Conservation Laws: Matter and Antimatter Annihilation in a Sealed Box

[Phrak]

Place equal amounts of matter and anitmatter in a box on a scale. It's a very good box; it's very reflective, and light doesn't get in or out. Allow all the stuff to annihilate to photons. The box floats away. Has a conservation law been violated?

 

[lzkelley]

No, for lots of reasons.
First and foremost, the box will not float away. Oddly enough, although light is "massless," energy is not. To my understanding, if the energy is perfectly contained inside the box - then the box overall will still be massive and weighted down.
Now, even if I'm wrong on that account -> there is no such thing as conservation of mass (or gravity etc). So if the box does float away, then energy would still be conserved.

Can anyone comment on the massivity of the box after annihilations?

 

[Ich]

Can anyone comment on the massivity of the box after annihilations?

It has exactly the same mass. Other than one might think, the mass of a system is not the sum of the masses of the components. It is rather the sum of the energies of the components, when measured in a frame where the system (its center of gravity) is at rest.

 

[pmb_phy]

No, for lots of reasons.
First and foremost, the box will not float away. Oddly enough, although light is "massless," energy is not. To my understanding, if the energy is perfectly contained inside the box - then the box overall will still be massive and weighted down.
Now, even if I'm wrong on that account -> there is no such thing as conservation of mass (or gravity etc). So if the box does float away, then energy would still be conserved.

Can anyone comment on the massivity of the box after annihilations?

Actually there is such a thing as conservation of mass. And the reason the box won't float away is due to the impact of the photons on the wall. Such impacts cause a force to be exerted on the walls. The lower wall is in contact with the supporting struction (e.g. a scale) while the upper wall also has photons imacting on it but for which the energy of the photons, as measured locally, have a different energy due to gravitational redshift. The over all result is that the weight of the box does not change. Neither will the mass of the box.

As far as the massivity of the box after annihilations, the mass is constant since the total mass will be the sum of the (relativistic/inertial) mass of each particle of the system. Since the energy of the system does not change then neither will the systems mass change. This holds even if you intrpret "mass" to mean "rest mass" since "rest mass" of a system equals "the systems energy as measured in the zero momentum frame"/c². Since energy and momentum is conserved then so too is the rest mass.

Pete

 

[atom888]

lol First, after annilhilation, some will form back into matter and antimatter. After a while, it will establish equilibrium(like a chemical) with lights and matters. Lights and mass are different. Lights and mass are energy of different form. In my opinion, it will have less weight. It's contradicting to say that that box of heat weight the same as a box of mass (assuming heat and mass amount are equal in term of energy). It's like one have temperature and weight and one have only weight. It's unfair. :)
The question is it violate the conservation? I don't think it is. It's just like a helium balloon floating in air (assuming that initally box+matter density = surrounding gas density). Now one may ask but floating away means that's extra energy, where is it come from? I'm not sure. Use the helium balloon example on this one. I know that once it go out to vaccum, it's at constant speed and constant speed doesn't require any energy.

 

[lzkelley]

pmb_phy; you're a little off. Conservation of mass does not hold (look it up). It hasn't since SR and QM. Especially the rest mass, is not going to be the same. Very clearly.

Also, the collisions of the photons with the walls will be Boltzmann like, and randomly distributed -> therefore not exerting any net force in any direction (also necessitated by conservation of momentum - which is an actual conservation). Gravitational redshift is also going to have absolutely no appreciable effect (try looking at a flashlight pointed away from Earth vs towards the earth).

 

[peter0302]

If the box were truly perfectly sealed from emitting radiation (obviously impossible) then the mass of the box would be indistinguishable after the m/a-am reaciton.

 

[yuiop]

pmb_phy; you're a little off. Conservation of mass does not hold (look it up). It hasn't since SR and QM. Especially the rest mass, is not going to be the same. Very clearly...

Not so clearly. The energy momentum equation $$m_0c^2 = \sqrt{(m_0c^2\gamma)^2 - (pc)^2}$$ says otherwise (for any closed system). The equation can be expressed as :

$$(Rest Mass Energy)^2 = (Total Energy)^2-(Momentum Energy)^2$$.
In the example of the box containing matter and anti matter assume the initial momentum is zero. After anhilation the total momentum is still zero. Each anhilation pair creates a pair of photons moving in opposite directions conserving the (zero) momentum. The total energy of the system remains the same (No free lunch theorem). Therfore the rest mass of the box system remains the same before and after the anhilation process and therefore the created photons have rest mass when considered as a total system rather than individually.

Mass, energy and momentum for a system is always conserved individually over time in the rest frame of the system, as long as the system is closed. Mass, energy and momentum are not conservered individually under transformation to a different reference frame but the energy-momentum relationship of a system is conserved under transformation. Sometimes the relationship is complicated a little bit by pressure adding an effective mass component and tension providing a negative component according to the stress-energy tensor but overall, as far as I am aware, there are conservation rules that regulate the universe even in post Newtonian physics.

 

[lzkelley]

I dig kev; but i feel like mass is variant even in a non-rest reference frame (even without a transformation to that reference frame). At the same time, even though energy is always equivalent to mass - and energy is always conserved; there isn't always energy in mass form. Does that make any sense?

 

[yuiop]

I dig kev; but i feel like mass is variant even in a non-rest reference frame (even without a transformation to that reference frame). At the same time, even though energy is always equivalent to mass - and energy is always conserved; there isn't always energy in mass form. Does that make any sense?

Yes .. and no I understand what you are getting at. If you fire a projectile its inertial mass appears to increase. Inertial mass is synonymous with total energy or dare I whisper it "relativistic mass". The mass of a single object can be considered to be "variant" over time, even within a single reference frame, but when the system as whole is taken into account it is conservered. In the example of the projectile when we add up the mass, momentum and energy of projectile, the cannon, the explosive fuel and and propellant particles then everything is conserved. I also understand that there is "mass without mass" which is better stated as motion can be mass even when it is the motion of something that has no rest mass such as a photon which is all "motion mass". If we consider mass as a quantity that endows an object with inertia and the ability to attract other uncharged objects then we have to accept that photons have all these qualitites despite have no rest mass individually. In this respect, rest mass is not really the mass we think of in everyday terms and total energy or inertial mass which is the same thing is more representative of what we commonly think of as mass. Rest mass can be thought of, as that part of the "total mass" that is not motion. Consider a box that contains a radiactive substance. After a period of time the substance hass decayed to beta particles that are bouncing around the box. If we add up the total rest mass of each beta particle we will find that the total is less than than the rest mass of the radioactive substance we started with. However, the total rest mass of the closed box remains unchanged because the motion of the beta particles bouncing around in the box contributes motion mass to the system. Now you might have noticed I said rest mass is that part of the mass that is not motion mass and now I am saying that the motion of the particles contributes to the total rest mass of the system. Well, the apparent contradiction is resolved when we realize that the motion of the system as whole is still zero and the energy-momentum equation adds up all the individual energies of the individual particles of the system but only considers the momentum (motion) of the system as whole. In that sense the rest mass of a closed system is always conserved over time in the rest frame. Hope that made sense ..

 

[Phrak]

lol First, after annilhilation, some will form back into matter and antimatter. After a while, it will establish equilibrium(like a chemical) with lights and matters. Lights and mass are different. Lights and mass are energy of different form. In my opinion, it will have less weight. It's contradicting to say that that box of heat weight the same as a box of mass (assuming heat and mass amount are equal in term of energy). It's like one have temperature and weight and one have only weight. It's unfair. :)
The question is it violate the conservation? I don't think it is. It's just like a helium balloon floating in air (assuming that initally box+matter density = surrounding gas density). Now one may ask but floating away means that's extra energy, where is it come from? I'm not sure. Use the helium balloon example on this one. I know that once it go out to vaccum, it's at constant speed and constant speed doesn't require any energy.

I hadn't considered trading back and forth between stuff and emf.

So if both 1)the box floats away and 2) the reaction is thermodynamically reversible, then the Earth and the box system imply a perpetual motion machine as the box would do work as it drops when containing the matter and antimatter.

 

[Dale]

What Pete and Ich were describing can be summed up in the conservation of the http://en.wikipedia.org/wiki/Four-momentum" for any isolated system. In one nice package the conservation of four-momenum gives you the classical conservation of momentum, conservation of energy, and conservation of mass.

The relevant quote from the Wikipedia link:

"Note that the mass of a system of particles may be more than the sum of the particles' rest masses, since kinetic energy in the system center-of-mass frame counts as system mass. As an example, two particles with the four-momentums {-5 Gev, 4 Gev/c, 0, 0} and {-5 Gev, -4 Gev/c, 0, 0} each have (rest) mass 3 Gev/c2 separately, but their total mass (the system mass) is 10 Gev/c2. If these particles were to collide and stick, the mass of the composite object would be 10 Gev/c2."

 

[Phrak]

What would Hawking say of two merging, nonrotating, uncharged, black holes; one originating from collapsing matter, the other in collapsing antimatter?

 

[pmb_phy]

pmb_phy; you're a little off. Conservation of mass does not hold (look it up). It hasn't since SR and QM. Especially the rest mass, is not going to be the same. Very clearly.

That is far from clear. I'm assuming that you have "proper mass" in mind when I used the term "mass." However that would be incorrect. The context in which a term is used will tell you what "mass" means just as the context of "momentum" depends on context. E.g. when the term "momentum" is used in quantum mechanics it does not refer to linear mechanical momentum. It refers to canonical momentum. In the context I used it in the term "mass" refers to "inertial mass" aka "relativistic mass" which is the time component of the objects 4-momentum. And it is possible to show that it the spatial part of 4-momentum (which is 3-momentum) is conserved in all inertial frames then so too is the temporal component. You suggest that I look it up. I have, many many many many many times. I suggest that you look it up. I recomment any of Rindler's texts on this point.

Pete