Implication
of Contiguous Equivalence
On
Inter-Bodies' Time and Distance
C P Viazminsky
Department of Physics
University of Aleppo
Syria
Abstract
We introduce
first the concept of inertial frames that are contiguous to an inertial frame.
It is argued that an inertial frame is physically equivalent to the set of
inertial frames that are contiguous to it. Utilizing this equivalence in
conjunction with Galilean transformation of velocity requires on one hand adopting
besides the familiar time and distance new m-time and m-distance, and on the
other hand leads to a contradiction that is resolved in accord with the
symmetric status of the inertial frame and that which is contiguous to it. A
major conclusion from this formulation is that the Euclidean geometry of the
physical space must be reserved only to the static space, i.e. to the space
composed of all points that are stationary with respect to each other. The
distance at some instant t between any two points that have a non-zero relative
velocity is no more equal to the distance separating their stationary locations
at that instant. The new m-distance equal to the product of a factor 
by the stationary
distance. It shown that the m-distance shrinks by a factor 
if the points are
approaching each other, and expands by the factor 
if the points are receding from each other. Similar statement
holds for the time associated with this distance. The factor 
tends at both
infinities to the corresponding relativistic Doppler's factors. As the law of
relativistic velocity addition is a direct consequence of the transformation
between the m-time and distance and the s-time and distance, the drag effect is
readily explained. The Doppler's and aberration effects are explained in a
straightforward manner. The independence of the velocity of light of the
relative motion between the source and observer is a consequence of the theory,
and accordingly all experiments which require for its explanation this
assumption are also explicable by the current theory.
1.
Introduction
We begin with a
simple example in which we consider the Galilean transformations from a
slightly different point of view. This will lead us eventually to the concept
of contiguous inertial frames. Let us imagine a long train passing by a station
deck with a uniform velocity 
Let 
be a rectangular
Cartesian frame of coordinates attached to the deck, with 
is a fixed point of
the deck and 
is along the train’s
motion. Let 
be a point of the
train that coincides with 
at 
, and 
be a rectangular
Cartesian system of coordinates fixed to the train and in standard
configuration with 
The Galilean transformations between 
and 
are
Let 
be a point with
coordinates 
in the deck frame 
Instead of specifying
the coordinates of 
relative to the train
frame 
i.e. relative to the
train frame with origin at 
we specify it
relative to the frame 
which is a frame moving with the train, but it has its origin
at a point 
that is contiguous to
at the instant t. According to this way of coordinatization,
an infinite set of adjacent inertial frame all moving with the train is
employed, and of which one at a time is used to specify the coordinates of the
point 
If the coordinates of
are 
in S and 
in 
then
The Galilean transformations are recovered
on noting that the coordinates of 
or 
with respect to 
are (-ut,0,0), so
that
It is clear that the set of measurements 
gathered by the
observers 
is identical to the set 
gathered by 
and consequently, the
velocity, acceleration, and force related to the material point 
are the same.
Assuming 
is a point fixed to
the train, then its velocity relative to 
is u. Although 
is at rest relative
to each frame 
its velocity with
respect to the systems of frames 
is
We may imagine 
as a point of a train
that is composed of extremely tiny compartments, and the position of 
is determined at each
instant with respect to the compartment that is contiguous to the point 
of the deck. The
point 
is at rest relative
to each compartment contiguous to 
but has velocity u
when more than one compartment is involved. On the other hand, a particle which
is at rest in the deck frame, has velocity (-u) relative to each compartment
contiguous to 
, but is stationary when many compartments are involved. We
should keep in mind that the compartments’ size is as small as we please. Also,
it is important to note that the velocity v of a particle that is measured in 
is the same as that measured by the set of observers 
However, and relative to one contiguous frame the velocity of
the particle will be 
Definition: Let 
be an inertial frame. The set of inertial frames 
having the
properties:
(i) 
is coincident with 
at the instant t, and
(ii) At each instant t, the measurements of
the position of the particle and the corresponding time are made with respect
to 
shall be called a u-contiguous frame to 
Note
that if 
is u-contiguous to 
, Then 
is –u-contiguous to 
.
We
shall assume that the velocity of light within each inertial frame equals to a
constant c. By this we shall mean that if light is emitted at an instant 
from the point 
and received at an
instant t at the point 
, where the points 
are stationary in the
inertial frame S, then 
We have assumed implicitly that clocks in each inertial frame
can be synchronized by the usual procedure of light signals (Rindler,1977 ;
Mould, 1994) so that time is absolute in each inertial frame. This latter
assumption regarding the constancy of the light’s velocity within each inertial
frame may hardly be counted as a postulate. It may be instead, considered as a
natural consequence that follows from the equivalence of inertial frames. In
comparison, the special theory of relativity assumes that the velocity of light
emitted from a source is the same in all inertial frames, and thus, is
independent of the state of relative motion between the source of light and the
observer. In our assumption the source is stationary relative to the observer.
For start we shall apply the Galilean law of velocity addition to light signals
emitted from a moving source and show that this results in a contradiction,
which requires for its elimination, adopting a new transformation on one hand
and introducing two types of time intervals on the other hand.
2.
Longitudinal Motion
Consider a source of light 
moving relative to the inertial frame 
along the x-axis with
constant velocity 
, where 
is the unit vector of the x-axis, and suppose that 
is approaching 
from left. The light
emitted by the source 
and received latter
by 
at some instant 
, may be envisaged by the observer 
, as well as by any S-observer, in either of the following
ways:
(i) Light has been emitted from the source 
and in accord with
the Galilean law of velocity addition, it should acquire a velocity 
relative to 
, and to any S-observer.
(ii) Light has been emitted from the
stationary point 
in S, which
was occupied by the source at an earlier instant 
, where 
is the coordinate of 
. The pulse takes by the first point of view a duration 
to reach 
and 
by the second view.
Thus we have 
, and hence
(2.1) 
This current view is unaltered had we
introduced a true source of light fixed at 
and sends a pulse of
a very short duration as 
passes through the
point 
We shall describe the
latter source as virtual although it may be a true source.
Consider
the inertial observer 
who is moving
relative to S with velocity 
and just passing by 
when light reaches 
(and of course 
). Suppose that the observer 
is endowed with an inertial
frame 
in standard
configuration with S. Relative to 
the source 
is stationary,
whereas 
is moving away from 
with velocity (-u).
Using the Galilean law of velocity addition, the observer 
deduces that the time
intervals 
, considered above, are related by the equation 
or
(2.2)
Being physical quantities for both
observers 
, the intervals 
refer to the same
quantities in equations (2.1) and (2.2). Indeed 
is the time interval
taken by light if emitted from 
to reach 
whereas 
is such time interval
if light is emitted from 
However, and if we
substitute for 
from one equation
into the other, we obtain
(2.3)
,
where 
To resolve this
contradiction and maintain at the same time the symmetric status of the
observer 
, we have to scale the right hand-sides of equations (2.1)
and (2.2) through multiplying by 
. This scaling process yields the relations
(2.4) 
Setting
(2.5) 
,
we write the latter equations as
(2.6) 
Since the true source is stationary relative to 
, its distance form 
(and 
when light reaches
these two points is 
. Similarly, and since the virtual source is stationary relative to 
, its distance from 
when light reaches these points is 
. Thus we have
(2.7) 
.
When light reaches 
the observer 
describes the process of light emission and reception as
follows: At an earlier instant 
the true and virtual
sources were adjacent at 
and each emitted a pulse of light. At the moment 
was contiguous to me(which I have already taken as 
), light has reached both of us. At this moment (t=0) the
virtual source is at a distance 
from me, whereas the
true source is at a distance 
. The observer 
describes the same
process as: at an earlier instant 
the true and virtual
sources were adjacent at 
and each emitted a
pulse of light. At the moment 
was contiguous to me (which I take t=0) light has reached
both of us. At this moment 
the true source is at a distance 
from me, whereas the virtual source is at a distance 
The relations (2.6)
and (2.7) hold as long as 
approaches 
However, as 
bypasses 
and travels away,
these relations become
(2.8) 
(2.9) 
We have seen that the nature of 
as a true or a virtual
source has no effect on the results we have obtained. Thus our treatment
may be considered to involve a system
of two sources, one of each is stationary in one frame and moving in the other.
The moving (true) source in S approaches 
while the moving
(virtual) source moves away from 
The observers 
and 
associate with each
body (source), or more precisely with the distance separating him from the
source, a certain time interval 
so that the same time
interval is associated with the same body. Both observers 
agree that: the time
intervals 
are equal to 
the time intervals 
are equal to 
and that 
provided that 
is approaching 
An equivalent
alternative view is the following: denote by 
the
distance between 
and by 
the distance
from 
, then 
The observer 
finds that the moving body 
(towards 
is closer than its
location 
by a factor 
Similarly, 
finds the distance of
the hypothetical source 
, which is moving away from 
is larger than the
distance of its location 
by a factor 
Instead
of considering the body 
is moving relative to
we may assume that 
is moving relative to
the frame 
in which 
is momentarily
stationary, and we get the same result: the body 
seems closer to 
than its location 
.
We
shall refer to 
as the body’s
stationary and moving coordinates respectively, or in brief, the s- and
m-coordinate respectively. Note that if we let 