Minimum Length Candidate

[SimonsCat]

Spacetime uncertainty is constructed from the metric and it is given as


$$\delta L \delta t\ g_{tt} \geq 1 - \frac{2 G \hbar}{c^4} = \frac{L^2_P}{c}$$


I showed how this might be suggestive of a relationship between redshift and quantum mechanics via:


$$\delta L \delta t \geq \frac{(1 - \frac{2 G \hbar}{c^4})}{g_{tt}}$$


The metric is just


$$g_{tt} = 1 + \frac{\phi}{c^2}$$


expanding we get


$$1 + \phi_1 - \phi_2 - \frac{1}{2} \phi^2_1 - \phi_1\phi_2 + \frac{3}{2}\phi^2_2 +...$$



To first order we get



$$\delta L \delta t \geq \frac{(1 - \frac{2 G \hbar}{c^4})}{[\frac{\phi_1}{c^2} - \frac{\phi_2}{c^2}]} = \frac{(1 - \frac{2 G \hbar}{c^4})}{\frac{\delta \phi}{c^2}}$$



adopting $$\delta s \equiv c \delta t$$ we can theorize the fundamental length as



$$\delta L \geq \frac{(1 - \frac{2 G \hbar}{c^3})}{g_{tt}c\delta t} = \frac{(1 - \frac{2 G \hbar}{c^3})}{\sqrt{\delta s^2 - g_{xx}\delta x^2}}$$



this can be constructed by using


$$\delta s^2 = g_{tt} c^2\delta t^2 + g_{xx} dx^2$$


and so


$$g_{tt}c \delta t = \sqrt{\delta s^2 - g_{xx}\delta x^2}$$


Since this formalism is NOT coordinate free, there is clearly work to be done.

 

[paddoboy]

Since this formalism is NOT coordinate free, there is clearly work to be done.

First congrats for at least posting your hypotheticals in the proper forum.
Secondly, as yet I'm unable to reasonably comment on them, but would suggest that if you had anything of any concrete nature in any of your hypotheticals, then writing up and publishing a proper scientific paper for professional peer review is the reasonable, logical way to go.

 

[SimonsCat]

First congrats for at least posting your hypotheticals in the proper forum.
Secondly, as yet I'm unable to reasonably comment on them, but would suggest that if you had anything of any concrete nature in any of your hypotheticals, then writing up and publishing a proper scientific paper for professional peer review is the reasonable, logical way to go.

Thank you for being concerned, but I am not interested in such things. I float the internet trying to teach what I have learned, I don't have any intentions of publishing any of these thoughts, any time soon. Maybe, just not soon.

 

[paddoboy]

Thank you for being concerned, but I am not interested in such things. I float the internet trying to teach what I have learned, I don't have any intentions of publishing any of these thoughts, any time soon. Maybe, just not soon.

I'm not concerned in the least.
The facts are forums such as this are open to any Tom, Dick or Harry, with whatever agenda and baggage each carry, to propose, deny, or fabricate whatever crap and quackery he or she likes, and this forum has many such irrelevant nonsense that have all died a natural death, lost on a remote science forum, in cyber space..
Again, the only proper way is for yourself to learn, that way of which I have already expressed for proper peer review.
The choice is yours.

 

[SimonsCat]

Fabricate?

Ok, I think my work proves none of it is fabricated. Just a simple scan of the four or five posts I have made in this sub-forum (armed with knowledge of physics) will prove this.

 

[paddoboy]

Fabricate?

Ok, I think my work proves none of it is fabricated. Just a simple scan of the four or five posts I have made in this sub-forum (armed with knowledge of physics) will prove this.

Your work actually proves nothing other than you probably do have a knowledge of physics...It's how you apply it that counts.
Otherwise just another alternative hypothetical, that will be forever lost on a small sliver of cyber space on a remote science forum.

 

[SimonsCat]

No not lost... just not fully appreciated... and it's ok, I am ready to deal with that.

 

[paddoboy]

No not lost... just not fully appreciated... and it's ok, I am ready to deal with that.

That's beautiful...*sniff*

 

[SimonsCat]

 

[paddoboy]

Since this formalism is NOT coordinate free, there is clearly work to be done.

OK, could you put this alternative hypothetical you have in reasonably simple terms, devoid as much as possible of the mathematics please.
Could you also do as best as you can to your other alternative hypotheticals.
Not that I have anything against maths.....of course any scientific model must certainly align with the mathamatics validation....The point is that I, and I dare say many others are not up with the mathematics.
No rush, as I'm off anyway for a musical saturday night!

 

[SimonsCat]

Ok, so what we have done is take the spherically-symmetric Schwarzschild solution an applied a commutation algebra: which is the same thing as applying a spacetime uncertainty with two products:

$$\delta L_P c \delta t$$

The more you locate the length, the less you know of the time, but the more you define the time, naturally, like all complimentary observables, will lead to an ill-defined length. Notice also, by writing out in this way gave us a metric definition that depends on the Planck length^2! No matter how good my guessing work is, we need to expand the gravitational potential for higher values of the redshift. No theory today really does well approximated the gravitational field to only first order, so I expect the true form of the minimum length will require higher corrections to the potential. Near the end we construct the final parts of the fundamental length by distributing the $$c\delta t$$ to the denominator which further allows us to express it in the following way:

$$ds^2 = g_{tt}c^2\delta t^2 + g_{xx} \delta x^2$$

So what we have done is create a very simple line element in the denominator, rearranging and taking the square root gives;

$$\sqrt{ds^2 - g_{xx} \delta x^2} = g_{tt}c\delta t $$

Which gives us the final aspect of the hypothetical length.

 

[James R]

Thank you for being concerned, but I am not interested in such things. I float the internet trying to teach what I have learned...

Welcome.

I suggest that a little more background/introductory material might be a good idea when you're posting this kind of thread on sciforums.

 

[SimonsCat]

Being coordinate free however,

Welcome.

I suggest that a little more background/introductory material might be a good idea when you're posting this kind of thread on sciforums.

Thank you but I don't actually have a great biography, you'd find me very boring.

 

[paddoboy]

Which gives us the final aspect of the hypothetical length.

Thanks for that...makes it slightly clearer, although I'm still unable to comment......

Thank you but I don't actually have a great biography, you'd find me very boring.

This forum as well as most others that allow it, and as I iterated to you earlier, has many "pretenders" that claim GR is wrong, gravitational waves are not what they seem, and a generally dismissal of 21st century cosmology....Most of these are driven by "god bothering" and "god of the gaps" agendas. Perhaps James like I, was trying to eliminate you from that kind.
I have made a few enemies in my time here, in revealing such pretenders.
You already know my details for what they are worth, although I believe James was more referring to your actual alternative claim and exactly what you are trying to refute and what you replace that with.

 

[SimonsCat]

Only on rare occasions do I try and refute something, because its not really the scientific way. It's like when someone says their objective goal is to debunk something, I tend to think of it another way: The scientist should be objective and using the scientific method, deduct whether something holds credibility or not.

Just going back, I am not refuting anything really, but I will tell you there are possibly several fundamental length candidates over the years and this post was just suggesting another way to look at it.

 

[SimonsCat]

But again, its not coordinate free and this is a problem so far. It can be made to suit the coordinate free case but I haven't investigated this yet.

 

[James R]

Thank you but I don't actually have a great biography, you'd find me very boring.

I wasn't referring to your personal biography. I meant you ought to explain what you're trying to show in the thread, in terms that a layman has some chance of understanding, if possible. Failing that, an explanation of the relevance of what you are presenting, in terms that somebody with a reasonable education in physics and/or mathematics would reasonably understand, would be nice.

On the other hand, maybe all I'm really saying here is that I don't appear to be operating mathematically at the same lofty level you are presenting at. Or something.

 

[SimonsCat]

I wasn't referring to your personal biography. I meant you ought to explain what you're trying to show in the thread, in terms that a layman has some chance of understanding, if possible. Failing that, an explanation of the relevance of what you are presenting, in terms that somebody with a reasonable education in physics and/or mathematics would reasonably understand, would be nice.

On the other hand, maybe all I'm really saying here is that I don't appear to be operating mathematically at the same lofty level you are presenting at. Or something.

Well originally I started working on these equations a few months ago to try and find a natural quantum solution to the redshift. Interestingly there appears to be evidence for a quantized redshift.

Then as I started looking at the equations further, I decided that the final form actually does good for a minimum length candidate.

Would you like me to go through each equation, blow by blow as it where?

 

[James R]

Would you like me to go through each equation, blow by blow as it where?

That's up to you, SimonsCat. You need to decide for yourself who your audience is, and at what level you think it most appropriate to pitch your ideas to the membership (and other readers) of this forum.

On a personal level, I'll be sure to ask you questions if I'm interested enough to follow up on something you have posted.

 

[rpenner]

Spacetime uncertainty is constructed from the metric and it is given as
$$\delta L \delta t\ g_{tt} \geq 1 - \frac{2 G \hbar}{c^4} = \frac{L^2_P}{c}$$

The Uncertainty Relation:
$$ \sigma_{x_i} \sigma_{p_i} = \sqrt{ \left< \hat{x}_i^2 \right> - \left< \hat{x}_i \right>^2 } \sqrt{ \left< \hat{p}_i^2 \right> - \left< \hat{p}_i \right>^2 } \geq \frac{\hbar}{2} $$
arises from wave-particle duality, the notion that it's not correct to say things are particles or waves (or particles and waves) but that they have particle-like and wave-like properties, which in concrete form shows up throughout quantum physics.

Assuming $$\delta L$$ is a standard deviation in measurement of relative position and $$\delta t$$ is standard deviation in measurement of elapsed time, not due to precision of measurement, but a statistical expectation from multiple iterated experiments on identical setups, there would be some parallelism here. But in flat-space-time, the product $$\delta L \delta t$$ is not an invariant of choice of standard of rest. (I'm using 3+1 spacetime. The product is preserved in 1+1 spacetime the trivial reason that $$\delta L$$ and whatever change in motion represented by the Lorentz transform are always parallel.) Therefore the theory doesn't look informed by special relativity. A more sensible left-hand part would be an element of space-time-volume: $$\delta x \delta y \delta z \delta t$$ but that would a completely different theory than proposed here.

$$\ell_{P} \equiv \sqrt{ \frac{G \hbar}{c^3} }$$ https://en.wikipedia.org/wiki/Planck_length
Therefore $$ \frac{2 G \hbar}{c^4} = 2 \ell_{P}^2 / c $$ has units of [LENGTH] × [TIME] which means it cannot be sensibly subtracted from 1.
It is unclear if you wish to define $$L_P$$ as a new length or is using an alternate symbol for the Planck length.
In SI units, tensor element $$g_{tt}$$ has dimensions of [LENGTH]² × [TIME]⁻². Since you bother to call out factors of c, it looks like you are working in units where $$c \neq 1$$, one might assume your conventions for the metric were similar, but this appears not to be the case.
In geometric units, with sign convention (+---), where time has units of length, tensor element $$g_{tt}$$ is dimensionless and therefore $$\delta L \delta t g_{tt}$$ presumably has units of [LENGTH] × [TIME] which cannot be sensibly compared to any fixed number because the choice of coordinate remains arbitrary in GR and spacetime is curved enough that there are no nice patches which cover everywhere, so one may not so easily place a lower bound on a quantity times $$g_{tt}$$.

In short, there is nothing sensible about the presentation of first given equation that conveys expertise with the subject matter.