Black and White Holes

[LaurieAG]

On 12/20/2024 at 12:19 AM, OceanBreeze said:

Nothing real can be ever shown to be infinite.

A calculation for infinite energy, (which I have seen made) is one such example. The calculation was made by two mathematicians with advanced degrees and they insisted it was correct. I found the error and showed them how ridiculous their calculation was. Infinite energy! More energy than in an infinite amount of universes in an infinite time and they believed it because of a math mistake!

OceanBreeze, there is one way it can creep into the mathematical mix physics wise. The following is a rehash of another post of mine here.

Nina Byers goes into Emmy Noether and her contribution to the conceptual structures of the mathematics in modern physics in detail in her paper "E. Noether s Discovery of the Deep Connection Between Symmetries and Conservation Laws" in 1998.

https://arxiv.org/abs/physics/9807044v2

Quote

Though the general theory of relativity was completed in 1915, there remained unresolved problems. In particular, the principle of local energy conservation was a vexing issue. In the general theory, energy is not conserved locally as it is in classical field theories - Newtonian gravity, electromagnetism, hydrodynamics, etc.. Energy conservation in the general theory has been perplexing many people for decades. In the early days, Hilbert wrote about this problem as  the failure of the energy theorem. In a correspondence with Klein [3], he asserted that this  failure  is a characteristic feature of the general theory, and that instead of  proper energy theorems  one had  improper energy theorems in such a theory. This conjecture was clarified, quantified and proved correct by Emmy Noether.

https://en.wikipedia.org/wiki/Emmy_Noether

At a conceptual structural level improper integrals in physics can be piecewise continuous integrals, with limits from +infinity to -infinity, that converge. Refer H.J. Keisler, p367, Definition to p369, examples 7, 8, and 9. If they are continuous and don't converge then they are indefinite integrals which are entirely different. Refer H.J. Keisler, p370, example 10, diagram 6.7.10 "It is tempting to argue that the positive area to the right of the origin and the negative area to the left exactly cancel each other out so that the improper integral is zero. But this leads to a paradox... So we do not give the integral ... the value 0, instead leave it undefined."

That doesn't mean that indefinite integrals don't play a part in our physics as an indefinite integral that cycles between +infinity and -infinity at its limits, as a sub function of a higher level function, is a valid proper use of indefinite integrals as definite integrals by change of variables. Refer H.J. Keisler, p224-5, Definition and example 8, diagram 4.4.6 second equation with u and substitute infinite limits. "We do not know how to find the indefinite integrals in this example. Nevertheless the answer is 0 because on changing variables both limits of integration become the same."

Reference H.J.Keisler "Elementary Calculus an Infinitessimal Approach"
 

 

[OceanBreeze]

On 12/23/2024 at 8:56 AM, LaurieAG said:

OceanBreeze, there is one way it can creep into the mathematical mix physics wise. The following is a rehash of another post of mine here.

Nina Byers goes into Emmy Noether and her contribution to the conceptual structures of the mathematics in modern physics in detail in her paper "E. Noether s Discovery of the Deep Connection Between Symmetries and Conservation Laws" in 1998.

https://arxiv.org/abs/physics/9807044v2

https://en.wikipedia.org/wiki/Emmy_Noether

At a conceptual structural level improper integrals in physics can be piecewise continuous integrals, with limits from +infinity to -infinity, that converge. Refer H.J. Keisler, p367, Definition to p369, examples 7, 8, and 9. If they are continuous and don't converge then they are indefinite integrals which are entirely different. Refer H.J. Keisler, p370, example 10, diagram 6.7.10 "It is tempting to argue that the positive area to the right of the origin and the negative area to the left exactly cancel each other out so that the improper integral is zero. But this leads to a paradox... So we do not give the integral ... the value 0, instead leave it undefined."

That doesn't mean that indefinite integrals don't play a part in our physics as an indefinite integral that cycles between +infinity and -infinity at its limits, as a sub function of a higher level function, is a valid proper use of indefinite integrals as definite integrals by change of variables. Refer H.J. Keisler, p224-5, Definition and example 8, diagram 4.4.6 second equation with u and substitute infinite limits. "We do not know how to find the indefinite integrals in this example. Nevertheless the answer is 0 because on changing variables both limits of integration become the same."

Reference H.J.Keisler "Elementary Calculus an Infinitessimal Approach"
 

 

While this is interesting, I don’t see where it is strongly related to my statement that “nothing real can ever be shown to be infinite.” Specifically I was referring to a mathematical solution of “infinite energy” arrived at by two  mathematicians of questionable degree.

As far as using infinity as a limit in improper integrals this is perfectly acceptable as long as the integration converges to a finite result. (although there are some who debate even this)

There are several ways to make sure that convergence happens.

For example, consider two particles that are 3 m apart that gravitation-ally attract each other with a Force of  2.5 Newton, and calculate the work needed to move them to be an infinite distance apart.

We know that the force of gravity varies inversely with the square of distance, F = K / s^2 where K is a constant of proportionality.

Since F = 2.5 N when s = 3 m, K = F s^2 = (2.5 N) (9 m^2) = 22.5 N∙m^2

Work =_s1^s2 ∫ F  ds, where s1 is 3 m and s2 is ∞

Work = _s1^s2 ∫ 22.5 / s^2 ds

Integrating this, for the limit infinity, 1/s = 0

Work =  22.5 N∙m^2 [0 + 1/3m] =  7.5 N∙m

This may remind you of escape velocity where it takes a finite amount of work to move an object an infinite distance.

This result is correct because an infinite distance is being used as a limit only.

The result of 7.5 N∙m is a finite number even though one of the limits was infinity and is an example of convergence.

This is not the same exact problem I was discussing with Halc, where the wrong result  obtained by the two mathematicians was infinite energy; it is not entirely unrelated to that problem, which was much more advanced than this simple example.

Bottom Line: Using infinity as a limit in integration is perfectly acceptable, but steps must be carefully followed to avoid getting an infinite result, such as infinite energy, which is absurd!

 

 

 

 

[LaurieAG]

13 hours ago, OceanBreeze said:

Bottom Line: Using infinity as a limit in integration is perfectly acceptable, but steps must be carefully followed to avoid getting an infinite result, such as infinite energy, which is absurd!

I agree OceanBreeze, even Microsoft changed their calculators divide by zero error from 'Error: negative infinity' in Windows 98a to 'Error: cannot divide by zero' in Windows 98b. I also suspect that using change of variables as a valid proper use of indefinite integrals as definite integrals is not actually valid if there is not at least one complete cycle in the higher level function, regardless of there being infinite limits or not.