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John Byl bio
On Deriving Special Relativity from
Electromagnetic Clocks
(published in
Galilean
Electrodynamics
Vol.14 (no.5, Sept/Oct 2003): 8992)
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 ("em" 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 em 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 em 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  β^{2})
r /(r^{3}[1 (β sinθ)^{2}]^{3/2})
(1)
where β º v/c and A = 1/(4πε_{0}). 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)/c^{2}]
(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 xdirection
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 xdirection, then
F = (v dm/dv +
m) d^{2}x/dt^{2} º m_{1} d^{2}x/dt^{2} (4)
where the subscript 1 refers to an orientation
parallel to the direction of motion, so that m_{1} is the effective
longitudinal mass. On the other hand, if the clock velocity v is
perpendicular to the xdirection we obtain
F » md^{2}x/dt^{2}
º m_{2} d^{2}x/dt^{2} (5)
where the subscript 2 refers to an orientation
perpendicular to the direction of motion, so that m_{2} is the
effective transverse mass. Since equation (5) indicates that m_{2}
= m, equation (4) implies that the longitude and transverse masses are
related according to the equation
m_{1} =
v dm_{2}/dv +m_{2} (6)
(a) Clock #1
Consider first a clock consisting of two
positive charges q, separated by a fixed distance L_{0}, 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 « L_{0} 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 xaxis),
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 = q^{2}A(1
 β^{2})[(L_{1} + x)^{2}  (L_{1}  x)^{2}]
= m_{1} d^{2}x/dt^{2} (7)
Since the axis is oriented in the direction of
motion and still assuming x « L, this reduces to
F » q^{2}A(1
 β^{2}) 4x L_{1}^{3} = m_{1} d^{2}x/dt^{2}
(8)
This equation can easily be solved for x, which
undergoes sinusoidal oscillation with a period
T_{1} =
2π [m_{1} L_{1}^{3}/4Aq^{2}(1  β^{2})]^{½}
(9)
or, in terms of the rest period T_{0},
T_{1}/T_{0}
= (L_{1}/L_{0})^{3/2}(m_{1}/m_{0})^{½}(1
 β^{2})^{½} (10)
(b) Clock #2
A second clock, described by Jefimenko [3], has
a ring of radius L_{0} 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 T_{0} 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 = q^{2}A(1
 β^{2})x(L_{2}^{2} + x^{2})^{3/2}[1
β^{2}/(1 + x^{2}/L_{2}^{2})]^{3/2}
(11)
Assuming x « L_{2}, this simplifies to
F » q^{2}AxL_{2}^{3}(1
 β^{2})^{½} = m_{1} d^{2}x/dt^{2}
(12)
Again, this yields a sinusoidal oscillation in
x, this time with a period
T_{1}/T_{0}
=(L_{2}/L_{0})^{3/2}(m_{1}/m_{0})^{½}(1
 β^{2})^{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
L_{1}/L_{2}
= (1  β^{2})^{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 = q^{2}
A4x L_{2}^{3}(i +
v´(v´i)/c^{2})(1
 β^{2})^{½} (15)
or
F = q^{2}
A4x L_{2}^{3}(1  β^{2})^{½} = m_{2}
d^{2}x/dt^{2} (16)
This equation again results in sinusoidal
motion, now with period
T_{2}/T_{0}
= (L_{2}/L_{0})^{3/2}(m_{2}/m_{0})^{½}(1
 β^{2})^{1/4} (17)
(d) Clock #4
A fourth clock consists of a particle of rest
mass m_{0} and negative charge q in a circular orbit of radius L_{0}
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 m_{0} L^{2}_{0} /T_{0}
= 2π m_{2} L^{2}_{2}/T_{2}
(18)
This in turn gives
T_{2}/T_{0}
= (L_{2}/L_{0})^{2} m_{2}/m_{0
}(19)
(e) Clock #5
Our fifth and final clock is a modification of
clock #3. The end charges are replaced with perfectly elastic,
chargeneutral 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
em 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
m_{0} L_{0}/T_{0}
= m_{2} L_{2} /T_{2} (20)
or
T_{2}/ T_{0}
= (m_{2} /m_{0})(L_{2} /L_{0})
(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
L_{2} =
L_{0} (22)
Hence, equation (14) yields
L_{1} =
L_{0}(1  β^{2})^{½} (23)
Applying this result and comparing equations
(17) and (21), it is found that
m_{2} =
m_{0} (1  β^{2})^{½ }(24)
This implies, using equation (6), that
m_{1} =
v dm_{2}/dv +m_{2} = m_{0} (1  β^{2})^{3/2
}(25)
Substituting the above results for m and L into
equations (10) and (21), we find that
T_{1} =
T_{2} = T_{0} (1  β^{2})^{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 em 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
T_{0} =
2 L_{0}/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 = L_{0}(1
 β^{2})^{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' = [c^{2}
 (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 L_{0}
(1  β^{2})^{½}/c (31)
This is the time as measured by S's clock. Since
M's clock is slow by a factor (1  β^{2})^{½}, M measures
the time interval to be 2 L_{0}/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 L_{0}, 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.
Although our Lorentzian approach ends up with the
same equations as special relativity, their interpretation is
quite different. The special relativistic effects are no longer symmetric for
observers in relative motion. Instead, they are determined solely by motion with
respect to the background space. Also, the effects are not merely apparent but
real: lengths are actually shortened, clockrates are actually reduced and mass,
as a measure of resistance against acceleration, is actually increased.
This method has the advantage of avoiding the
counterintuitive aspects of Einstein's special relativity. For example, the
twin paradox is resolved since one no longer expects symmetric aging. Further,
it is now only the measured, rather than the actual, speed of
light that is the same for all observers.
In conclusion, we have shown that the equations of
special relativity can be derived as a natural consequence of classical physics,
thereby providing a more intuitive framework for interpreting special
relativity.
REFERENCES
[1] J. Byl, "Special Relativity Via Electromagnetic
Clocks", Galilean Electrodynamics 10 (No.6):107110 (1999).
[2] O. Heaviside, "The Electromagnetic Effects of a
Moving Charge", The Electrician 22, 147148 (1888).
[3] O.D. Jefimenko, "Direct Calculation of Time
Dilation", Am. J. Phys. 64(6), 812814 (1996).
[4] F. Selleri, "On the Meaning of Special
Relativity If a Fundamental Frame Exists", in Progress in New Cosmologies:
Beyond the Big Bang (edited by H.C. Arp et al, Plenum Press, New York,
1993), pp.269284.
