Is it permissible to use infinity, which is not defined in physics, to assume the impossibility of traveling at the speed of light?

[Z.10.46]

 

Is it permissible to use infinity(infinity mass or infinity energy), which is not defined in physics, to assume the impossibility of traveling at the speed of light?

 

Here is an example of using regularization via the $ zeta$ function in the Casimir effect:

Mathematical calculations lead to the famous divergent series $1+2+3+\ldots=+\infty$.

However, the physical results do not correspond to infinite values for the energy of the moving plates.

To resolve this issue, we applied regularization through the Riemann zeta function. Eventually, we obtain a finite value of $-1/2$ for the divergent series $1+2+3+…$, and this result provides a good explanation for the Casimir effect.

 

The regularization of the expression of the relative mass M(v) at v=c yields M(c)=-M(c-1), which can have a physical significance, as in the Casimir effect. For more details, please refer to the attached file.

Vitesse de lumiere.pdf

Edited July 22, 2023 by Z.10.46

[Genady]

50 minutes ago, Z.10.46 said:

Here is an example of using regularization via the $ zeta$ function in the Casimir effect:

Mathematical calculations lead to the famous divergent series $1+2+3+\ldots=+\infty$.

However, the physical results do not correspond to infinite values for the energy of the moving plates.

To resolve this issue, we applied regularization through the Riemann zeta function. Eventually, we obtain a finite value of $-1/2$ for the divergent series $1+2+3+…$, and this result provides a good explanation for the Casimir effect.

There are many regularization schemes leading to the same result. AFAIK, Riemann zeta function is one of them and is not necessary.

Regularization is not just a mathematical trick. It has physical basis. As explained in Zee, A. Quantum Field Theory in a Nutshell: Second Edition (p. 72). Princeton University Press,

Quote

Instead of panicking when faced with the divergent sum in (1), we remind ourselves that we are proud physicists, and that physics tells us that the sum should not go all the way to infinity. In a conducting plate, electrons rush about to counteract any applied tangential electric field. But when the incident wave oscillates at sufficiently high frequency, the electrons can’t keep up. Thus, the idealization of a perfectly conducting plate fails. We regularize (such an ugly term but that’s what field theorists use!) the sum in a mathematically convenient way by introducing a damping factor. The single parameter a is supposed to summarize the unknown high frequency physics that causes the electron to fail to keep up. In reality, a⁻¹ is related to the plasma frequency of the metal making up the plate.

 

[Z.10.46]

Yes, I agree with you regarding the Casimir effect. However, the question here is why we retained infinity to assume that no object with mass can exceed the speed of light, even though when attempting to regularize this infinity using the zeta function, we obtain a finite value for the expression of the relative mass M(v)=m0/sqrt(1-v^2/c^2), which equals -M(c) and can explain the expansion of the universe, for example..

[Genady]

2 minutes ago, Z.10.46 said:

Yes, I agree with you regarding the Casimir effect. However, the question here is why we retained infinity to assume that no object with mass can exceed the speed of light, even though when attempting to regularize this infinity using the zeta function, we obtain a finite value for the expression of the relative mass M(v)=m0/sqrt(1-v^2/c^2), which equals -M(c) and can explain the expansion of the universe, for example..

I didn't download the attached paper.

But I think exceeding speed of light contradicts causality, regardless of what happens to mass/energy.

[Z.10.46]

Here, we're not talking about surpassing the speed of light but being equal to the speed of light. This particle would no longer have a positive relative mass but a negative one, -M(c-1). It is uncertain whether this contradicts causality, but these particles could potentially explain dark energy and dark matter. Dark matter and dark energy exist without contradicting causality.

[exchemist]

1 hour ago, Z.10.46 said:

Yes, I agree with you regarding the Casimir effect. However, the question here is why we retained infinity to assume that no object with mass can exceed the speed of light, even though when attempting to regularize this infinity using the zeta function, we obtain a finite value for the expression of the relative mass M(v)=m0/sqrt(1-v^2/c^2), which equals -M(c) and can explain the expansion of the universe, for example..

But can't we reach the same conclusion about c being a speed limit just by considering limits? I'm no mathematician, but I understood the point about limits is you can see where a function is going without resorting to infinities. 

[Z.10.46]

However, in relativity, we have concluded that as an object's velocity approaches the speed of light (v=c), its mass becomes infinite, or its energy becomes infinite. Yet, we have not observed a particle with mass traveling at the speed of light. However, in the Casimir effect, we encounter the sum of positive integers resulting in infinity (1+2+3...=infinity). However, the observation yields a sum of (-1/12) for this divergent series 1+2+3... The regularization by the zeta function justifies the equivalence of infinity to (-1/12) for this series. Here, the concept of mass becoming infinite at v=c (-M(c)) could explain why we do not observe particles with mass moving at the speed of light, as they might transform into undetectable dark energy or dark matter.

[mistermack]

51 minutes ago, Z.10.46 said:

However, in relativity, we have concluded that as an object's velocity approaches the speed of light (v=c), its mass becomes infinite, or its energy becomes infinite. Yet, we have not observed a particle with mass traveling at the speed of light.

I don't think it's quite like that. As I understand it, objects with rest mass DON'T approach the speed of light. There isn't a mechanism for accelerating a massive object to that sort of speed. But theoretically, if there WAS such a mechanism, it would take infinite energy to do it. It's a thought experiment that can't happen in reality. 

[Z.10.46]

It is here that the problem of infinite energy or infinite mass arises, indicating that it is impossible not to reach the speed of light. However, we cannot draw conclusions here because we step outside the context of the theory by assuming the existence of infinity within this theory. When we try to give meaning to this infinity, similar to the Casimir effect, we find -M(c-1), which justifies why we do not observe a particle with mass moving at the speed of light, as it becomes undetectable and transforms into dark energy or dark matter with a negative relative mass -M(c-1).

Maybe particles are already transforming if you accelerate it with finite energy(And this infinite energy can also be calculated using the zeta regularization method E=-c^2*M(c-1)) but you don't even realize it because in the end the energy balance would be preserved.

Edited July 22, 2023 by Z.10.46

[swansont]

1 hour ago, Z.10.46 said:

However, in relativity, we have concluded that as an object's velocity approaches the speed of light (v=c), its mass becomes infinite, or its energy becomes infinite. Yet, we have not observed a particle with mass traveling at the speed of light.

“yet” implies we expect to. We don’t.

We have added large amounts of energy to particles - many times their rest energy - and they travel at speeds close to, but not meeting or exceeding c. As expected by the theory of relativity. 

At c, the gamma is undefined.

 

1 hour ago, Z.10.46 said:

 

However, in the Casimir effect, we encounter the sum of positive integers resulting in infinity (1+2+3...=infinity). However, the observation yields a sum of (-1/12) for this divergent series 1+2+3... The regularization by the zeta function justifies the equivalence of infinity to (-1/12) for this series.

IIRC the Casimir effect can be solved without encountering infinities; you don’t sum the series. You look at the terms left over when you exclude a finite number of modes of the vacuum. 

1 hour ago, Z.10.46 said:

Here, the concept of mass becoming infinite at v=c (-M(c)) could explain why we do not observe particles with mass moving at the speed of light, as they might transform into undetectable dark energy or dark matter.

Transforming anything into dark energy or dark matter is a huge leap that would require some testable mechanism rather than a waving of the hands.

[Genady]

If you accelerated a particle to a speed which is arbitrarily close but not equal the speed of light, then there is a reference frame in which the particle is at rest. To accelerate in this frame, you have to start all over again.

Edited July 22, 2023 by Genady

[studiot]

6 hours ago, Z.10.46 said:

Is it permissible to use infinity(infinity mass or infinity energy), which is not defined in physics, to assume the impossibility of traveling at the speed of light?

Quote

 

Is it permissible to use infinity(infinity mass or infinity energy), which is not defined in physics, to assume the impossibility of traveling at the speed of light?

Surely your title is contradicted by your first line.

If infinity is not defined in Physics how could it be permissible to use it ?

 

Of course it is well defined.

 

And why to you need such an exotic example as the Casimir Effect ?

What is wrong with schoolboy Physics.

Density = mass/volume

So what is the density at a point, which has zero volume ?

[Z.10.46]

19 hours ago, swansont said:

“yet” implies we expect to. We don’t.

We have added large amounts of energy to particles - many times their rest energy - and they travel at speeds close to, but not meeting or exceeding c. As expected by the theory of relativity. 

At c, the gamma is undefined.

 

IIRC the Casimir effect can be solved without encountering infinities; you don’t sum the series. You look at the terms left over when you exclude a finite number of modes of the vacuum. 

Transforming anything into dark energy or dark matter is a huge leap that would require some testable mechanism rather than a waving of the hands.

@Genady

Yes, but it is not certain that if we accelerate a packet of particles, some particles will transform with M(c)=-M(c-1) without even realizing it.

That doesn't prevent the regularization by the zeta function from solving the problem. If the Casimir effect is not measured, one would assume it to be an infinite energy.

Here, it's not just speculation but a formula based on mathematics, and we can test these formulas well. I would be interested, for example, in creating negative energy E=-c^2*M(c-1) and with m0 for object , to see if it would disappear and transform into dark energy or dark matter.

 

@studiot

Could you give an example of a physical theory that keeps infinity in its equations without assigning it a finite value to draw conclusions, similar to how the theory of relativity deals with the impossibility of reaching the speed of light without transforming the infinity generated at v=c by its finite value?

Edited July 23, 2023 by Z.10.46

[Genady]

6 minutes ago, Z.10.46 said:

it is not certain that if we accelerate a packet of particles, some particles will transform with M(c)=-M(c-1) without even realizing it

Do they transform before or after reaching c?

[Z.10.46]

 

 Experiments need to be conducted to test and find out exactly what happens. Perhaps the phenomenon is very brief on the time scale, like the appearance of quantum vacuum fluctuations with negative energy.

Here, for example, in physics, when energy is negative. Do you know that in quantum solutions, when they had negative energy, it was a sign that antiparticles existed?

https://fr.lambdageeks.com/when-energy-can-be-negative/

And here, there is a way to make the energy of an object negative.

https://fr.lambdageeks.com/can-energy-be-negative/

Edited July 23, 2023 by Z.10.46

[exchemist]

4 minutes ago, Z.10.46 said:

 

 Experiments need to be conducted to test and find out exactly what happens. Perhaps the phenomenon is very brief on the time scale, like the appearance of quantum vacuum fluctuations.

Here, for example, in physics, when energy is negative. Do you know that in quantum solutions, when they had negative energy, it was a sign that antiparticles existed?

https://fr.lambdageeks.com/when-energy-can-be-negative/

And here, there is a way to make the energy of an object negative.

https://fr.lambdageeks.com/can-energy-be-negative/

But surely in the gravitational case it is only -ve relative to infinite separation of the bodies concerned, which we arbitrarily set to "zero" by convention. 

[Z.10.46]

 We provide work greater than the energy of the object to give it negative energy. There's also the case of the electron, which already has negative energy. There's nothing preventing us from conceiving this experiment with an object of mass m0 so that it has an energy E=-c^2M(c-1) and see if it transforms into dark matter or invisible dark energy. We can also look for natural phenomena capable of doing this, such as black holes or quantum vacuum, which could also be potential sources of this transformation.

 

Edited July 23, 2023 by Z.10.46

[swansont]

37 minutes ago, Z.10.46 said:

it is not certain that if we accelerate a packet of particles, some particles will transform with M(c)=-M(c-1) without even realizing it.

Provide some evidence that this happens, or some reason - backed by some physics -  to think it would happen

10 minutes ago, Z.10.46 said:

We provide work greater than the energy of the object to give it negative energy.

That does not result in negative energy.

28 minutes ago, Z.10.46 said:

And here, there is a way to make the energy of an object negative.

Trivially so; we define a zero-energy condition (typically potential energy), and remove some energy. But that zero energy is an arbitrary choice - we choose zero for convenience - and are usually interested in energy differences between states, so the negative sign doesn’t matter. 

Some energies are positive definite, such as mass energy and kinetic energy.

[Z.10.46]

I can suggest an experiment instead: if we accelerate two packets to collide and annihilate each other, we can verify that all the particles are present before the impact. If some particles are missing, but the measured energy remains the same, it means that some particles have already transformed without being detected.

[swansont]

2 minutes ago, Z.10.46 said:

I can suggest an experiment instead: if we accelerate two packets to collide and annihilate each other, we can verify that all the particles are present before the impact. If some particles are missing, but the measured energy remains the same, it means that some particles have already transformed without being detected.

You mean like the proton-antiproton collider that operated at CERN for a decade, with no hint of any missing particles?

[Z.10.46]

 Yes, but we don't have the test before the impact just to see if all the particles are there. We conduct the test only after the impact to observe any newly emerged particles, like the Higgs boson, and to ensure that the total energy is conserved.

 

[Genady]

17 minutes ago, Z.10.46 said:

we can verify that all the particles are present before the impact. If some particles are missing, but the measured energy remains the same, it means that some particles have already transformed without being detected.

If some particles are transformed before reaching the speed of light, then in some reference frame they are transformed while being at rest. Physics is the same in all reference frames. We don't even need to accelerate particles. Just to observe if particles get spontaneously transformed sometimes. Which of course have never been observed. The result of the experiment is negative. Done.  

[Z.10.46]

Could you observe dark matter or dark energy?

The main purpose of the experiment is to observe any missing particles before the impact and to confirm that the total energy is conserved, thereby confirming that these particles have indeed reached the speed of light to transform with M(c)=-M(c-1).

Edited July 23, 2023 by Z.10.46

[Genady]

1 minute ago, Z.10.46 said:

Could you observe dark matter or dark energy? The main purpose of the experiment is to observe any missing particles before the impact and to confirm that the total energy is conserved, thereby confirming that these particles have indeed reached the speed of light to transform with M(c)=-M(c-1).

If they were missing BEFORE the impact, then they transformed BEFORE reaching speed of light. Then, they transformed while being at rest. We don't need to accelerate them to refute this.

[Z.10.46]

bove all, we need negative energy E=-c^2M(c-1). There are many different ways to obtain such energy, even with acceleration, and it can manifest a few moments before the impact.

 

Exemple : If we consider an atom, we can say that the formation and breakup of an atom can define negative energy. The amount of energy absorbed during the breakup of an atom is the same as the amount of energy released during the formation of that atom. The energy supplied to break an atom is considered as negative energy.

Edited July 23, 2023 by Z.10.46