Nature of Light

[Ed3]

Hi Farsight,

A topology formally is an ordered pair consisting of a set S and a class of open subsets of S, which by definition includes S and the null set. Informally, we can speak of the class of open sets as a topology because S is included in the open sets.

A given set, in general, may have a number of different topologies. Therefore a given set, such as a moebius strip without a definition of its’ open sets, is not formally part of the study of Topology.

The reason that Topology and metrics are related is because a given metric on a set S can define open balls in that set, which in turn can uniquely generate the class of open sets of S.

Once the topology is generated, frequently by using a specified metric, two different topological sets can be related.

It turns out that the concept of continuous functions can be generalized by showing that a function maps open sets to open sets. Most interestingly, functions that are not continuous can be made continuous by defining the image of open sets to be open sets in the range of the function.

Two sets, with their specified topologies, are defined to be topologically equivalent if and only if there exists a continuous bijection (a one to one and onto map) of the two sets.

At this point, equivalency classes exemplified by certain well known objects such as the Real plane can be established. It turns out that the moebius strip, a doughnut, and a cylinder and others are all topologically equivalent to the Real plane.

Personally, I think examining various topologies to see if they can generate metrics, and vice versa is more interesting.

In any case this brings us back to the metric that you are using for your space.

I have made three drawings to illustrate the problem of simply saying that you use motion for your metric.

Figure 1:

In this picture we are assuming that there is no passage of time.

The classic Euclidean answer, and the engineers will get testy if you don’t give very nearly this answer, is that the distance from Particle 1 to Particle 2 is + square root (14).

However your answer, if taken literally, is 0.

Figure 2:

Here I have changed z to t.

The relativistic answer is that the distance from Particle 1 to Particle 2 is + square root (5 – 9c^2).

Again, unless you assume that the particles are moving, your answer is 0.

Even if you assume that the particles are stationary and only passing through time, the change in the spatial dimensions would be 0 and again your answer would be 0.

Figure 3:

Here, assuming that Particle 1 is stationary and Particle 2 is moving with velocity v, there is actually a possibly meaningful interpretation. If we let v[size=85]x[/size] be the coefficient of the vector v[size=85]x[/size] and v[size=85]y[/size] be the coefficient of the vector v[size=85]y[/size] then one possible metric could be d = square root (v[size=85]x[/size]^2 + v[size=85]y[/size]^2).

Additionally, we could be looking at space as a curved surface, embedded in a Euclidean space, where x = f[size=85]1[/size](q, r, s), y = f[size=85]2[/size](q, r, s), z = f[size=85]3[/size](q, r, s) and t = f[size=85]4[/size](q, r, s).

However, there are numerous possible metrics and it could get complicated after that. I am left uncertain about what you mean.

If you have a rigorous formal definition, I would appreciate it, if you would share it with me.

Thanks Ed

[Farsight]

Perhaps there’s some confusion here between space and topological space. The latter is a mathematical construct, where the word “space” means something different to the space of physics or cosmology that I’m referring to. But if we then turn to metric space and Euclidean space, I think I can see the problem:

“The Euclidean metric defines the distance between two points as the length of the straight line connecting them”.

Looking at your figure 1, we use simple Pythagoras’ theorem to work out distance. We’ve got one triangle with sides of length 1 and 3, so the hypotenuse is √10. Then we’ve got a further triangle with a side of length 2, so the final hypotenuse is √14. No problem. The problem comes with “the passage of time” and how you actually measure distance. Imagine you are at point 1 in real space, and I’m at point 2. You want to measure the distance from you to me.

If there was no motion in this universe, you and I would be totally frozen, along with all aspects of our brain and eyes. We wouldn’t be able to think or observe, and we are thus unable to define anything. Distances aren’t zero so much as undefinable. This is a very unusual and uncomfortable universe, so let’s introduce motion. Points 1 and 2 are not moving apart, but now we can both think and observe and measure and calculate. All these things require motion of some sort. You might measure the straight-line distance between point 1 and point 2 using radar, which employs the motion of electromagnetic radiation, effectively light. Alternatively you might simply use a ruler. But how did you calibrate that ruler? Look at the definition of the metre:

The metre is the length of the path travelled by light in vacuum during a time interval of 1/299792458 of a second.

This definition is using the motion of light. It also refers to the second, which has its own definition:

The second is currently defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom.

To define the second, you have to sit there counting microwaves coming at you, and when you get to 9,192,631,770 you tick off a second. The second is defined using the motion of light, as is the metre. This means that the standards for both time and space rely upon motion. Not only do you calibrate your rulers using the motion of light, but you also calibrate your clocks using the motion of light. If light doesn’t move, we can assume the same for all electromagnetic phenomena including the hyperfine transition and electron spin, and we’re in that very unusual and uncomfortable universe again.

Let’s explore Euclidean space with a little general relativity thrown in. You’re on the surface of the earth, and I’m up at the altitude of a typical GPS satellite, see en.wikipedia.org/wiki/Global_Pos … relativity and note where it says “the gravitational time dilation that makes a satellite run about 5 parts in 10¹º faster than an Earth based clock”. You send out a radar pulse from point 1 reflecting off point 2, and you measure the time for the signal to get back to you, and then determine the distance between us. I also send out a radar pulse from point 2 to point 1 such that our pulses are traversing the same space, and I also measure the time for the signal to come back and determine the distance between us. The two pulses follow the same path at the same time albeit in opposite directions, and they take the same absolute or universal time to travel. However we don’t agree on what that time was, because compared to me you’re time dilated by a factor of about a billionth. Hence we don’t agree on the distance. We have to agree that according to our measurements, the distance from point 1 to point 2 is not the same as the distance from point 2 to point 1! Hence Einstein’s interest in the non-symmetric field.

This isn’t a rigorous formal definition, but I hope it’s been useful.

[Farsight]

If you want to find out more about what I’ve been saying, a good primer is this article about “black swans” at physicsworld.com/cws/article/news/40714 and look at the bottom right hand corner where you see an advert for this:

amazon.co.uk/RELATIVITY-Theo … 0956097804

If you can see the dedication, you can see that I take pains to stress that it isn’t all my own work. This is not “my theory”, instead it’s a synthesis where I’ve joined a few dots. It’s easy reading, intended for the layman, painting a coherent simple picture. There will be a few things wrong with it, because nothing is ever perfect, but it is in line with Einstein and others such as Feynman, Schrodinger, Dirac, Maxwell, Faraday, and Newton, and it’s so very simple and logical that nobody seems able to offer any rational argument as to why the thrust of it is incorrect. What’s particularly interesting is that it’s somewhat similar to en.wikipedia.org/wiki/Quantum_topology, which Ed Witten was working on in the nineties. I’m not sure why he dropped it, but I suspect that’s the sort of rigor that now needs to be applied. Another interesting approach is David Hestenes’ geometric calculus. I really ought to learn about it, for example by buying this:

amazon.co.uk/Clifford-Algebr … 9027725616