# Is B always the curl of A in curved space time?

• A
yourWitten
By definition of the vector potential we may write

\nabla \times A =B

at least in flat space. Does this relation hold in curved space? I am particularly interested if we can still write this in a spatially flat Friedmann-Robertson-Walker background with metric ds^2=dt^2-a^2(dx^2+dy^2+dz^2) and if not, how it should be modified.

I know this question is extremely simple but I'm still developing intuition on GR.

Staff Emeritus
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Curl as such only exists in 3-dimensional Euclidean space. You need to look at a generalisation in the 4d case of both flat and curved spacetimes, namely the exterior derivative. In relativity, both the electric and magnetic fields together form a second rank tensor F called the electromagnetic field tensor. This in turn can be written in terms of the exterior derivative as F=dA where A is the 4-potential containing both the electric scalar and the magnetic vector porential. Of course, what is what (electric/magnetic) depends on the reference frame.

• vanhees71 and PeroK
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