Do gravitational fields interact with electric or magnetic fields?

We know that Electric Fields interact with Magnetic Fields, but do Electric Fields or Magnetic Fields interact with Gravitational Fields, and if so how?

• – Sam Dec 31 '19 at 7:25
• Electric fields do not interact with magnetic fields. The two cannot always be told apart as expressed by Faraday's law. – my2cts Dec 31 '19 at 7:40

When you talk about the electric and magnetic fields interacting I would guess you are thinking of Maxwell's equations that give us the electric field $$\mathbf E$$ and magnetic field $$\mathbf B$$ for any arrangement of charges and currents. However it is not correct to say the electric and magnetic fields are interacting. Instead they are both components of the same single electromagnetic field, and Maxwell's equations are the equations that describe this single electromagnetic field described by the electromagnetic field strength tensor $$\mathbf F$$. The fields $$\mathbf E$$ and $$\mathbf B$$ are components of this tensor and Maxwell's equations tell us how they are related rather than how they interact.

The gravitational field is not related to the electric and magnetic fields in the same way i.e. there is not (as far as we know) a single "gravitoelectromagnetic" field described by some set of equations analogous to Maxwell's equations. However gravitational and electromagnetic fields do affect each other. This happens in two ways:

Firstly EM fields are affected by spacetime curvature. This is why we see gravitational lensing. Light waves, like all EM waves, follow geodesics in a curved spacetime so they are deflected by the curvature. When calculating this we typically take the spacetime curvature as a fixed background i.e. the EM fields do not affect the curvature. This is generally a good aproximation because the energies present in EM fields are generally far too small to have any significant effect on the curvature. We take the spacetime curvature into account by writing the Maxwell's equations for a curved spacetime.

Secondly, although I have just said that the energies present in EM fields are generally far too small to have any significant effect on the curvature in principle strong enough EM fields could affect the curvature. Einstein's equation uses a quantity called the Einstein tensor $$\mathbf G$$ to describe the curvature, and this is related to the stress energy tensor $$\mathbf T$$ by:

$$\mathbf G = \frac{8\pi G}{c^4} \mathbf T$$

Under most conditions the stress energy tensor is dominated by the mass present, so the curvature just depends on the mass (and the angular momentum if the mass is rotating). We expect this to be the case for all the black holes we expect to see through our telescopes. However EM fields do contribute to the stress-energy tensor and when EM fields are present the equation becomes:

$$\mathbf G = \frac{8\pi G}{c^4} (\mathbf T + \mathbf T_{EM} )$$

where $$\mathbf T_{EM}$$ is the electromagnetic stress-energy tensor. For example if we include the effect of charge in calculating the spacetime curvature of a black hole we get the Reissner-Nordström black hole.

However I should emphasise that in real life it would take ridiculously strong EM fields to have any significant effect on the spacetime curvature, and in practice we don't expect this to happen.

• That makes sense. Thanks for the explanation and fixing a wrong concept in my head! – Sanjit Sarda Dec 31 '19 at 16:22

Yes, Gravitational Fields do interact with magnetic and electric fields. It can be termed as Gravitoelectromagnetism. From the law of conservation of energy we obtained these equations as,

$$\Delta g = \sqrt{fG \epsilon_0 E}$$

and

$$\Delta g = \sqrt{\frac{fg}{\mu_0 B}}$$ Here the variation of gravitational acceleration Δg is determined with the magnetic flux density B and the electric field intensity E. Where f is the gravitational redshift parameter.

• Gravitoelectromagnetism is the gravitational analog of electromagnetism, not an interaction with it. – my2cts Dec 31 '19 at 7:43
• Please, try to make formulas more readable by using mathjax formatting: math.meta.stackexchange.com/questions/5020/… – GiorgioP Dec 31 '19 at 7:49