Disclosure: I don’t know much about physics and this may well be a very stupid question.

I heard, once again, that the faster an object goes the more mass it attains. Reaching the speed of light an object would obtain infinite mass (yes I realize that it’s not possible). But what does that really mean? Mass how?

A steel ball at rest has X amount of atoms. If you accelerate that ball to 1/2 the speed of light I presume it gains a lot of mass. Does it gain more atoms? I would think not. If not, where does the extra mass come from?

So I have a croquet ball. It has X mass. I hit it with a mallet. Moving, it has more mass. Is the added mass transferred from the mallet into accelerating the ball?

First of all, a single object does not gain mass no matter how fast it’s moving. Yes, yes, I know that that’s taught in all the schools, but it depends on a silly definition of mass which simplifies nothing, complicates several other things, is in no way necessary, and is at odds with how the term is used by all relativists.

That said, however, there are a number of ways that a system can gain mass. For instance, a system consisting of two objects, moving in opposite directions, has more mass than a system consisting of the same two objects at rest. Mass is not just a sum of the number of atoms in a system, and it is not additive: If you combine two systems into one, the mass of the total system is not, in general, equal to the sums of the masses of the two subsystems.

Ok, I’m confused. I appreciate your post (and realize you are well versed in these matters) but…

It’s often said (in lay terms) that the reason objects can’t accelerate to the speed of light is because they will then attain infinite mass which is impossible and will somehow defy the laws of physics.

If a single object can’t actually attain mass by accelerating why is that so often repeated (lay wise)?

In Newtonian physics, the expressions for kinetic energy and momentum, etc., are pretty simple: KE is .5 my squared, and momentum is mv. They turn out to be low speed approximations of the true relativistic expressions. Rather than using the true relativistic expressions, some people chose to put all the complications into the “m” term (leading to the idea that mass increases with speed), so they could keep using the old expressions - but that is not really a productive way of talking about the real physics of relativity.

Slightly more detail. The relativistic expressions for KE and momentum are .5gammamvv (sorry about the typo in my earlier post) and mgammav, where gamma is a function that’s small when velocity is small and infinite when velocity is the speed of light. So there was the temptation to take the m*gamma part and call it “relativistic mass” just so the equations were similar to the ones that people were used to. Obviously when velocity gets close to c, relativistic mass gets really big.

This is an explanation that doesn’t explain why you can’t do this with a rocket (which is probably the most feasible way to try to do it anyway). Suppose we accept that things that are going fast actually do have more mass. But a rocket engine is attached to the thing it is accelerating – as far as it’s concerned it is always at rest and always has the same mass, so why would some uninvolved observer who sees it moving really fast have an effect on how the rocket works?

Basically, if you see faster-than-light travel in one reference frame, there is another reference frame that sees backwards-in-time travel. So one is just as impossible as the other.

F = ma, i.e. force equals mass times acceleration, is well-known from Newtonian physics. This means that the more massive an object is, the greater the force is required, i.e. the greater the mass of a body the greater its inertia, which is its resistance to a change in its state of motion.

In special relativity F = ma becomes, for an object travelling along the x-axis becomes:

Where f[sub]x[/sub], f[sub]y[/sub] and f[sub]z[/sub] are components of the force along the x, y and z axes, a[sub]x[/sub], a[sub]y[/sub] and a[sub]z[/sub] are components of the acceleration along the x, y and z axes and m[sub]0[/sub] is the rest mass of the object.

γ is relativistic gamma which is equal to [1 - (v[sup]2[/sup]/c[sup]2[/sup])][sup]-0.5[/sup], where v is the speed of the object and c is the speed of light in a vacuum

Now as γ[sup]3[/sup]m[sub]0[/sub] is a measure of the object’s inertia along its direction of motion, in the early days of relativity, it was called the “longitudinal mass”. Equally, as γm[sub]0[/sub] is a measure of the object’s inertia in directions at right angles to its direction of motion it was called the “transverse mass”. Sometimes the transverse mass is called the relativistic mass, though both longitudinal and transverse mass are regarded as out-dated concepts and “mass” these days is taken to mean rest mass.

As γ (relativistic gamma) goes to infinity as v goes to c, both the longitudinal and transverse masses also go to infinity as v goes to c, providing m[sub]0[/sub] > 0. Or in other words becomes more difficult to accelerate a massive object the faster it is going such that there is no way to accelerate it to speeds of c or above with finite forces.

And, the difficulty of accelerating a massive object is, in some sense, the very definition of what mass is. Mass is sometimes defined (or explained) simply as that property, possessed by physical objects, of being “difficult” (that is, requiring force) to accelerate.

The mass is sorta, kinda, very indirectly transferred from the mallet to the ball.

Energy comes from the motion of the mallet. That energy is (in part) transferred to the ball. This causes the mallet to slow down (a little) and the ball to speed up (quite a lot.) Under these new velocities (from the point of view of the guy holding the mallet) the mallet does, in fact, have a little less mass…and the ball has a bit more. But the mass wasn’t transferred: energy was.

(But…of course…Einstein points out the two are the same!)

If that were true, then there might be some justification to the notion. But it’s not. The relativistic formula for momentum is as you stated it, but the relativistic formula for kinetic energy is (gamma - 1)*mc^2. So just trying to stick all of the relativity into the mass part doesn’t actually work.

So, when it is said that the Higgs mechanism imparts a small amount of an atoms mass, and that most of it is ‘relatavistic’ mass, is this:

a) a casual but inaccurate characterization of the situation, utilizing outmoded concepts?

or

b) a fair and accurate appraisal of the situation; after all, there is a lot of energy tied up in the bonds that hold the quarks and gluons together, and that gets counted up in the total mass-energy of the atom, because E=mc**2 and all?*

I have to say, I often find myself confused by the b) situation- it seems to me that the ‘bound’ state should be further ‘down a hole’ and lower in energy than the unbound state.

More this than the other one, I think. But in addition to the binding energy, the constituent quarks have a lot of kinetic energy, which, looking at the system from the outside gets experienced as mass.

This is why the proton and neutron have such similar mass. It’s not because the constituent quarks have similar masses. It’s because the masses of the constituent quarks do very little to determine the mass of these particles. The binding energy and the kinetic energy of the particles determine that, and that is governed by the strong force, which is identical between the particles.

The binding energy is negative, but the kinetic energy of the sub-particles of the system is *very *positive. (Some more details).

All of the energy that you can’t make go away by switching to another reference frame. So binding energy yes, as well as all of the kinetic energy that doesn’t go away when the proton is overall at rest (i.e. the kinetic energy of the constituent particles moving around inside the proton, but not any overall motion of the proton).

Yeah, I guess that is saying that the “mass” is equal to the “rest mass”, which makes sense, since we’ve discarded the term “relativistic mass” (I think). I’m guessing that we have discarded the term “rest mass” as well?

You still do include the kinetic energy resulting from the motion of the quarks relative to the centre of momentum of the proton, though.

Basically, if you are looking at an individual quark, that energy is kinetic energy and not mass because there is a reference frame in which that quark is stationary. But looking at the system of quarks (and gluons) that same energy is part of the mass, because there is no reference frame in which all of the quarks are stationary.