introduced in Einstein's special relativity according to the GLT, for example, time dilation, length contraction, and the relativity of simultaneity, are not completely convincing.(2)-(5),(14),(~s) Einstein's concept of space-time is far beyond the comprehension of this author. Here, we cannot present the meaning of Einstein's space-time concept without any controversy. Readers should refer to the original work of Einstein. (i) However, we present the concepts of space and time adopted by this new theory of relativity.
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RelativisticKinematicsI: A Theory of RelativisticKinematicsBasedon PhysicalReality As mentioned before, we begin by presupposing the coordinate systems of the reference frames which have the same stipulation of space and time units - in order to avoid the problem of logical circularity and to make the relativistic transformations physically meaningful. This new theory of relativity entails the Newtonian concept of universal time. Two events simultaneously happening in one reference frame have to happen simultaneously in every other reference frame. The relativistic transformation is an instantaneous transformation of the infinitesimal displacement vector, instead of the space-time coordinates. We describe the relativistic motion of particles without introducing such extraneous concepts as time dilation and length contraction, as in Einstein's special relativity. 6.5 The Unified Scheme of Modifying Classical Force Laws vs the Unification of Fields Attempts have been made, for the usual theory of relativity, to unify the gravitational field and the electromagnetic field, through the construction of the new space-time concept of Einstein. So far, this approach has not succeeded. In contrast, in the theory herein, the gravitational field and the electromagnetic field have different characters. This theory does not attempt to unify these fields. We know that the gravitational force law and the electromagnetic force law hold for the motion of low-speed particles. However,when the speeds of particles are comparable to the speed of light, these classical force laws need to be modified. This theory provides a unified method for such a relativistic modification to given classical force laws: the gravitational force law and the electromagnetic force law. 7. CONCLUSIONS Readers may have the impression that there is strong experimental support of Einstein's theory of relativity. Therefore, any theory of relativity that is different from Einstein's relativity theory must be refuted. However, some scientists still seriously question those experiments that are claimed to have confirmed Einstein's space-time concept; the interpretations of experiments may not be sound, systematic errors are treated in a questionable way, the data are on the limits of detectability, and the techniques of data reduction are seldom made clear. (s),(16)-(2~ Therefore, in order to diminish the speculations and criticisms, a clean experimental test is needed, avoiding ambiguities of interpretation and other theoretical bones of contention. The theory of relativity presented here is based on the assumptions that are also accepted by the usual theory of relativity. Nevertheless, some of the predictions made by this theory turn out to be different from that of the usual theory of relativity.(12),(i3) We hope that this series of papers of relativistic kinematics will provoke the reader to carefully examine this new theory of relativity, as well as to closely reexamine the physical meaning
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of the mathematical symbols used in Einstein's theory of relativity and to critically evaluate the reliability of the relativistic experiments. Acknowledgment This author gratefully acknowledges the critical comments of Dr. C.M.L. Leonard in the preparation of this paper. This author sincerely thanks Dr. T.E. Phipps, Jr., for his encouragement and endorsement. APPENDIX The generalization of the assumption given in Sec. 5 is as follows. For any particle moving with respect to a reference frame, at any arbitrary instant, there exists at least one reference frame with respect to which the particle moves with a constant velocity, but zero acceleration, zero time derivative of acceleration, and so on. There is always a set of infinite number of such frames with uniform velocities relative to the rest frame of the particle. This assumption is not the same as Einstein's equivalence principle. (1),(21) First, consider a test particle moving in a force field, for example, a uniform gravitational field, in an inertial reference frame. At any instant there exists an instantaneous reference frame with respect to which this test particle moves uniformly, that is, with zero acceleration, zero time derivative of acceleration, and so on. Suppose that there are other particles infinitesimally close to this test particle, but with different velocities. According to this assumption, this test particle moves uniformly with respect to that instantaneous reference frame, but the other particles do not necessarily move uniformly with respect to that instantaneous reference frame. However, according to the equivalence principle, a local reference frame (freely falling frame) can be introduced such that with respect to that freely falling frame, all these particles move uniformly. Second, this assumption states that at any instant, the acceleration of the test particle due to any force field can be virtually eliminated with respect to a chosen reference frame, and its associated set of frames, at that instant. However, according to the equivalence principle, the complete physical equivalence of the uniform gravitational field and the accelerated reference frame is assumed. Third, this assumption is based on the point of view of pure kinematics. Therefore, this assumption can be employed to describe the motion of particles in any force field, not only for the gravitational field. However, the equivalence principle is applied to describe the motion of particles in the gravitational field only. Furthermore, the equivalence principle is logically broader than this assumption in applications describing the motion of particles in the gravitational field. That is, in applications in the gravitational field, the equivalence principle includes this assumption. In that case, if this assumption is not true, then the equivalence principle is not true. Received on 14 September 1989.
Young-Sea Huang
R6sum~ Ceci est le premier d'une s&ie d'articles sur une nouvelle cyndmatique relativiste ("Relativistic Kinematics If, IlL and IV,," soumis ~ Phys. Essays). Une nouvelle th&rie de la relativitd est d~duite d'hypoth&es dent la validitb est bas& strictement sur des exp&iences g&dralement accept& Le formalisme de cette thborie offre un sch&na thdorique qui modifie relativistiquement les lois classiques de la force: la loi de la force ~lectromagndtique et celle de la force gravitationnelle ("Relativistic Kinematics II and 111'). No,~" mettons l'acvent sur le fair que les transformations rdativiste, de Lorentz, devraient btre interprbt&s comme transformations dans l'espace des vitesses, et non pas dans l'espace des coordonn& de l'espace-temps comme il est fait dans la thdorie d'Einstein [A. Einstein, The Meaning of Relativity, 5th ed. (Princeton UniversiO~Press, 1956)]. Le concept d'espace et de temps qui decoule de cette th&rie est diff&ent de celui de l'espace-temps d'Einstein.
References 1. A. Einstein, The Meaning of Relativi~, 5th ed. (Princeton University Press, 1956). 2. H. Nordenson, Relativity, Time and Reality (George Allen and 11nwin, London, 1969). 3. L. Essen, The Special Theory of Relativily: A Criticaldnalysis (Clarendon Press, Oxford, 1971). 4. H. Dingle, Nature 216, 119 (1967); idem, Science at the Crossroads (martin Brian & O'Keeffe, London, 1972). 5. T.E. Phipps, Jr., Heretical Verities: Mathematical Themes in Physical Description (Classic Non-fiction Library, Urbana, IL, 1986). 6. F. Christensen, Philos. Science 48, 232 (1981). 7. J.D. Jackson, Classical Electrodynamics (lohn Wiley, NY, 1975), Chaps. tl, 12. 8. W. Kantor, Relativistic Propagation of Light (Coronado Press, 1976). 9. H.E. Ives and G.R. Stilwe]l, J. Opt. Soc. Am. 28, 215 (1938). 10. M. Kaivola, O. Poulsen, E. Riis, and S.A. Lee, Phys. Rev. Lett. 54, 255 (1985).
11. Young-Sea Huang, "Relativistic Kinematics IV: The Compatibility of Differential Lorentz Transformations and Heisenberg's Uncertainty Principle," submitted to Phys. Essays. 12. Idem, "Relativistic Kinematics II: The Electromagnetic Force Law Relativistically Reexamined," submitted to Phys. Essays. 13. Idem, "Relativistic Kinematics III: k Relativistic Modification for Newton's Gravitational Force Law," submitted to Phys. Essays. 14. W. Kantor, Czech. J. Phys. B 22, 1029 (1972). 15. D.T. macRoberts, Spec. Sci. Tech. 3, 365 (1980). 16. L. Essen, Wireless World 84, 44 (Oct. 1978). 17. W. Kantor, Found. Phys. 4, 105 (1974). 18. R.A. Waldron, Spec. Sci. Tech. 3, 385 (1980). 19. D.T. Wilkinson, in Some Strangeness m the Proportion: A Centennial 3),mposium to Celebrate the Achievements of Albert Einstein, edited by H. Woolf (Addison-Wesley, NY, 1980), p. 137. 20. W.A.S. Murray, Electron. Wireless World 92, 28 (Dec. 1986). 21. H.C. Ohanian, )an. J. Phys. 45, 903 (1977).
Young-SeaHuang Department of Physics Soochow University Shih-Lin, Taipei Taiwan
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