**Preliminary Assumptions**

We shall begin by assuming all operations to occur in vacuum and far from any object capable of exerting a measurable gravitational force.

We shall use the word "object" to refer to an elementary particle or other thing which may be localized and which may be affected as a whole by some other object, the interaction occurring at an identifiable location but without regard for rotation or resolution of substructure of any object.

We allow for clocks to synchronize events, an event being the coincidental observation of one or more objects in the same place as the clock. We allow distance to be measured (by units of length) between events.

We assume no object can move at infinite speed, or causality (ordering of effects) would not be possible.

We shall allow any object either to be accelerable or inaccelerable but never both.

**Definition of Accelerability**

An accelerable object is any one which can be made to move at different velocities, depending on what is done to affect its motion.

**Definition of an Inertial Frame**

An inertial frame, or briefly, a *frame*, is a
coordinate system attached to the motion of an
accelerable object so that the object is at rest
in that frame. Given such an inertial frame,
then, if a force be exerted on the object, it will
be accelerated so it is in motion in that frame;
if no force be exerted, the object will remain at
rest in that frame.

**Definition of Inaccelerability**

Objects that are inaccelerable always travel at a fixed speed, possibly zero, relative to every accelerable object, and at a fixed velocity relative to any given inertial frame.

**Observation: Light is Inaccelerable**

The experiment of Michelson and Morley, and the calculations of Maxwell, show that light (photons) is inaccelerable. We assume here, that when light is reflected or refracted, photons may be affected by the reflecting or refracting object so that they vanish and are replaced after some little time by others with the same speed but different velocity (direction).

**An Inaccelerable Object Must Be Faster than any Accelerable One**

**Proof:**
Suppose an inaccelerable object *I* which travels at some speed *v*_{I}.
Let *I* travel in some direction * v*, passing close to an accelerable
object

Now, let us repeat this observation but this time suppose
that there might exist an accelerable object *A* allowing us
exactly to copy the previous observation, with *A* substituted
for *I* and travelling at speed *v*_{A} which
is greater than, or equal to, *v*_{I}. If so, we
could repeat yet a third time with both *I* and *A* starting
at *x*_{i} together. But, *I* must travel at speed
*v*_{I} in all inertial frames, including
the rest frame of *A*, and so *I* must arrive at
*x*_{f} before *A*.
This contradicts the assumption that *I* travels at the same
speed in all frames; therefore, no accelerable object can
travel at a speed as great as that of an inaccelerable object.

**The Speed of Every Inaccelerable Object is the Same**

First, proof aside, if there were more than one speed for inaccelerable objects, they could change speed and therefore would not be inaccelerable. However, one might imagine that inaccelerable objects, like photons, never change between interactions and might then retain different speeds if created with different speeds. This very reasonable possibility is what requires the following:

**Coarse Proof:**
Suppose two inaccelerable objects *I* and *J* could travel at
different speeds *v*_{I} and *v*_{J},
with *v*_{I} greater than *v*_{J}.
Start *I*, *J*, and an
accelerable object *A* at the same time at *x*_{i}
as above and in the direction * v* as above. Now, there is no reason
why

**More Rigorous Proof:**
Let there exist two accelerable objects *A*_{i} and *A*_{f}
initially at points *x*_{i} and *x*_{f} respectively
in the same frame, such that the distance,
*L* = *x*_{f} - *x*_{i}
might be chosen to be large and well-defined in that frame. Let the direction of
*x*_{f} - *x*_{i} be represented by the vector __ x__.

Let there exist a different two accelerable objects *B*_{i} and *B*_{f}
separated by the same distance *L* along the same distance vector but in a different
frame which is in motion in the direction of __ x__ at some arbitrarily high speed
relative to the frame of the

Representing *B*_{f} as to the right of *B*_{i}, let *B*_{f}
pass close to *A*_{i} and continue on, moving to the right. When *B*_{i}
then passes close to *A*_{i}, let there be an emission event as follows:

At the same instant in both frames, *A*_{i} emits a pair of inaccelerable objects
in the direction of *A*_{f}, and *B*_{i} emits an identical pair
in the direction of *B*_{f}. Of course, the two direction vectors are the same.
Let each pair of inaccelerable objects be called *I* and *J*, with, by hypothesis,
the speed of *I* greater than that of *J*.

Now, because the distance between the *A*'s is equal to that between the *B*'s,
and because inaccelerable objects travel at the same speeds in every inertial frame,
both of *A*_{f} and *B*_{f} must receive a pair of *I* and
*J* separated by the same time interval, delta-*t*, as measured in their respective
frames.

Because by hypothesis the speed of an *I* is greater than that of a *J*,
in both frames, the *I*'s must arrive first. But, the frame of the *B*'s is in motion
relative to that of the *A*'s, in the direction of the inaccelerable propagation,
so the distance from *A*_{i} to *B*_{f} must increase (in
both frames) while the inaccelerable objects are propagating. Therefore, the difference
in arrival times between *I* and *J* at *B*_{f} must be greater
than delta-*t*. See the figure.

The same quantity can not both be equal and not equal to delta-*t*, so the assumption
implying delta-*t* must be invalid, and inaccelerable objects can not exist which
travel at different speeds. All differences between such speeds must be identically 0.
This makes delta-*t* equal to 0 regardless of relative motion of inertial frames,
preventing the contradiction just derived.

We conclude that all inaccelerable objects move at the
same speed *c*, in the frame of every accelerable object.
And no object, accelerable or inaccelerable, can exceed
the speed *c* in any frame.

The speed of light then equals this same *c*.

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The Pulfrich Effect, SIU-C. Last updated 2007-04-10