# General relativistic effects on an electro-magnet

1. May 7, 2014

### jartsa

Let's say there's a current going around in a superconducting loop in vacuum. Then the loop gets buried in huge amount of matter, which has the same magnetic susceptibility as vacuum. (when not affected by gravity the matter has the same magnetic susceptibility as vacuum)

Will the magnetic field change near the loop?
Will the magnetic field change far away from the loop?

2. May 7, 2014

### WannabeNewton

Yes. The magnetic field is coupled to the connection which is itself coupled to the gravitational field.

3. May 8, 2014

### Bill_K

Model the "huge amount of matter" as a spherical shell of mass M and radius R, and consider the situation before the current loop is introduced. Outside the shell the field will be Schwarzschild, with metric

ds2 = (1 - 2M/r) dt2 - (1 - 2M/r)-1 dr2 - r2 d2Ω

Inside, the field will be flat. The boundary condition at the shell is that its intrinsic curvature must be the same as viewed from inside and outside. Choose inner coordinates to match the outer ones. What's different is grr. The inner metric will be

ds2 = (1 - 2M/R) dt2 - dr2 - r2 d2Ω

which is just the Minkowski metric with a rescaled t coordinate.

Now introduce the current loop. The B field will be continuous across the shell, so both inside and outside, the B field is that of a magnetic dipole. However the dipole moments differ, because an inside observer will rescale his t coordinate to match Minkowski.

The result is simply time dilation - from the outside the currents appear slower. The external effect of the mass shell is a reduced apparent magnetic dipole, and correspondingly a B field that is reduced.