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I am suddenly struck by the question of whether gravitation affects magnetism in some way. On the other hand, gravity is a weak force, but magnetism seems to be a strong force, so would magnetism affect gravity?

Or do they "ignore" each other, being forces which do not interact?

The answer to this is related to this question: If the earth's core were to cool so that it were no longer liquid, no longer rotated, and thus produced no magnetic field, would this do anything to earth's gravity?

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The electromagnetic field tensor $F_{\mu\nu}$ which encodes all the information about the electric and magnetic field, certainly contributes to the energy-stress tensor $T_{\mu\nu}$, which appears in the Einstein Field Equations: $$G_{\mu\nu}= 8\pi G T_{\mu\nu}$$ The left hand side of this equation encodes the geometry of spacetime, while the right hand side describes the 'sources' of gravity. Therefore, we can say that magnetism does have an effect on the geometry of spacetime i.e. gravity.

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    $\begingroup$ I think the questioner is interested in levels of field strength easily created on (or by) the Earth. Talking about engery stress tensors is like answering the question "Will running make me lose weight?" with the equation for relativistic mass. $\endgroup$
    – Foo Bar
    Commented Sep 22, 2014 at 19:32
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    $\begingroup$ @FooBar that made me chuckle :) If you feel a lower level answer is in order, I invite you to provide one! However, you have to keep in mind that this site aims to be of use to more people than just the asker, so answers at all levels are more than welcome, and I don't feel inclined to alter my own response. $\endgroup$
    – Danu
    Commented Sep 22, 2014 at 19:36
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    $\begingroup$ @FooBar: Actually electric and magnetic fields constitute a significant part of the rest mass of ordinary matter. The fields inside the nucleus are extremely intense. The relative contribution is highest for elements with a high atomic number. $\endgroup$
    – user4552
    Commented Sep 22, 2014 at 19:59
  • $\begingroup$ Nice! This answer suggests that if you could somehow generate a magnetic field that was strong enough, you could create artificial gravity. That would have to be an unimaginably strong field to produce anything practical, however. And where would you get the energy to do it with? Wow. $\endgroup$
    – Cyberherbalist
    Commented Sep 22, 2014 at 22:11
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    $\begingroup$ @Cyberherbalist How current loops and solenoids curve space-time, A. Füzfa "We finally propose an experimental setup, achievable with current technology of superconducting coils, that produces a phase shift of light of the same order of magnitude than astrophysical signals in ground-based gravitational wave observatories." $\endgroup$
    – endolith
    Commented Jan 15, 2016 at 4:29
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Magnetic effect on gravity: In technical terms: yes. For practical purposes: no.

As Danu states, a magnetic field is a form of energy, and Einstein showed that energy can be equivalent to mass. A sufficiently large amount of energy (of any kind) collected together will produce measurable gravity (aka bending of space-time).

This source puts the total energy of Earth's magnetic field at $10^{19}$ joules. If that's correct, it's equivalent to 111 kilograms, which is about $10^{-23}$ of Earth's mass (and gravity).

Caveat: I am not a "real" physicist; I might be calculating the wrong thing. But I'm pretty confident that the number is negligible, relative to many other things that affect Earth's gravity.

As Ben states, the electromagnetic energy of protons and electrons is worth a significant amount of the mass in every atom. But commonplace magnetic fields at the human scale expose such a tiny fraction of this energy as to be irrelevant gravitationally.

Gravitational effect on magnetism: It depends on how you look at it.

Unlike an electromagnet (such as the Earth's core), you can't "turn off" a gravitational field and compare the difference.

But in a grand sense, the magnetic fields produced by large convecting masses only exist because of the great pressure and heat generated by self-gravity. Whether that counts is more a question of philosophy than physics.

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