# Can a sufficiently strong magnetic field create gravity?

• I
This has been noodling in the back of my mind ever since I read about the field strength in magnetars etc.

Magnetic fields store energy.

Magnetic fields in magnetars are inconceivably large (~10^11T).

This inconceivably large magnetic field contains ~4x10^27J/m3

From that famous guy whose name escapes me this works out to 4x10^10kg/m3 mass energy equivalence.

There is no material we can touch even close to that mass density.

Therefore, does a 10^11T mag field exert gravity?

More specifically another way of looking at it, is the thing we describe as mass, actually the effect of an extremely high concentration of energy (ie m=E/c2), then does a different high concentration of energy (eg mag field) also start to exhibit properties of mass, like inertia?

PeterDonis
Mentor
There is no material we can touch even close to that mass density.

We can't here on Earth, but this density is about six orders of magnitude smaller than the typical density of a neutron star, so it is by no means exceptional in cosmological terms.

does a 10^11T mag field exert gravity?

Everything that has energy acts as a source of gravity in GR. The source of gravity is the stress-energy tensor, which includes all forms of energy density (not just rest mass). It also includes pressure, and a magnetic field will have pressure as well as energy density, which does not have the same form as the pressure of ordinary matter; that means the gravity produced by a magnetic field will not be quite the same as the gravity produced by ordinary matter.

is the thing we describe as mass, actually the effect of an extremely high concentration of energy

This question is not well-defined; "energy" can have many meanings. But as far as being a source of gravity is concerned, rest mass appears in the stress-energy tensor along with other forms of energy.

does a different high concentration of energy (eg mag field) also start to exhibit properties of mass, like inertia?

Inertia is a different property from "source of gravity". But yes, energy has inertia--or more precisely, if a body has energy other than rest mass, that energy contributes to its inertia.

PeterDonis
Mentor
Moderator's note: Thread moved to the relativity forum.

pervect
Staff Emeritus
This has been noodling in the back of my mind ever since I read about the field strength in magnetars etc.

Magnetic fields store energy.

Magnetic fields in magnetars are inconceivably large (~10^11T).

This inconceivably large magnetic field contains ~4x10^27J/m3

From that famous guy whose name escapes me this works out to 4x10^10kg/m3 mass energy equivalence.

There is no material we can touch even close to that mass density.

Therefore, does a 10^11T mag field exert gravity?

More specifically another way of looking at it, is the thing we describe as mass, actually the effect of an extremely high concentration of energy (ie m=E/c2), then does a different high concentration of energy (eg mag field) also start to exhibit properties of mass, like inertia?

Yes. The source of gravity in GR is the stress energy tensor. Wiki writes this tensor in terms of the E and B fields in <<link>>

wiki said:
$$T^{\mu\nu} =\begin{bmatrix} \frac{1}{2}\left(\epsilon_0 E^2+\frac{1}{\mu_0}B^2\right) & S_\text{x}/c & S_\text{y}/c & S_\text{z}/c \\ S_\text{x}/c & -\sigma_{xx} & -\sigma_\text{xy} & -\sigma_\text{xz} \\ S_\text{y}/c & -\sigma_{yx} & -\sigma_\text{yy} & -\sigma_\text{yz} \\ S_\text{z}/c & -\sigma_{zx} & -\sigma_\text{zy} & -\sigma_\text{zz} \end{bmatrix}$$

S is the Poynting vector, which will be zero if E=0. The ##\sigma_{ij}## are the presssure terms Peter mentioned, if E=0 they will be given by
$$\sigma_{ij} = \frac{1}{{\mu _0}}B_i B_j - \left( \frac{1}{2\,\mu _0}B^2 \right)\delta _{ij}$$

If we chose the B-field to be oriented around the X axis, I get:

$$T^{\mu\nu} = \begin{bmatrix} K & 0 & 0 & 0 \\ 0 & -\frac{K}{2} & 0 & 0 \\ 0 & 0 & \frac{K}{2} & 0 \\ 0 & 0 & 0 & \frac{K}{2} \end{bmatrix}$$

with ##K = \frac{B^2}{\mu_0}##

The term in the upper left is the energy density term, the diagonal terms are the pressures Pete mentioned. So we can identify the terms according to ##\rho = K, P_x = -K/2, P_y=k, P_z=K/2## - if I haven't made a sign error, which is unfortunately quite possible. This is important because the sum of ##\rho + P_x + P_y + P_z## determines what happens to a ball of coffee grounds surrounding the area containg the "stuff" described by the stress-energy tensor. It determines whether the coffee ground wind up shrinking in volume, changing it's shape without changing volume, or expanding, as explained in Baez's paper https://arxiv.org/pdf/gr-qc/0103044.pdf.

I would loosely describe the state of affairs where the volume of the coffee grounds shrinks with time (the second derivative of the volume with time is less than zero) as being a gravitational contribution to collapse. If I haven't made a sign error, the relevant sum is positive and equal to 2K = ##2 \, B^2 / \mu_0## and a ball of uncharged coffee grounds in a strong magnetic field will shrink due to gravitational effects.

For instance, if you have a ball of coffee grounds surrounding a mass m, they're attracted to the mass, and the ball shrinks. If you have a ball of coffee grounds in empty space, away from any mass, the volume doesn't change. If the grounds are in empty space but close to a large mass, tidal forces will distort the shape of the ball of coffee grounds, but the volume won't change - not in the same way as it will due to gravity. More preceisely, the second derivative of the volume with respect to time will be zero in empty space, while the second derivative of the volume with respect to time will be negative if the grounds surround some mass.

We can see the importance of pressure - in general relativity, pressure makes the ball of coffee grounds shrink just as mass does. So it's a mistake to think of only mass causing gravity if one is doing GR. That used to be true in Newtonian gravity, but GR is not Newtonian gravity, the rules are slightly different.

Unfortunately, I'm not quite sure I didn't get the sign of the pressure terms wrong.

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We can't here on Earth, but this density is about six orders of magnitude smaller than the typical density of a neutron star, so it is by no means exceptional in cosmological terms.

It may not be exceptional in cosmological terms, however given that 22x10^3kg/m3 is the densest "stuff" we have here on earth, the fact that a magnetic field, which to us at least, is not visible or touchable, can have mass density seven orders of magnitude higher is a difficult thing to conceptualize. If you could create such a field strength with out killing everything within a few light years, what would this look/feel like?

Everything that has energy acts as a source of gravity in GR. The source of gravity is the stress-energy tensor, which includes all forms of energy density (not just rest mass). It also includes pressure, and a magnetic field will have pressure as well as energy density, which does not have the same form as the pressure of ordinary matter; that means the gravity produced by a magnetic field will not be quite the same as the gravity produced by ordinary matter.

I think need to understand more about what this "pressure" is, its likely not the more earthy based gas pushing on stuff, but the more generalized "force per area" kind of thing?

This question is not well-defined; "energy" can have many meanings. But as far as being a source of gravity is concerned, rest mass appears in the stress-energy tensor along with other forms of energy.

This was kind of from another interesting article I read how Mass has been demoted as an intrinsic property in modern physics. To paraphrase the article: As we go and dig deeper and deeper into what "matter" is, eventually the mass of the things making up the previous thing no longer adds up. That happens when you break up a proton or neutron into quarks, apparently 99% of the "mass" is unaccounted for and is due to red green and blue "color charge" or something, I'll state here that my knowledge of quantum physics is limited.

In the end the position the article was presenting is that there is actually no such thing as mass and all the effects we see as "mass" at our classical scale is due to to the energy of the gluons and quark-anti-quark pairs in this color field. (again willingly admit my knowledge in this is weak and do not know how valid this claim is)

Inertia is a different property from "source of gravity". But yes, energy has inertia--or more precisely, if a body has energy other than rest mass, that energy contributes to its inertia.

After doing some reading, it is implied even in special relativity, that rest mass is due to rest energy and inertia is no longer purely mass/momentum based, but more generally to "the force needed to change a bodies velocity", which would include both inertia due to "rest mass" and inertia due to "energy".

In retrospect had I actually looked at the equations of special relativity the inclusion of Electric and magnetic fields in the calculations would have been obvious and my momentary confusion unnecessary. :)

I guess E=mc2 is really completely substituteable and anywhere "m" is used, can be replaced with E/c2 and visa versa, eg force between two masses F=G*(m1m2/r^2) is also force between a mass and energy or between two blobs of energy etc.

PeterDonis
Mentor
If you could create such a field strength with out killing everything within a few light years, what would this look/feel like?

I'm not sure there is an easy intuitive way to visualize it. But see below.

I think need to understand more about what this "pressure" is

Take two ordinary kitchen magnets and try pushing them together. You will feel a repulsive force between them resisting. That is a (heuristic) example of pressure from magnetic fields.

The full stress-energy tensor of an EM field is more complicated, but these pages will at least give you a start:

https://en.wikipedia.org/wiki/Electromagnetic_stress–energy_tensor

http://farside.ph.utexas.edu/teaching/em/lectures/node128.html

In the end the position the article was presenting is that there is actually no such thing as mass and all the effects we see as "mass" at our classical scale is due to to the energy of the gluons and quark-anti-quark pairs in this color field.

This is not using "mass" in the sense of "energy divided by c squared"; it's using "mass" in the sense of "invariant mass" or "rest mass" of the elementary particles that make up ordinary matter. This is a different question from the question of what contributes to inertia or acts as a source of gravity; even if it turns out that, at the most fundamental level, all of the elementary particles we know of are massless (which is basically what our current Standard Model of particle physics says--particles that appear to have rest mass, like electrons and quarks, are actually acquiring it due to interactions like the Higgs at low enough temperatures), they will still have inertia and act as sources of gravity just the same, since that depends on the total stress-energy content, not on whether some of it is "packaged" as particles with rest mass or not.

To give an example: consider a single proton. It has a rest mass of about 936 MeV (note that I just gave a "mass" in energy units--this is commonly done in particle physics since it's much more convenient than having to worry about factors of ##c## or ##c^2## all the time). As far as considering that proton's inertia or its contribution as a source of gravity, that's all you need to know about it. The fact that, according to our best current model, only about 30 MeV or so (I'm pulling these numbers from memory so they might be off a little, but they are the right order of magnitude) of that 936 MeV is due to the rest masses of the three valence quarks (two up and one down), while the rest is due to what you describe as "energy in the color field", makes no difference to the proton's inertia or its behavior as a source of gravity.

Ibix