Conservation of Particle Number

[thehangedman]

Because E=mc^2, when we increase the energy of a system we can introduce new particles. What about accomplishing this simply via a change of coordinates via Special (or even General) Relativity?

If I have a system of one electron sitting still (E = E0), then I change to a coordinate system moving at near the speed of light, is it possible that particles could pop out or will there only ever be one electron just with increased energy? I'm not talking about particle collisions, but strictly from the change in energy coming from the new coordinates...

 

[Dale]

E=mc^2 is for a particle at rest. If you transform then the particle is no longer at rest.

 

[thehangedman]

Well, actually that's not true. m changes when it's in motion, as does E. Regardless of that, you're not answering my question. Since the energy of the particle in the moving reference frame is significantly higher, do particles appear? Is the particle number conserved under coordinate change or not? I would expect, IF particles are created, that they come in particle / antiparticle pairs, but I'm not sure if the total particle number (operator N) is conserved or not.

 

[Dale]

Actually, it is true. Energy and momentum form a four-vector. The invariant norm of the four-vector is mass. The timelike component of the four-vector (energy) is only equal to mc^2 in the rest frame.

There is an old concept of relativistic mass, which is no longer used. That is probably what you are thinking of.

Also, you need to distinguish between frame invariance and conservation. Particle numbers are not conserved, but they are frame invariant.

 

[pervect]

Because E=mc^2, when we increase the energy of a system we can introduce new particles. What about accomplishing this simply via a change of coordinates via Special (or even General) Relativity?

Adding energy to a system via a physical process is a different thing totally from adding energy to a system via changing coordinates.

In spite of this, I believe it is true that photon number in particular is not conserved in General Relataivity. (It's conserved just fine in SR).

The effect responsible is called Unruh radiation.

This is on my list of thigns I want to understand better someday.

 

[thehangedman]

There is an old concept of relativistic mass, which is no longer used. That is probably what you are thinking of.

Yes, and like I said, it's not relevant to my question. Thank you however for the commentary. I'll make a note to not use relativistic mass anymore.

Also, you need to distinguish between frame invariance and conservation. Particle numbers are not conserved, but they are frame invariant.

Semantics. Ok, sounds like the answer to my question is that particle number in invariant under coordinate change. So, despite the extra energy, no new particles will appear simply because we are in a moving reference frame compared to the other.

That would mean though that a frame falling in a gravitational field would ALSO show in-variance in particle number.

 

[Dale]

That would mean though that a frame falling in a gravitational field would ALSO show in-variance in particle number.

Locally, yes, but I am not sure that holds over regions where curvature becomes significant.

 

[thehangedman]

Locally, yes, but I am not sure that holds over regions where curvature becomes significant.

Sure that would make sense since curvature means the space is filled with non-zero energy ( Energy Momentum tensor non-zero ). That, I would guess, acts like an interaction and thus particles can be created as the interaction occurs.

 

[pervect]

Yes, and like I said, it's not relevant to my question. Thank you however for the commentary. I'll make a note to not use relativistic mass anymore.

A good idea. The fact that photon number is conserved when you have a linear Lorentz boost pretty much demonstrates to total irrelevance of the issue in any event.

Semantics. Ok, sounds like the answer to my question is that particle number in invariant under coordinate change. So, despite the extra energy, no new particles will appear simply because we are in a moving reference frame compared to the other.

While the particle number is invariant under some coordinate changes (i.e. Lorentz boosts), it is not in general invaraint under ALL coordinate changes.

Specifically, an eternally accelerating observer WILL see "the vacuum" turn into a sea of thermal particle radiation, which explicitly demonstrates the non-conservation of particle number.

See for instance Unruh's paper: http://prd.aps.org/abstract/PRD/v14/i4/p870_1

The behavior of particle detectors under acceleration is investigated where it is shown that an accelerated detector even in flat spacetime will detect particles in the vacuum. The similarity of this case with the behavior of a detector near the black hole is brought out, and it is shown that a geodesic detector near the horizon will not see the Hawking flux of particles.

Or the wiki article for an overview:
http://en.wikipedia.org/w/index.php?title=Unruh_effect&oldid=512133286

You'll find similar remarks in some GR textbooks, for instance Wald, though I'm a bit too lazy to track them down at the moment.

That would mean though that a frame falling in a gravitational field would ALSO show in-variance in particle number.

It's not particularly clear if that is true or not, sorry. The results are subtle enough that you have to be very careful not to jump to conclusions about cases that seem similar at first glance.

http://www.springerlink.com/content/d582g4784283x603/fulltext.pdf "On the Physical Meaning of the Unruh Effect" and http://arxiv.org/abs/hep-ph/0610391 "On the relation between Unruh and Sokolov--Ternov effects" discuss the rather interesting case of an electron orbiting in a magnetic field as an Unruh-type detector.

The second reference of the two above says the following:

All these arguments give a clearer picture of the conditions under which an observer will
or will not detect particles/radiation and the character of the radiation. However, there are
confusing aspects of this topic which require a more detailed investigation of the quantization
of fields in curvilinear coordinates. For example, one would like to know the details of the
complete basis of harmonics in spaces with horizons and how these basis harmonics transform
under various coordinate changes. The main question one would like to be able to answer in
all these different cases is: “What is the criteria for the existence of radiation due to some
particular gravitational background and/or detector motion?”

The presence of a horizon is not a necessary criteria; there is no horizon for the orbiting
observer. In fact, there is a crucial difference between the Rindler and the orbiting observers.
The Rindler observer has a horizon, since depending on the acceleration there is some point
behind the observer beyond which even a massless particle can not reach the observer. On
the other hand the orbiting observer can always be reached by particles from outside his light
surface. Note, however, that an observer accelerating linearly for a finite time does not have a
horizon.

Furthermore, it is not correct to say that any non–inertial moving observer sees radiation,
while any inertial moving observer does not. Indeed, a free falling detector in the black hole
background does see radiation (e.g. an observer orbiting around a black hole). It is only the
free falling observer in a homogeneous gravitational field (i.e. with zero Riemann tensor) which
does not encounter radiation. At the same time an observer fixed above a gravitating body
without a horizon (such as the Earth) does not see any radiation. Of course in the latter
case the radiation would be so small that it could not be excluded experimentally. However,
theoretically there is no means for such an object as the Earth to create particles and lose mass.
Thus, at this stage we propose the following criteria for the existence of radiation from a
gravitational background and/or detector motion: a gravitational background and/or detector
motion will have radiation associated with it if there exist negative energy states for the Hamiltonian
of the QFT in the non–inertial reference frame [17]. Rephrasing, the criteria is based
on the existence of a non–trivial saddle point contribution in the analogs of eq.(38) and eq.(44)
for general motions and/or backgrounds.

I'm not sure how to reconcile the statements made here with the one's in the abstract of Unruh's paper, specifically
"it is shown that a geodesic detector near the horizon will not see the Hawking flux of particles. ".

In any event , the question here is mostly QFT rather than classical GR. You might get better answers in another forum.