Monday, November 22, 2004

The Article

LIGHT CONE DYNAMICAL LAWS
WRITTEN BY MICHAEL WEIGHT



ABSTRACT
Light cones are thought of to be space-time diagrams. This is true, but they are useful for calculation of times, and null surfaces and singularities in space-time, and in General Relativity. Out of anything, light cones are just objects of mathematics that create a way to calculate space-time. In this article we will discuss some of the laws that govern the "new" idea of light cones, and how to properly calculate them.

LIGHT CONE DYNAMICAL LAWS-SECTION 1
The set of laws that govern light cones is 7 laws, each of which describes the geometry of a light cone.
Here are the laws:
The geometry of a light cone consists of one geometrical singularity, and a past and a future cones.
A light cone bends topologically with any bends or curves of space-time. (Which would mean that it could bend non-topologically).
3. All inertial or outside coordinates=0 at the end of calculation.
4. A world line will be described as a 4-scalar position.
5. A light cone may take the form with n-genus, or N-D (depending upon the space-time that you are working with).
6. Any null surface on a light cone has to be 45 degrees or spatially 90 degrees.
7. Time on a light cone can go along any axis (even null surfaces).
Now, what you find is that something is new here. These laws were never predicted by Minowski, nor Einstein himself. But, let’s look at some of the consequences of these laws, starting with law 1, and then going down to 7.

LAW 1 CONSEQUENCES-SECTION 2

According to law one, a light cone consists of one geometrical singularity. In the case of the classical model of a light cone- it would be the "present." We will adopt this original idea, because the geometry is describing a classical light cone. But, the singularity would make something interesting. According to the law, the singularity is geometrical, not time related. What this means is that this singularity isn’t found in reality, just in mathematical form. Also, the null surfaces (described by dynamical law 6) are also just present in a mathematical form. Any calculations of a light cone is just mathematics, which makes like cones just mathematical objects.
Now, the future light cone is symmetrical to the past cone. The only thing that separates them is that the null surface is the additive opposite of the past surfaces. This can be represented by saying that the null surface t+ in the past would then be represented by t-. Now, what this means is that if you add them together you get zero, which satisfies dynamical law 3. But, according to dynamical law 6 states that the only null surfaces is at 45 degrees and spatially 90 degrees. But, according to dynamical law 3, everything has to equal zero. So, this would mean that all trajectories are null surfaces when they reunite with the singularities, there for an evolution in coordinates. But, since a trajectory is usually described by linear equations, we can augment this evolution with an augmented matrix. But, how do we derive of this augmented matrix, and what are the linear equations?
First off, using classical laws, we are able to figure out an equation that can describe the world line (but, to contradict dynamical law 4, a 4-scalar position, we’ll get to that later, and fix the augmentation.
The linear equation is:
Equation 1
T0=t1+vrm +Dm =0

In this, T0 is "present" or the mathematical singularity, and t1 is the time of the event when we started the evaluation. Then, we find that v is the velocity vector of the object or something that is moving in time, and rm is the radius 4-vector position of the inertial calculation, and Dm is the distance between the past cone, or past event and the mathematical singularity. Now, it would appear that r=D, and why not just make them equal? For that exact reason, we make them 4-vector positions to set it strait, to separate the inertial and general coordinates. Now, to the augmentation.
Calculation 2
11110


With this first section of the augmented matrix, we can assume a pattern. Since dynamical law 3, we have to have the resulting section of the matrix to be just zero’s. But, how could we do this? Simply, as we move up the time axis, the world line evolves. Now, all we have to do is evolve the above equation. This is an easy task, for all we have to do is just account for the certain coordinates, and give some certain coordinates certain jobs. In other words, we just have to take off some of the coordinates and assume new coordinates. For example, the next row of numbers will have to be 111100, and so on until we get 000000. So, what is the linear equation that we are looking for? Well, since we are assuming a local description, we can take off the Dm (the reason why it’s there in the first place is because it is describing a general view of both inertial and outside observers). We’re not going to do 4-scalar analyst in this case, but we should end up with the same result, there for it could be plausible that both methods of computation are correct. But, this is only a local calculation. If we use this method, we’re gong have to have two sets of linear equations, but yet one augmented matrix. If we use 4-position scalars (that will come later) you have one equation, and one augmented matrix. Now, with the other linear equation:
Equation 3
T0=t1+vrm =0


So far our calculation is going good. Because we have the missing Dm , that is equal zero. But, it appears that actually it’s not necessary, but mathematically it is. This is because it still is required (remember we still have to do the outside observer, so we have to take everything in account). Now, the matrix looks like this:
Calculation 4
11110
11100


Now that our calculations are going on smoothly, we can assume that there is symmetry in both the inertial and outside observers. But, this doesn’t necessarily mean that they are the same coordinates, they just follow a follow dynamical law 3. They both end up at T0=0.
Up to this point, we find that the null surface is evolving. We have a partial null surface. How this works is that we can assume that t=t+(v), where t+ is the null surface on the light cone (which can be pin pointed on any area of the light cone, just that dynamical law 6 states that there has to be null surfaces at 45 degrees and 90 degrees (spatially, of course).
Up to this point, it gets tricky. If we want to continue the matrix, we have to consider and change the coordinates and variables that we were using. For example, mathematically speaking it would be logical to stop now (or to keep the calculations to appear correct). But, in order to keep them correct, mathematically speaking, we have to assume that some physical change happens when we are at this point on the evolution of t+. But, what is the physical change? What is the linear equation that purposes this type of change? Mathematically speaking, we could easily just say that the evolution of t+=dt/rm . But, this isn’t even part of the set of linear equations that will be presented in the augmented matrix. The point I was making was that the null surface t+ (on any area on the cone), is only local when compared with the local time frame. In other words, the outside observer is only seeing the general picture, while if you go locally, you get the evolved picture. If the above argument was true, that would mean that even though we end up with 000000, and the matrix are symmetrical, it doesn’t mean that the equations are the same. Remember that we just use the co-effiencients.
Now to continue. Remember that physical change? -Well, mathematically we find that actually the linear equation is:
Equation 5
T0=t1+v=0

So, now we have 111000. Right here is symmetry in the null surface and the non-null surface. What are the physical implications of this? I mean, as you run down the street pondering this, how are you part of a null surface, and the other half your on a non-null surface? Simply, there isn’t any physical implication except for that you’re able to calculate the evolution in time from point A to point B from both observers. Now, what does our matrix look like?
Calculation 6
11110
11100
11000


As you can see, our calculations appear to be going according to dynamical law 3.
But, these null surfaces have some effects on the actual space-time (we have to take these into account, since a light cone is a map of space-time). What are the effects? Simply, the null surfaces would represent a mathematical calculation of the present space-time. For example, let’s say that I wanted to calculate something between point A and point B, and we found out that the total distance between them isn’t as what it seems, we found a null surface. Of course, this example isn’t very applicable with actual calculation, for with standard space-time equations that was first set out by Minowski and Einstein himself. But, this new method brings out a new mathematical analyst for space-time itself. The point that I make is that actually, mathematically space-time is different than from a physical point of view. This is known between mathematicians and physicist, which is why they hardly ever intercourse. But, the physical aspects of space-time presented by these light cones will be read at the end of the article.
Let’s complete our calculations. It appears that in order to get 00000 our equations wouldn’t work. But, just take a look. The next linear equation, which states:
Equation 7
T0=dt=0
In this, we find that actually the geometrical singularity is equal to that of the differential of time. What this means is that the relationship between the geometry and the physical aspects is in perfect harmony as we near the singularity (and travel even further along the null surface). All and all, we are starting to see that at the singularity things tie themselves together, instead of destroying. That may not sound very logical, but it appears that way. Now what does our matrix look like? It looks like this:
Calculation 8
11110
11100
11000
10000

Now, as you can see, as we evolve on the null surface, more evident the singularity. Now, it’s only evident what the last linear equation is. But, before we get to that, let’s do some logical thinking about the evolution of the null surface. If we were to take the null surface t+, and then add it to the dt, what would the effects be? Well, the format would look like t+ + dt, then it would be something of a singularity, or where r=0 (or in this case, t=0), so we get t+ + dt=t=0. This is an interesting equation, because notice that the null surface, however you increase it, you still have a singularity. This is logically consistent with the original thought that all null surfaces lead to a singularity. So, what is the last linear equation? Because of the equation (t+ + dt=0), I would think that we’re at the end of the line. So, what we just say is that:
Equation 9
To=0

It appears that we are at the train station- the beloved singularity. Now, our matrix is complete! The final calculation is:
Calculation 10
11110
11100
11000
10000
00000

With this complete matrix, we now have all of the coordinates (locally, speaking), to describe the space-time around and leading a singularity.

LAW 2 CONSEQUENCES-SECTION 3
Now, with the argument presented above in Section 2, it would appear that dynamical law 2 would disappear. But, this is the most unique law that is presented. So, what are some of the consequences?
According to the law, a light cone will be bent with the fabric of space-time. That would mean that the light cone would be bent and distorted just like space-time, in the presence of matter in space-time, or gravitation. What this means is that the geometry of the light cone would fall under the laws of Riemman. And, the proof of it is:
Since light bends with gravitation, and the 45 degrees represents the speed of light, so it would bend at the null surface. And, since a light cone is a map of space-time, presence of matter would distort it. It only seems logical. Now, what is the geometry?
Since we did some of the basic ( we could calculate the temporary non-singular curves, or if any more singularities occur) geometry, we’ll just figure out the "excess radius" of the light cone. In order to do this, we need an equation, which is:
Equation 11
Rtotal=rexcess-(rpred./8p )1/2


This is the familiar formula that Riemman used. Same applies to space-time, and to light cones. Now, the reason why I use radius r is because just imagine that I drew a circle on the curved light cone. Now, all of the other equations that I presented above would apply, because topologically the coordinates are in the same neighborhood, or locally and from an outside observer. But, the null surfaces and curves would be effected, but by having "radius excess".
Let’s get down to what 4-scalar positions are. According to dynamical law 4, all world lines are described by 4-scalar positions. What are 4-scalars positions?
A four scalar position will follow basic scalar rules, except for some other things. For example, it still follows the scalar field law. But, if you just divide the scalar with the dt, then you have a four scalar position. For example:dt/k*(x)=dt'/k*(x)' Equation 12where k(x) is the scalar, and it is going under a transformation. Now, since according to dynamical law 3, all calculations end up as 0. But see, the scalar can't=0. So, it's not the complete picture. It's only the world line. In order to sustain dynamical law 3, we will have to augment some of the inertial coordinates with sets of equations, which we did.
Now, what are 4-Scalar equations? As with the equation dt/k*(x)=dt'/k*(x)' (or a 4-scalar field law), this will have to sustain a certain equation. Much like when working with light cones, we have to follow the general guidelines that the Light Cone dynamical laws bring. For example, the scalar law will only sustain the general coordinate transformation of x.But, first let's describe all of the scalars and 4-Scalars locally. If we had an event E at a certain time t, then the scalar would be:E=t(k*(x))/dt Equation 13In this case, the only local coordinate would be (x). But, notice that the 4-scalar position is the only thing that is keeping it from just saying that E=t. This is logically consistent with the scalar law, for what would the transformation look like? Locally, it would look something like this:E=t(k*(x))/dt=t'(k*(x)')/dt' Equation 14 Notice, that we reversed the law by saying (k*(x))/dt other than dt/(k*(x)). Why is that? Well, first notice that t=t'-dt, and that dt=dt'-t. This provides symmetry. But, why we switched the law is because of a simple mathematical rule. In this case, if we used the format t(dt)/k*(x) would only provide that t is the dependant variable, while it should be (in this case) the independent. But, it is not what it seems. The 4-Scalar position is the key to this calculation. Since the 4-Scalar is dependent off the dt, that makes the scalar the dependant variable of this equation! So, now, we find that the argument presented above isn't valid in this case! So, t and the dt are independent of each other.

PHYSICAL ASPECTS THAT EMERGE OUT OF LIGHT CONE DYNAMICS-SECTION 4
Now, with the general overview that we presented above, with a very general and "skimming of the surface", we find it necessary to say what physical happenings come out of this.
For example, there are some pure mathematical subjects that we talked about, null surfaces, which have no physical application, but what they bring is the point. What the physical consequences of that is that when calculating a velocity and time, you will see that no matter what you do- it will equal zero. That would be a physical null surface and curve, but just not in the way that we used it. And, if you used not the space-time presented by Einstein, but the space-time by Minowski and Einstein, and now the space-time that is being presented. The calculation would bring something that would never be proven-a singularity in space-time in a globular light cone. Surely as I sit here and type this article I surely don’t feel a singularity, because it’s just a mathematical happening. A physical happening is that an event is only calculated through the 4-scalar position, and is just a greater proof that space-time is just a 4-dimentional fabric. In fact, a light cone is something of mathematics, but physics has brought it up. Like, the null surface at 45 degrees is the speed of light, there for all coordinates in a photon would be zero ( which brings us to the theory of displant, but we’ll get to that at a later time, another article). And, the other physical happenings, like the geometrical 90 degree null surface described by dynamical law 6 means that as I just sit here and type, and you just sit there doing nothing, time is moving foreword, but yet it still equals zero. That is the beauty. It doesn’t make sense, but yet it’s true. Physics is crazy, it’s confusing, it’s diabolical, but yet, it’s simple, so simple, it’s complex. And, mathematics just makes it even more evident. Discoveries is just over the horizon, and may this be the first step to finding this out.
REFERENCES
[1] Feynman, Richard, Six Not So Easy Pieces: Einstein’s Relativity, Symmetry, and Space-time, Perseus Books Group, March 1, 1998.
[2] Gribbin, John, In Search of the Edge of Time: Black holes, White holes, Worm holes, Penguin Books, 1995, reprinted with introduction, 1998.
[3] Misner, Charles W., Thorne, Kip S., Wheeler, John Archibald, Gravitation, W. H. Freeman and Company, 1970.
[4] Peebles, P. J. E., Principles of Physical Cosmology, Princeton University Press, 1993.

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