Anisotropy of Space: Implications for Lorentz Invariance and Quantum Mechanics

[mhill]

is really space isotrope ?? , for example in SR the metric is Minkowsky metric and there are no special reference frames, but what would happen at quantum level ? does still Lorentz transformation hold if space is not isotrope and Lorentz invariance is false ??

 

[Dale]

No, by definition the Lorentz transform would not hold if Lorentz invariance were false.

If space were anisotropic then by Noether's theorem momentum would not be conserved. Since the conservation of momentum is so well-established I think it is safe to say that space is isotropic.

 

[atyy]

Modern Tests of Lorentz Invariance
David Mattingly
http://relativity.livingreviews.org/Articles/lrr-2005-5/index.html

 

[Aether]

If space were anisotropic then by Noether's theorem momentum would not be conserved. Since the conservation of momentum is so well-established I think it is safe to say that space is isotropic.

There are three experiments to establish that space is isotropic (Michelson-Morley, Kennedy-Thordike, and Ives-Stilwell)...there are some first-order experiments that may also apply. However, the conservation of momentum can not be established by experiment, rather it is assumed implicitly when one chooses to work with isotropic coordinate systems.

 

[mhill]

the question is .. could Lorentz invariance fail at low distances ?? , perhaps if we measured more accurately the space we would find tha Lorentz invariance does not hold.

 

[Aether]

the question is .. could Lorentz invariance fail at low distances ?? , perhaps if we measured more accurately the space we would find tha Lorentz invariance does not hold.

Yes, it could fail at a small scale (some unified field theories predict that it will), but it could also still hold there. See: V. Alan Kostelecky (http://www.physics.indiana.edu/~kostelec/) and the Standard Model Extension.

http://www.physics.indiana.edu/~kostelec/faq.html]Our[/PLAIN] basic premise is that minuscule apparent violations of Lorentz and CPT invariance might be observable in nature. The idea is that the violations would arise as suppressed effects from a more fundamental theory.

 

[Naty1]

Space better be isotropic otherwise one direction will be different from another and virtually all of physics will be impacted, not just lorentz invariance. . Light might move twice as fast in the x direction,say, as the y,z directions...screwing up observations of all types.

At quantum small distances an interesting theoretical question remains...if Planck size is the "minimum" size in this universe, is it a constant? Would an observer moving at high velocity still see Planck length, for example, at the same length as an observer in the same inertial frame as the observed Planck length? As recently as a few years ago this remains yet another unanswered question.

 

[Aether]

At quantum small distances an interesting theoretical question remains...if Planck size is the "minimum" size in this universe, is it a constant? Would an observer moving at high velocity still see Planck length, for example, at the same length as an observer in the same inertial frame as the observed Planck length?

The Planck length (as a dimensionful constant) should transform between inertial frames in the same way as any other length.

As recently as a few years ago this remains yet another unanswered question.

Can you cite a published paper where someone said that they consider this to be an unanswered question?

 

[DrGreg]

if Planck size is the "minimum" size in this universe, is it a constant?

I don't know a huge amount about quantum theory, but that isn't really what "planck size" is. It's the uncertainty with which an observer is able to measure size. If an observer makes multiple measurements of the length of the same object, those measurements could vary by up to a Planck length. (Note I say "up to", not "exactly".)

Different observers make different measurements, so there's no contradiction in all observers agreeing on the value of Planck length.

 

[Naty1]

As recently as a few years ago this remains yet another unanswered question.

Can you cite a published paper where someone said that they consider this to be an unanswered question?

Different observers make different measurements, so there's no contradiction in all observers agreeing on the value of Planck length.

This is discussed in Lee Smolin's THE TROUBLE WITH PHYSICS, 2007, beginning page 227:

In essence, I believe what Smolin says is that another physicst, Amelion-Camelia, modified the Einstein equations of special relativity in such a way that Planck Length remains a constant from all intertial frames ... as does the speed of light in Einstein's formulation. So maybe Camelia did "solve" it in a sense...depending on which formulation of special relativity you prefer.

The theory is referred to as "DSR" Double Special Relativity in which both the speed of light and Planck length are constants...independent of inertial frame observations. However, Smolin also says it is expected that relativity breaks down at Planck length, so I'm unsure of the actual significance of this formulation.

Smolin says:

Planck length is in some sense the smallest size that can be observed. The question is will all observers agree on what this shortest length is?

He discusses a number of issues simultaneously such as whether photons of different energies all travel at precisely the same speed...it's not easy to sort them all out. But a varying Planck length, dependent on inertial frames, seems to raise fundamental questions about the discrete (quantum) nature of space++. Smolin also notes that he came across a version of the paradox on his own some years earlier, could not figure it out, and so moved on. Within calculations on loop quantum gravity:

...Our calculations seemed to contradict Einstein's special theory of relativity..

(p 229)

++My own impressions: Two things I believe may be implied but not overtly expressed in Smolins discussion. One is "T" duality in string theory where a decreasing radius R begins to grow as 1/R...it's a duality between different string theory formulations. It seems to imply a minimum size in string theory. The second is that it is currently believed that each Planck area of 10^-66 cm is the minimum area capable of containing one bit of information...a common reference to this the surface area of a black hole event horizon and it's information content. Is this a real fixed minimum or is it inertial frame dependent??

 

[Naty1]

I checked Planck length in Wikipedia

http://en.wikipedia.org/wiki/Planck_length

and found defined as Planck Length = sq rt (h(bar)G/c^3) ...

I did not get much from the description, no surprise, but if c is a constant, then for Planck length to vary dirac's constant h or the gravitational constant G or both might vary...

Anybody know the significance of that, if anything??

 

[Aether]

I checked Planck length in Wikipedia

http://en.wikipedia.org/wiki/Planck_length

and found defined as Planck Length = sq rt (h(bar)G/c^3) ...

I did not get much from the description, no surprise, but if c is a constant, then for Planck length to vary dirac's constant h or the gravitational constant G or both might vary...

Anybody know the significance of that, if anything??

Why would c be a constant while Planck length varies? They are both isotropic within every inertial frame, and they are both anisotropic between inertial frames.

 

[Naty1]

Why would c be a constant while Planck length varies?

That is the question Smolin discusses in post #10...Does Einstein's formulation of special relativity conflict with a fixed Planck length...Smolin sez "yes". Camelia has another formulation to "fix" that "problem"...I hypothesized that IF Planck size varies, there might be some HUGE consequences...

I'm not claiming one nor the other as I don't research...I'm only reflecting the comments of another as best I can...I have enough trouble understanding my wife!

 

[Aether]

That is the question Smolin discusses in post #10...Does Einstein's formulation of special relativity conflict with a fixed Planck length...Smolin sez "yes". Camelia has another formulation to "fix" that "problem"...I hypothesized that IF Planck size varies, there might be some HUGE consequences...

I'm not claiming one nor the other as I don't research...I'm only reflecting the comments of another as best I can...I have enough trouble understanding my wife!

You said:

In essence, I believe what Smolin says is that another physicst, Amelion-Camelia, modified the Einstein equations of special relativity in such a way that Planck Length remains a constant from all intertial frames ... as does the speed of light in Einstein's formulation.

However, Planck length was already a constant within all inertial frames in the same way as the speed of light in Einstein's formulation. There must be something more, maybe subtle, to what they were saying about this. Please quote the relevant passage directly.

 

[Fredrik]

If space were anisotropic then by Noether's theorem momentum would not be conserved. Since the conservation of momentum is so well-established I think it is safe to say that space is isotropic.

It's also safe to say that it's angular momentum.

However, the conservation of momentum can not be established by experiment, rather it is assumed implicitly when one chooses to work with isotropic coordinate systems.

This is a strange thing to say. If you mean that you can't do an experiment that forces us to conclude that momentum (or angular momentum) is conserved, then yes, of course. Experiments never lead to conclusions like that. That's not how science works.

If you mean that science can't test the part of the theory that says that (angular) momentum is conserved, then my answer is: Of course it can.

 

[Aether]

It's also safe to say that it's angular momentum.

What do you mean by "it's angular momentum"? That angular momentum is conserved, but linear momentum is not?

This is a strange thing to say. If you mean that you can't do an experiment that forces us to conclude that momentum (or angular momentum) is conserved, then yes, of course. Experiments never lead to conclusions like that. That's not how science works.

If you mean that science can't test the part of the theory that says that (angular) momentum is conserved, then my answer is: Of course it can.

I mean that inertial frames are constructed, by definition, so that linear momentum is conserved, but other equally valid coordinate systems can be constructed where linear momentum is not necessarily conserved. This may or may not also apply to angular momentum, we can consider that if you agree that linear momentum is only conserved if one chooses to use a special set of coordinate systems to make this so.

 

[Fredrik]

What do you mean by "it's angular momentum"? That angular momentum is conserved, but linear momentum is not?

The discussion at that point was about the isotropy of space, i.e. it's rotational invariance. Rotational invariance implies conservation of angular momentum, but says nothing about (linear) momentum.

If we take spacetime to be Minkowski space, then space (as defined by an inertial frame) also has translational invariance, and that implies that (linear) momentum is conserved.

 

[Aether]

This is discussed in Lee Smolin's THE TROUBLE WITH PHYSICS, 2007, beginning page 227:...In essence, I believe what Smolin says is that another physicst, Amelion-Camelia, modified the Einstein equations of special relativity in such a way that Planck Length remains a constant from all intertial frames ... as does the speed of light in Einstein's formulation.

I read that chapter, and he seems to be saying (he doesn't show any math in this chapter) that on a ten-dimensional manifold the Planck length can be made to be invariant under all coordinate transformations (like the line interval ds). He doesn't say whether or not this becomes true for all lengths. This would be a little bit stronger than saying "that Planck length remains a constant from all inertial frames" in the context of Lorentz symmetry acting on a 4D manifold.

 

[Aether]

The discussion at that point was about the isotropy of space, i.e. it's rotational invariance.

Okay.

Rotational invariance implies conservation of angular momentum, but says nothing about (linear) momentum.

I was thinking only of linear momentum when I made that statement above, so I will restrict my claim to linear momentum for now. Can you cite an experiment where the conservation of angular momentum has been used to probe the isotropy of space?

If we take spacetime to be Minkowski space, then space (as defined by an inertial frame) also has translational invariance, and that implies that (linear) momentum is conserved.

Inertial frames are defined in part by an assumption that linear momentum is conserved. Do you agree that equally valid coordinate systems can be defined in which linear momentum is not generally conserved?

I often have limited internet access while traveling, so any reply may be quite delayed.

 

[Aether]

The discussion at that point was about the isotropy of space, i.e. it's rotational invariance. Rotational invariance implies conservation of angular momentum, but says nothing about (linear) momentum.

Rotational invariance implies conservation of angular momentum only if one first assumes that linear momentum is conserved. For example: if angular momentum is mvXr, and rotational invariance implies that r is isotropic, then (assuming constant m) the conservation of angular momentum is dependent on the isotropy of v in exactly the same way as is the conservation of linear momentum.