Quantization of Length? Heresy or Merit?

[Daedalus]

After some considerable thought about a post I made in response to, Check my equation of I x F = E, I decided to follow up with a question about the possibility of quantized length. The question that is posed is one that asks whether there is merit in the notion of quantized space versus analog space and uses the Lorentz factor in regards to length contraction as a basis for the argument.

 

We know that we cannot reach the speed of light due to the definitions in relativity. The Lorentz contraction states that if a body moved at the speed of light relative to an observer, then it would appear to have its length contracted to zero along the direction of motion as seen by the observer.

 

[math]L'=L \sqrt{1-\frac{v^{2}}{c^{2}}}[/math]

 

When [math]v=c[/math], we get:

 

[math]L'=L \sqrt{1-\frac{c^{2}}{c^{2}}}=0[/math]

 

However, we also know that a body in motion would reflect or emit light along each and every position along its trajectory. Therefore, the photons emanating from a body along the collapsed length would still exist and be observable. I realize that it is an extreme leap to go from this view to one where the Lorentz factor is not entirely correct. But if there does exist some smallest quantized length, then a body would not necessarily contract to a zero length at the speed of light. The mathematics that could support this is as follows:

 

If [math]L'\ne0[/math] when [math]v=c[/math], then the Lorentz factor would be off by some factor [math]F[/math], such that:

 

[math]L'=L \sqrt{1-\frac{c^{2}}{c^{2}}\times F}=L\times l[/math]

 

Solving for [math]F[/math] we get:

 

[math]F=1-l^{2}[/math]

 

Substituting this result back into the modified equation we arrive at:

 

[math]L'=L \sqrt{1-\frac{v^{2}}{c^{2}}\times (1-l^{2})}=L \sqrt{1-\frac{(v+v l)(v-v l)}{c^{2}}}[/math]

 

We can see that if [math]l=0[/math], then we get back the original equation:

 

[math]L'=L \sqrt{1-\frac{(v+v\times 0)(v-v\times 0)}{c^{2}}}=L \sqrt{1-\frac{v^{2}}{c^{2}}}[/math]

 

And when [math]v=0[/math], the modified equation yields the same result as the original equation:

 

[math]L'=L \sqrt{1-\frac{(0+0\times l)(0-0\times l)}{c^{2}}}=L[/math]

 

The only difference lies in how it treats space in regards to relativistic velocities near or at the speed of light such that when [math]v=c[/math] we get:

 

[math]L'=L \sqrt{1-\frac{(c+c\times l)(c-c\times l)}{c^{2}}}=L\times l[/math]

 

Thus if [math]l[/math] is extremely small, then it would not even be observed as it would fall outside of our current abilities to detect and measure this hypothetical quantized length. If we solve for [math]l[/math] algebraically we get:

 

[math]l=\frac{\sqrt{v^{2}-c^{2}(1-\frac{L'^{2}}{L^{2}})}}{v} [/math]

 

From this result we can infer two things:

 

1.) We can measure [math]l[/math] by evaluating the Lorentz contraction for a body that is approaching the speed of light such that:

 

[math]l=\frac{\sqrt{c^{2}-c^{2}(1-\frac{L'^{2}}{L^{2}})}}{c}=\frac{L'}{L} [/math]

 

2.) Having the velocity ending up in the denominator would suggest that it is impossible to truly be at rest. This is in complete agreement with our observations because all things are in motion. This result seems logical in the aspect that it is easier for me to accept a consequence that I can never truly be stationary versus one that states I would be contracted to an improbable zero length.

 

The question now remains if this idea has any merit?

Edited July 8, 2011 by Daedalus

[Daedalus]

Continuing with this discussion, let us see how this modification effects relativistic mass. The original equation is:

 

[math]m_{r}=\frac{m_{0}}{\sqrt{1-\frac{v_{r}^{2}}{c^{2}}}}[/math]

 

Substituting this hypothetical correction to gamma we get (I am going to ignore our relative temporal velocity because it will always be zero for all observable objects moving with us through time):

 

[math]\gamma=\frac{1}{\sqrt{1-\frac{(v_{r}+v_{r} l)(v_{r}-v_{r} l)}{c^{2}}}}[/math]

 

[math]m_{r}=m_{0}\gamma=\frac{m_{0}}{\sqrt{1-\frac{(v_{r}+v_{r} l)(v_{r}-v_{r} l)}{c^{2}}}}[/math]

 

When relative velocity is equal to the speed of light we get:

 

[math]m_{r}=\frac{m_{0}}{l}[/math]

 

If [math]l=0[/math] it would take an infinite amount of energy to reach the speed of light. However, I have shown that we do move at the speed of light in my thread on Temporal Uniformity (Time) and AJB showed me that the four vector of a massive particle in the rest frame is equal to the speed of light. Since we move at the speed of light through space-time, then it seems as though [math]l[/math] does exist and that it does not require an infinite amount of energy to reach the speed of light. This actually makes sense because it seems improbable that the universe has an infinite amount of physical space and mass-energy. If you do believe that there is an infinite amount of mass-energy and space, then my argument for temporal uniformity would be true because the universe would still be exploding at time zero at the center of the big bang as I type these words. So my theory suggests a finite universe. But we shouldn't get our hopes up of reaching light speed as it would probably take all of the energy in our universe to do so.

Edited July 13, 2011 by Daedalus

[ajb]

Quantisation of area is quite standard in noncommutative geometry. If we suppose that the space-time coordinates [math](x^{\mu})[/math] no longer commute, lets say they satisfy [math]x^{\mu}x^{\nu}- x^{\nu}x^{\mu} = i k J^{\mu \nu}[/math] with [math]k[/math] being a fundamental area scale, which we suppose is the order of the Planck area, then space-time will be dived up into Planck cells.

 

That is near every "point" there will be some fuzziness, a small region in which the classical notions of space are missing. The volumes of these Planck cells is of the order [math]2 \pi k^{2}[/math].

 

If space-time is at some small scale noncommutative then one would expect Lorentz symmetry to be violated. So, one would expect the length contraction formula to be modified near the Planck scale. One would need to know details of the quantum nature of space-time to such more.

[Daedalus]

Quantisation of area is quite standard in noncommutative geometry. If we suppose that the space-time coordinates [math](x^{\mu})[/math] no longer commute, lets say they satisfy [math]x^{\mu}x^{\nu}- x^{\nu}x^{\mu} = i k J^{\mu \nu}[/math] with [math]k[/math] being a fundamental area scale, which we suppose is the order of the Planck area, then space-time will be dived up into Planck cells.

 

That is near every "point" there will be some fuzziness, a small region in which the classical notions of space are missing. The volumes of these Planck cells is of the order [math]2 \pi k^{2}[/math].

 

If space-time is at some small scale noncommutative then one would expect Lorentz symmetry to be violated. So, one would expect the length contraction formula to be modified near the Planck scale. One would need to know details of the quantum nature of space-time to such more.

 

Thank you AJB for stepping in and showing how this concept does have some merit. I realize that [math]l[/math] probably has a lot more mathematics behind it, but at leasts it is a start.

[ajb]

Thank you AJB for stepping in and showing how this concept does have some merit. I realize that [math]l[/math] probably has a lot more mathematics behind it, but at leasts it is a start.

 

I have no idea if your particular idea is of any merit, but modifications of space-time on the smallest scales is an active area of research. It has also fuelled developments in mathematics and in particular the question of "what is geometry?".

[Daedalus]

I have no idea if your particular idea is of any merit, but modifications of space-time on the smallest scales is an active area of research. It has also fuelled developments in mathematics and in particular the question of "what is geometry?".

 

Point taken and well made. I didn't mean to actually imply that you agree with my theories. My intention was to show how scientists are looking for a more accurate and precise metric for space-time and to point out that my theory aligns itself along this view.

Edited July 13, 2011 by Daedalus

[Daedalus]

I thought I would post an update to this thread in regards to a possible finding at CERN which may have actually found neutrinos that moved faster than the speed of light. This finding is still being analyzed by physicist and is currently being talked about here at ScienceForums.net in the following thread:

 

Faster than lightspeed achieved?

 

I will quote Swansont regarding this finding because he has made an excellent point:

 

The scientists aren't claiming to have toppled relativity and are calling for independent confirmation. Just as one would expect of responsible physicists.

 

With that being said, I would like to discuss the point I have made in this thread regarding a hypothetical correction to the Lorentz factor. This correction was meant to imply that it is impossible for length to contract to zero, but it can also allow for the possibility of a particle to travel faster than light. I am not suggesting that the mathematics proposed here is correct or that the physicist have in fact confirmed this finding. Instead, I am making this post to entertain the notion of how a particle could move faster than light using the proposed mathematics.

 

My proposed modification to the Lorentz factor:

 

[math]\gamma=\sqrt{1-\frac{v^{2}}{c^{2}}\times (1-l^{2})}[/math]

 

We know that SR has a value for [math]l[/math] which is equal to zero. However, if [math]l\ne 0[/math] then a particle could hypothetically move faster than light:

 

[math]\gamma=\sqrt{1-\frac{v^{2}(1-l^{2})}{c^{2}}}[/math]

 

We can clearly see that if [math]l\ne 0[/math], it would act to reduce the numerator in [math]\frac{v^{2}}{c^{2}}[/math] and allow for higher relative velocities than [math]c[/math].

Edited September 23, 2011 by Daedalus

[Pincho Paxton]

Lots of theories make their way into the speculations forum. I never know when to reply to posts with my own theory, because I wonder if everyone just want to associate their theory with current physics. But anyway from my point of view...

 

You have mass, and inside that mass are spaces, and inside those spaces is Aether. The Aether flows through those spaces. The Aether is spherical, and it's flow rate is determined by a queuing system. It can't bound over its neighbour unless that neighbour changes to negative mass, and quite a lot of it will change to negative mass.

 

A moving body gathers a certain amount of Aether ahead of itself.. the bow shock. The Aether is very light, and gathers momentum with the moving body, and also folds into figure 8 loops... time...

 

The flow direction is important. Like streams, the Dark Flow of the Aether is directional, and you can hit a fast stream, or a head on current.

 

Worst case scenario, you hit a head on current. What would happen to the mass? When you reach the limit of mass speed the Aether is folded to negative mass. But too high a speed will also fold the atoms to negative mass, because they are also made of Aether. The difference with atoms is that the shell can slide over the Aether until it is breached. I suppose you can think of an atom shell as streamlined for Aether up to a point.

 

That's my reply. Ignore it if you like.

Edited September 23, 2011 by Pincho Paxton

[Daedalus]

Lots of theories make their way into the speculations forum. I never know when to reply to posts with my own theory, because I wonder if everyone just want to associate their theory with current physics. But anyway from my point of view...

 

I made this post to "entertain" the possibility of FTL in response to the "possible" discovery at CERN. Also, I originally created this thread in the speculation forum back in July 2011.

 

You have mass, and inside that mass are spaces, and inside those spaces is Aether. The Aether flows through those spaces. The Aether is spherical, and it's flow rate is determined by a queuing system. It can't bound over its neighbour unless that neighbour changes to negative mass, and quite a lot of it will change to negative mass.

 

A moving body gathers a certain amount of Aether ahead of itself.. the bow shock. The Aether is very light, and gathers momentum with the moving body, and also folds into figure 8 loops... time...

 

The flow direction is important. Like streams, the Dark Flow of the Aether is directional, and you can hit a fast stream, or a head on current.

 

Worst case scenario, you hit a head on current. What would happen to the mass? When you reach the limit of mass speed the Aether is folded to negative mass. But too high a speed will also fold the atoms to negative mass, because they are also made of Aether. The difference with atoms is that the shell can slide over the Aether until it is breached. I suppose you can think of an atom shell as streamlined for Aether up to a point.

 

That's my reply. Ignore it if you like.

 

It is not that I choose to ignore you Pincho Paxton. I simply have not considered an Aether. So any response that I could make would most certaintly be ignorant of your ideas.

Edited September 23, 2011 by Daedalus