Does mass really increase with speed

[DrStupid]

You do know that Newton's theory of gravity is wrong, right? That it is experimentally falsified? Including his law of gravitation? So if you are using his laws to define "gravitational mass", you are defining something that is going to give you false predictions in regimes where his laws are known to be wrong.

I am aware of this problem but there is no other definition of gravitational mass.

If you use the correct relativistic formulas, you get that "inertial mass" always equals "gravitational mass", in so far as those terms even have useful definitions.

That makes no sense because in GR there is no such thing like "gravitational mass". In GR the source of gravitation is not mass but the stress-energy tensor. If you talk about gravitational mass you are talking about Newton's law of gravitation (even if you are not aware of it). If you do not want to refer to Newton's law of gravitation you must not talk about gravitational mass.

 

[PeterDonis]

I am aware of this problem but there is no other definition of gravitational mass.

Then why bring up the term? As I've already noted, the effect you were trying to describe, bending of light by the Sun, can be described without even talking about "gravitational mass" or "inertial mass" at all.

That makes no sense because in GR there is no such thing like "gravitational mass". In GR the source of gravitation is not mass but the stress-energy tensor. If you talk about gravitational mass you are talking about Newton's law of gravitation (even if you are not aware of it). If you do not want to refer to Newton's law of gravitation you must not talk about gravitational mass.

Well, you were the one who brought up the term "gravitational mass"; I put the term in scare-quotes precisely because of the issue you describe. A better way of expressing the point I was making would be to say that the trajectory of a body that is in free fall is independent of the rest mass of the body. The trajectory does depend on the body's initial velocity relative to the source of gravity, but given two objects of different rest mass with the same initial velocity, they will both follow the same trajectory (as long as no other forces are acting).

 

[DrStupid]

Then why bring up the term?

Try to quantify the effect of velocity on gravity without it and you will see why. Gravitational mass is a well defined value that allows to describe this effect with a few words and a very simple equation (see the abstract quoted by timmdeeg).

The trajectory does depend on the body's initial velocity relative to the source of gravity, but given two objects of different rest mass with the same initial velocity, they will both follow the same trajectory (as long as no other forces are acting).

That's not what we are talking about because:

1. It applies to trajectories in static gravitational field only but a real gravitational field will be influenced by the bodies and as this interaction depends on the mass two objects with different mass will not follow the same trajectory.

2. We are not talking about two different bodies in the same static gravitational field but about one body in the dynamic gravitational field of another body moving with different velocities.

3. If we want to talk about two bodies with different velocities in an almost static gravitational field we should not compare their trajectories but their accelerations (not 4-accelerations).

 

[lvirany]

This is where Minkowski differs from Einstein & Lorentz. Minkowski explains the observed length contraction without having to use the concept of mass. Space-time as measured by the moving observer is uniformly dilated in a sheet-like way through the concept of 'proper space' as well as 'proper time'.

 

[PeterDonis]

Try to quantify the effect of velocity on gravity without it and you will see why. Gravitational mass is a well defined value that allows to describe this effect with a few words and a very simple equation (see the abstract quoted by timmdeeg).

As an abstraction used to simplify the understanding of one particular phenomenon, I have no real objection; I personally would not use the term "increased gravitational mass" to describe what's going on; I would say that the force exerted by the moving massive body is not a pure Newtonian static force but has a "magnetic" component, as I said before. But that interpretation leads to the same equation as is given in the abstract, so it's an issue of terminology, not physics.

But the abstraction does not generalize well, and it certainly does not qualify IMO as a "fundamental property" of objects that needs a fully general explanation. It's just a particular abstraction that happens to work well in a particular restricted set of cases.

1. It applies to trajectories in static gravitational field only but a real gravitational field will be influenced by the bodies and as this interaction depends on the mass two objects with different mass will not follow the same trajectory.

In principle this is true, to have a fully self-consistent solution one must take into account the "self-interaction" of anybody with its own field. This raises the same issues that it does in electromagnetism: for a "point particle" the self-interaction is infinite. Most of the time we can avoid this issue altogether by idealizing all bodies but one as "test bodies" whose mass is negligible and whose effect on the overall field is therefore also negligible. That is the idealization I understood us to be using in this discussion. Even if we consider the body that is the source of the field to be moving, the other bodies in the scenario we are considering, as I understand it, are still "test bodies" in this sense.

In practice, we find that bodies as large as planets appear to follow geodesics in the overall background field of the solar system. By that I mean that we can compute their trajectories without having to know their individual masses, just the overall mass that produces the field. So any "self-interaction" effects are not enough to disturb the trajectories even of objects of significant size in this particular case.

There are cases (e.g., binary pulsars) where we do see significant effects due to interaction between two massive bodies, but the key piece of evidence for such interaction is the emission of gravitational waves by the system as a whole, and consequently the gradual inspiral of the two objects towards each other, which in at least one case has been measured for (IIRC) more than 30 years and matches the predictions of GR. This effect is not even predicted at all by Newtonian theory, which predicts that such binary systems should maintain the same orbital parameters indefinitely.

2. We are not talking about two different bodies in the same static gravitational field but about one body in the dynamic gravitational field of another body moving with different velocities.

In other words, in the rest frame of the body producing the gravitational field, you are talking about two different "test bodies" of negligible mass with different initial velocities, rather than two different "test bodies" with different masses but the same initial velocity. Fair enough.

If we want to talk about two bodies with different velocities in an almost static gravitational field we should not compare their trajectories but their accelerations (not 4-accelerations).

Why? What makes these "accelerations" (which are just coordinate accelerations in a particular frame and have no direct physical meaning) the right things to compare, as opposed to 4-accelerations which correspond to a direct physical observable (e.g., the reading on an accelerometer).

Please note, I'm not asking why we *can* talk about these coordinate accelerations. I don't disagree that we can. But you are saying we *should* talk about them, which to me means that there is something physically fundamental about them, something that has to appear in *any* physical model of what's going on. I disagree; I can give a physical model that explains everything without ever using these coordinate accelerations, but only using 4-accelerations (and other covariant or invariant geometric objects).

 

[DrStupid]

What makes these "accelerations" (which are just coordinate accelerations in a particular frame and have no direct physical meaning) the right things to compare, as opposed to 4-accelerations which correspond to a direct physical observable (e.g., the reading on an accelerometer).

What would be the reading on an free falling accelerometer?

 

[PeterDonis]

What would be the reading on an free falling accelerometer?

Zero.

 

[DrStupid]

Zero.

Correct. And an observable that is alway zero does not provide any useful information.

 

[PeterDonis]

Correct. And an observable that is alway zero does not provide any useful information.

An "observable" that is always zero because it is identically zero conveys no useful information, yes. But an observable that is zero precisely because some physical condition is met, and could just as well be nonzero if that condition is not met, certainly does convey useful information. Free fall is a definite physical state of motion; you can enter and leave it at will, simply by shutting off your rocket or turning it back on again, and seeing that your accelerometer reading changes accordingly.

Put another way, 4-acceleration is zero for a freely falling object, but not all objects are freely falling, so the fact that 4-acceleration is zero for a particular object does, in fact, convey useful information.

 

[DrStupid]

Free fall is a definite physical state of motion; you can enter and leave it at will, simply by shutting off your rocket or turning it back on again, and seeing that your accelerometer reading changes accordingly.

But leaving free fall requires a force and I am afraid as soon as we do that your next question would be "Where does this force comes from and what about the involved energies?" Therefore I prefer a setup with gravitational interactions only.

 

[PeterDonis]

But leaving free fall requires a force and I am afraid as soon as we do that your next question would be "Where does this force comes from and what about the involved energies?" Therefore I prefer a setup with gravitational interactions only.

Fine. What does that have to do with whether looking at such a scenario using coordinate acceleration is *required*, as opposed to one possible method but not the only one?

 

[DrStupid]

What does that have to do with whether looking at such a scenario using coordinate acceleration is *required*, as opposed to one possible method but not the only one?

Nothing.

 

[timmdeeg]

My problem is to understand why "active gravitational mass of a moving object" isn't a priori in contradiction with 'mass is invariant'.

Because you're using the wrong definition of "active gravitational mass"; you're plugging numbers into the Newtonian formula for gravitational "force" and trying to read off what the "active gravitational mass" is by applying F = ma, but the Newtonian formula for F is not correct; it doesn't fully describe the actual "force" exerted by a massive object.

Ok, there is no contradiction. And it is probably simply wrong to compare 'invariant mass' (a term within SR) with 'active gravitational mass' (GR).

But how about this reasoning:

A heated body has increased mass and thus an increased gravitational field due to the increased kinetic energy of the particles from which it is composed. A heated body weighs more. Now let's imagine a cold spherical mass M, which's radius exceeds 2GM very slightly. Would it form a black hole on heating (assuming the coefficient of thermal expansion low enough)?

The szenario of DrStupid is much different, but kinetic energy is involved as well.

 

[PeterDonis]

A heated body has increased mass and thus an increased gravitational field due to the increased kinetic energy of the particles from which it is composed.

Yes. But bear in mind that this assumes that the net momentum of the particles composing the mass is unchanged by the heating process (for example, it could be heated by radiation falling on it from all directions in a spherically symmetric manner). This is similar to the stipulation in DrStupid's scenario where the rockets' momenta are equal and opposite so they sum to zero.

Now let's imagine a cold spherical mass M, which's radius exceeds 2GM very slightly. Would it form a black hole on heating (assuming the coefficient of thermal expansion low enough)?

In principle, yes, you can cause an object to collapse into a black hole by heating it. Technically, the exact scenario you describe cannot be realized because it is impossible to have a body in stable equilibrium with a radius less than 9/8 of the Schwarzschild radius (i.e., 9/4 GM / c^2). So a body whose radius exceeds 2 GM / c^2 only very slightly would not be stable, it would already be collapsing into a black hole. But you could take a body that was just at the stable limit and add heat to it, and that would push it "over the edge" into collapsing (because its radius would now be slightly *less* than the minimum for stability for its new, slightly higher mass).