Why are gravitational mass and inertial mass equivalent?

[ZenBeam]

@ZenBeam

In Weinberg’s “Gravitation and Relativistic Cosmology”*, he showed that, at least under certain assumptions, the equivalence principle falls out of a quantum theory of gravity (in the classical limit).

• It’s at work, so I can’t check the correct title, or look for what assumptions were made. Or even check if I’m recalling correctly.

The full title is “Gravitation and Cosmology Principles and Applications of the General Theory of Relativity”. In Chapter 10, section 8, on page 289, he writes:

Despite these difficulties, there is one very important conclusion […] It is quite impossible to construct a Lorentz invariant quantum theory of particles of mass zero and helicity 2 [i.e. gravitons] without building some sort of gauge invariance into the theory, […]. However […] the theory of gravitational radiation is guage invariant because general relativity is generally covariant, and […] general covariance is but the mathematical expression of the Principle of Equivalence. It therefore appears that the Principla of Equivalence, on which the whole of classical general relativity is based, is itself a consequence of the requirement that the quantum theory of gravitation should be Lorentz invariant."

I also found an online source, QUANTUM THEORY OF GRAVITATION by Martin J.G. VELTMAN (PDF), which comes to the same conclusion (on page 296):

From the above it may be observed that gauge invariance implies a unique coupling between gravitons and scalar particles. This is unlike in the case of electrodynamics, where the covariant derivative involves the charge of the particle which is still arbitrary (otherwise stated, the constant e is not related to something appearing already in the photon part of the Lagrangian). In the case of gravity gauge invariance forces all couplings to be given by the same k. This is essentially the principle of equivalence. Given the free Lagrangian (i.e. the inertial mass of a particle) the coupling to the gravitational field is fixed.

[Leo_Bloom]

@Pleonast

Yes. Gravity is a pseudo-force. That is, it is an artifact of a frame of reference, just like a centrifugal force or coriolis force. Asking why gravitational mass is the same as inertial mass is exactly equivalent to asking why a centrifugal force is proportional to inertial mass.

This Introduction to Physics was just posted on YouTube. It’s nice on plain old inertial frames (although the teacher constantly says “if you believe in inertia,” which I understand is pedagogic shorthand, but sounds weird I think). 13:00 on has nice demos and analytical recaps.

He also mentions centripetal force, not centrifugal, as a fictitious force. Depends on whose point of view (inertial reference frame) is what I take away. But everyone talking about phsics always mentions only centrifugal force as essentially fictional, when in fact it depends whose ox is getting gored, right?

[Asympotically_fat]

@Leo_Bloom

He also mentions centripetal force, not centrifugal, as a fictitious force. Depends on whose point of view (inertial reference frame) is what I take away. But everyone talking about phsics always mentions only centrifugal force as essentially fictional, when in fact it depends whose ox is getting gored, right?

It’s not obvious as he pronounces it centrifical, but he does talk about the centrifugal force as the fictitious force.

[Leo_Bloom]

Ok, I wondered if it was just a garbled hearing thing myself.

[Alan_Smithee]

I can’t believe I get to be the first one to post this. (On of my favorite XKCDs.)

ETA: Not the centrifugal force one. That’s here.

[tracer]

@Leo_Bloom

He also mentions centripetal force, not centrifugal, as a fictitious force. Depends on whose point of view (inertial reference frame) is what I take away.

A rotating reference frame is not inertial.

[moriah]

Doesn’t a object that gets spun around obtain (extra or a new type of) inertia while not gaining gravity?

[AaronX]

Does gravitational mass depend only on rest mass, or relativistic mass? If it’s only rest mass, isn’t that a sign they’re not equivalent?

[leahcim]

@AaronX

Does gravitational mass depend only on rest mass, or relativistic mass? If it’s only rest mass, isn’t that a sign they’re not equivalent?

Gravity depends on the stress-energy tensor. In a lot of cases the most important component of the stress energy tensor is the energy density. As Chronos will be by shortly to point out, “relativistic mass” is just a synonym for “total energy”. So it’s not too much of lie to say relativistic mass is what gravitates.

Or to be a little more loose: Mass doesn’t gravitate, energy gravitates – it just happens that most objects we think of as gravitating have the vast majority of their energy locked up as mass so we can ignore the non-mass energy without making too much of an error.

[Chronos]

As Chronos will be by shortly to point out, “relativistic mass” is just a synonym for “total energy”. So it’s not too much of lie to say relativistic mass is what gravitates.

Actually, I would consider that “too much of a lie”. That would seem to imply, for instance, that a particle accelerated up to a sufficiently-ludicrous speed would collapse into a black hole due to its total energy, but this doesn’t happen. I would be comfortable with saying that gravity depends on the entire stress-energy tensor, and I would (usually) be comfortable with approximating the entire stress-energy tensor by just the rest mass portion and saying that gravity depends on mass, but neither of those is the same thing as the total energy, and I can’t think of any situation where it would be reasonable to approximate the entire stress-energy by the total energy, or to say that “gravity results from the total energy”.

[Peaton_of_peat]

@Chronos

Actually, I would consider that “too much of a lie”. That would seem to imply, for instance, that a particle accelerated up to a sufficiently-ludicrous speed would collapse into a black hole due to its total energy, but this doesn’t happen.

Just for giggles I poked around on Wolfram Alpha, and unless I seriously screwed up somewhere you’d need to get a particle up in the neighborhood of 10^9 kilograms before you get a black hole with the same radius as a proton. That’s equivalent to around 10^44 electron volts, which is way outside the range of anything that’s ever been observed.

It’s such a weird scenario that I have no intuition whether it makes more sense for that particle to become a black hole or not, so I’ll trust you that it wouldn’t. It’s always humbling to find out that the universe is even stranger than you imagined.

[Pasta]

@Peaton_of_peat

Just for giggles I poked around on Wolfram Alpha, and unless I seriously screwed up somewhere you’d need to get a particle up in the neighborhood of 10^9 kilograms before you get a black hole with the same radius as a proton. That’s equivalent to around 10^44 electron volts, which is way outside the range of anything that’s ever been observed.

It’s such a weird scenario that I have no intuition whether it makes more sense for that particle to become a black hole or not, so I’ll trust you that it wouldn’t. It’s always humbling to find out that the universe is even stranger than you imagined.

Chronos’s point wasn’t that black holes can’t be formed from lots of energy. It’s just that it needs to be right kind of energy, namely the energy from the mass of the particle (in the “particle at sufficiently-ludicrous speed” scenario). The total energy of a such a particle is reference-frame dependent, so whether it forms a black hole had better not be governed by the total energy.

[Chronos]

Right, because something either is or is not a black hole. You couldn’t have something that is a black hole in one reference frame, but is not a black hole in some other reference frame.

Note, by the way, that if you took two ludicrous-speed particles and aimed them at each other, you could get a black hole out of that one.

[Peaton_of_peat]

Ah right, I forgot that within your reference frame, you don’t gain any energy from accelerating. I was just wondering if we had experimental evidence that this didn’t happen or if it was just theoretical, and it turns out it’s only theoretical but the theory is pretty damn solid. Thanks for clarifying!

[leahcim]

@Chronos

Actually, I would consider that “too much of a lie”. That would seem to imply, for instance, that a particle accelerated up to a sufficiently-ludicrous speed would collapse into a black hole due to its total energy, but this doesn’t happen.

Yeah, now that you spell it out, I grok how what I was saying doesn’t make sense. I should think harder before opening my cake-hole. I blame the booze.

Would you agree with the revised statement of: “For normal things, like boulders moving much slower than the speed of light, the S-E tensor (for points within the boulder) is dominated by its (0,0) component, which is in turn dominated by the energy density due to mass density, which is why we get away with thinking of gravity as being generated by mass in such situations”?

[Asympotically_fat]

@leahcim

Yeah, now that you spell it out, I grok how what I was saying doesn’t make sense. I should think harder before opening my cake-hole. I blame the booze.

Would you agree with the revised statement of: “For normal things, like boulders moving much slower than the speed of light, the S-E tensor (for points within the boulder) is dominated by its (0,0) component, which is in turn dominated by the energy density due to mass density, which is why we get away with thinking of gravity as being generated by mass in such situations”?

It’s clear that when there exists a frame in which all components of the stress-energy tensor T can be neglected but T[sup]00[/sup] then T[sup]00[/sup] is the mass density. Also there must be a limit where the mass M enclosed in a volume V is:

M ≈ ∫[sub]V[/sub] T[sup]00[/sup]dV

Otherwise Newton’s law of gravity wouldn’t work. But of course first of all M must be defined in general relativity and all the conditions for it to reduce to the above must be worked out.