The Speed of Light as the Rate of Propagation of Time

This is a interactive presentation. In the text that follows, some words are links, and when you click on them things happen in the applet. It should be self explanatory.

We will try to show that there is such a thing as a "rate of propagation of time" along the space dimension, and that this is closely connected to the speed of light.

For this purpose, we will examine three spacetimes. A flat Galilean spacetime, an "intermediate" imaginary "Inclined" spacetime, and a Lorentzean spacetime. The characterizations "Galilean", "Inclined", and "Lorentzean", refer to the equations of motion that hold true for each spacetime. These equations determine its behavior.

In the text that follows, we will be using the terms "timeline" and "spaceline", so a short definition is in order:

A spacetime is the set of all points of a space in all its moments of time, and it is depicted by the whole two-dimensional 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.

 

Galilean Spacetime

Let's consider first a flat Galilean spacetime. The equations of motion for this spacetime are:

`x = upsilon* T`

`t = T`

As we explain in "Proper Time Adjusted Special Relativity", the first of these equations determines the timeline of the Moving Body (its x coordinate as a function of its t coordinate). The second equation determines the shape of the space dimension (essentially, it is the inverse of the previous equation, it defines the t coordinate of the spaceline as a function of its x coordinate). For a flat Galilean spacetime, this equation is "zero-degree", that is, it equates t' with a plain number that increases with the passage of time.

This defines a straight line perpendicular to the time axis, like this. This spaceline moves upwards with the passage of time like this. The intersection of this spaceline (space dimension) with the timeline of a Moving Body such as this, gives us the position of the Body in each moment of time like this (the solution of the system of the two equations of motion is graphically represented as the intersection of the graphs of the equations; see Proper Time Adjusted Special Relativity).

Since the spaceline is perpendicular to the time axis, all its points are projected on the same point of the time axis. This means that all its points have the same time coordinate. When t changes taking its next value, it changes for the whole length of the space dimension (for all its points simultaneously). This can be seen as an infinite rate of propagation of time along the length of the space dimension.

Now, in such a flat Galilean spacetime, the speed of light can also be considered infinite. This will become clear when we examine next the Inclined Spacetime.

 

Intermediate "Inclined Spacetime"

In order to make things clearer, before we examine the spacetime with Lorentz transformation, we will examine first a simpler intermediate spacetime. For this spacetime, the equations of transformation giving the coordinates measured by the Moving Body on the basis of those measured by the Stationary Body are:

`x' = x-upsilon*T`

`t' = T-x/c`

In order to derive the equations of motion for this spacetime, we set x' = 0, which gives us the position of the Moving Body itself.

So, for the specific cases of the Stationary Body, the Moving Body, and the Photon, the equations of motion are:

Now, we see that in this spacetime the equation that determines the shape of the spaceline is first-degree with respect to x. This defines an inclined line that meets the time axis at an angle, like this. (We will also keep the Galileo spaceline toggled on for comparison. The Galileo spaceline is also the x axis of the Stationary Body situated at x = 0). The Inclined Spaceline moves upwards with the passage of time like this. The intersection of this spaceline (space dimension) with the timeline of a Moving Body such as this, gives us the position of the Body in each moment of time like this. (As stated above, the solution of the system of the two equations of motion is graphically represented as the intersection of the graphs of the equations.)

Now, let's set t = 4. According to the equations of motion above, the time coordinate of the photon is constant and equal to 0. This means that the photon is situated at this point, and its timeline is a straight line perpendicular to the time axis at t = 0 like this. However, the point where the photon is situated, is also the point of intersection of the space dimension with the 0 time coordinate. So, with the passage of time, the photon moves towards the right with the speed of light, but the same holds true for the position of time moment t = 0. So, at t = 4, time moment t = 0 is at x = 4 (for c = 1). At t = 5, time moment t = 0 (along with the photon) is at x = 5, etc. So, as the space dimension moves upwards with the passage of time like this, the x coordinate of time moment t = 0 (its position) moves towards the right with the speed of light. That is, time propagates towards the right, with the speed of light.

You will notice that the angle at which the Inclined Spaceline meets the t axis is determined by c (or determines c – the two quantities are identical). If we increase or decrease c, the angle changes. The smaller the angle, the slower the light and the rate of propagation of time, while the more we approach the perpendicular to the time axis, the speed of light and the rate of propagation of time increase.

Finally, if this angle were to reach 90°, the speed of light, and the rate of propagation of time would become infinite. In such a case, the Inclined spaceline would become perpendicular to the time axis, and the spacetime would become Galilean. This is also evident from the equations of motion of the Inclined Spacetime, which as c tends to infinity, the `x//c` part tends to 0 and the proper time t' tends to the Stationary Body time T, so that the equations are reduced to the equations of the Galilean spacetime.

This suggests that the light and time are closely connected, if not identical phenomena.

What we said about the Inclined Spacetime, also holds true for the Lorentzean Spacetime.

 

Lorentzean Spacetime

For a spacetime with Lorentz transformation, the equations giving the coordinates measured by the Moving Body on the basis of those measured by the Stationary Body are:

`x' = (x-upsilon*T)/sqrt(1-upsilon^2/c^2)`

`t' = (T-x upsilon/c^2)/sqrt(1-upsilon^2/c^2)`

In order to derive the equations of motion for this spacetime, we set x' = 0, which gives us the position of the Moving Body itself.

So, for the specific cases of the Stationary Body, the Moving Body, and the Photon, the equations of motion are:

Now, we see that in a spacetime with Lorentz transformation the equation that determines the shape of the spaceline is second-degree with respect to x. This defines an expanding circle of radius t (or ellipse if c ≠ 1), which for t = 4 is this. (We will also keep the Galileo spaceline toggled on for comparison. The Galileo spaceline is also the x axis of the Stationary Body situated at x = 0). The Lorentzean Spaceline expands outwards with the passage of time like this. The intersection of this spaceline (space dimension) with the timeline of a Moving Body such as this, gives us the position of the Body in each moment of time like this. (As stated above, the solution of the system of the two equations of motion is graphically represented as the intersection of the graphs of the equations.)

Now, let's set t = 4. According to the equations of motion above, the time coordinate of the photon is constant and equal to 0. This means that the photon is situated at this point, and its timeline is a straight line perpendicular to the time axis at t = 0 like this. However, the point where the photon is situated, is also the point of intersection of the space dimension with the 0 time coordinate. So, with the passage of time, the photon moves towards the right with the speed of light, but the same holds true for the position of time moment t = 0. So, at t = 4, time moment t = 0 is at x = 4 (for c = 1). At t = 5, time moment t = 0 (along with the photon) is at x = 5, etc. So, as the space dimension expands outwards with the passage of time like this, the x coordinate of time moment t = 0 (its position) moves towards the right with the speed of light. That is, time propagates towards the right, with the speed of light.

The speed of light c, as well as the rate of propagation of time, is determined by the ratio between the two semi-axes of the ellipse. When they are equal, c = 1 and the spaceline has the form of a circle. If we increase or decrease c, the shape of the space dimension changes. The smaller c is, the slower the light and the rate of propagation of time are. As c increases, the speed of light and the rate of propagation of time increase.

Finally, if the ratio between the semi-axes of the ellipse tends to infinity, the upper part of the ellipse tends to to coincide with the perpendicular to the time axis at t = T, and the spacetime tends to become Galilean, with an infinite speed of light and rate of propagation of time. This is also evident from the equations of motion of the Lorentzean Spacetime, which as c tends to infinity, the `x^2//c^2` part tends to 0 and the proper time t' tends to the Stationary Body time T, so that the equations are reduced to the equations of the Galilean spacetime.

This suggests that the light and time are closely connected, if not identical phenomena.