I Maxwell's Equations & GR: How Scientists Got Away With It

[Sorcerer]

Maxwell’s Equations are Lorentz invariant, so they are valid in inertial reference frames, right? However, the surface of Earth is not truly an inertial reference frame, yet the experiments that led to Maxwell’s equations were all done on the surface of Earth.

Does that not pose a small problem?

(a) How were scientists able to get away with this and still have an accurate theory? Is it because the Earth is big enough compared to the experimental set ups that spacetime was close enough to being flat that the divergence in results was too small to notice?

(b) I take it there are general relativistic modifications to Maxwell’s Equations? Could anyone point me to a source that explains the differences between the flat spacetime Maxwell Equations and the GR versions?Thanks!

 

[sweet springs]

Maxwell equations survive replacing usual derivatives by covariant derivatives.

 

[Sorcerer]

Maxwell equations survive replacing usual derivatives by covariant derivatives.

Oh my, new operations to look up. Thanks!

 

[bhobba]

Oh my, new operations to look up. Thanks!

A better view IMHO is the Lagrangian view - it also explains why:
https://en.wikipedia.org/wiki/Lagrangian_(field_theory)#Electromagnetism_in_general_relativity

Thanks
Bill

 

[Orodruin]

I don't think you need to go to curved spacetime to understand the basics of (a). An accelerated observer in Minkowski should do fine locally to deal with the OP's question regarding the effect of acceleration. You should get something like $\nabla \to \nabla + \vec \alpha/c^2$ in the sourced Maxwell's equations. If the correction term $\sim \vec \alpha/c^2$ is small compared to the field derivatives you can forget about the effects of acceleration and $c^2/\alpha$ gives you the typical length scale over which the effects would be relevant. For $\alpha \sim 10\ \mbox{m/s}^2$, you would find a relevant length scale of about a lightyear (coincidentally, since a year is roughly $3\cdot 10^7$ seconds and therefore $c/\alpha \sim 1$ year).

So:

(a) How were scientists able to get away with this and still have an accurate theory? Is it because the Earth is big enough compared to the experimental set ups that spacetime was close enough to being flat that the divergence in results was too small to notice?

No. It is because the Earth gravity is way too weak for the effect to be noticeable at relevant length scales.