__Learning to Love the Lorentz Transformation__

__Abstract.__**
A rather thorough review of the effects that result in the Lorentz
Transformation. **

__1. A Macroscopic Bohr Hydrogen Atom.__

Fig. 1_1 depicts a negatively charged satellite, q, passing a resting, positively charged particle in inertial frame K’.

Figure 1_1

System at t’=0 in Frame K’

We shall assume that any radiation reaction force on the satellite is mechanically counteracted. If this is done, then the satellite’s motion is solely attributable to the Coulomb force.

The satellite will travel in a circle if q is such that

. (1_1)

Viewed from frame K, which moves toward
–x’ at speed v’, the satellite is momentarily at rest and the central body
moves in the +x direction at speed v_{Q}=v’. Under a *Galilean* transformation
the satellite’s path is a *cycloid*.
This path can also be deduced by applying Coulomb’s law and Newton’s 2^{nd} law
in its non-relativistic form:

. (1_2)

Unfortunately Coulomb’s law does not produce
a *magnetic* field
in K. And of course Eq. 1_2 does not take into account the dependence of
inertial mass on speed. What is required in K is a correct expression for the *Lorentz* force
acting on the satellite, equated to the relativistically correct expression for
the Newtonian force. These two requirements are addressed in the next two
sections.

__2. The Lorentz Force Law in K.__

The electric and magnetic fields of a point charge (e.g., Q), moving with constant velocity, are well understood. In component form,

, (2_1)

, (2_2)

. (2_3)

Figure 2_1

Point Charge Q Moving with Constant Velocity

The Lorentz force experienced by the satellite thus has components

, (2_4)

. (2_5)

Note here that v_{x} and
v_{y} are the *variable* components
of the *satellite’s* velocity
relative to K.

__3. The Relativistically Rigorous Newtonian
Force.__

Newton’s 2^{nd} law
states relativistically that

. (3_1)

Or, since

, (3_2)

Eq. 3_1 can be rewritten in component notation as

, (3_3)

. (3_4)

Solving Eq. 3_4 for a_{y} and
substituting into Eq. 3_3 produces

, (3_5)

. (3_6)

__4. Computing the Motion of q Relative to K.__

The satellite initial conditions in K at time t=0 are (ref. Fig. 1_1)

, (4_1)

, (4_2)

, (4_3)

, (4_4)

. (4_5)

Referring to Eqs. 3_5 and 3_6, we may use
these values to compute a_{x} and
a_{y} at time t=0.
Having done this, we can then compute

, (4_6)

, (4_7)

, (4_8)

, (4_9)

where dt is an adequately small increment of time, say

. (4_10)

An interesting question is: What is the moving
system’s *shape*? We can
certainly place *Q* at
the origin of K by subtracting v_{Q}t from its position at each time
epoch t. Subtracting the same value from the computed satellite position then
shows its location relative to Q. The Python program, that computes satellite
positions relative to Q, is provided in Appendix A at this article’s end.

Fig. 4_1 shows the satellite position, relative to Q, when v_{Q}=.001c.
As might be expected, the path is practically the same circle as in K’.

Figure 4_1

q’s Path (adjusted) in K, v_{Q}=.001c

Fig.
4_2 shows the satellite path, relative to Q, when v_{Q}=.95c. Note the
expected length contraction of the circle in Fig. 4_1 (by a factor of (1-v_{Q}^{2}/c^{2})^{1/2}).
The time for one quasi-cycloid in K is also (2pR/v_{Q})/(1-v_{Q}^{2}/c^{2})^{1/2},
manifesting the expected time dilation.

Figure 4_2

q’s Path (adjusted) in K, v_{q}=.95c

__5. The Lorentz Transformation.__

Fig. 4_2 and the time-dilated cycle time in K,
when v_{Q}=.95c, suggest that the Galilean transformation requires
adjustment. Indeed the length-contracted "Bohr Orbit," coupled with
the time-dilated cycle time in K, suggest that *all* moving
systems might be length-contracted and time-dilated. This would presumably
include the rigid grid of K’, as viewed from K, and the (moving) clocks of
K’. It is a bold generalization considered by Einstein and others. And as
Einstein pointed out, the distributed clocks of K’ would also not be
synchronized in the opinion of K (and vice versa), assuming K’ synchronized
its clocks in the belief that the speed of light was c in all directions
relative to that frame (as experiment indicates is the case). In essence we end
up with two points of view:

**K Point of View: **(1)
Grid of K’ is length-contracted in __+__ x’
direction; (2) Distributed clocks at rest in K’ run slowly; (3) Distributed
clocks at rest in K’ are not synchronized with one another.

**K’ Point if View: **(1)
Grid of K is length-contracted in __+__ x
direction; (2) Distributed clocks at rest in K run slowly; (3) Distributed
clocks at rest in K are not synchronized with one another.

It is worth noting that these points of view
can be corroborated by *actual
measurements*. Of course each frame believes that *its* measurements
are correct, and *the
other’s* are erroneous,
owing to the length contraction of the other’s grid, etc. The elegant thing is
that a complete symmetry exists among *all* inertial
frames. And as demonstrated above, given an electrodynamic system’s specifics
in one frame, Newton/Maxwell/Lorentz can be applied in other frames to obtain
the system component motions in those other frames.

Taking into account the length contraction,
etc., of K’ from the point of view of K, it is a straightforward problem in
algebra to answer the following question: if an event (say an explosion) occurs
at space-time coordinates (x,y,z,t) in K, then what will the space-time
coordinates of the *same* event
be in K’? A problem solution is suggested in Appendix B. The Lorentz
transformation provides the answer. It is of course

, (5_1)

, (5_2)

, (5_3)

. (5_4)

From these transformations one can obtain
transformations for velocity, acceleration, etc. And, provided the dependence of
mass upon speed is taken into account, one can derive the transformation of ** F**=d/dt(m

It is an impressive result that the Lorentz
force transforms precisely as Newton’s d(m** v**)/dt
does. Indeed it might be

__6. "Snapshots" and
Non-Synchronicity.__

In the Galilean transformation, one universal
time for all observers is implied. In effect all inertial observers agree that
if clocks were distributed throughout *all* inertial
frames, then the clocks would all run at the same rate and could be
synchronized. Among other things this would imply that if K observes events at x_{2} and
x_{1}<x_{2} to
occur simultaneously (say at time t), then K’ will agree on the time and
simultaneity of the same events.

It is clear in Eq 5_4 that this agreement does
not apply under the Lorentz transformation. In the first place, K and K’ may
not agree about the time of an event at nonzero values of x and x’.
Furthermore, spatially discrete events that occur simultaneously in K may be
found to occur at different times in K’. More generally any
"snapshot" of the world, at a given instant t in K, will not
constitute a "snapshot" in K’. Rather it will constitute a virtual
continuum (in space *and* time)
of events in K’.

An excellent example of such non-synchronicity
is the electromagnetic field. The fields specified in Eqs. 2_1 – 2_3 and Fig.
2_1 actually pertain to all spatial points at a single instant in K. One should
not fall into the trap of thinking that the fields in K’, at a single instant,
can be obtained by application of the general field transformations. For these
transformations provide field values at a *continuum* of
times in K’.

Fortunately, if one has a formula for ** E** and

__7. Concluding Thoughts.__

Owing to the Lorentz transformation, which
among other things implies that *all* systems
are length contracted when they move, an astonishing and unexpected symmetry
emerged. Based on *actual
measurement*, inertial observers K and K’ each justifiably concluded that
clocks in the other frame ran slowly. Each agrees that the other makes careful
measurements. But each contends that the other’s conclusions are based on
faulty assumptions (that the other’s clocks are synchronized, etc.)

Historically, before such inherent symmetries
were understood, it was believed that there existed some one special inertial
frame of reference in which (for example) moving charge really *does* engender
a ** B** field.
Attempts to find this frame of course failed. It seemed as though nature’s
length contraction of moving systems, etc., conspired to place

Have we unequivocally ruled out the existence
of a "primary" (or "ether" or "dark matter" or
...) frame? In truth we have not, although we must acknowledge the possibility
that no experiment can differentiate such a frame from all the other inertial
frames. Perhaps, with advances in astronomy, we can attempt to narrow the search
and *define* the
"primary" frame to be the frame in which the center of mass of the
known universe is at rest. Perhaps. But what will such an exercise gain us?

One thing does seem certain. The length contraction of moving systems appears not to be an illusion, even though K and K’ each measures such effects for systems moving relative to himself. Such effects are real, and are predicted by the remarkable fact that the physics of Newton, Maxwell and Lorentz work equally well in every inertial frame of reference.

***Appendix A***

"""A program that demonstrates length contraction and time dilation.

Given a charged satellite, orbiting an oppositely charged central body,

use Maxwell, Lorentz and Newton to compute the satellite's motion

relative to inertial frame K. Output data reflecting the system shape

in K for plotting purposes."""

import math

c=299792000. #Speed of light

eps0=8.85e-12 #Permittivity constant

Steps=1000000 #Steps in an iterative loop

Selection=int(input("Speed of satellite in K'=1 for .01c, 2 for .95c "))

if(Selection==1):

SatelliteSpeedInKprime=.01*c

else:

SatelliteSpeedInKprime=.95*c

CentralBodySpeedInK=SatelliteSpeedInKprime #Central body constant speed in K

OrbitalRadiusInKprime=1. #Orbital radius in K'

SatelliteRestMass=1. #Satellite rest mass

SatellitePeriodInKprime=2*math.pi*OrbitalRadiusInKprime/SatelliteSpeedInKprime #Orbital period in K'

#Try an educated trial value for the quasi-cycloid period in K.

gammap=1/math.sqrt(1-CentralBodySpeedInK**2/c**2)

SatellitePeriodInK=gammap*SatellitePeriodInKprime #Trial value for period in K

deltat=SatellitePeriodInK/Steps #Time increment

#Find what size charge will cause circular motion in K'.

SatelliteCharge=math.sqrt(4*math.pi*eps0*OrbitalRadiusInKprime*gammap*SatelliteRestMass*SatelliteSpeedInKprime**2)

#Initialize satellite variables for t=0 in K.

#Satellite starts out at rest on SatelliteYCoordinate axis of K.

SatelliteXCoordinate=0.

SatelliteYCoordinate=[]

SatelliteYCoordinate.append(OrbitalRadiusInKprime)

SatelliteXVelocityCoordinate=0.

SatelliteYVelocityCoordinate=0.

SatelliteSpeed=0.

gamma=1/math.sqrt(1-SatelliteSpeed**2/c**2)

AdjustedSatelliteXCoordinate=[]

for index in range(Steps-1):

#1. Compute the central body fields at the satellite using the

#Electric field components for a charge moving with constant velocity.

t=index*deltat

#Use previously computed satellite positions.

AdjustedSatelliteXCoordinate.append(SatelliteXCoordinate-CentralBodySpeedInK*t)

#R is always the satellite displacement from the central body

Rx=AdjustedSatelliteXCoordinate[index]

Ry=SatelliteYCoordinate[index]

R=math.sqrt(Rx**2+Ry**2) #R might vary in K

if(Rx==0.):

if(t==0.):

theta=math.pi/2

else:

theta=3*math.pi/2

else:

if(Rx<0. and Ry>=0.):

theta=math.pi/2+math.atan(-Rx/Ry)

if(Rx<0. and Ry<0.):

theta=math.pi+math.atan(Ry/Rx)

if(Rx>=0. and Ry<0.):

theta=3*math.pi/2+math.atan(Rx/-Ry)

if(Rx>=0. and Ry>=0.):

theta=math.atan(Ry/Rx)

Ex=SatelliteCharge/(4*math.pi*eps0)*math.cos(theta)/R**2*(1-CentralBodySpeedInK**2/c**2)/(math.sqrt(1-(CentralBodySpeedInK/c*math.sin(theta))**2))**3

Ey=SatelliteCharge/(4*math.pi*eps0)*math.sin(theta)/R**2*(1-CentralBodySpeedInK**2/c**2)/(math.sqrt(1-(CentralBodySpeedInK/c*math.sin(theta))**2))**3

Bz=1/c**2*CentralBodySpeedInK*Ey

#2. Compute the Lorentz force acting on the satellite.

Fx=-SatelliteCharge*(Ex+SatelliteYVelocityCoordinate*Bz)

Fy=-SatelliteCharge*(Ey-SatelliteXVelocityCoordinate*Bz)

#3. Compute the relativistically rigorous acceleration.

ax=(Fx*(c**2-SatelliteXVelocityCoordinate**2)-Fy*SatelliteXVelocityCoordinate*SatelliteYVelocityCoordinate)/(SatelliteRestMass*c**2*gamma)

ay=(Fy*(c**2-SatelliteYVelocityCoordinate**2)-Fx*SatelliteXVelocityCoordinate*SatelliteYVelocityCoordinate)/(SatelliteRestMass*c**2*gamma)

#4. Update the position and velocity.

SatelliteXCoordinate=SatelliteXCoordinate+SatelliteXVelocityCoordinate*deltat+1/2*ax*deltat**2

#This step avoids division by zero in computing theta.

if(index<Steps-1):

SatelliteYCoordinate.append(SatelliteYCoordinate[index]+SatelliteYVelocityCoordinate*deltat+1/2*ay*deltat**2)

SatelliteXVelocityCoordinate=SatelliteXVelocityCoordinate+ax*deltat

SatelliteYVelocityCoordinate=SatelliteYVelocityCoordinate+ay*deltat

SatelliteSpeed=math.sqrt(SatelliteXVelocityCoordinate**2+SatelliteYVelocityCoordinate**2)

gamma=1/math.sqrt(1-SatelliteSpeed**2/c**2)

#Compare final time with the SatellitePeriodInK guesstimate.

print('t= ',t,' SatellitePeriodInK= ',SatellitePeriodInK)

#Output the AdjustedSatelliteXCoordinate and SatelliteYCoordinate satellite positions for shape plotting.

f=open('c:/Python33/Website/PLOTVALS.dat','w')

f.close()

f=open('c:/Python33/Website/PLOTVALS.dat','a')

for index in range(int(Steps/1000-1)):

f.write(str(AdjustedSatelliteXCoordinate[1000*index]))

f.write(',')

f.write(str(SatelliteYCoordinate[1000*index]))

f.write('\n')

f.close()

print('Ready for plotting')

"""A program that demonstrates length contraction and time dilation.

Given a charged satellite, orbiting an oppositely charged central body,

use Maxwell, Lorentz and Newton to compute the satellite's motion

relative to inertial frame K. Output data reflecting the system shape

in K for plotting purposes."""

import math

c=299792000. #Speed of light

eps0=8.85e-12 #Permittivity constant

Steps=1000000 #Steps in an iterative loop

Selection=int(input("Speed of satellite in K'=1 for .01c, 2 for .95c "))

if(Selection==1):

SatelliteSpeedInKprime=.01*c

else:

SatelliteSpeedInKprime=.95*c

CentralBodySpeedInK=SatelliteSpeedInKprime #Central body constant speed in K

OrbitalRadiusInKprime=1. #Orbital radius in K'

SatelliteRestMass=1. #Satellite rest mass

SatellitePeriodInKprime=2*math.pi*OrbitalRadiusInKprime/SatelliteSpeedInKprime #Orbital period in K'

#Try an educated trial value for the quasi-cycloid period in K.

gammap=1/math.sqrt(1-CentralBodySpeedInK**2/c**2)

SatellitePeriodInK=gammap*SatellitePeriodInKprime #Trial value for period in K

deltat=SatellitePeriodInK/Steps #Time increment

#Find what size charge will cause circular motion in K'.

SatelliteCharge=math.sqrt(4*math.pi*eps0*OrbitalRadiusInKprime*gammap*SatelliteRestMass*SatelliteSpeedInKprime**2)

#Initialize satellite variables for t=0 in K.

#Satellite starts out at rest on SatelliteYCoordinate axis of K.

SatelliteXCoordinate=0.

SatelliteYCoordinate=[]

SatelliteYCoordinate.append(OrbitalRadiusInKprime)

SatelliteXVelocityCoordinate=0.

SatelliteYVelocityCoordinate=0.

SatelliteSpeed=0.

gamma=1/math.sqrt(1-SatelliteSpeed**2/c**2)

AdjustedSatelliteXCoordinate=[]

for index in range(Steps-1):

#1. Compute the central body fields at the satellite using the

#Electric field components for a charge moving with constant velocity.

t=index*deltat

#Use previously computed satellite positions.

AdjustedSatelliteXCoordinate.append(SatelliteXCoordinate-CentralBodySpeedInK*t)

#R is always the satellite displacement from the central body

Rx=AdjustedSatelliteXCoordinate[index]

Ry=SatelliteYCoordinate[index]

R=math.sqrt(Rx**2+Ry**2) #R might vary in K

if(Rx==0.):

if(t==0.):

theta=math.pi/2

else:

theta=3*math.pi/2

else:

if(Rx<0. and Ry>=0.):

theta=math.pi/2+math.atan(-Rx/Ry)

if(Rx<0. and Ry<0.):

theta=math.pi+math.atan(Ry/Rx)

if(Rx>=0. and Ry<0.):

theta=3*math.pi/2+math.atan(Rx/-Ry)

if(Rx>=0. and Ry>=0.):

theta=math.atan(Ry/Rx)

Ex=SatelliteCharge/(4*math.pi*eps0)*math.cos(theta)/R**2*(1-CentralBodySpeedInK**2/c**2)/(math.sqrt(1-(CentralBodySpeedInK/c*math.sin(theta))**2))**3

Ey=SatelliteCharge/(4*math.pi*eps0)*math.sin(theta)/R**2*(1-CentralBodySpeedInK**2/c**2)/(math.sqrt(1-(CentralBodySpeedInK/c*math.sin(theta))**2))**3

Bz=1/c**2*CentralBodySpeedInK*Ey

#2. Compute the Lorentz force acting on the satellite.

Fx=-SatelliteCharge*(Ex+SatelliteYVelocityCoordinate*Bz)

Fy=-SatelliteCharge*(Ey-SatelliteXVelocityCoordinate*Bz)

#3. Compute the relativistically rigorous acceleration.

ax=(Fx*(c**2-SatelliteXVelocityCoordinate**2)-Fy*SatelliteXVelocityCoordinate*SatelliteYVelocityCoordinate)/(SatelliteRestMass*c**2*gamma)

ay=(Fy*(c**2-SatelliteYVelocityCoordinate**2)-Fx*SatelliteXVelocityCoordinate*SatelliteYVelocityCoordinate)/(SatelliteRestMass*c**2*gamma)

#4. Update the position and velocity.

SatelliteXCoordinate=SatelliteXCoordinate+SatelliteXVelocityCoordinate*deltat+1/2*ax*deltat**2

#This step avoids division by zero in computing theta.

if(index<Steps-1):

SatelliteYCoordinate.append(SatelliteYCoordinate[index]+SatelliteYVelocityCoordinate*deltat+1/2*ay*deltat**2)

SatelliteXVelocityCoordinate=SatelliteXVelocityCoordinate+ax*deltat

SatelliteYVelocityCoordinate=SatelliteYVelocityCoordinate+ay*deltat

SatelliteSpeed=math.sqrt(SatelliteXVelocityCoordinate**2+SatelliteYVelocityCoordinate**2)

gamma=1/math.sqrt(1-SatelliteSpeed**2/c**2)

#Compare final time with the SatellitePeriodInK guesstimate.

print('t= ',t,' SatellitePeriodInK= ',SatellitePeriodInK)

#Output the AdjustedSatelliteXCoordinate and SatelliteYCoordinate satellite positions for shape plotting.

f=open('c:/Python33/Website/PLOTVALS.dat','w')

f.close()

f=open('c:/Python33/Website/PLOTVALS.dat','a')

for index in range(int(Steps/1000-1)):

f.write(str(AdjustedSatelliteXCoordinate[1000*index]))

f.write(',')

f.write(str(SatelliteYCoordinate[1000*index]))

f.write('\n')

f.close()

print('Ready for plotting')

*****Appendix B*****

A Derivation the Lorentz Transformation of Space and Time Coordinates

Given two inertial frames of reference (say K and K’) with the usual relative motion, three of the basic tenets of Special Relativity Theory are (1) an observer, using the rectangular coordinate grid and clocks of K, will measure the grid of K’ to be contracted in the x direction; (2) the same observer will measure the clocks at rest in K’ to run more slowly than those of K; and (3) the clocks in K’, with different x’ coordinates, will not be synchronized.

The intriguing thing is that the same things will be found for the grid and clocks of K, when the grid and clocks of K’ are used to make the measurements. A "symmetry of disagreement" exists, and it is not possible to definitively demonstrate that one party is correct and the other is wrong.

The contraction of another frame’s grid, and the slower rate of its clocks are both specified by the factor

. (B_1)

The amount by which the other frame’s clocks are out of synch is less obvious, but derivable. Such a derivation is provided in this article.

We begin by imagining that we are an observer in K, watching an observer in K’ attempt to synchronize the K’ clocks. (K’ moves in the positive x direction of K at speed v.) We shall consider three K’ clocks: (1) the clock at x’=D; (2) the K’ origin clock; and (3) the clock at x’=-D. The K’ observer sends out a pulse of light in the positive and negative x’ directions. Since (according to him) the speed of light is c in all directions, relative to K’, he posits that the clocks at D and –D are synchronized with the origin clock if they are set to D/c upon reception of the pulse.

Let us call (1) the sending out of the pulse(s) "Event 1"; (2) the reception of the pulse at x’=D "Event 2"; (3) the reception of the pulse at x’=-D "Event 3." The K’ space-time coordinates of Event 1 are

, (B_2a)

. (B_2b)

And let us begin by considering the arrival of the pulse at x’=+D.

The K’ observer says that the K’ space-time coordinates of Event 2 are

, (B_3a)

. (B_3b)

From the perspective of frame K, the same events have the coordinates

, (B_4a)

. (B_4b)

At
time t=0, the distance to the K’ clock at x’=D is only g^{-1}D
(owing to length contraction). And since the pulse speed is finite, the clock
moves to the right while the pulse is en route. Specifically, if the time for
the pulse to propagate from the K’ origin clock to the targeted K’ clock is Dt
then when the pulse arrives that K’ clock will be at

. (B_5)

We can solve for Dt by also noting that

. (B_6)

Thus

. (B_7)

And
of course t_{2} equals Dt:

. (B_8)

Substitution into Eq. B_5 produces

. (B_9)

Now while the pulse of light was en route, the K’ origin
clock has been running. According to K, when the pulse arrives at the targeted
x’ clock, the K’ origin clock will have advanced from t_{o}’=0 to

. (B_10)

According to K, this is what the targeted K’ clock * should* be
set to if it is to be truly synchronized with the K’ origin clock. The

. (B_11)

And this lack of synchronization will (in the opinion of K) persist as time passes.

Let us now consider a random event, with space-time coordinates x and t. Our only requirement for now will be that x is greater than vt (the location of the K’ origin clock when the event occurs). This being the case, the distance between the event and the K’ origin clock will, in the opinion of K, be (x-vt). But using the grid of K’, this maps to

. (B_12)

At
time t the K’ origin clock will read g^{-1}t.
And the K’ clock at x’ will read x’v/c^{2} behind
that. Therefore

. (B_13)

Or, in terms of K parameters,

. (B_14)

Let us now consider the synchronization of the K’ clock at x’=-D. We shall again say that Event 1 is the sending out of a light pulse from the K’ origin clock. Thus once again

, (B_15a)

. (B_15b)

Event 3, the reception of the light pulse by the K’ clock at x’=-D, has coordinates

, (B_16a)

. (B_16b)

Again we have

, (B_17a)

. (B_17b)

And
the distance between the K’ origin clock and the targeted K’ clock is
(according to K) only g^{-1}D.
If Dt
again denotes the pulse’s time of flight, then in this case

. (B_18)

And now

, (B_19)

so that Dt solves to

. (B_20)

Once
again, t_{3} equals Dt:

. (B_21)

Substitution for Dt in Eq. B_18 produces

. (B_22)

When the pulse reaches the targeted K’ clock, the K’ origin clock will have advanced from zero to

. (B_23)

According to K, this is what the targeted K’ clock * should* be
set to in order to be synchronized with the K’ origin clock. But in this case
the actual setting, D/c, is

. (B_24)

Let
us again consider a random event with coordinates x and t. This time we address
the case where x is* less* than
vt (the location of the K’ origin clock when the event occurs). The distance
between the event and the K’ origin clock is (vt-x). In K’ terms this maps
to a distance of g(vt-x).
Or, since x’ is negative,

. (B_25)

At
K time t the K’ origin clock will read g^{-1}t.
But the x’ clock will read –xv/c^{2} * more* than
that. Thus

(B_26)

which is identical to Eq. B_13. Therefore

. (B_27)

In conclusion, given an event with K space-time coordinates x and t, the event’s K’ space-time coordinates will be

, (B_28a)

. (B_28b)

These two equations are the celebrated Lorentz transformations for space and time coordinates.

Of course from the perspective of K’, it is the * K* clocks
that are not synchronized. K’ contends that the K clocks on the

, (B_29a)

. (B_29b)

***Appendix C***

A Few Useful Transformations

Velocity:

If ** u** is
the measured velocity of a particle in K, then the same measurement in K’ has
components

(C_1)

(C_2)

Acceleration:

If ** a** is
the measured acceleration of a particle in K, then the same measurement in K’
has components

(C_3)

(c_4)

Force:

If ** F** is
the measured force on a particle in K, then the same measurement in K’ has
components

(C_5)

(C_6)