# Does the equivalence between inertial and gravitational mass imply anything about the Higgs mechanism?

For example: the role it might play in a theory of quantum gravity (ie causing space-time curvature)?

I realize that inertial mass can result from binding energy alone. Has the equivalence principle been tested on elementary particles (like the electron) whose mass would be entirely due to the Higgs coupling?

It seems like an obvious question, but I have never heard it discussed before.

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Higgs mechanics is used just to explain the mass in the context of Standard Model where there is no gravity. So the question doesn't really make sense. Are you interested in beyond-SM theories (specifically SUSY and SUGRA) that resolve some Higgs issues? – Marek Jan 15 '11 at 16:06
I understand the role of the Higgs in the SM, and I understand that the SM does not concern itself with gravity. I did not use the term "Standard Model" in my question, because the SM does not have a monopoly on mechanisms of electroweak symmetry breaking. Inertial and gravitational mass are supposedly equivalent, so any mechanism by which inertial mass is generated must have some relation to the generation of gravitational mass. Right? – user1247 Jan 15 '11 at 17:59
@user1247 "any mechanism by which inertial mass is generated must have some relation to the generation of gravitational mass" - If I'm not mistaken, the relation is simply that one takes the stress-energy tensor from whatever field theory you believe in for your matter content and puts it into Einstein's equation. Perhaps your question is just one about tests of general relativity? Please edit your question to make more precise what you're looking for. – j.c. Jan 15 '11 at 18:10

Dear user, the equivalence between the inertial mass and gravitational mass tells us the following thing about the Higgs mechanism:

Any inertial mass produced or modified by the Higgs mechanism also has to produce or modify a source of gravity of the same magnitude. And vice versa, a gravitational mass produced by the Higgs mechanism also has to produce an inertial mass of the same magnitude.

The reason why others - and myself - and confused about your question is that the equivalence principle tells us the very same thing not only about the Higgs mechanism but also about confinement, electrostatic attraction, spinning gyroscopes, or any other process, object, or mechanism that is taking place, has been taking place, or will take place. The equivalence principle is a totally universal law that holds for all objects and all processes in this Universe - and beyond.

It is not true that the Higgs mechanism has a more special relationship to the equivalence principle than any other mechanism in non-gravitational physics.

Concerning your more specific question about the experimental tests - of course that it has been tested. The available experimental tests of the equivalence principle show that all materials have the same ratio of inertial and gravitational mass up to the precision of $10^{-15}$ or so. Different materials have a different percentage of their mass coming from the electrons - the more neutrons a material has, the smaller the fraction is. So the mass stored in the electrons may go from 0.02% to 0.05%. While this is much smaller than 100%, it's surely enough to exclude the conjecture that the electron mass - as produced by the Higgs mechanism - doesn't obey the equivalence principle.

The percentages above are just 3.5 orders of magnitude below 100%. So you still have 12 orders of magnitude left that prove that the mass produced by the electrons' interactions with the Higgs is exhibited both as inertial mass and gravitational mass - with the same ratio (one - in normal units) - as all other objects have. So once again, yes, all methods to obtain energy/mass i.e. $E=mc^2$ obey the equivalence principle and this fact has been tested with an amazing accuracy. The equivalence principle is true for masses produced by the Higgs mechanism, confinement, or anything else.

This fact is a problem for some "cheap" methods to solve the cosmological constant problem. The problem with the cosmological constant is that even things such as virtual objects in atomic physics, QCD, or anywhere create sources of vacuum energy density. It is not possible to throw them away because such a procedure would ultimately contradict the equivalence principle. So the mystery is why the cosmological constant is so tiny even though we may enumerate lots of possible sources that are much bigger and that appear at any conceivable scale.

Cheers LM

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 Thanks! One more thing to clarify. The equivalence principle may be a universal law, but the source of spacetime curvature (due to a stationary, neutral particle, for simplicity) is very restricted: its rest mass. Its rest mass can come from two distinct things: a lagrangian mass term, or confined energy. Unless the lagrangian mass term is an effective description of confined energy, then it seems like it really does have a special relationship to the equivalence principle. But I guess I just answered my own question: the Higgs must be an effective description of some form of confined energy. – user1247 Jan 15 '11 at 19:20 Thanks for your interest but: not at all, user1247. The spacetime curvature is always caused by the total mass, not the rest mass, this is the whole point of Einstein's equations. They relate the Einstein tensor (of curvature) to the stress-energy tensor. The 00-component of the stress-energy tensor is not related to the rest mass: it is related to the total mass, including all interaction energy (including one with Higgs), kinetic energy, or any other form of energy in space (divided by $c^2$ to get the mass). The other components are completing this density of total energy to a full tensor. – Luboš Motl Jan 15 '11 at 19:47 And no: it doesn't follow from anything that the Higgs has to be an effective description of anything else - whatever you mean by "confined energy". In non-gravitational physics (or limit of small $G$), the gravitational mass becomes unphysical, and so does the whole equivalence principle. So the equivalence principle itself can't possibly constrain anything about non-gravitational physics itself. (Consistency of quantum gravity does say something, but that's a different issue.) It just says that gravity has to behave properly and reuse the same inertial mass as the gravitational mass, too. – Luboš Motl Jan 15 '11 at 19:49 This is the reason why I said: "(due to a stationary, neutral particle, for simplicity)". In the neutral particle's rest frame, its only contribution to the stress-energy tensor is its rest mass. – user1247 Jan 15 '11 at 19:54 Dear user, whether the particle is neutral is totally secondary. There are lots of other forms of interaction energy - aside from the electrostatic energy - that such a neutral particle can have. The neutrality has no relationship to the problem. The adjective "stationary" is more relevant, but it is still incorrect. If you have a neutron star, it can be thought of as a collection of (nearly...) stationary neutrons etc. But there is still a strong, QCD-like interaction energy between the neutrons and be sure that the interaction energy is still contributing to the gravitational field. – Luboš Motl Jan 15 '11 at 19:56