Special Relativity and momentum

[W.D.Clinger]

Quoting a freshman-level textbook on special relativity, with italicized English words as in the original:

[size=+2]7 More about relativistic dynamics[/size]

In this chapter we shall be discussing two main topics. The first of these is a more extended discussion of momentum and energy, with particular emphasis on the transformation of these quantities between two inertial frames. The second topic is the concept of force in relativistic dynamics—the way in which it is defined, its transformations, and the limitations on its usefulness. We begin with an important invariant that can be constructed from the measured values of momentum and energy in a given frame.

Now a simple way of restating [ E² = (cp)² + E₀² ] is that, if a particle has rest energy E₀ (i.e., total energy E₀ as measured in the frame in which its momentum is zero), then its energy and momentum as measured in any other frame can be combined to form an invariant quantity as follows:

E² - (cp)² = E₀²​

....Since this holds for E and p as measured in any frame, the measures of energy and momentum for a particle in any two frames are related according to the equation

E² - (cp)² = (E')² - (cp')² = E₀²​

....It turns out that [the equation above] applies not merely to a single particle, but to any arbitrary collection of particles, in the following way. If, as measured in any given frame of reference, the sum of the energies of the particles is E and the vector sum of all their momenta is of magnitude p, then the value of E² - (cp)² has the same value as the corresponding combination (E')² - (cp')² as measured in any other frame. This invariant value is equal to the square of the total energy E₀ of all the particles as measured in a frame in which the vector sum of the momenta is zero.

Note especially that, in this extended form..., the energy E₀ is not, in general, merely a sum of rest energies. The collection of particles considered may have all kinds of motions relative to one another; there need not exist any frame in which they are all at rest.

....Consider, for example, an argon atom containing numerous electrons in states of rapid motion, and having at its center a nucleus, itself a composite of neutrons and protons with large kinetic energies. We have no hesitation in describing this atom, from the standpoint of the kinetic theory of gases, as a single particle endowed with a certain velocity. And the theorem of the inertia of energy makes it all the easier to think of this complicated structure as being describable in terms of a single mass possessed of a certain momentum, despite our awareness of its internal structure.

In Newtonian mechanics we are accustomed to thinking of measures of space as being definable without reference to time, and vice versa. Likewise, we are accustomed to thinking of momentum and energy as representing essentially different (although to some extent related) properties of a body. We have now seen how these distinctions, both kinematic and dynamic, are blurred in special relativity. The specification of time in one system involves both position and time in another system; the specification of energy involves both energy and momentum in another system.

In the above results one can discern...that in general force and acceleration are not parallel vectors....Only in the instantaneous rest frame of a body...can one guarantee that F, as defined by the time derivative of momentum, is in the same direction as the acceleration.

 

[The Man]

Let us assume there is a stationary Hydrogen atom in ground state (1,0,0) in the intergalactic space.
After a long time the Hydrogen atom is going to be attracted by a galaxy gravity and the atom starts to fall towards the galaxy.
The Hydrogen atom achieves let's say 0.5c speed towards the galaxy.
What is the shape of the Hydrogen atom at this moment in the rest frame of the atom and the galaxy center inertial frame?

To my limited understanding, in a purely special relativity vein the, spherical orbital would be length contracted along the axis of travel.

Hint: Does binding energy change in the galaxy center inertial frame? Does the binding energy change in the Hydrogen atom rest frame?

No, while kinetic energy would increase potential energy would decrease over the area of the length contracted depressions on the orbital. The difference would come with photo ionizing the atom along the axis of motion in the same and opposing directions of that motion. The opposing direction would take less energy while the photon and atom moving in the same direction would take more. Though that difference would average out and do likewise proportionally at obscure angles.

The thing is it's probabilistic that's what those orbital diagrams represent the probability of interacting with an electron over that surface area at some time. So basically the spherical area of uniform probability gets depressed along the axis of travel.

Again just my unprofessional surmise.

 

[Ziggurat]

Still, the biggest problem is the 4-force change is frame dependent.

Why is that a problem? It isn't.

Very good example is the falling Hydrogen atom, my
Hint: Does binding energy change in the galaxy center inertial frame? Does the binding energy change in the Hydrogen atom rest frame?

Of course the binding energy is reference frame dependent. Energy is always frame dependent.

 

[Ziggurat]

No, while kinetic energy would increase potential energy would decrease over the area of the length contracted depressions on the orbital. The difference would come with photo ionizing the atom along the axis of motion in the same and opposing directions of that motion. The opposing direction would take less energy while the photon and atom moving in the same direction would take more. Though that difference would average out and do likewise proportionally at obscure angles.

Nope. Angles don't matter. The change in energy is independent of the orientation of the orbital relative to the direction of motion. It might seem like it should be, but it doesn't work out that way when you crunch the number. I did that at one point years ago on this forum, maybe I'll try to dig that post up later.

The thing is it's probabilistic that's what those orbital diagrams represent the probability of interacting with an electron over that surface area at some time. So basically the spherical area of uniform probability gets depressed along the axis of travel.

Nope. No averaging is necessary. Averaging wouldn't work anyways, because you can make crystals with macroscopically aligned orbitals.

 

[The Man]

Nope. Angles don't matter. The change in energy is independent of the orientation of the orbital relative to the direction of motion. It might seem like it should be, but it doesn't work out that way when you crunch the number. I did that at one point years ago on this forum, maybe I'll try to dig that post up later.



Nope. No averaging is necessary. Averaging wouldn't work anyways, because you can make crystals with macroscopically aligned orbitals.

OK thanks, knew my back of the brain analysis might be wrong and why I stuck the caveat at the end.

 

[Ziggurat]

OK thanks, knew my back of the brain analysis might be wrong and why I stuck the caveat at the end.

I can't find the post, the search function for old stuff doesn't seem to really be working. But I remember some of the basics, and while the basic principles are not all that complicated, there are some subtleties that make it non-obvious why the angle shouldn't matter. I'll try to summarize even though I won't redo the calculations.

So the really naive approach is to note that the orbitals get squished along the direction of motion, and assume that if the orbital is oriented that way (for example, an S orbital), that will push the electron closer to the nucleus and this be a lower energy state compared to an orbital aligned sideways. And the first problem this naive approach runs into is that the electric field of a moving charge (in this case we're talking about the field from the nucleus) is not spherical. The electric field is also compressed along the direction of motion, which compresses the potential as well. You need to get closer in along the direction of motion in order to reach the same potential as you do from the sides. Problem solved, right?

Wrong. If we then try to recalculate the potential based on this squished electric field, we actually find that the electric potential for orbitals aligned with the motion and sideways won't be equal. The squishing isn't the same for the orbitals and the electric potential, so the effects of squishing the electric field doesn't quite cancel the effect of squishing the orbital. We have reduced the gap in our calculation of the energy for these two orientations, but we haven't totally closed it. "Aha!" you might say, "so there IS a direction dependence!"

Wrong again. We've still forgotten something, and this one is even easier to miss. Our electron is co-moving with our nucleus. We calculate the potential by figuring out how much energy it would take (ie, the path integral of force*displacement) to bring our charge from infinity to the location of interest near our nucleus. But while we only needed to worry about the electric field for our stationary hydrogen atom, now we need to worry about the magnetic field as well. Our hydrogen nucleus has a magnetic field due to its motion, and our co-moving electron will feel a force from this magnetic field as we drag it from infinity to sit next to it. So we've basically got a potential energy contribution from the magnetic field due to motion, and it's also not isotropic (in fact, it's zero in the forward and backward direction). And this magnetic field contribution closes the remaining gap. So when you account for both the addition of a magnetic field and the compression of the electric field, the final result is a potential which compresses the same way as the orbital does, regardless of the orientation angle of the orbital. Which is how it should be, and in fact how it must be. And all is right with the world again.

 

[SDG]

Why is that a problem? It isn't.



Of course the binding energy is reference frame dependent. Energy is always frame dependent.

This is the problem:

How Fast a Hydrogen Atom can Move Before Its Proton and Electron Fly Apart?


The relativity predicts hydrogen atom should exist at high speeds based on no change in the rest frame while falling towards a gravity source but at the same time the hydrogen atom cannot exist based on the galaxy center inertial reference frame binding energy values.

Here is the experiment to detect falling in the space.
If we are in a spaceship with no signal from the outside we can observe separation of electron from proton in a hydrogen atom and we know we reached threshold velocity.

 

[The Man]

I can't find the post, the search function for old stuff doesn't seem to really be working. But I remember some of the basics, and while the basic principles are not all that complicated, there are some subtleties that make it non-obvious why the angle shouldn't matter. I'll try to summarize even though I won't redo the calculations.

So the really naive approach is to note that the orbitals get squished along the direction of motion, and assume that if the orbital is oriented that way (for example, an S orbital), that will push the electron closer to the nucleus and this be a lower energy state compared to an orbital aligned sideways. And the first problem this naive approach runs into is that the electric field of a moving charge (in this case we're talking about the field from the nucleus) is not spherical. The electric field is also compressed along the direction of motion, which compresses the potential as well. You need to get closer in along the direction of motion in order to reach the same potential as you do from the sides. Problem solved, right?

Wrong. If we then try to recalculate the potential based on this squished electric field, we actually find that the electric potential for orbitals aligned with the motion and sideways won't be equal. The squishing isn't the same for the orbitals and the electric potential, so the effects of squishing the electric field doesn't quite cancel the effect of squishing the orbital. We have reduced the gap in our calculation of the energy for these two orientations, but we haven't totally closed it. "Aha!" you might say, "so there IS a direction dependence!"

Wrong again. We've still forgotten something, and this one is even easier to miss. Our electron is co-moving with our nucleus. We calculate the potential by figuring out how much energy it would take (ie, the path integral of force*displacement) to bring our charge from infinity to the location of interest near our nucleus. But while we only needed to worry about the electric field for our stationary hydrogen atom, now we need to worry about the magnetic field as well. Our hydrogen nucleus has a magnetic field due to its motion, and our co-moving electron will feel a force from this magnetic field as we drag it from infinity to sit next to it. So we've basically got a potential energy contribution from the magnetic field due to motion, and it's also not isotropic (in fact, it's zero in the forward and backward direction). And this magnetic field contribution closes the remaining gap. So when you account for both the addition of a magnetic field and the compression of the electric field, the final result is a potential which compresses the same way as the orbital does, regardless of the orientation angle of the orbital. Which is how it should be, and in fact how it must be. And all is right with the world again.

Well, thanks for the details. Perhaps best if I explained my thinking, it was more along the lines of how the probability distribution gets compressed for specifically the hydrogen atom in the ground state. That distribution is uniform so in the direction of travel that uniform distribution is compressed. You'll tend to interact with the electron closer to the nucleolus along the axis of travel. Moving further out to the orthogonal directions. Basically the probability distribution would be dimpled along the direction of travel, normal (like at rest) on the orthogonals and varying in between. Those were the angles I was referring to. Putting that in terms of the field potentials and binding energy is why I concluded that the binding energy must be the same. Though from your explanation the magnetic field contribution is zero in the forward and reverse directions (at the probability distribution dimples) so that's all electrical field. While both electrical field and magnetic field contributions off the axis of travel. Making, If I'm getting this right the binding energy uniformly higher over the entire distorted (dimpled) probability distribution.

 

[The Man]

This is the problem:

[qimg]https://i.imgur.com/zNM4Ean.png[/qimg]

How Fast a Hydrogen Atom can Move Before Its Proton and Electron Fly Apart?


The relativity predicts hydrogen atom should exist at high speeds based on no change in the rest frame while falling towards a gravity source but at the same time the hydrogen atom cannot exist based on the galaxy center inertial reference frame binding energy values.

Here is the experiment to detect falling in the space.
If we are in a spaceship with no signal from the outside we can observe separation of electron from proton in a hydrogen atom and we know we reached threshold velocity.

A gravitational field is non-inertial though it may appear locally inertial. Locally the curvature of space time may be insufficient to affect calculations to a meaningful degree of error.

ETA: It is the tidal forces (local variance of the curvature of space time) that will rip things apart when falling in a gravitational field.

See Spaghettification^WP

 

[SDG]

A gravitational field is non-inertial though it may appear locally inertial. Locally the curvature of space time may be insufficient to affect calculations to a meaningful degree of error.

ETA: It is the tidal forces (local variance of the curvature of space time) that will rip things apart when falling in a gravitational field.

See Spaghettification^WP

What tidal forces for hydrogen atom size in an intergalactic space?

 

[SDG]

To my limited understanding, in a purely special relativity vein the, spherical orbital would be length contracted along the axis of travel.



No, while kinetic energy would increase potential energy would decrease over the area of the length contracted depressions on the orbital. The difference would come with photo ionizing the atom along the axis of motion in the same and opposing directions of that motion. The opposing direction would take less energy while the photon and atom moving in the same direction would take more. Though that difference would average out and do likewise proportionally at obscure angles.

The thing is it's probabilistic that's what those orbital diagrams represent the probability of interacting with an electron over that surface area at some time. So basically the spherical area of uniform probability gets depressed along the axis of travel.

Again just my unprofessional surmise.

My post #147 address your comments.

 

[W.D.Clinger]

This is the problem:

[qimg]https://i.imgur.com/zNM4Ean.png[/qimg]

How Fast a Hydrogen Atom can Move Before Its Proton and Electron Fly Apart?

Lawrence factor?

That paper refers to "Lawrence" twice. It does not mention "Lorentz" even once.

Optics and Photonics Journal for the win loss:

In short, all manuscripts submitted for publication in our journals are strictly and thoroughly peer-reviewed. The review process is single blind. If the manuscript is accepted for full review, it will be reviewed by a minimum of two external reviewers.

Reviewed by "a minimum of two external reviewers", yet no reviewer noticed such obvious errors, and no editor insisted upon changes to this sophomoric first paragraph of the paper:

Einstein’s theories of special and general relativities change our opinion about the universe [1,2]. The new concepts such as time inflation and curved spacetime frequently appeared in scientific publications. Some idea developed from Einstein’s theory even causes the imagination of the fiction novel writer and they write a lot of books regarding the time travel [3,4]. Meantime, some scientists mainly focus on how to make the time travel theoretically possible. The reason why human beings are so interested in time travel is in that based on the Einstein’s theory, the people can live much longer by time travel. This dream for long life ignites the human beings’ speculation on the universe we lived in.

Does it get better? Consider this sentence from the second paragraph of the paper:

H. G. Well even designed the time machine which can be used for time travel, just like space shuttle [7].

Oh, and if you're wondering about that reference [7]:

H. G. Well, "The Time Machine," William Heinemann, London, 1895.

That's right, folks: H. G. Well [sic] "designed the time machine which can be used for time travel, just like space shuttle".

ETA: It seems the author got equation (8) from equation (7) by assuming (x+y)²=x²+y² and similar mistakes. Then there's a blank page between pages 37 and 38. (I'm still looking at the equations, but I've seen enough to know this paper is garbage.)

 

[Ziggurat]

How Fast a Hydrogen Atom can Move Before Its Proton and Electron Fly Apart?

https://en.wikipedia.org/wiki/Scientific_Research_Publishing
"Scientific Research Publishing (SCIRP) is a predatory[1][2] academic publisher of open-access electronic journals, conference proceedings, and scientific anthologies that are considered to be of questionable quality."

So the source is garbage. The paper itself is garbage too, and fails spectacularly very early on. On the first page, they make this claim:

"If we accelerate the hydrogen atom, total energy of system increases. When total energy of system reaches or more than g'(m_n⁰+m_e⁰)c², then the proton and electron in hydrogen atom will fly apart,"​

I have used g for gamma. Basically, the claim is that if the energy of the bound and moving atom exceeds the energy of a free electron plus proton, then the atom will come apart because that's a lower energy state.

But this is trivially wrong, because that process would not conserve momentum. Any unbound state of an electron and proton that you could form that maintained the momentum of the bound atom would have higher energy. That momentum requirement basically puts a floor on the kinetic energy of the unbound state which they are ignoring. They're basically saying you can transition from a moving bound state to an unmoving unbound state, and that this transition will happen spontaneously.

But it cannot happen spontaneously, because momentum IS conserved. They have not only ignored but actually violated momentum conservation.

Now, their calculation does still have some meaning. I haven't read the entire paper because that mistake is fatal to their claim, but if you had a way to dump momentum, then you could indeed transition from a bound state to an unbound state by trading kinetic energy for potential energy. And there is, in fact, a way to do that: with a collision. So what they have found is not a speed limit to hydrogen. What they have found, without understanding it, is the minimum velocity needed to ionize hydrogen through collision. Which is of mild interest, but it's not what they claim.

tl;dr: you found a garbage paper in a garbage source that makes a trivially obvious mistake that any Freshman physics student should learn to avoid.

 

[The Man]

My post #147 address your comments.

No, it doesn't.

 

[SDG]

Quoting a freshman-level textbook on special relativity, with italicized English words as in the original:

...

In the above results one can discern...that in general force and acceleration are not parallel vectors....Only in the instantaneous rest frame of a body...can one guarantee that F, as defined by the time derivative of momentum, is in the same direction as the acceleration.

I encourage everybody to study this.

Yes, agreed, "This is a messy problem".
It does not end well for the relativity and momentum.

 

[The Man]

https://en.wikipedia.org/wiki/Scientific_Research_Publishing
"Scientific Research Publishing (SCIRP) is a predatory[1][2] academic publisher of open-access electronic journals, conference proceedings, and scientific anthologies that are considered to be of questionable quality."

So the source is garbage. The paper itself is garbage too, and fails spectacularly very early on. On the first page, they make this claim:

"If we accelerate the hydrogen atom, total energy of system increases. When total energy of system reaches or more than [g'(m_n⁰+m_e⁰)c², then the proton and electron in hydrogen atom will fly apart,"​

I have used g for gamma. Basically, the claim is that if the energy of the bound and moving atom exceeds the energy of a free electron plus proton, then the atom will come apart because that's a lower energy state.

But this is trivially wrong, because that process would not conserve momentum. Any unbound state of an electron and proton that you could form that maintained the momentum of the bound atom would have higher energy. That momentum requirement basically puts a floor on the kinetic energy of the unbound state which they are ignoring. They're basically saying you can transition from a moving bound state to an unmoving unbound state, and that this transition will happen spontaneously.

But it cannot happen spontaneously, because momentum IS conserved. They have not only ignored but actually violated momentum conservation.

Now, their calculation does still have some meaning. I haven't read the entire paper because that mistake is fatal to their claim, but if you had a way to dump momentum, then you could indeed transition from a bound state to an unbound state by trading kinetic energy for potential energy. And there is, in fact, a way to do that: with a collision. So what they have found is not a speed limit to hydrogen. What they have found, without understanding it, is the minimum velocity needed to ionize hydrogen through collision. Which is of mild interest, but it's not what they claim.

tl;dr: you found a garbage paper in a garbage source that makes a trivially obvious mistake that any Freshman physics student should learn to avoid.

Also they smack it with photons changing the state of the electron and proton. Noting in the summary "Our work here reveals that the frequency shift depends on both the speed of initial and final state of hydrogen atom." So non-inertial.

 

[SDG]

...
But this is trivially wrong, because that process would not conserve momentum. Any unbound state of an electron and proton that you could form that maintained the momentum of the bound atom would have higher energy. That momentum requirement basically puts a floor on the kinetic energy of the unbound state which they are ignoring. They're basically saying you can transition from a moving bound state to an unmoving unbound state, and that this transition will happen spontaneously.

But it cannot happen spontaneously, because momentum IS conserved. They have not only ignored but actually violated momentum conservation.
...

Bound system, hydrogen atom, has less energy than free electron and free proton.
If hydrogen atom is in a ground state energy has to be added to excite the atom.
When the hydrogen atom goes to lower state a photon/energy is released, leaving the hydrogen atom system with less energy.
The equation (2) is correct in this meaning.

 

[The Man]

What tidal forces for hydrogen atom size in an intergalactic space?

Small, like the gravitational forces of the surrounding galaxies. Heck, those forces could even cancel out. The fact remains that a gravitational field is no-inertial and if your going to make it such that the gravitational influences and that non-inertial aspect don't matter then just stick with purely inertial considerations. If your whole point is that general relativity ain't the same a special relativity then congratulations, you've learned something.

 

[Ziggurat]

The equation (2) is correct.

You aren't paying attention. The mistake I pointed out was right AFTER equation 2. It's equation 3 which fails.

 

[SDG]

...
But this is trivially wrong, because that process would not conserve momentum. Any unbound state of an electron and proton that you could form that maintained the momentum of the bound atom would have higher energy. That momentum requirement basically puts a floor on the kinetic energy of the unbound state which they are ignoring. They're basically saying you can transition from a moving bound state to an unmoving unbound state, and that this transition will happen spontaneously.

But it cannot happen spontaneously, because momentum IS conserved. They have not only ignored but actually violated momentum conservation.
...

Right, now you talking like Einstein. There is no way God plays dice.
What is the hidden variable? The speed through space.