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Special Relativity via Electro-Magnetic Clocks(Published in Galilean Electrodynamics Vol.10 (no.6, Nov/Dec/1999): 107-110) John Byl, Ph.D.
Abstract
The basic special relativistic effects of length contraction, time dilation, and mass increase are all simply derived by examining the effect of motion on a number of electro-magnetic clocks. The derivations are based on classical mechanics and electromagnetism plus the assumptions that there exists an ether and that any time dilation due to motion is the same for all electromagnetic clocks.
1. Introduction In physics it is often possible to build a theoretical structure using a variety of different postulates. Although these models are often observationally equivalent, the choice of postulates is important because it may lead to theories that differ in their simplicity, beauty, or intuitive appeal.
Special relativity is usually presented in terms of Einstein's two famous postulates:
From these postulates one can derive the Lorentz transformations and, from them, the relativistic effects of time dilation, length contraction, and mass increase.
One objection that has been voiced against these postulates, particularly the second one, is that they are ad hoc and counter-intuitive. A further concern is that the equivalence of all inertial frames implies the absence of an underlying ether, a medium for light to propagate in. It is interesting to note that Einstein (1920) himself, after have first rejected the notion of an ether, later reverted back to the conception of an ether:
"According to the general theory of relativity space without ether is unthinkable; for in such space there would not only be no propagation of light, but also no possibility of existence for standards of space and time." Thus, seeing that general relativity again makes use of an ether, in the form of the background space, special relativity's rejection of an ether seems to be a disadvantage rather than an advantage.
An alternative approach, the Lorentz theory, retains the ether and develops relativity from the concepts of length contraction, time dilation, and mass increase. Already in 1889 Fitzgerald (and Lorentz in 1892) proposed that objects moving in the ether are contracted in their direction of motion. Such contraction explained the null result of the famous Michelson-Morley experiment of 1887.
Ether theories have been criticized for making ad hoc assumptions regarding length contradiction, time dilation and mass increase. In particular, the Fitzgerald contraction is often derided as a mere contrivance invented to explain the Michelson-Morley experiment. Actually, however, it was based on Heaviside's (1888) formula for the electric field of a moving charge, which predicts that the electric field is contracted in the direction of motion by the same factor as the Fitzgerald contraction.
Prokhovnik (1965) used Heaviside's equation to infer length contraction (although he did not actually show this). Then he used a rod-clock (a rod with mirrors at both ends to a reflect a light beam back and forth along the rod, the interval between successive reflections at one of the mirrors defining the unit of time) to show that length contraction implied time dilation. The same method is used more recently by Selleri (1993). A drawback, according to Erlichson (1973), is the assumption that all clocks behave like the rod-clock, which is not obvious. Also, to attain complete equivalency with Special Relativity one must still assume the mass-increase formula or an equivalent relativity principle.
Bell (1987) presented a rough sketch of how to obtain special relativity from Heaviside's field equation for a moving charge. This involved the hypothetical numerical integration of an electron orbit about a nucleus, wherein the orbit is distorted in the direction of motion by the Fitzgerald factor and the period of the orbit is similarly changed. Bell, too, had to add an explicit assumption in the form of the equation for relativistic momentum.
Recently Jefimenko (1995) showed that the Lorentz transformations could be derived from Maxwell's equations using retarded potentials. The derivation is, however, somewhat involved and, here too, one still needs to assume the mass-increase formula or its equivalent.
2. Basic Assumptions In this paper we shall follow Prokhovnik and others in developing special relativity via Heaviside's equation. Our approach, however, involves only two assumptions, both of which seem quite plausible: (1) There is an ether with respect to which the speed of light (in a vacuum) is constant in all directions. (2) The rates of electromagnetic clocks moving with constant speed v relative to the ether all vary with v in the same manner.
An electromagnetic ("e-m" for short) clock is one based on oscillations of a charged particle in accordance with Heaviside's equation for the force of a moving charge and the Lorentz force law. Recently Jefimenko (1996) provided a causal explanation of time dilation by considering the behaviour under motion of some simple e-m clocks. Our procedure builds on this technique. From an analysis of four e-m clocks we shall derive all three basic relativistic effects - time dilation, length contraction, and mass increase.
Note that no assumptions are made about the precise formula for time dilation, or even that time dilation actually occurs, but only that if it does occur then it should effect all e-m clocks the same way. This assumption seems plausible since such clocks, being all based on the same equations, can be expected to behave in the same manner. By assuming that all e-m clocks exhibit similar behaviour, it suffices to consider a few specific cases, from which more general conclusions can then be drawn.
3. Dynamic Electromagnetic Clocks The basic equation is Heaviside's equation for the electric field of a moving point charge:
E = qA (1 - β2)/(r3[1- (β sinθ)2]3/2) r (1)
where β º v/c and A = 1/(4 πε0). Here r is the vector from the charge to the point of observation, v is the velocity of the charge, and θ is the angle between r and v. This equation was first derived by Heaviside (1888) and is well-established, although its derivation is somewhat complicated. Since the moving electric field generates a magnetic field B = v´E, the total force acting upon a charge q in the region is given by the Lorentz force
F = q[E + v´(v´E)/c2](2)
As will soon be shown, the period T of the clocks to be considered depends on the mass of a moving charge and the length of the clock. The assumption of equal rates of moving clocks will be seen to imply a dependence of both length and mass on the velocity and orientation of the clock. To accommodate a possible dependence of mass on velocity we write Newton's third law in the form
F = d(mv)/dt = v dm/dt + mdv/dt = v(dm/dv)(dv/dt) + mdv/dt (3)
If the charge oscillates in the direction of v then
F = (v dm/dv + m) d2x/dt2 º m1 d2x/dt2 (4)
where the subscript 1 refers to an orientation parallel to the direction of motion. For oscillations perpendicular to v we obtain
F = md2x/dt2 º m2 d2x/dt2 (5)
where the subscript 2 refers to an orientation perpendicular to the direction of motion. Note that
m1/m2 = [v dm/dv +m]/m (6)
(a) Clock #1 The first clock to be considered consists of two positive charges q separated by a fixed distance L0 plus a third positive charge q which is constrained to move along the line joining the other two charges. (Imagine the outer two charges to be fixed at the ends of a thin hollow cylindrical insulator and the third charge, with a radius slightly less than that of the cylinder, free to slide about inside the cylinder).
The inner charge has an equilibrium position at the middle of the cylinder. If moved a small distance x « L0 from equilibrium the inner charge will oscillate about the equilibrium point; the period of oscillation is independent of x and provides the time unit for the clock.
Let the clock move at a speed v in a direction parallel to its axis (i.e., r = xi and v = vi), where v is much greater than the maximum speed of the inner charge relative to the clock. In this case the inner charge experiences no magnetic force (i.e., v´E = 0) and the electric force from equations (3) and (4) becomes:
F = q2A(1 - β2)[(L1 + x)-2 - (L1 - x)-2] = m1 d2x/dt2 (7)
Since the axis is oriented in the direction of motion and assuming x « L, this reduces to
F » -q2A(1 - β2) 4x L1-3 = m1 d2x/dt2 (8)
This force causes sinusoidal motion with a period
T = 2π [m1 L13/4Aq2(1 - β2)]1/2 (9)
or, in terms of the rest period T0
T/T0 = (L1/L0)3/2(m1/m0)1/2(1 - β2)-1/2 (10)
(b) Clock #2 Consider now a second clock, described by Jefimenko (1996). This clock consists of a ring of radius L0 and charge q. A negative charge -q is constrained to move through the axis of the ring. If perturbed a small distance x above the plane of the ring the moving charge will oscillate with a period T0 independent of x. Let this clock, too, move in a direction parallel to its axis. Again, since v and E are parallel, there is no magnetic force on the inner charge. The electric force on the inner charge is given by:
F = q2A(1 - β2)x(L22 + x2)-3/2[1- β2/(1 + x2/L22)]-3/2 (11)
which simplifies (assuming x « L2) to
F » -q2AxL2-3(1 - β2)-1/2 = m1 d2x/dt2 (12)
This results in a period
T/T0 =(L2/L0)3/2(m1/m0)1/2(1 - β2)1/4 (13)
Note that the only significant difference between the two clocks is the orientation of the fundamental length L. Equations (10) and (13) indicate that the periods of both clocks can change by the same ratio only if
L1/L2 = (1 - β2)1/2 (14)
Thus, while the absolute values of length contraction are not yet known, it is clear that the assumption of equal clock rates leads to a dependence of length contraction on the orientation of the object to its direction of motion.
(c) Clock #3 Next, consider again clock #1, but this time with its axis oriented perpendicular to its direction of motion (i.e., take r = xi and v = vj). Now there is also a magnetic force on the moving charge and the force equation (4) becomes:
F = -q2 A4x L2-3(i + v´(v´i)/c2)(1 - β2)-1/2 (15)
or
F = -q2 A4x L2-3(1 - β2)1/2 = m2 d2x/dt2 (16)
This yields
T/T0 = (L2/L0)3/2(m2/m0)1/2(1 - β2)-1/4 (17)
Since the periods of clocks 2 and 3 are assumed to be the same, equations (13) and (17) imply that
m1/m2 = 1/(1 - β2) (18)
Substituting this result into equation (6), we find
v dm/dv = m/(1 - β2) - m = mβ2/(1 - β2) (19)
or,
dm/m = vdv/(c2 - v2) (20)
Integrating, this yields
m = m2 = m0 (1 - β2)-1/2 (21)
where, in order for m to reduce to the rest mass m0 when β = 0, we have taken the integration constant to be zero. Substituting this into equation (17), we find that
T/T0 = (L2/L0)3/2 (1 - β2)-1/2 (22)
(d) Clock #4 We consider one final clock. Imagine a particle of rest mass m0 and negative charge -q in a circular orbit of radius L0 about a positive charge q. The central charge is gently accelerated in a direction perpendicular to the plane of the orbit until it is moving with speed v. Since, according to equation (2) the force is central, angular momentum will be conserved. Hence, since the circular speed is 2πL/T, this leads to
2π m0 L20 /T0 = 2π m2 L22/T (23)
Solving for T and applying equation (21), this yields
T/T0 = (L2/L0)2 (1 - β2)-1/2 (24)
Finally, comparing equations (22) and (24), we conclude that
L2 = L0(25)
and
T = T0/(1 - β2)1/2 (26) Equations (14) and (25) together give
L1 = L0 (1 - β2)1/2 (27)
4. Discussion We have derived the special relativistic effects of length contraction, time dilation, and mass increase quite simply from Heaviside's equation of classical electromagnetism. Since the standard derivation of Heaviside's equation involves retarded potentials, our analysis implies that the relativistic effects are due primarily to distortions in the electric field caused by the finite speed of electromagnetic effects.
From the equations for time dilation (26) and length contraction (27) one can easily derive the Lorentz transformations (see Erlichson (1973), provided a suitable clock synchronization scheme is applied (e.g., the moving observer can adopt the Einstein convention of assuming the one way speed of light in his frame to be c or, equivalently, synchronize other clocks at rest with respect to the observer by slowing moving a clock).
Since our calculations are independent of the actual size, mass, charge, or material composition of the clocks, it would seem that our conclusions about mass decrease and length contraction should hold for all objects moving within e-m fields, depending only on their velocity and orientation to the fundamental reference frame.
Prokhovnik (1965) has shown that time dilation, length contraction and the synchronization convention result in all inertial observers measuring the speed of light to be the same. Also, since Maxwell's equations are known to be Lorentz-invariant, the form of these equations is the same for all inertial observers. Thus Einstein's two relativity postulates are now no longer mysterious assumptions but are readily explicable in terms of our more plausible initial assumptions regarding the ether and e-m clocks.
Does time dilation effect other types of clocks? Another type of clock commonly considered in special relativity is the simple rod-clock, as described above. If this clock is at rest in a fundamental reference frame in which the speed of light is a constant c in all directions then its time unit is T0 = 2L0/c. If the rod is moving with a speed v with respect to the fundamental reference frame and oriented parallel to the direction of motion, the time unit becomes
T1 = L1/(c + v) + L1/(c - v) = 2L1c/(c2 - v2) = T0 (L1/L0)/(1 - β2) (28)
If we now orient the rod perpendicular to its direction of motion we find
T2 = 2L2/(c2 - v2)1/2 = T0 (L2/L0)/(1 - β2)1/2 (29)
Applying the length contraction equations (25) & (26), it is evident that
T1 = T2 = T0/(1 - β2)1/2(30)
Thus, regardless of its orientation, the rod-clock dilates in the same manner as the e-m clocks.
Note that, for the rod-clock, equations (28) and (29) lead to the relation
T1/T2 = (L1/L2 ) (1-β2) -1/2 (31)
Applying equation (14), it follows that T1 = T2 (i.e., the rate of the rod-clock is independent of its orientation). Equation (14) was based on an analysis only of clocks #1 and #2, both of which have the same orientation of the oscillating charge. The fact that rod-clock rates are independent of orientation renders more plausible the assumption that the same is true of e-m clocks."
This result lends some plausibility to the further hypothesis that the time dilation equation applies to all clocks. This has some interesting implications if we extend our analysis to gravity clocks. Consider, for example, a clock similar to the above clock #2, but replacing the charges with masses, suitably chosen so that the clock has the same period as clock #2. If both clocks behave the same for any given velocity and orientation, then the gravitational field should have the same form as that of the electric field for a moving charge (i.e., equation (1)) and the gravitational force should have a form similar to the Lorentz force (i.e., equation (2)). Such modifications were suggested already by Heaviside in 1893 and have been proposed by others more recently. Here, however, it follows quite simply from our clock considerations.
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