Does a Magnet Curve Spacetime More Than a Non-Magnetic Mass?

[Will Learn]

TL;DR Summary

When constructing the stress-energy tensor, various forms of mass-energy can be considered. I've not seen magnetic energy or magnetic potential energy being considered but is it reasonable to do so?
Specifically, would a permanent magnet of mass m actually curve spacetime more than a non-magnetic material of mass m?

Hi.
My question is described in the summary.
I'm seeking some advice.
The Reissner-Nordstrom solution for charged spherical bodies seems to indicate that electrostatic fields will be a source of gravitation. I've not seen anything similar for magnetic fields but I can't imagine how it could be different.

Before running too far I would just like to check a simple and concrete example:

Considering a universe which is empty apart from a standard permanent bar magnet vs. a universe that is empty apart from a non-magnetic material of the same shape and mass as the magnet and only contributes mass density - is it fair to say that gravitational differences can be observed intrinsically? For example, if we sent a group of test particles through these universes, would there be a difference in the divergence of their geodesic paths?

Thanks.

Late editing: Found an old thread with a similar question, which I'm reading now.

 

[.Scott]

I believe a non-magnetic iron bar still consists of magnets at the atomic level.
But since they are not aligned, they may be at a lower energy level.
So, when an iron magnet is demagnetized, does it give off photons - and thus mass.
I suspect it does.

In a sense, it would be like chemical energy. Water has less mass than the equivalent amounts of H2 and O2.

 

[PeterDonis]

When constructing the stress-energy tensor, various forms of mass-energy can be considered. I've not seen magnetic energy or magnetic potential energy being considered but is it reasonable to do so?

It is not only reasonable, it is done. What you are calling "magnetic energy" or "magnetic potential energy" is part of the stress-energy tensor of the electromagnetic field:

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

would a permanent magnet of mass m actually curve spacetime more than a non-magnetic material of mass m?

Spacetime curvature is not a single number, so it doesn't really make sense to ask which curves spacetime "more". But a magnet would certainly have a different stress-energy tensor than an otherwise identical non-magnetic material, because of the stress-energy tensor of the electromagnetic field.

 

[Will Learn]

Hi and thanks Scott.
Most of this seems reasonable.
If I've understood what you said correctly then, what you're stating directly implies that the inertial mass of an Iron bar changes when it is magnetised (all magnetic regions aligned) compared to when it isn't (random orientation of magnetic regions). I may need some time to check on this.

I can also side-step this issue entirely. In the two situations it was stated that the masses were the same. Even if we need to shave off some more material from the magnet, we will ensure the masses are the same. Additionally the second situation involves a non-magnetic material, so it's not Iron in a de-magnetised state where you could argue that all the small magnetic regions were still there anyway - it's some hypothetical object that only contributes mass. It doesn't matter if no such material exists in reality, in the second example I will be choosing to construct the stress-energy by considering the mass as the only contribution.

I think I'm beginning to answer my own question. I believe that in the first situation I need to consider the B field (the magnetic field) as a contribution to the energy density. So the over-all stress energy tensor will be different and then the curvature of spacetime is expected to be different (unless by some amazing co-incidence the B field is distributed through space in exactly the right way to negate the fact that there just is more mass-energy appearing in the stress-energy tensor).

I'm really grateful for your time Scott and I could be wrong about a lot of what I've just said, so I'd appreciate more opinions.

**Looks like someone else has just already replied. I can only apologise if my post overlaps.

 

[Will Learn]

OK. I've just read the reply from Peter Donis. Thank you for your time.

 

[PeterDonis]

what you're stating directly implies that the inertial mass of an Iron bar changes when it is magnetised (all magnetic regions aligned) compared to when it isn't (random orientation of magnetic regions).

The stress-energy tensor of the object includes a contribution from the electromagnetic field. "Inertial mass" is not the same thing as "stress-energy tensor".

For an iron bar that is magnetized vs. not, the electromagnetic stress-energy tensor would differ, but the difference would not be as simple as just "changed inertial mass" or even changed energy density, since energy density is only one component of the stress-energy tensor.

I will be choosing to construct the stress-energy by considering the mass as the only contribution.

You can't. You don't get to choose what the stress-energy tensor of an electromagnetic field is. It's dictated by physics. Once you say, for example, "magnetized iron bar", you have no further freedom to specify the electromagnetic stress-energy tensor (except for the obvious freedom of choosing a single constant to represent the average field strength).

 

[Dale]

For completeness, the EM stress energy tensor is described here:
https://en.wikipedia.org/wiki/Electromagnetic_stress–energy_tensor

It is given by $T^{\mu\nu} = \frac{1}{\mu_0} \left[ F^{\mu \alpha}F^\nu{}_{\alpha} - \frac{1}{4} \eta^{\mu\nu}F_{\alpha\beta} F^{\alpha\beta}\right]$

 

[Will Learn]

Thanks everyone.
I think the replies have overlapped a bit here.

If I've understood what you said correctly then, what you're stating directly implies that the inertial mass of an Iron bar changes when it is magnetised (all magnetic regions aligned) compared to when it isn't (random orientation of magnetic regions).

I did say the above but I was referring to something Scott said not something Peter Donis said. It's my fault for not making that clear. At the time I started my post Scott was the ONLY reply and it never occurred to me that Peter Donis would get a reply up before I had finished writing. Many of the comments in your (Peter Donis) second reply seem to have assumed my comments were aimed at Peter Donis' first reply. It doesn't matter too much but there is one issue I would seek clarification on.

Can I seek some clarification on the following:

For an iron bar that is magnetized vs. not, the electromagnetic stress-energy tensor would differ, but the difference would not be as simple as just "changed inertial mass".

Does the inertial mass of a magnetised Iron bar change compared to the same bar in a de-magnetised condition? I don't know and I can't find any references for this.

None of this is too important, no one has to reply but all replies are appreciated.

 

[PeterDonis]

Does the inertial mass of a magnetised Iron bar change compared to the same bar in a de-magnetised condition?

One would expect it to, yes, based on the fact that the stress-energy tensor changes. "Inertial mass" is not a fundamental quantity in GR; you have to derive it from the stress-energy tensor.

 

[Will Learn]

Thanks Peter Donis.

"Inertial mass" is not a fundamental quantity in GR; you have to derive it from the stress-energy tensor.

I'm seeing a problem here:
I need the stress-energy tensor to derive the inertial mass BUT
I can't derive the stress-energy tensor without knowing what the mass is.

Nevermind, perhaps there isn't a clear answer. Thanks for your time.

 

[PeterDonis]

I need the stress-energy tensor to derive the inertial mass

Yes.

I can't derive the stress-energy tensor without knowing what the mass is.

Sure you can. Have you looked at the formulas for the stress-energy tensor of the electromagnetic field? Where do you see inertial mass in them?

 

[Will Learn]

Where do you see inertial mass in them?

I don't - but those formulae don't describe the stress-energy tensor for a permanent bar magnet. They describe the tensor for a source-free electromagnetic field.
I want to include the bar of Iron itself, this has mass.

 

[PeterDonis]

those formulae don't describe the stress-energy tensor for a permanent bar magnet. They describe the tensor for a source-free electromagnetic field.

No, they describe the stress-energy tensor due to the electromagnetic field, for any electromagnetic field.

If you have other stress-energy present besides the EM field, then of course you need to add that to the overall stress-energy tensor as well. For the iron in a magnetized iron bar, for example, you will have a term for the energy density of the iron, which adds to the overall energy density component (the 0-0 component) of the tensor. But that doesn't have inertial mass in it either; it's just the density of iron times $c^2$.

 

[Will Learn]

Yes I have looked at both wikipedia articles suggested by Peter Donis and Dale. I'm not trying to waste your time and just ignoring the links you provide.

This formula (screenshot 1) taken from Wikipedia (https://en.wikipedia.org/wiki/Electromagnetic_stress–energy_tensor)

- - - - - - -

- - - - - - - -is the same as this formula (screenshot 2) from Wikipedia (https://en.wikipedia.org/wiki/Stress–energy_tensor#)

- - - - -

- - - - - -

The first one is in flat space, so g = Eta . The second Wikipedia article directly states "of a source-free electromagnetic field" as shown in the screenshot above.

I can only apologise if I got this wrong. I was probably looking at the other article at the time.

But that doesn't have inertial mass in it either; it's just the density of iron times c (squared).

Yes, I agree with this and it makes sense, thank you. But, does the density of a lump of Iron change after it has been magnetised?

 

[PeterDonis]

does the density of a lump of Iron change after it has been magnetised?

I don't see why it would. The magnetization reorients individual current loops inside the iron, but it doesn't change how the atoms are packed.

 

[Will Learn]

Thank you to everyone. I'll leave you all in peace now.

 

[PAllen]

I don't see why it would. The magnetization reorients individual current loops inside the iron, but it doesn't change how the atoms are packed.

As I understand it, the lowest energy state of a ferromagnetic material is with domains randomly oriented. Forcing permanent magnetization adds energy, but this is a metastable state with enormous lifetime. Thus, it seems to me, that a magnetized ferromagnetic material would have slightly greater inertial and gravitational mass than the same body unmagnetized at the same temperature.

 

[PeterDonis]

Forcing permanent magnetization adds energy

Yes, and that energy is stored in the electromagnetic field, so it would appear as extra terms in the appropriate components of the stress-energy tensor, matching the form of the SET components of an EM field.

a magnetized ferromagnetic material would have slightly greater inertial and gravitational mass than the same body unmagnetized at the same temperature.

Yes, I agree, and the reason is that the appropriate averaged quantities used to calculate inertial and gravitational mass will have additional contributions from the EM field stress-energy tensor terms.

What I do not think will be the case is that the matter terms in the SET will change. The energy density due to iron atoms themselves won't change, nor will the pressure due to inter-atomic forces. I think the only change in the SET will be the addition of the EM field terms in the appropriate components, because the energy added to magnetize the iron ends up stored in the EM field.

 

[PAllen]

Yes, I think @PeterDonis is right on this. The local energy state of one domain cannot “care” what its spin alignment is; following cooling from above the Curie temperature, it can be any direction. Thus, it seems the only place the energy added to align domains can be stored is outside any domain, i.e. In the EM field itself.

 

[256bits]

Yes, I think @PeterDonis is right on this. The local energy state of one domain cannot “care” what its spin alignment is; following cooling from above the Curie temperature, it can be any direction. Thus, it seems the only place the energy added to align domains can be stored is outside any domain, i.e. In the EM field itself.

some potential energy is stored within.
The elastic energy of rotating the dipole moment in alignment with the external field, and the movement and rotation of domain walls. There is a thing called magnetorestriction where the dimensions of the material does change, either positive, or negative, in the direction of the field as well as a volume change in particular materials.
http://magnetism.eu/esm/2007-cluj/questions/magnetostriction.pdf

this movement is responsible for the hum from transformers, and the heating of iron core.

other aspects --

Section 4
The total internal energy of the domain structure in a ferromagnetic material is made up from the following contributions.

Exchange energy (or) Magnetic field energy.
Crystalline energy (or) Anisotropy energy.
Domain wall energy (or) Bloch wall energy.
Magnetostriction energy
https://www.brainkart.com/article/Magnetic-Materials_6823/

 

[PAllen]

So I guess my initial intuition was more correct, but I won’t pretend I knew any of this before. All I knew was that the magnetized state was a higher energy meta stable state than the unmagnetized state.