How do magnetic fields curve spacetime?

[vibe3]

According to the Einstein field equations, matter and energy both curve spacetime. I'm wondering how magnetic fields contribute to the curvature of spacetime. I have a few questions:

1. Does a magnetic field in a current-free region of a curved spacetime still satisfy Laplace's equation? Or is there an adjustment needed to Laplace's equation?

2. If there is a magnetic dipole sitting in space, how strong would it need to be to create stable orbits for some object nearby?

3. Do there exist any naturally occurring magnetic fields in the universe strong enough to cause something to orbit it without additional matter?

4. Can humans generate magnetic fields strong enough to cause observable gravitational effects, ie an artificial gravity field?

5. Does there exist a nice solution for the metric due a general potential magnetic field? I did a literature search but only found specialized solutions for Swarzschild/Kerr metric with a magnetic dipole.

 

[PeterDonis]

I'm wondering how magnetic fields contribute to the curvature of spacetime.

Take a look at the Einstein-Maxwell equations:

https://en.wikipedia.org/wiki/Einstein_field_equations#Einstein.E2.80.93Maxwell_equations

They are the Einstein Field Equation including the stress-energy tensor of a pure electromagnetic field (i.e., no charges or currents present) plus the source-free Maxwell Equations.

 

[vibe3]

Take a look at the Einstein-Maxwell equations:

https://en.wikipedia.org/wiki/Einstein_field_equations#Einstein.E2.80.93Maxwell_equations

They are the Einstein Field Equation including the stress-energy tensor of a pure electromagnetic field (i.e., no charges or currents present) plus the source-free Maxwell Equations.

I'm aware of them and a better discussion is here:

https://en.wikipedia.org/wiki/Maxwell's_equations_in_curved_spacetime

I'm hoping an expert can let me know if a non-flat metric will cause a B field to satsify some other equation than Laplace. From staring at the equations for $D^{\mu\nu}$ and $J^{\nu}$ on that page, it seems that perhaps the answer is yes, but I'd like someone with more knowledge to chime in.

If B satisfies a different equation, I'd appreciate any links to solutions or further discussion

 

[rootone]

4. Can humans generate magnetic fields strong enough to cause observable gravitational effects, ie an artificial gravity field?

I call a NO on that one.
All animals have a nerve system which involves tiny electrical currents, and therefore tiny magnetic fields.
Can a human generate a magnetic field similar to that produced by the the supercooled magnets used by the LHC?
No, although evidence to the contrary would be very interestng.

 

[vibe3]

I call a NO on that one.
All animals have a nerve system which involves tiny electrical currents, and therefore tiny magnetic fields.
Can a human generate a magnetic field similar to that produced by the the supercooled magnets used by the LHC?
No, although evidence to the contrary would be very interestng.

I don't understand the analogy - we wouldn't need to generate a complex field, a simple dipole field, of sufficient strength would probably cause enough curvature of spacetime to cause observable gravitational effects no?

 

[PeterDonis]

I'm hoping an expert can let me know if a non-flat metric will cause a B field to satsify some other equation than Laplace.

For the field to be a pure B field you have to choose a particular frame; the Laplace equation isn't covariant to begin with.

 

[PeterDonis]

Can humans generate magnetic fields strong enough to cause observable gravitational effects, ie an artificial gravity field?

To answer this, look at typical components of the stress-energy tensor of the EM field and compare them to typical energy densities for, say, a planet.

 

[PeterDonis]

Do there exist any naturally occurring magnetic fields in the universe strong enough to cause something to orbit it without additional matter?

AFAIK no, not even close.

 

[rootone]

I don't understand the analogy - we wouldn't need to generate a complex field, a simple dipole field, of sufficient strength would probably cause enough curvature of spacetime to cause observable gravitational effects no?

Magnetic fields produced by any animal on Earth are of no consequence to the Universe.
Just don't go swimming in seas with high voltage eels/.

 

[jbriggs444]

Magnetic fields produced by any animal on Earth are of no consequence to the Universe.
Just don't go swimming in seas with high voltage eels/.

Inconceivable!