Guest post: George Barouxis – “Extrinsic” Relativity November 16, 2007Posted by dorigo in Blogroll, mathematics, physics, science.
First, let me thank Tommaso for the opportunity to write a guest post for his blog. My name is George Barouxis (aka Gebar), and I am a professional translator from Athens, Greece. (The photo is not very recent, but it gives you some idea). This is my translation site where you can see most of the books I have translated into Greek over the years. I also have the “physics bug“, and I often build Java applets in order to understand and explain my ideas. So, about those ideas. They concern lowly Special Relativity, but bear with me for a while.
“Proper” Special Relativity
Let’s start with some incontrovertible mathematical facts. In the equations of the Lorentz transformation, if we set x’ = 0 we derive the equations of motion for the Moving Body (see derivation here):
Equations 1a, 1b
where t’ is the proper time of the Moving Body and T is the time of the Stationary Body. Now, the equation for proper time t’, with some rearranging becomes
which is the equation of a circle of increasing radius T if c = 1 (or an ellipse with increasing semi-axes T and cT if c ≠ 1) as shown below:
So the Moving Body, if placed on its “proper” position on the graph (the position indicated by its proper time), is always situated on the circumference of this circle of increasing radius T.
Let’s see what this means. The usual picture in Special Relativity is this:
The Stationary Body is the blue point at x = 0, T = 10, the Moving Body is the red point at x = 6, T = 10, and the Photon is the yellow point at x = 10, T = 10 (this is for a Moving Body with velocity v = 0.6 and for c = 1). The gray axes are the ones calculated for the Moving Body on the basis of the Lorentz transformation.
You will notice that the Moving Body is at T = 10, although its proper time, as indicated by the gray number on its left, is t’ = 8. Now, if we place the Moving Body at T = 8, and we add its x and t axis, and the expanding circle of radius T, we get this picture.
The red point at x = 6, T = 8 is the Moving Body in its “adjusted” position and the yellow point at x = 10, T = 0 is the Photon in its “adjusted” position. The relationship between the usual Lorentz axes (gray) and the new axes proposed here for the Moving Body (cyan) is this:
And this concludes the facts of the case. (You can see a more detailed presentation at my site here, with a Java simulation that among other things shows the formation of the (gray) Lorentz axes by the projection of the (cyan) adjusted Lorentz axes on the x axis of the Stationary Body with the passage of time, as well as algebraic derivations for the above points.)
Now the question is what we can make of all this. Quite a few people who have seen this have concluded that I have just come up with a geometric construction or a geometric interpretation of the Lorentz transformation, and they point out that since it predicts the same things as Special Relativity, it probably is just Special Relativity.
Well, I have to agree that this is definitely one possible scenario. Let’s see now another one that may prove more interesting and useful.
Extrinsic Definition of the Shape of the Space Dimension
First we have to talk a bit about spacetime curvature. Now, everybody knows that spacetime in Special Relativity is flat, so we have no curvature there. Spacetime in General Relativity is curved, and curvature in GR is usually described intrinsically, that is without reference to some higher dimensional space within which we imagine our spacetime to be embedded.
Here we will use the extrinsic definition of curvature, in the sense of having an equation that gives us the shape of the space dimension. Let’s see how we could go about it. We start with a Stationary Body and its coordinate system, this:
This is just a mathematical space. We add the space dimension on which the Stationary Body is situated, like this.
This space dimension (orange line) is a straight line perpendicular to the time axis of the Stationary Body, described by a “zero degree” equation of the form t = T. By having T take successive values 1, 2, etc., we can depict the passage of time, through which the space dimension moves “upwards” taking successive positions (time coordinates) at each successive time moment. So at T = 0 it is at the position shown in the graph, at T = 1 it will be at time coordinate 1, etc. If we have a Moving Body with an equation of x = v*t, for v = 0.6 and T = 10 we will have the following picture:
The space dimension is at T = 10, and intersecting the “worldline” of the Moving Body, it gives us its position at that time moment. The solution of a system of two simultaneous equations is depicted graphically as the intersection of the graphs of the two equations. The two equations here are the equations of motion in a flat Galilean spacetime:
Equations 3a, 3b
No curvature here. In such a spacetime the speed of light can be considered infinite. And notice something important, that the orange space dimension coincides with the x axis of the Stationary Body. Essentially, the first equation of motion x = v*T defines the movement of the body in time, and the second equation defines the shape of the space dimension. In this case of a zero degree equation, it is a straight line perpendicular to the time axis of the Stationary Body, so we have no curvature there. The intersection of the two graphs (the solution of the system of simultaneous equations) gives us the position of the Body at each time moment.
Now let’s see what happens if we use the equations of motion derived from the Lorentz transformation.
Equations 4a, 4b
Here the equation of the worldline of the Moving Body is the same as previously, but the second equation that gives us the shape of the space dimension is second degree. So instead of a straight line space dimension that moves upwards with the passage of time, we have a cyclic space dimension of radius T that expands “outwards” with the passage of time (reminds you of something?). The intersection of the graph of the Moving Body worldline equation with the graph of the space dimension equation gives us again the position of the Moving Body like this:
Now, here we do have curvature, which is extrinsically described by the second equation of motion, the one that gives the proper time of the Moving Body for each point of the space dimension. (This equation defines the shape of the space dimension, essentially how it is embedded in the two dimensional spacetime represented by the whole surface of the graph.) Because we have curvature, the space dimension does not coincide with the x axis of the Stationary Body, and this has some important consequences: although the proper time of the Moving Body is 8 (in the specific case depicted above), we place it at T = 10 (because that’s where are x axis is), and this affects accordingly its worldline like this:
This essentially means that we see the projection of the Moving Body on our x axis. This produces the phenomena of Special Relativity as described by the Lorentz transformation, and this may mean that these phenomena are due to the fact that we live in a “spherically” expanding universe but we describe it through a straight linear coordinate system.
The usual objection here is that spacetime in Special Relativity is flat so this can’t be right. Well, spacetime in Special Relativity is definitely flat –after you do the projection mentioned above. Specifically, it is this:
You will notice that the darker gray line at T=10 is straight, so, yes, there is no curvature there. However, this darker gray line is not the space dimension in which the Stationary Body, the Moving Body and the Photon exist. It is just the x axis of the Stationary Body, an abstract mathematical entity with no physical existence. The space dimension here is the expanding cyan circle, and the Moving Body (and its axes) and the Photon get projected on the x axis of the Stationary Body like this:
So I do not claim that spacetime in Special Relativity is curved. I claim that there is an underlying curved spacetime, that when is mathematically described through this projection that makes it flat, we have as a result the phenomena of Special Relativity. The difference between the two previous figures is that Figure 10 shows things as they “seem” from the point of view of the coordinate system of the Stationary Body, while Figure 11 shows things as they “really are”, or as they seem from a “higher” point of view that includes the one of Figure 10 and, in a manner of speaking, “explains” them.
Let’s take it inversely. If we live in a curved expanding spacetime as shown by General Relativity, we should observe the effects shown in the above graph (if we describe the curved expanding universe extrinsically). And we did observe them, in the Michelson-Morley experiment, which was explained through the Lorentz transformation and Special Relativity. In other words, the effects of Special Relativity are due to the curved expanding universe of General Relativity. But you need the extrinsic description of curvature in order to see this.
Now, if we describe the shape of the space dimension extrinsically, we have to specify the dimension in relation to which the space dimension is curved. This is done by the second equation of motion, that defines the shape of the space dimension. We saw above what happens when this equation defines the time coordinate as a function of velocity. The “objects” of our spacetime now have their own “proper” time depending on their velocity, instead of having a universal time that is the same for all, as in the case of a flat Galilean spacetime. Again, if the space dimension is a straight line perpendicular to the time axis, all its points have the same time coordinate. Then the speed of light is infinite and time is absolute. This is a flat Galilean spacetime. If the space dimension is a curve (or even a straight line not perpendicular to the time axis), then different points will have different time coordinates. The speed of light will be finite and time will be relative (the time coordinate of a Moving Body will depend on its velocity).
So this is the second scenario I was talking about, the one that says that this is not a geometrical construction or interpretation of the Lorentz transformation, and that the circle defined by the second equation of motion derived by the Lorentz transformation is not just a geometrical construct, it is the space dimension of our spacetime extrinsically defined by that equation. (Let me repeat here that I use the term “extrinsic description of the shape of a dimension” in the sense of having an equation that for each x coordinate of the space dimension gives us its coordinate along another dimension.) So how could one determine which of the two scenarios is the correct one? I am open to suggestions.
To be frank, I haven’t worked much in trying to prove that this is a correct description of physical reality and not a geometrical construction. I just treat it as a working hypothesis, and if it’s wrong I expect I will end up at an impasse sooner or later. In the meantime I try to extend its applications in order to see where it will lead me. For instance…
Other applications of the extrinsic definition of the shape of the space dimension
Let me add a few more conjectures that I also treat as working hypotheses, to be expressed mathematically and checked against experimental facts. They are extensions of the basic concept used above, the extrinsic description of the shape or embedding of the space dimension. So in what follows, the aim is again to arrive at the equations of motion, that is, an equation that defines the movement of a particle as a function of time (worldline) and an equation that defines the shape of the space dimension (cyan line) in relation to another dimension, or its embedding in the 2-d or 3-d space formed by all the dimensions. We talked previously about the shape of the space dimension in relation to the time dimension. This referred to the whole length of the space dimension, as described by the second equation of motion derived by the Lorentz transformation. However, we can also imagine local areas of curvature like this:
I think that this will produce gravitation. We see often such pictorial depictions of spacetime curvature produced by gravity, usually with a disclaimer that “this is not how things really are, but it does give some idea of the situation”. This is because spacetime curvature is described intrinsically by General Relativity, while such depictions use an extrinsic representation to make things easier to understand. However, we can very well arrive at a precise extrinsic mathematical description of this spacetime curvature, which will give it a physical, as well as a mathematical, meaning. Spacetime curvature in the picture above has a definite physical meaning if described extrinsically, it is an area of the space dimension that its points have different proper time coordinates that produce the above form in the space dimension (or the form that will produce the acceleration of gravitation as observed in the real world, if this one is wrong).
Additionally, some kind of configuration of the space dimension at the apex of this gravity well may represent physically a fundamental particle. This configuration may be a simple or complex loop or other structure that acts as a photon trap, with the spin of the particle being a partial description of the form of this configuration. If at the apex the space dimension doubles upon itself (somehow intersects itself), this “knot” may give the appearance of a “point” particle, while the loop created by this self-intersection will have a non zero size and may allow the point particle to have properties.
As we said, when we use the extrinsic description of curvature we specify the dimension in relation to which the space dimension is curved. The above concerns curvature in relation to time. However, we can also imagine the space dimension curving within an additional space or time dimension, and this curvature may represent charge:
So the curvature along the t axis represents the mass of the particle, while the curvature along the z axis represents its charge, with the opposite charge being like this:
The propagation of a disturbance of the space dimension in relation to this additional space or time dimension, which is produced by the movement of a charged particle, may represent electromagnetic waves, like this.
I will not hazard a guess as to how this disturbance may be quantized in respect to the energy it carries, but this may become more obvious if we have a clear (extrinsic) mathematical description that defines its shape as a function of time. Right now an explanation that seems reasonable would be to look at the “pixels” that constitute the space dimension. (In a curved space dimension, different points have different t and z coordinates, but what is the physical size of these “points” along the dimensions of “z-length” and “t-length”, and how is the propagation of a disturbance along such a medium affected by the size of these lengths? There may be some kind of “stair step” there.)
Let me point out again that these “working hypotheses” are highly conjectural (meaning I could be wrong). However, even so, they show something important, that when you use the extrinsic definition of the shape of the space dimension your statements acquire physical meaning and this is very useful on many fronts.
I appreciate the reasons that made physicists, starting with Einstein himself, to prefer the intrinsic description of curvature. I believe however that the extrinsic description of the embedding of the space dimension can be used at least in a complementary way to the intrinsic description of curvature because it gives physical meaning to our formulations and in this way it can show us more easily where to look, and what to look for. Besides, the Nash embedding theorem may mean that the “transfer of knowledge” between the two descriptions is a legitimate operation.
Well, that’s about it. I hope that what I have presented here, although rudimentary (especially on the math side), will at least give some food for thought.