Euclidean Special Relativity

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bda said:
By the way, this brings me to one of my strong points of divergence with Rob. Rob says that photons travel on null geodesics of [itex]X_4[/itex] but null geodesics need mixed signature spaces; as far as I can understand Rob's spaces have all plus signatures and so they cannot have null geodesics. Maybe Rob will like to clarify this point.
Hello Jose,
Excuse me for reacting so late but I only now returned from work.

As I see it, the "null geodesic" is defined as the path along which the tangent vector has norm 0 in a geometry with metric (-+++), hence the word "null". The norm 0 results from the [itex]ds^2=0[/itex] in the geodesic.

When this is translated to Euclidean relativity with metric (++++), [itex]ds^2[/itex] is not zero (it then equals [itex](cdt)^2[/itex] for the photon which >0) but the displacement in the [itex]\tau[/itex] dimension is zero. So strictly spoken one could not speak of a "null geodesic" any more; it is a timelike geodesic according to the original definition from Minkowski space-time. I have maintained the use of the familiar term in Euclidean relativity because it is generally associated with the path of massless particles and referring to that as "timelike" in Euclidean relativity would likely cause a lot of confusion.

It would probably be best however to introduce a whole new term in Euclidean relativity for the path of massless particles and I am inclined to suggest something like "3D geodesics" versus "4D geodesics" of mass carrying particles.

Best regards,
Rob
 
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  • #52
Hurkyl
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As I've suggested, many of my problems are just with the advertising/naming/etc -- I've made those objections and I'll try to stop harping on them.


One thing I wanted to point out is that there is no natural way to map worldlines in 3+1-space to worldlines in 4-space. What you need to work with are "pointed" worldlines; that is, you have to choose a point on the worldline acts as the origin. (e.g. for a worldline in 3+1-space, you have to choose what point corresponds to [itex]\tau = 0[/itex] before you can map it to 4-space)


bda said:
By placing myself on the null cone
...
I believe if one wants to be formally correct there is a lot more to be said about these operations but that will not alter the substance that we have a means of mapping 4D Minkowski geodesics to Eucliedean 4D ones; so far I am not claiming anything else.
I have a problem with the journey, but not the destination. I have no problem with mapping pointed geodesics back and forth between 3+1-space and 4-space (and up to geodesics in 4+1-space)... I just have a problem with the way you go about doing it. I don't know if it will matter, so I won't say any more about it right now.
 
  • #53
CarlB
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Hurkyl said:
One thing I wanted to point out is that there is no natural way to map worldlines in 3+1-space to worldlines in 4-space. What you need to work with are "pointed" worldlines; that is, you have to choose a point on the worldline acts as the origin. (e.g. for a worldline in 3+1-space, you have to choose what point corresponds to [itex]\tau = 0[/itex] before you can map it to 4-space).
That is also one of my problems with that way of doing it. That is, if you are going to talk about a Euclidean relativity, it should map to the coordinates I plug into my dad's GPS device.

My solution for this is to make the tau dimension be very small, so that errors in converting coordinates from one version to the other may be ignored. Sometimes it seems that Euclidean relativity is the sort of heresy that reminds one of Tolstoy. Everyone happy with relativity is the same, while everyone unhappy with it is different.

Carl
 
  • #54
bda
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I believe we are on the right track now, so let us take a few more steps.

Hurkyl said:
One thing I wanted to point out is that there is no natural way to map worldlines in 3+1-space to worldlines in 4-space. What you need to work with are "pointed" worldlines; that is, you have to choose a point on the worldline acts as the origin. (e.g. for a worldline in 3+1-space, you have to choose what point corresponds to [itex]\tau = 0[/itex] before you can map it to 4-space)
Quite right. I would like to rephrase that to make sure we understand exactly where we are: Although geodesics can be mapped from 3+1- to 4-space and back, the same thing cannot be done with points, that is, if three geodesics cross at one point in 3+1-space they will normally not have a common crossing point in 4-space; this has important consequences.

What is known as an event in special relativity, a set of particular values for [itex]\langle t, x, y, z \rangle[/itex], cannot usually be univocally translated into a set of particular values for [itex]\langle x, y, z, \tau \rangle[/itex]. This is, I think, why Carl proposes that the [itex]\tau[/itex] be curled up in a tight helix, but he will have to explain that himself; I am sticking to flat spaces. Furthermore, it must also be noted that only the spacelike part of 3+1-space is mapped to the full 4-space. We could, if needed, make a separate mapping from the timelike part to another full 4-space.

I said above that points in 3+1-space don't usually map to points in 4-space but now remember how I modified Bondi's approach to define both [itex]t[/itex] and [itex]\tau[/itex]:

[tex] t = (t_0 + t_2)/2 [/tex]

[tex] \tau = \sqrt{t_0 \ast t_2} [/tex]

As we move the distant object closer to the observer [itex]t_0 \rightarrow t_2 [/itex] and so [itex] \tau \rightarrow t [/itex]. So, for the observer, there is no problem in translating from one space to the other. This means, for instance, that I can map a collision event from one space to the other, because it happens at one single point in spacetime and I can place myself, as observer, at that point. Please see ArXiv: physics/0201002 for a collision discussion.

Now let us think a little about how we translate distant or moving objects. We must restrict the discussion to objects moving on geodesics (straight lines); I will say a little bit about dynamics below. The observer's worldline is the [itex] t [/itex] axis in 3+1-space and the [itex] \tau [/itex] axis in 4-space; as we have seen the two measurements coincide for the observer. If an object's worldline crosses the observer's at any point, this point can be mapped from one space to the other; since the worldlines are also mapped, we just need to translate time intervals measured in 3+1-space into distances measured on the worldline in 4-space.

The problem is a bit more tricky if the object's worldline never crossed the observer's. The two definitions above should always solve the problem but there is one philosophical argument I would like to advance: If the Universe is expanding from a big bang it can be argued that all worldlines must have crossed at some time in the past, so justifying the synchronism of all clocks.

Dynamics is a problem and I would say it does not map from one space to the other. I've given some attention to the different dynamics in the two spaces and you can read about in the paper I mentioned above. However I don't think we should care too much about those differences because special relativity, as we know, is not the final answer to dynamics. When we come to discussing equivalence between general relativity and a generalization of ESR, I will deal with dynamics problems.

Jose
 
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  • #55
bda
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CarlB said:
Sometimes it seems that Euclidean relativity is the sort of heresy that reminds one of Tolstoy. Everyone happy with relativity is the same, while everyone unhappy with it is different.
I can see your point but I don't think it apllies here. I wrote about an alternative way of looking at problems but I did not say relativity was wrong. My position about this is that no physical theory is ever final and so no physical theory is ever absolutely right, which does not mean it is wrong. We must accept that every theory has an application domain; sometimes we may think that the application domain is wider than it really is and I belive that to be the case with a large number of physicists in relation to GR.

Jose
 
  • #56
Hurkyl
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bda said:
[tex] \tau = \sqrt{t_0 \ast t_2} [/tex]
I just noticed a problem with this -- it has absolutely nothing to do with the proper time along a worldline! There are several ways to see it (such as looking at two worldlines that cross twice), but I'll address it this way:

bda said:
The two worldlines are related by

[itex]d t^2 = dx^2 + d \tau^2[/itex]
Let's compute it:

[tex]d t_1 = d\left( \frac{t_0 + t_2}{2} \right) = \frac{dt_0 + dt_2}{2}[/tex]

[tex]d x_1 = d\left( \frac{t_2 - t_0}{2} \right) = \frac{dt_2 - dt_0}{2}[/tex]

[tex]d \tau = d \sqrt{t_0 t_2} = \frac{1}{2\sqrt{t_0 t_2}} (t_0 dt_2 + t_2 dt_0)[/tex]

So [itex](dt_1)^2 - (dx_1)^2[/itex] looks nothing like [itex](d \tau)^2[/itex]! (this is not a rigorous disproof, but I was merely going for a demonstration)


What your [itex]\tau[/itex] computes is the proper time along the straight-line path from the origin of your coordinate system to the reflection event. (Complete with the problems when the event is space-like separated from the origin. I.E. when [itex]t_0 < 0[/itex] and [itex]t_2 > 0[/itex])
 
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  • #57
bda
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This message was edited to correct a serious mistake; the editing is clealy marked with bold.:redface:

Dear Hurkyl,

I'm very happy because finally someone is really paying attention to what I write

(...)

Hurkyl said:
I just noticed a problem with this -- it has absolutely nothing to do with the proper time along a worldline! There are several ways to see it (such as looking at two worldlines that cross twice), but I'll address it this way:
Put in those terms I can't but agree with you. Actually [itex]\tau[/itex] is a line integral in 3+1-space and so its value depends on the specific worldline where it is evaluated. Conversely, the same happens with [itex] t [/itex] in 4-space. But remember I am applying to specific objects moving on geodesics (...);
for simplicity let us consider a geodesic going through the origin, for which it will be x = v t ; using the definitions for x and t we can then write

[tex] t_2 - t_0 = v (t_2 + t_0) [/tex]

now square that

[tex] (t_2)^2 + (t_0)^2 -2 t_2 t_0 = v^2 (t_2)^2 + (t_0)^2 + 2 t_2 t_0[/tex]

arrange the terms

[tex] (t_2 t_0)^2 = (1 - v^2) \frac{(t_2)^2 + (t_0)^2}{4} [/tex]

now replace back with the definitions

[tex] \tau^2 = (1 - v^2) x^2 [/tex]

This shows that [itex] \tau [/itex] is indeed proper time. Now, if the worldlines don't cross at the origin we can always make a vertical translation of the [itex] x [/itex] axis and it will still be [itex] (d \tau)^2 = (1- v^2)(d x)^2 [/itex]. In two dimensions
the worldlines will always cross but in 4D they may not cross. As I argued in a previous post, all worldlines will have crossed at some time in the past if the Universe is expanding from a big bang.

However, since we don't have access to that privileged origin, we must find a way of synchronizing time measurements on non-crossing worldlines. This is done by third worldline that crosses both. So, if I want to find out the time over another worldline I send out a radar pulse to interact with the distant object and time the send
and receive instants; then use the definitions to get [itex] t [/itex] and [itex] \tau [/itex].

I will soon have to clarify how I deal with curved worldlines but this is realm of general relativity, not special relativity. Before going into that I want to make sure that the mapping method in flat space is clear.


Best regards,
Jose
 
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  • #58
Hurkyl
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You don't need GR to handle curved worldlines...


What is the point of #57? I think you're still trying to describe how to go from 3+1-space to 4+1-space... but I think it's much easier than you're making it.

When we're working in 3+1-space, if we pick any pointed worldline, we already know how to assign a proper-time to any point on that worldline. Then, IMHO it's straightforward to lift that worldline into 4+1-space simply by making the new coordinate to be equal to the proper time at that point.

(If I've misunderstood your intent, let me know)
 
  • #59
bda
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Hurkyl said:
You don't need GR to handle curved worldlines...
If you allow non-geodesic movement you have then different paths linking any two points in 3+1-space, so you get different evaluations for the proper time difference between them. Each path could then be separately mapped to 4-space but their endpoints would not coincide. I want to make sure I am dealing only with geodesic movement (straight lines).

There are only 4 interactions in nature, but restricting ourselves to the macroscopic world we only need to consider gravity and electromagnetism. The former is described by geodesics if you go to GR and I want to do something similar. EM can also be made geometric by a procedure similar to Kaluza-Klein. So, in short, I want to consider only geodesics when I do dynamics. This is looking ahead to forthcoming posts.

Hurkyl said:
What is the point of #57? I think you're still trying to describe how to go from 3+1-space to 4+1-space... but I think it's much easier than you're making it.

When we're working in 3+1-space, if we pick any pointed worldline, we already know how to assign a proper-time to any point on that worldline. Then, IMHO it's straightforward to lift that worldline into 4+1-space simply by making the new coordinate to be equal to the proper time at that point.
Yes, as long as we stick to geodesics; we will leave curved worldlines to be handled by the metric.

Jose
 
  • #60
bda
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Introducing General Euclidean Relativity

I believe we are now ready to start addressing GR and its Euclidean counterpart; this post will introduce the subject.

In a curved space displacements are evaluated by a quadratic form which we write as

[tex]

(d \tau)^2 = m_{\mu \nu} d x^\mu d x^\nu

[/itex]

In this expression we use a few conventions. First of all there is the Einstein summation convention which says that repeated indices below and above imply a summation over all possible values. The second convention is that Greek letter indices chosen from [itex]\langle \mu, \nu, \lambda \rangle[/itex] take values in the interval 0 to 3. The expression above thus implies a sum of 16 terms on the rhs. For the non-diagonal elements it is always [itex]m_{\mu \nu} = m_{\nu \mu}[/itex]


We are also assuming that [itex]m_{00} >0 [/itex] and [itex]m_{mm} <0 [/itex] (indices m, n, o take values from 1 to 3). This assumption is essential to ensure that tangent space is Minkowski. I've used the letter m for the GR metric tensor, rather than the more usual g, because I am very short of letters working in different spaces at the same time.

If we go one dimension up, as we did in special relativity, we have new indices that go from 0 to 4, chosen from [itex]\langle \alpha, \beta, \gamma \rangle[/itex] and also from 1 to 4, chosen from [itex]\langle i, j, k \rangle[/itex]. In 5D the distance between two points is evaluated by the quadratic form

[tex]

(ds)^2 = g_{\alpha \beta} dx^\alpha d x^\beta

[/tex]

where now the second member has 25 terms, [itex]g_{00}<0[/itex] and [itex]g_{ii} >0[/itex].

In the most general case all the [itex]g_{\alpha \beta}[/itex] can be different from zero and they can be functions of all the 5 coordinates. I don't know if we will ever look at this most general case but for now I want to make two restrictions:

1 - [itex]g_{0i} = g_{i0} = g_{4i} = g_{i4} = 0[/itex]

2 - All the [itex]g_{\alpha \beta}[/itex] are functions only of the 3 coordinates [itex]x^m[/itex].

Now suppose we are interested in null geodesics [itex](ds)^2 =0[/itex]. We can expand the rhs as follows

[tex]

0 = g_{00} (dx^0)^2 + g_{mn} dx^m dx^n + g_{44}(dx^4)^2

[/tex]

Now pull [itex](dx^4)^2[/itex] to the lhs

[tex]

(dx^4)^2 = -\frac{1}{g_{44}}(g_{00} (dx^0)^2 + g_{mn} dx^m dx^n)

[/tex]

This has obviously the form of a GR metric and we can assign [itex]dx^4 \equiv d \tau[/itex] and [itex]dx^0 \equiv dt[/itex]. Instead of pulling out [itex](dx^4)^2[/itex] we can pull [itex](dx^0)^2[/itex] to the lhs

[tex]

(dx^0)^2 = -\frac{1}{g_{00}}(g_{mn} dx^m dx^n + g_{44} (dx^4)^2)

[/tex]

This has the form of a pseudo-Euclidean metric, that is, a metric with all the diagonal terms positive. Let us write it down

[tex]

(dt)^2 = e_{ij} dx^i dx^j

[/tex]

where the 4 [itex]e_{ii}[/itex] are positive.

I think this is enough for now. If it is unclear I will add more explanations before we proceed.

Best regards,

Jose
 
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  • #61
jcsd
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Hurkyl said:
You don't need GR to handle curved worldlines...


What is the point of #57? I think you're still trying to describe how to go from 3+1-space to 4+1-space... but I think it's much easier than you're making it.

When we're working in 3+1-space, if we pick any pointed worldline, we already know how to assign a proper-time to any point on that worldline. Then, IMHO it's straightforward to lift that worldline into 4+1-space simply by making the new coordinate to be equal to the proper time at that point.

(If I've misunderstood your intent, let me know)
This is precisely how it seems to be and my beef is that 5 coordinates does not necessarily mean 5 dimensions, 5 independent coordinates mean 5 dimensions.

For example in a Cartesin coordinate system describing a Euclidena plane you could insert an extra axis say 45 degrees to the other two axis and assign every point in the plane 3 coordinates. These 3 coordinates are not indepent though, knowing any two coordinates describing a point will allow you to work out the third one.

(Unless I've too misunderstood the intentions) that is precisely what is being done here. In this case the extra axis is the worldline of the object we're descrbing and knowing any four cooridnates of an event will allow you to calculate the fifth.

It seems to me we don't have a 5 dimensional structure, instead we've got a 4 dimensional structure being described by a quirky coordinate system that uses 5 coordinates.
 
  • #62
CarlB
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jcsd said:
This is precisely how it seems to be and my beef is that 5 coordinates does not necessarily mean 5 dimensions, 5 independent coordinates mean 5 dimensions.
...
In this case the extra axis is the worldline of the object we're descrbing and knowing any four cooridnates of an event will allow you to calculate the fifth.

It seems to me we don't have a 5 dimensional structure, instead we've got a 4 dimensional structure being described by a quirky coordinate system that uses 5 coordinates.
Your complaint is that any single world line does not fully utilize all 5 dimensions. But the set of all possible worldlines uses them, so they are fully utilized.

Consider a 3 dimensional substance, with one dimension curled up, which carries quantized waves that all happen to travel at the same speed c. If you know the speed of the particle in two of those dimensions you can compute the speed in the third dimension, so by your argument, there are actually only two dimensions.

What I'm saying here is just because you can mathematically eliminate a redundant piece of information from the description of a physical object certainly does not prove that that piece of information is not a part of the physical object. And eliminating these things can bring you a world of hurt in terms of making your physical intuition more difficult and your mathematics more complicated.

At the moment, if you are unfamiliar with the hundreds of papers written under the assumption of Euclidean relativity, you are not in a position to pass judgement on the efficacy of the technique. If you are not intimately familiar with both techniques you are not in a position to judge one against the other.

You can see that there are people who have studied this thing carefully for years and thought about it deeply, made calculations, rewrote the foundations of physics to fit the new assumption, etc. Having done this, we tell you that the grass is greener on this side of the fence.

To see the world from this side you will have to relearn the relativity that you already learned once. I admit that this is a mountain to climb. I admit that the only reason I had the time available to waste on this was because the economy turned down and it looked like a good time to take a vacation from my usual employment. Maybe you don't have this luxury. Life is short.

However, if you do decide to make the effort, the view from up here is beautiful and the weather is fine. The road was very difficult, more especially for a particle physicist than most, because it required that I rethink almost everything I thought I knew about particle physics. It looked like a real stupid idea many times to me, but eventually I worked out new ways of explaining the contradictions and they were simpler and more beautiful than the paradoxes that I was taught before.

Having arrived here at great effort, and experiencing great enjoyment, I will never leave, but I recognize that the effort to get here is beyond the capability of most at this time. To leave the fields where 99.99% of physicists live will give you solitude that may be unwelcome for most, and the path will require a nose for elegance that will evade most. It just ain't easy.

In the example of bringing 3 dimensions down to 2 in an unphysical way, one would notice that in 2 dimensions, there actually are two different choices for that undetermined 3rd velocity coordinate. The corresponding choice in Euclidean relativity is the choice of going forwards and backwards in time, as in the Feynman description of antiparticles. See Hans Montanus for a description of this in classical relativity.

Carl
 
  • #63
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CarlB said:
You can see that there are people who have studied this thing carefully for years and thought about it deeply, made calculations, rewrote the foundations of physics to fit the new assumption, etc. Having done this, we tell you that the grass is greener on this side of the fence.
To see the world from this side you will have to relearn the relativity that you already learned once.
Although my experience and probably also the effort done in the field so far is only a fraction of Carl's yet, I can only fully agree! Euclidean special relativity has really opened my eyes.
 
  • #64
bda
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jcsd

jcsd said:
It seems to me we don't have a 5 dimensional structure, instead we've got a 4 dimensional structure being described by a quirky coordinate system that uses 5 coordinates.
You are right in saying that we have a 4 dimensional structure. By imposing a null displacement condition in 5D we are reducing by 1 the number of independent dimensions. The advantage is that we are left with a choice for describing this 4D structure in either Minkowski or Euclidean geometry and we have the mathematical machinery to translate between the two geometries.

The advantage of starting off in 5D will be even bigger when we consider quantum behaviour, which we are neglecting for the moment.

Best regards,

Jose
 
  • #65
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Hi and hallo to all enthusiasts of Euclidean Relativity. I’ve read the whole post and some papers of some of you – those of Rob and Jose. Although I don’t understand some of your thesis and ideas I’m an enthusiastic supporter of the ER concept.

Studding the geometry based on the equation (ct)^2 = s^2 + x^2 + y^2 +z^2 , I found it possible to derivate the Lorentz transformation, equation of time dilatation and Lorentz contraction that are identical in comparison to those of Special Relativity. Also the composition of velocities equivalent to SR makes no problem. Unfortunately I have no drawing that can be posted here, but if you “play” more with the geometry you should have no problem with adding velocities. BTW, I don’t like the term “adding velocities”. It is rather looking for the answer to the question: “what is the velocity of a moving body measured by the observer, who is moving with a known velocity with respect to us, if we also know the velocity of the moving body in our reference frame. If we ask this way it is obvious why the “sum” can not exceed ‘c’.

Cheers
Martin
 
  • #66
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Hello Martin,
You are very welcome. I'm glad you are enthousiastic about Euclidean relativity. It may be my wishful thinking but it seems like Euclidean relativity is beginning to attract more attention in widening circles, thanks amongst others to the efforts of Jose Almeida, Hans Montanus and Carl Brannen who recently brought the topic on the agenda of some physics conferences.
Best regards,
Rob
 
  • #67
CarlB
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One of the social difficulties of Euclidean relativity is that few physicists are ready to accept an alternative to the special theory of relativity. But perhaps things are changing. Here are some quotes from Lee Smolin's new book, "The Trouble With Physics -- The Rise of String Theory, the Fall of a Science, and What Comes Next":

(p. 218) When the ancients declared the circle the most perfect shape, they meant that it was the most symmetric: Each point on the orbit is the same as any other. The principles that are hardest to give up are those that appeal to our need for symmetry and elevate and observed symmetry to a necessity. Modern physics is based on a collection of symmetries, which are believed to enshrine the most basic principles. No less than the ancients, many modern theorists believe instinctively that the fundamental theory must be the most symmetric possible law. Should we trust this instinct, or should we listen to the leson of history which tells us that (as in the example of the planetary orbits) nature becomes less rather than more symmetric the closer we look?

(p 221) These events [i.e. AGASA events over GZK limit] may be a signal that special relativity is breaking down at extreme energies.

(p 226) I mentioned at the beginning of this chapter that there were two possiblities. We have already discussed one, which is that the principle of the relativity of motion is wrong -- meaning that we could indeed distinguish absolute motion from absolute rest. This would reverse a principle that has been the linchpin of physics since Galileo. I personalll find this possibility abhorrent, but as a scientist I must acknowledge that it is a real possibility.

(p 256) What could that wrong assumption be? My guess is that it involves two things: the foundations of quantum mechanics and the nature of time. We have already discussed the first; I find it hopeful that new ideas about quantum mechanics have been proposed recently, motivated by studies of quantum gravity. But I strongly suspect that the key is time. More and more, I have the feeling that quantum theory and general relativity are both deeply wrong about the nature of time. It is not enough to combine them. There is a deeper problem, perhaps going back to the origin of physics.

(p 314) Fine, you might say, but who are the seers? They are by definition highly independent and self-motivated individuals who are so committed to science that they will do it even if they can't make a living at it. There should be a few out there, even though our professionalized academy is unfriendly to them. Who are they and what have they managed to do to solve the big problems?

They are hiding in plain sight. They can be recognized by their rejection of assumptions that most of the rest of us believe in. Let me introduce you to a few of them.

I have a lot of trouble believing that special relativity is false; if it is, then there is a preferred state of rest and both the direction and speed of motion must be ultimately detectable. But there are a few theorists around who have no trouble with this concept. Ted Jacobson ... Joao Maguiejo ... Robert Laughlin ... Grigori Vilovik ... Xiao-Gang Wen ...

(p 354) To put it more bluntly: If you are someone whose first reaction when challenged on your scientific beliefs is "What does X think?" or "How can you say that? Everybody good knows that ...," then you are in danger of no longer being a scientist. ...
Carl
 
  • #68
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I would like to mention that the article "Dimensions in Special Relativity Theory", that was presented to initiate this thread will be published in the Jan/Feb 2007 issue of the peer-reviewed journal Galileon Electrodynamics. Not a top-rated journal like e.g. Physical Review, but nevertheless encouraging for an amateur without any affiliations.

Rob
 
  • #69
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A discussion is going on in thread https://www.physicsforums.com/showthread.php?t=232693, "Prove that 4 vector potential is really a 4 vector?". The conclusion of samalkhaiat in message #13 is that it is not a 4-vector.
Remarkably, this is also the conclusion I got, reasoning from principles of Euclidean special relativity. Section 6 of the article "http://www.euclideanrelativity.com/4vectors/node6.html" [Broken]" suggests that the classical potential 4-vector is in fact an Euclidean 4-vector instead of a Minkowski 4-vector.
 
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  • #70
robphy
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A discussion is going on in thread https://www.physicsforums.com/showthread.php?t=232693, "Prove that 4 vector potential is really a 4 vector?". The conclusion of samalkhaiat in message #13 is that it is not a 4-vector.
Remarkably, this is also the conclusion I got, reasoning from principles of Euclidean special relativity. Section 6 of the article "http://www.euclideanrelativity.com/4vectors/node6.html" [Broken]" suggests that the classical potential 4-vector is in fact an Euclidean 4-vector instead of a Minkowski 4-vector.
samalkhaiat's conclusion is that there is a gauge-related term in the transformation.
 
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  • #71
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As CarlB suggests, the fact that Mortimer's theory suggests changes to GR may well be an attractive aspect. I think we are at a point in physics that is similar to the transition between the Ptolemaic view and the post-Copernican view. The Ptolemaic model was an exquisitely complicated system that, while remarkably functional, turned out not only to be more complicated than necessary but flawed exactly where it was most complicated. It was dramatically simplified by Copernican models explanations of retrograde aspects. I think the same thing will happen to string theory (and perhaps inflation, too) in the near future by the recognition of a more plausible broad-scale disposition of the material universe.

But what it won't be is a unified theory of everything. It will be a temporary consolidation and simplification that will continue to evolve into more complicated theories just like the Copernican model continued to evolve. Young Kmarinas' perspective kind of speaks to the point.
I had the same idea. I have a theory of a fractal universe (so far mostly qualitative) which agrees with these statements, which proposes that our visible universe of galaxies and stars is a boson (specifically a gluon) and that by looking at the "edge of the universe" we may be looking at the surfaces of very large black holes (specificially the surfaces of fermions (quarks)).

Presuming a fractal symmetry in the universe is, in my estimate, an area rich in potential for evolving the standard model because it suggests and ongoing hierarchy. But to presume that hierarchy to be identically repeating (ie, the quark can be identically found at both extra-visible-universe scales and sub-nuclear scales, etc, etc) is similar in many ways to a common failing found in most of our prior cosmologies.

Trying to limit the scalar diversity of the universe to what we humans can see of it at any given time is typically where our prior cosmologies were corrected by their succeeding cosmologies. The Copernican crystal sphere terminus was expanded by Milky Way island universe terminus, which was expanded by the contemporary "finite but unbounded" space/time model, which is less egregiously mitigated by an infinite presumption of the cosmological principle. All of these terminating criteria try to depict a universe that can be completely characterized (if not examined) using only the data at hand. With the exception of the most recent version of an infinitely homogeneous universe, they all failed in precisely the extent to which they strived to to limit the scalar diversity of the universe as it might evolve beyond our ability to examine at any given time. We should presume that the universe cannot be completely depicted from the perspective of something stuck inside it.

In kmarinas' fractal disposition above, I would suggest that the fractal symmetry he depicts as absolute and repeating represents this same type of effort to have the universe conform completely to the data at hand. The fractal behavior of the universe is more likely to be only locally transmitted up and down the scale before evolving into new symmetries across an ongoing, open-ended material hierarchy.

That is, if the enigmatic obits of electrons around the atomic nucleus is only loosely reflected by the very deterministic obits of the planets around stars, so too might the fractal symmetry kmarinas suggests as identically repeating be found evolve its symmetry across any scalar interval. (the black hole may be only vaguely "quark-like" at even greater mega-scales beyond.)

Change is permanent, evolution is inherent. But change is also symmetrical and continuous across all spectra. So while we will always be able to expand our knowledge, we will probably never be able to assert a final deposition and should formally recognize this prospect in our models. It might seem very depressing, but imagine how we'd feel if one day we found out we'd figured it all out and there was nothing left to discover. Now, that would be depressing.

-Mike
 

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