OK, so this equation is found by squaring the four-vector velocity and and the four-vector momentum (which are both invariant) and combining both equations. I still do not see how a photon can have mass…
A system of photons I am willing to take your word for (though I am still trying to figure out the anti-parallel photon case), can you refer me to a site that has more info?
Sorry, this does not help me as this just defines the rest mass (and energy) of a system. The kinetic energy of a particle is defined by
KE=gamma m c [sup]2[/sup]
Where gamma is 1/Sqrt(1-v[sup]2[/sup]/c[sup]2[/sup]).
As v goes to zero we are left with the rest energy mc[sup]2[/sup].
This still does not convince me that a photon (or even two) can have mass. Let me do some four-vector calculations and get back to you… I’ve dusted off the mechanics book (and was it dusty!) and should probably read more before I spout…
I’m not going to get into a ‘do photons have mass?’ debate. Yes they have relativistic mass, no they don’t have rest mass. You have to clearly define exactly what you mean by mass first.
E=mc^2, defines the total energy for a system, kinetic and the energy contained in the rest mass. It only defines the rest mass if you ignore the relatvistic mass.
look at the slightly expanded equation: E=m(0)c^2 + (pc)^2
If you had some sort of universal measuring device you couldn’t measure kinetic energy, thermal energy, radiological energy or any other form of “energy” directly, either.
All we can measure are effects. A Geiger counter measures the decay of a small amount of particles at its own sensor to get a general idea of environmental radiation levels. A themocouple absorbs or transmits a small amount heat to get an general idea of the surrounding temperature. They don’t capture the whole system. They’re just a means of keeping track of things.*
When you picked up the rock, you applied energy (mechanical work) to it. As you hold it, it itself is storing postential energy due to gravity, and when you drop it, that potential energy is converted to kinetic energy.
That may seem simple, but it isn’t. There are multiple systems in involved here. The mechanical relationship between the rock and the earth (gravity), your entire metabolism, and wind velocity.
The only process we care about is the “picking up the rock and dropping it” part. We don’t care about your heart rate or what you had for lunch. We don’t care how you got the rock that high.
All we care about is “what can we get out of this rock falling.” So we measure the height and mass to get the P.E. and figure out some way to harness it. But we won’t get as much out as you put in lifting it.
We’re human and we will put labels on anything, just to keep track of what’s going on.
I remember seeing a bumper sticker (or maybe it was written on a bathroom wall…not important) that said:
“Women – You too can put the toilet seat down! Potential energy is on your side!”
Unfortunately the few times I’ve tried this line on women it seems to sail right by them (in fairness many of the men nearby seem to miss it as well…I guess physics 101 doesn’t stick too well for most people).
Ring, I’m going to have to be a little bit more convinced before I can embrace your idea of “mass”. You seem to be using the formula m[sup]2[/sup] = E[sup]2[/sup] - p[sup]2[/sup] as the definition of mass, while everywhere I’ve seen it, it’s a result derived from the more fundamental Lorentz dilation of mass formula, which presupposes a definition of mass. I cannot recall ever having seen a Physics text or article which uses that definition for the mass of the system, although I am open to one being pointed out.
Well Achernar I’ve provided you with, what I consider, a relativistically correct mathematical definition of mass. Since you don’t buy this definition I think it devolves on you to present us with your mathematical definition so we can compare the two.
One thing I’m not going to do is scour the net looking for cites. If you choose to not to accept this definition… that’s certainly your prerogative.
I choose not to accept it. (I really should read this board more often. So many interesting threads pass me by.)
To some degree, this is just a question of semantics, but there are two quantities floating around here, and they are indeed different things.
Firstly, we have the mass of a particle. If you’d like, one definintion of a particle’s mass would be based on the strength of its coupling to the Higgs field. A photon has no mass. Two photons together have no mass.
Secondly, we can construct this certain Lorentz invariant quantity for any system. For the special case of a single particle system, this quantity equals the mass of the particle. Thus, we call this quantity the invariant “mass” (as opposed to the invariant “something else”).
An examples of these terms in use:
A neutral pion (p[sup]0[/sup]) has a mass of 135 MeV/c[sup]2[/sup]. It (usually) decays into two photons. When we see two photons in a particle detector, we talk about the invariant mass of that system of photons. If the invariant mass equals 135 MeV/c[sup]2[/sup], we can say that those photons came from the decay of a p[sup]0[/sup]. We are not claiming that the system of photons has actual mass – it has, in fact, none.
There is very strong evidence that neutrinos oscillate. This requires (among other things) that neutrinos have mass. If neutrinos did not have mass, you could still have a system of neutrinos that did have invariant mass. However, no amount of invariant mass is ever going to make those neutrinos oscillate. Mass is a property of the particles; full stop.
It’s clear that I’m approaching this with a particle physicists’ perspective, but in that field at least, the distinction is clear.
Regarding a GR comment made by Ring above (although I should step lightly here, as I cannot claim any GR expertise)…
Two photons travelling side-by-side do affect spacetime. A single photon certainly does, and it has the same stress-energy tensor as two photons of half the energy. Alternately, the linearized field equations are, well, linear, and switching from one photon to two will just double the curvature.
Y’know, I scoured my GR textbooks looking for corraboration for the idea of using “magnitude of the 4-momentum” as definition of “mass”, and I just can’t find it. But on the other hand, whenever a relativity course is taught, that definition is always used. And it’s certainly a self-consistent definition, and which gives the right Newtonian limit. So I’d say, roll with it.
I tried scouring the net for something, as well as a couple of textbooks, and I’m starting to get the opinion that it just doesn’t come up all that often. I admit that I’ve not had as many relativity courses as you, but I don’t remember any definition of mass given in the ones I’ve had - it was just something we didn’t bother with. I think that mass is not as useful a concept in relativity as it is in classical mechanics.
Maybe I should rephrase that statement. Two parallel photons are gravitoelectrically attracted to each other, but this effect is negated by an equal and opposite gravitomagnetic repulsion.
Or as Steve Carlip (Ph.D. gravitational physicist) puts it:
Yes it’s a physics question. Maybe it could have been stated differently, like so:
If I lift a rock off the groud, I store energy in the Earth/rock system, but WHERE is this energy stored, and what form does it take? Does it have a location?
In E&M, if we pull a negatively charged object away from a positively charged one, we store energy. Pulling the plates of a charged capacitor further apart will store energy in that capacitor. However, extremely detailed inspection of the capacitor plates doesn’t reveal any difference. Even a tricorder couldn’t find any extra energy stored inside the metal. Their charge does not change.
The extra capacitor energy is stored in the space between the plates. Measure the e-field in the whole volume surrounding the capacitor plates, take the integral of the field strength squared, and it tells you the amount of energy in that e-field. Pull the capacitor plates further apart and the stored energy rises. Let the plates fall together and the stored energy falls, but just enough to create the KE in the moving plates.
Perhaps the same is true of a rock sitting on the earth? Suppose we could measure the gravity field surrounding the Earth and the rock, then calculate the energy stored in that field. If we lift the rock up off the Earth, wouldn’t this distort the g-field and store some energy? The rock wouldn’t change (for basically the same reason that the above capacitor plates don’t change when energy is injected into the capacitor by forcibly widening the gap.)
bbeaty, I’ll think you’ll find that it is stored in the rock as extra mass. IIRC Einstein and Bohr had an argument, where Einstein attempted to show that Quatum physics was in flagrant violation of the second law of thermodynamics as it allowed the building of a perpetual-motion machine. But Bohr proved him wrong by showing he had forgot to take into account the mass gained and lost in the machine due to GPE.
Ring, thank you for the cite from Spacetime Physics. Now please consult Gravitation, by Misner, Thorne, and Wheeler, chapter 19. They explicitly give the mass M of a weak-field object as the integral over T[sub]00[/sub]d[sup]3[/sup]x (eq. 19.6a). They say that this definition is wrong for strong fields, however; in this case, they mean that what they’re calling the mass is not equal to the parameter M in the Kerr metric. So, for strong fields, do they define the mass M as the magnitude of the 4-momentum? Not really. They define what they call the “total mass-energy” M as the magnitude of the 4-momentum. To me, this supports my idea that mass is not that useful of a concept in relativity, because they abandon it.
Now, I do realize that your definition of mass is consistent, and that it’s good, in that it reduces to Newtonian in the proper limits. But I do want to point out that it’s not the only good, consistent definition of mass that reduces to the Newtonian in the proper limits, so I have a problem with acknowledging it as the definition of mass. Specifically, when someone says, “photons do not have mass” I have trouble saying “You’re wrong!” I would feel more comfortable saying, “Depending on the definition of mass you’re using, You’re wrong!”
(I really should read this board more often. So many interesting threads pass me by.)