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On Deriving Special Relativity from
Electromagnetic Clocks

(published in Galilean Electrodynamics Vol.14 (no.5, Sept/Oct 2003): 89-92)

John Byl, Ph.D.

Abstract

Using only classical physics, the basic special relativistic effects are derived by examining the effect of motion on a number of simple electromagnetic clocks. This paper improves upon an earlier derivation by proving, rather than assuming, that clock periods are independent of orientation.

 

Introduction

It was shown by Byl [1] that the special relativistic effects of length contraction, time dilation, and mass increase can all be derived by examining the effect of motion on a number of electromagnetic clocks. The derivations were based solely on classical physics. In that paper it was assumed that the rates of all electromagnetic clocks varied with speed in the same manner, regardless of their orientation. That assumption might be questioned as to its plausibility. Hence, in this paper, we drop this assumption and show that, by considering one additional clock, the clock rates can in fact be proven to be independent of orientation.

 

Basic Assumptions

In this paper we shall again apply only the classical physics of Newtonian mechanics and Maxwell's electromagnetic equations. This involves, in particular, Newton's laws of motion, the Lorentz force law, and Heaviside's equation for the electric field of a moving charge, which Heaviside [2] derived from Maxwell's equations.

Implicit in Heaviside's equation is the assumption that there exists a background space, a preferred frame of reference, with respect to which motion can be measured. The electric field of a point charge is spherically symmetric only if the charge is at rest with respect to the background space. Conversely, the asymmetry of the electric field is a measure of the motion of a point charge relative to the background space.

As before, we take an electromagnetic ("e-m" for short) clock to be based on the oscillations of a charged particle. No assumptions are made about the precise dependence of clock period, length, and mass on speed or orientation, or even that such dependencies actually exist. However, it is assumed that e-m clocks, if dependent on speed and orientation, all exhibit exactly the same dependencies. This assumption seems plausible, since these clocks are independent of any assumptions regarding their material constituents or scale and since these clocks are all subject to the same physical forces. 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.

From an analysis of five clocks, the first four of which are the same as considered in Byl [1], we shall derive all three basic relativistic effects - time dilation, length contraction, and mass increase. This implies, as we shall show, that all observes, whether at rest or in motion, find the same local value for the speed of light, as determined by their own measuring apparatus.

 

Electromagnetic Clocks in Motion

The basic equation is Heaviside's equation for the electric field of a moving point charge:

     E = qA (1 - β²) r /(r³[1- (β sinθ)²]^3/2)    (1)

where β º v/c and A = 1/(4πε₀). 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 [2] 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)

The clock period will depend, as will soon be shown, on the mass of the moving charge and the effective length of the clock. Allowance must thus be made for a possible dependence of both mass and length 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)

Suppose the charge oscillates in the x-direction and that its speed with respect to the clock is much smaller than the speed of the clock (i.e., |dx/dt| « v). If the clock has a velocity v in the x-direction, 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, so that m₁ is the effective longitudinal mass. On the other hand, if the clock velocity v is perpendicular to the x-direction we obtain

     F » md²x/dt² º m₂ d²x/dt²    (5)

where the subscript 2 refers to an orientation perpendicular to the direction of motion, so that m₂ is the effective transverse mass.  Since equation (5) indicates that m₂ = m, equation (4) implies that the longitude and transverse masses are related according to the equation

     m₁ = v dm₂/dv +m₂    (6)

 

(a) Clock #1

Consider first a clock consisting 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. For example, one could take the outer two charges to be fixed at the ends of a thin hollow cylindrical insulator with the third charge, having a radius slightly less than that of the cylinder, free to slide about inside the cylinder. Then the inner charge has an equilibrium position halfway along the cylinder.

The inner charge, upon being displaced a small distance x « L₀ from its equilibrium position,  will oscillate about the equilibrium point. The period of oscillation is independent of x and provides the time unit for the clock.

Now suppose the clock itself is moving at a speed v in a direction parallel to its axis (i.e., r = xi  and v = vi, where i is a unit vector along the x-axis), where v is much greater than the maximum speed of the inner charge relative to the clock. For this configuration the inner charge experiences no magnetic force (i.e., v´E = 0). Hence, applying equations (1) and (2), the electric force on the inner charge is given by:

     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 still assuming x « L, this reduces to

     F » -q²A(1 - β²) 4x L₁^-3 = m₁ d²x/dt²     (8)

This equation can easily be solved for x, which undergoes sinusoidal oscillation with a period

     T₁ = 2π [m₁ L₁³/4Aq²(1 - β²)]^½     (9)

or, in terms of the rest period T₀,

     T₁/T₀ = (L₁/L₀)^3/2(m₁/m₀)^½(1 - β²)^-½    (10)

 

(b) Clock #2

A second clock, described by Jefimenko [3], has a ring of radius L₀ and positive charge q. A particle of negative charge -q is constrained to move through the perpendicular axis of the ring. The moving negative charge, when displaced a small distance x above the plane of the ring, will oscillate with a period T₀ independent of x.

This clock, too, is set in motion in a direction parallel to its axis. Since v and E are again 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)

Assuming x « L₂, this simplifies to

     F » -q²AxL₂^-3(1 - β²)^-½ = m₁ d²x/dt²    (12)

Again, this yields a sinusoidal oscillation in x, this time with a period

     T₁/T₀ =(L₂/L₀)^3/2(m₁/m₀)^½(1 - β²)^1/4    (13)

The only significant difference between these first 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)^½    (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

Consider next a third clock, similar to clock #1, but this time with its axis oriented perpendicular to its direction of motion (i.e., take r = xi  and v = vj, where i and j are unit vectors along the x and y axes, respectively). In this case the inner charge experiences also a magnetic force. The force equation (2) then becomes:

     F = -q² A4x L₂^-3(i + v´(v´i)/c²)(1 - β²)^-½    (15)

          or

     F = -q² A4x L₂^-3(1 - β²)^½ = m₂ d²x/dt²    (16)

This equation again results in sinusoidal motion, now with period

     T₂/T₀ = (L₂/L₀)^3/2(m₂/m₀)^½(1 - β²)^-1/4    (17)

 

(d) Clock #4

A fourth clock consists of a particle of rest mass m₀ and negative charge -q in a circular orbit of radius L₀ about a fixed 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. The Lorentz force on the negative charge, as given by equation (2), will be central. Hence its angular momentum will be conserved. For a  circular speed of 2πL/T, this leads to

2π m₀ L²₀ /T₀ = 2π m₂ L²₂/T₂    (18)

This in turn gives

     T₂/T₀ = (L₂/L₀)² m₂/m₀    (19)

 

(e) Clock #5

Our fifth and final clock is a modification of clock #3. The end charges are replaced with perfectly elastic, charge-neutral walls, so that the inner particle merely bounces from wall to wall with constant speed u. The clock, oriented perpendicular to the direction of motion, is gently accelerated from rest to the final speed v, where u << v. Strictly speaking, this is a purely kinematic clock since no e-m forces enter into consideration. Since the accelerating force on the clock is perpendicular to the direction of oscillation of the inner particle, the linear momentum mu of the inner particle will be conserved. Taking the oscillation speed as u = 2L/T, this results in the relation

     m₀ L₀/T₀ = m₂ L₂ /T₂    (20)

          or

     T₂/ T₀ = (m₂ /m₀)(L₂ /L₀)    (21)

 

Discussion

From the above results for the various clock periods we can readily derive the equations for time dilation, length contraction, and mass increase. Comparing equations (19) and (21), we conclude that

     L₂ = L₀    (22)

Hence, equation (14) yields

     L₁ = L₀(1 - β²)^½    (23)

Applying this result and comparing equations (17) and (21), it is found that

     m₂ = m₀ (1 - β²)^-½ (24)

This implies, using equation (6), that

     m₁ = v dm₂/dv +m₂ = m₀ (1 - β²)^-3/2    (25)

Substituting the above results for m and L into equations (10) and (21), we find that

     T₁ = T₂ =  T₀ (1 - β²)^-½    (26)

Thus the clock rate is independent of the clock's orientation.

In short, we have derived length contraction, time dilation, and mass increase quite simply from classical mechanics and electromagnetism. The above calculations were independent of the actual size, mass, charge, or chemical composition of the clocks. Therefore the formulas for length contraction and mass increase should hold for any moving object.

One important consequence of this is that all moving observers, when using e-m clocks, measure the speed of light to be c. This can be shown as follows. Suppose the speed of light is determined by measuring the time T taken for a light ray to travel back and forth along a rod of length L with a mirror at one end. A stationary observer, S, measures the time interval to be

     T₀ = 2 L₀/c    (27)

Now consider another observer, M, moving with velocity v and with the rod inclined at an angle θ, as seen in M's system, with respect to v. Due to length contraction, as given by equations (22) and (23), the length of the rod becomes [4]

     L = L₀(1 - β²)^½/[1 - (β sinθ)² ]^½    (28)

In S's stationary frame the speed of light is taken to be c in all directions. Applying the usual Galilean vector addition, the speed of light in M's frame can readily be shown to be

     c' = [c² - (v sinθ)² ]^½ + v cosθ    (29)

The time taken for a light ray to travel back and forth along the rod is

     T = L/c'₊ + L/c'_-    (30)

where c'₊ is the speed of light one way, as given by equation (29), and c'_- is the speed of light on the return trip, v in equation (29) now being replaced by -v. This reduces to

     T = 2 L₀ (1 - β²)^-½/c    (31)

This is the time as measured by S's clock. Since M's clock is slow by a factor (1 - β²)^-½, M measures the time interval to be 2 L₀/c. In other words, the time taken, by M's clock, for a light ray to traverse M's moving rod is exactly the same as the time measured, by S's clock, to traverse S's stationary rod. Since M's standard meter has shrunk by the same fraction as his rod, he still measures the rod to be a length of L₀, in terms of his own standard meter. Hence all observers obtain the same numerical result for the speed of light, in terms of their own standard lengths and clocks, regardless of their speed or the orientation of their rods.

Thus we have derived, simply from classical physics, the relativistic postulate that the speed of light is the same for all observers. From this, along with the above results, the Lorentz transformations can readily be deduced.