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The following analogy might seem a little bit silly, but I believe it most clearly gets across what I want to ask.

The fundamental charges (mass, electric, strong and weak) of a physical object seem to act as "messengers" to other physical objects which basically tell them "hey, I am a physical object, I have this type of charge and I want to interact with you via its corresponding interaction". For example, electric charge generates electromagnetic forces and mass generates gravitational "forces" (in the more abstract sense that it dictates spacetime curvature). However, mass seems to be special because it also solely determines the response of the objects to this request for interaction by characterizing their resistance to motion, aka. inertia.

Where does this additional function of mass come from and why do the other charges lack it?

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Gravitation and inertia are fundamentally related according to general relativity, because they are considered to be two different aspects of the same thing.

Suppose that you tie a rock to the end of a piece of string, and use the string to whirl the rock around in a circle, above your head.

You can feel a tension in the string, and you have to grip it hard stop it from slipping out of your hand.

  • In your stationary inertial reference frame, we can say that the tension in the string is caused by the rock's inertial mass. The rock's inertia is trying to make it travel in a straight line, and the string tension is constantly deflecting the rock to force it to move in a circle.
  • But in a frame that rotates in "sync" with the circling rock, centred on your location, the "inertia" explanation doesn't work, because in this frame, the rock isn't moving. It's just hanging there, at the end of your piece of horizontal string, as the background stars whirl beneath it. What we say now, is that in the rotating frame, there exists a special sort of outward-pointing radial gravitational field, caused by the surrounding rotating shell of matter (the background stars). In this frame, we say that the field pulls the rock directly away from the central rotation axis, and the rock is suspended in the field by the string. Now, the tension in the string is due to the gravitational mass of the rock.

Both explanations relate to exactly the same physical situation, just described from two different points of view. The inertial explanation and the gravitational explanation are dual, and equivalent.

This gives us the Principle of Equivalence ("PoE") of inertia and gravitation.

Ernst Mach

In Mach's explanation, the inertia of a body is a measure of the interaction of the body's field with the background field. Since inertia is then not decided solely by the properties of the body in isolation, but in conjunction with all other bodies, we have the relativity of inertia, and Mach's principle. These tell us that the amount of forces needed to accelerate a baseball relative to the outside universe should be exactly the same as that needed to accelerate the outside universe (complete with all its fields) relative to the baseball (an idea proposed by Berkeley). This idea that a body's inertial resistance to acceleration and rotation is a relative rather than an absolute property, then gives us the General Principle of Relativity, which applies the relativity principle to all forms of motion whatsoever.

Einstein's alternative derivation of gravitational time dilation

In a modern implementation, the communication between masses regarding inertia is done by the gravitational field. We can double the strength of the interaction (and double a body's inertia), either by doubling the amount of matter in the body, or by doubling the amount of matter in the outside universe. Which is a little bit impractical.

However, since inertia is now being described as field-mediated, we can "fake" the effect of doubling the amount of background matter by doubling the background field intensity ... by simply piling up lots of matter in the immediate vicinity of the test object.

If we do this (said Einstein in his 1921 Princeton Lectures), the inertia of the test mass increases. If the test mass is the flywheel of a mechanical pocket-watch, the flywheel rocks from side to side more slowly for the same applied force, and the watch ticks more slowly. Since we require all idealised timepieces to tick at the same rate as each other, it is not just the pocket-watch that gets slowed, but also quartz resonance clocks, biological clocks, candle-clocks, other chemical clocks, atomic clocks, light-clocks and nuclear decay. By increasing the background inertial-gravitational field (said Einstein), we get the Machian description of gravitational time dilation.

Inertial stuff = gravitational stuff (again)

And, of course, if the Earth has an inertial field that causes a variation in timeflow and lightspeed, then Huyghens principle requires that light be deflected towards the region of slowest lightspeed (Einstein, 1911), and matter (if it contains EM energy in equilibrium), needs to be deflected similarly. So the Earth's field should attract matter and light. If we start with the relativity of inertia, and develop the idea into the idea of an "inertial field", what we end up with is gravitational effects and the gravitational field.

What makes gravitational and electric charge different

The reason why gravitational charges and electric charges have different cumulative behaviours is difficult to explain, but it relates to the fact that "electric" charges have two polarities, while "gravitational" charges only have one.

If the universe contains lots of gravitational charge, the effect is cumulative. Gravitational seems weak because the effect on space of adding a gravitational mass to a region, compared to the cumulative effects of all the other masses in the universe, is proportionally small.

On the other hand if we have astronomical amounts of electric charge in the universe, that fact that half of it seems to be positive and half negative means that, to a first approximation, the more you have, the more evenly it tends to cancel out. So the effect of adding an electron to a region, as a proportional change in the electric background field, is greater.

I appreciate that that last bit is not a very satisfactory answer, but it's probably the best you'll get without getting into hypotheticals such as the Dirac Large Numbers Hypothesis.

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    $\begingroup$ As far as I know, using a rotating frame won't explain centrifugal force by way of the distant planets. You still have to introduce an arbitrary metric / curvature of spacetime. GR just provides equations, that provided with a suitable - although unjustified - spacetime, will predict the proper movements. $\endgroup$ Commented Mar 9 at 10:15
  • $\begingroup$ The Principle of Equivalence - Einstein's happiest thought. $\endgroup$ Commented Mar 9 at 14:18
  • $\begingroup$ @ReinstateMonica : Well, explaining accelerational and rotational; forces was certainly the ==aim== of general relativity ... but it's true that there was often a gap between what Einstein ==said== the theory predicted, and what he could wring out of the math. He never did quite get the implementation to match the original design spec. In some ways what he delivered was not so much a general theory, as a //concept// of a general theory, plus some equivalence-principle math. By 1924 he'd pretty much given up on Mach's principle, and perhaps even the GPoR, and was embracing covariance instead. $\endgroup$ Commented Mar 9 at 14:24
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    $\begingroup$ Nice explanation, especially its historical aspect. But why should "the surrounding rotating shell of matter (the background stars)" cause an "outward-pointing radial gravitational field" with intensity proportional to the rotation speed? And could we link that to the fact that a moving charge attracts another moving charge (by way of the magnetic field, which is actually a consequence of relativity as it does not exist in the reference frame of the field source)? $\endgroup$ Commented Mar 11 at 17:20
  • $\begingroup$ @Jean-ArmandMoroni , If we sit on a children's playground roundabout, and it rotates, we feel an outward force that is stronger the faster the roundabout spins. If we throw a ball directly outward, radial to the rotation axis, we see it veer off to one side, in the direction of rotation of the background stars ... so the rotating shell needs to create an outward radial force proportional to distance from the axis, and also a sideways drag effect. ... $\endgroup$ Commented Mar 13 at 21:01
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A suggestion which may not completely resolve your problem but which at least is nice to move inertial mass to one side and charges/gravitational mass to the other.

You can think of things in terms of forces, like given the electromagnetic charge $q_1,q_2$ you can describe the force

$$F=k\frac{q_1q_2}{r^2},$$

which works perfectly fine. But you can also think in terms of charge-to-inertial-mass ratios,

$$\frac{F}{m_1m_2}=k\frac{Q_1Q_2}{r^2},$$

where $Q=q/m$ is a charge-to-inertial-mass ratio. You can still extract the acceleration perfectly fine if you know the masses of the particles involved (which you have to, in order to be able to describe their charge-to-inertial-mass ratios).

Now, we have ratios on one side and inertial masses on the other. In the case of protons, the ratios are on the order of $10^8 \text{ C/kg}$ so we can put those on one side and then insert one mass on the other to obtain acceleration. For gravity, we just say that the gravitational-mass-to-inertial-mass ratio of any particle is equal to 1. In this way, gravitational mass can be thought of still as a sort of "gravitational charge" separate from inertial mass, but they always end up equal.

More formally, this GM-to-IM = 1 ratio is enforced by the equivalence principle which is a well-established foundation of relativity and which I recommend you read more about.

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  • $\begingroup$ Higgs mechanism? $\endgroup$ Commented Mar 9 at 0:28
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    $\begingroup$ @kangermu what about it? $\endgroup$ Commented Mar 9 at 0:40
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    $\begingroup$ Where does the mass come from is the question. The Higgs mechanism. No amount of moving things to particular sides of equations or considering gravitational mass to inertial mass ratios can explain where the inertial mass comes from. So that is what still needs to be answered. $\endgroup$ Commented Mar 9 at 0:45
  • $\begingroup$ @kangermu True that that is important to recognize, but I interpreted the original question as asking why mass governs inertia and not, say, electric charge. Whether or not the Higgs coupling gives mass to certain particles doesn't affect whether or not or if or how that mass contributes to inertial vs. gravitational charge. $\endgroup$ Commented Mar 9 at 0:49
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It is a mistake to think of mass as the source of gravity (or of spacetime curvature). That's true in Newton's theory, but not in our current best theory of gravity, general relativity (GR). In GR the source of spacetime curvature is something called the stress-energy-momentum tensor. This includes mass, but it also includes other things like kinetic energy, momentum, and pressure.

As for the relationship between inertia and gravity: GR explains this by modeling gravity by curvature of spacetime, which makes gravity by definition an inertial or pseudo force. Another way to think of it is that gravity acts on all objects equally, unlike say electromagnetism in which there are neutral objects unaffected by the force.

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