Proper Time Adjusted Special Relativity:
The Phenomena of Special Relativity
as a Result of the Curved Expanding Universe
This is a interactive
presentation of the Proper Time Adjusted Special Relativity theory proposed in this site. In the text that follows,
some words are links, and when you click on them things happen in the applet. Most of them work as toggles. It
should be self explanatory.
We will try to show that the
phenomena of Special Relativity are produced by the curved expanding universe that was proposed for the first time
as a solution to the equations of General Relativity.
Let's start with the Lorentz
transformation. The axes of the Moving Body as calculated by the Lorentz transformation (for c = 1 and
v = 0.6) are
these. The left axis is the worldline of the Moving Body (its t axis), and at
t = 10 the Stationary Body is
here and the Moving Body is
here.
However, its proper time, as
indicated by the gray number beside it, is t' = 8. Now, let's see what will happen if we place the Moving
Body at the time coordinate given by its proper time. That is, instead of it being at t = 10, we place it
at
t' = 8. We will call this "adjusted" Moving Body. This of course defines a new worldline,
this.
Now, from the Lorentz
transformation, if we set x' = 0, which gives us the position of the Moving Body itself, we get the
following equations of motion for the Moving Body (click here to see derivation):
This equation for proper time
t' is in fact the equation of an ellipse:
And the ellipse is
this:

Ellipse with semiaxes T (time)
and cT (space)

Let's
see it on the graph (where it is a circle since c = 1). If we
clear all other elements, keeping only the adjusted Moving Body, and play around with its speed, increasing
or decreasing
it, we see that it moves along the circle (or ellipse if we increase
or decrease
c from its initial value of 1). (All this happens through the equation for t', it is not "made up".)
This means that the circle (or
ellipse) is indeed the space dimension along which the Moving Body moves. We can define the space dimension as the
set of all points of a spacetime that coexist in a specific moment of time (see below). Another way to define it,
which amounts to the same thing, is to say that the space dimension is the set of all points that can be reached in
a specific time interval by moving objects of all possible velocities, from zero to infinity, or the maximum
velocity allowed in the spacetime. So, when we increase or decrease the speed of the Moving Body, it essentially
traces the space dimension.
Now let's
toggle back on the elements that interest us. It is reasonable to put the x axis of the adjusted Moving Body
perpendicular to its worldline (tangent to the circle or ellipse) like this.
Now, what is the relationship
between the gray axes calculated by the Lorentz transformation, and the blue axes of the Moving Body in its
"adjusted" proper time position? We see right away that the Moving Body in the "unadjusted" position can be
considered a projection of the adjusted Moving Body on the dark gray line (which is the x axis of the Stationary
Body), like
this. (Click here to see a simple algebraic proof.)
What about the Lorentz x axis?
In fact, the point that the Lorentz x axis intersects the x axis of the Stationary Body is a projection of the
intersection of the blue x axis with the (light gray) x axis of the Stationary Body at t = 0. It sounds
complicated, but it is simply
this. (Click here to see a simple algebraic proof.)
So essentially the adjusted
Moving Body and its x axis are projected on the x axis of the Stationary Body, and this produces the axes of the
Lorentz transformation. And I think that this cannot be a coincidence.
You can see the formation of
the normal Lorentz axes by the adjusted Lorentz axes in "real time" using the contols below.
Now, the question is, "What do
we make of this?". I would say that the expanding circle looks like a "curved" expanding spacetime, the one that
was postulated as a solution to the equations of General Relativity.
Let me try to explain what I mean.
The Missing Space Dimension
Traditionally, motion graphs
depict movement with lines that give the position of a body as a function of time, according to the familiar
Cartesian system of representing graphically the values of a function. On the basis of this system, if we have a
Stationary Body, a Moving Body with velocity v = 0.5 and a Photon with velocity c = 1, we can graph their movement
as in the following diagram. The three orange lines are the worldlines of the three objects (Stationary Body,
Moving Body, and Photon). Here, instead of the term "worldline", we will propose the term "timeline", in its
familiar meaning of the evolution of an entity through time.

Worldlines ("timelines")of Stationary Body (v = 0), Moving Body
(v = 0.5)
and Photon (c = 1).

However, graphs of this form do not depict a very important element, the space
dimension in which these objects exist. We will use the term "spaceline" to denote this space dimension.
Thus, a spacetime is the set of all points of a space in all its moments of
time, and it is depicted by the whole twodimensional surface of the graph. A spaceline (space dimension) is
the set of all points of a spacetime that coexist in a specific moment of time, i.e. all points that have the same
time coordinate (isochronal line). The timeline of a spacepoint is the set of all points of a spacetime that
have the same space coordinate as the spacepoint. This is also the timeline of a stationary object. Obviously, we
can also define in a similar manner spacesurfaces and spacevolumes, and timesurfaces and timevolumes.
Thus a spacepoint (a space coordinate) defines a timeline (or a timesurface in a 2+1
spacetime, or a timevolume in a 3+1 spacetime) and a timepoint (a time coordinate) defines a spaceline (or a
spacesurface in a 2+1 spacetime, or a spacevolume in a 3+1 spacetime). Finally, we can define the timeline of a
moving object as the set of all points that satisfy the motion equation of the object.
Let us see how we can introduce the space dimension in our diagrams.
Shape of the Space Dimension
So, we can ask what is the form of the space dimension (of the
spaceline) in the graph above. This form can be determined by the type of coordinates transformation that holds for
a specific spacetime. We have two possible transformations, the Galileo transformation and the Lorentz
transformation. In each transformation, the motion of a body is defined by a system of two simultaneous equations.
For the Galileo transformation, these equations are:
while for the Lorentz
transformation (see derivation here) they are:
The first equation gives the x coordinate of the body, and it is essentially its "timeline". The second gives its time coordinate, and is
essentially the equation of the space dimension (the equation that defines the shape of the space dimension
or spaceline).
The space dimension, irrespective of its equation (and therefore its shape), takes different successive positions with the passage of
time, as T increases. At each one of these positions, the spaceline (space dimension) intersects the timeline of
the body at a specific point, which defines the position of the body for that moment of time (the solution of a
system of two simultaneous equations is depicted graphically as the intersection of the graphs of the two
equations).
Galileo Transformation
In a spacetime with Galileo transformation, the equations of motion of the Stationary Body, the Moving Body and the Photon are as
follows:



Stationary Body

Moving Body

Photon

In such a spacetime, the
spaceline is a straight line perpendicular to the t axis (as evidenced by its equation t' = T, with T being a
number), and it "moves upwards" with the passage of time as the value of T increases. All the points of such a
spaceline have the same time coordinate (t' is equal to T for all the length of the spaceline), and because of
this, time is absolute. This is depicted in the figure below.


Stationary body (blue, v = 0), Moving Body (red, v = 0.5) and Photon
(yellow, c = 1), at time moments 1 and 2 in a spacetime with Galileo
transformation.

The Speed of Light in the Galilean Spacetime
In such a spacetime, the speed
of light is not constant for the Stationary and the Moving Body (Stationary and Moving Observer). Click
here to see the measurement of the speed of light by the Moving Body. The values for t, v and c shown in the
textboxes on the right are the ones that the Stationary Body sees, while those shown on the graph in orange (xg,
tg, cg) are the ones measured by the Moving Body (the "g" is for "Galileo"). So, in this case, while for the
Stationary Body c = 1, the Moving Body measures c to be 0.4.
In a spacetime with Galilean
transformation the spaceline has the same time coordinate in all its length, and coincides with the x axis of the
Stationary Body (and the Moving Body). Obviously, this can happen only if its equation is of the form t = T
(T = 0, 1, 2...), a "zero degree" equation that describes a straight line perpendicular to the time axis.
In any other case, different points of the spaceline will have different time coordinates (for the coordinate
system of the Stationary Body).
This means they will literally
be at different points of time, and of course any objects situated at these points will also be at different points
of time. For instance, we could imagine a spaceline with a first degree equation t = ax + T (a
special case of which is the spaceline of the Galileo transformation if a = 0). Even this simple equation
will give us a spaceline that meets the time axis not perpendicularly but at an angle, and whose points will have
different time coordinates. A spaceline of this form, moving upwards with the passage of time, will transect the
timelines of the three objects at different time moments as measured by the Stationary Body, giving them different
time coordinates in its coordinate system. (You will find a fuller treatment of this point in the article Speed
of Light and the "Rate of Propagation of Time"  opens in new window.)
Adjusted Lorentz Transformation
Now, in a spacetime with
adjusted Lorentz transformation, the equations of motion are:
(See derivation here.) If
we plug the velocities of the Stationary Body, the Moving Body and the Photon, we get:



Stationary Body

Moving Body

Photon 
We pointed out previously that the second equation that defines the shape of the spaceline is the equation of an ellipse or circle, depending on the value of c (see derivation here), which "expands outwards" with the passage of time as T increases. The
points of such a space dimension do not have the same time coordinate (as measured by the linear coordinate
system of the Stationary Body, although they do have the same "time coordinate" in a polar coordinate system). In
such a spacetime, time is "relative" because every movement of a body changes not only its space coordinate, but
its time coordinate also. This is depicted in the figure below:


Stationary body (blue, v = 0), Moving Body
(red, v = 0.5) and Photon (yellow, c = 1),at time moments 1 and 2 in a
spacetime with adjusted Lorentz transformation.

Let us combine the two previous graphs in order to see the differences between the Galileo and the adjusted Lorentz transformation.


Stationary body (blue, v = 0), Moving Body
(red, v = 0.5) and Photon (yellow, c = 1),at time moments 2 and 4 in a
spacetime with Galileo transformation (orange),and a spacetime with adjusted Lorentz transformation
(blue).

We can easily see that the Photon remains at
the time moment it was emitted (for the coordinate system of the Stationary Body), since the Photon coordinate
system is rotated by 90° for the coordinate system of the Stationary Body. Also, it is obvious that the faster the
Moving Body moves, the "slower" its time will appear to the coordinate system of the Stationary Body (tending to
zero as it approaches the speed of light), because the further away it moves from the Stationary Body, the more it
will be able to "gain" time in relation to the time of the Stationary Body.
In fact, if we consider the equations of
motion of the Moving Body,
and we set the
factor, equal to g the equations will become,
In this form, the
product vT describes the rate of movement of the body along the space dimension, and the product gT its rate of "movement" in the time dimension, that
is how "fast" its time "flows".
In a Lorentz spacetime, the circular shape
of the spaceline is not perceivable. Being at the position of the Stationary Body, we assume that our spaceline is
straight (that our universe is "flat"), with all its points having the same time coordinate (our own), and in this
way we impose linear coordinates on a curved space dimension. This is due to the fact that the curvature of the
spaceline in relation to time is non perceivable and can only be deduced by the experimental evidence that lead to
the formulation of the Lorentz transformation.
This means that we do not perceive the
objects themselves, (the Moving Body and the Photon) situated at their correct proper time coordinate, but their
projection on our x axis (their projection on our time moment), and so we "transform" the circular spaceline
into a straight Galileo spaceline, something that is described by the Lorentz transformation. This produces the
phenomena of length contraction and time dilation, since this projection causes a scaling of the axes of the Moving
Body, changing the size of their units.
Obviously, if we went about it in the
inverse way, starting from such a circular expanding space dimension, and assumed that the positions of objects are
projected on the x axis of the Stationary Body, we would end up deriving the Lorentz transformation.
The Speed of Light in the
Adjusted Lorentz Spacetime
In a spacetime with adjusted
Lorentz transformation, the speed of light is constant for the Stationary and the Moving Body (Stationary and
Moving Observer). (As we saw, this is not true for spacetimes with Galilean transformation.) Click
here to see the unadjusted and adjusted Lorentz system.
The equations of motion for the
Photon are:
This means that the Photon's
proper time is always 0, that is, it remains on the moment of time it was emitted
So the Photon is in
fact here.
However, as we said, the fact that an "object" (Photon or Body) is situated in a different time coordinate than our
own is not perceivable, so we attribute to all objects our own time coordinate, which essentially means that we see
and measure their projections on our x axis.
So, the Stationary Body sees
the Photon
there and so it measures c to be 1 (it sees it at x=10, and according to its clock, 10 seconds have passed).
The unadjusted Moving Body sees the photon at the same point, and according to the Lorentz transformation makes
these measurements, so it also measures c to be 1. And the adjusted Moving Body, sees the Photon
there, and makes
these measurements, so c is 1 for it too.
Spacetime
curvature
So, why do I claim that the
phenomena of Special Relativity are produced by the "curved expanding spacetime" that was proposed for the first
time as a solution for Einstein's equations of General Relativity?
General Relativity is based on
a description of curvature through the metric of spacetime. What I am proposing here is based on a description of
curvature through the equation of the space dimension, that is, the second equation of motion that gives us the
proper time of the body. This equation contains all the information we need. The two descriptions are not
contradictory, they describe the same thing in different ways.
Generally we can say that if
all the points of dimension A are projected on the same point of dimension B, then dimension A does not
curve in relation to dimension B, and the plane they form is "flat". In such a case, the equation of dimension A is
of the form B=n, where n is a number, the coordinate of A along B.
If the points of dimension A
are projected to more than one point of dimension B, then we can say that dimension A "curves" within dimension B.
(Although there is also the theoretical case that A is "inclined" in relation to B, if its equation is first
degree.) If its equation is second degree or higher, then we have genuine curvature, and this equation defines
clearly the shape and curvature of the dimension in all its length.
This is what happens in
spacetimes with Lorentz transformation, where we have specifically the space dimension curving within the time
dimension, since the points of the space dimension do not all have the same projection on the time dimension, that
is, they do not all have the same time coordinate, as they do in Galilean spacetime.
The curved expanding universe
of General Relativity can be likened to the surface of a sphere with increasing radius, if we reduce the dimensions
to 3 (2 dimensions of space and 1 time), in order to be able to visualize this. Or, if we reduce the dimensions
further (to 1 dimension of space and 1 time), it can be likened to a circle with increasing radius – and this is
exactly what we have here.
And so we are at the position
of the Stationary Body, stationed at one point of this circle, and in observing the spacetime around us we use a
coordinate system with a straight x axis, like
this. However, the space dimension on which we are stationed is not straight, it is like
this, and so it does not coincide with our x axis. However, we do not perceive this, and attributing to all
other objects our own time coordinate, we essentially study their projections on our x axis. This produces the
phenomena of Special Relativity, like
this. (Click
here if you want to reset the simulation.)