# B and A in Curved Space Time: Does \nabla \times A =B?

• A
• yourWitten
In summary, the vector potential remains the same in curved space as long as the torsion term is zero.
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.

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
yourWitten said:
@Orodruin I am familiar with the covariant expressions of Maxwell's equations but having trouble working this out. In particular http://sedici.unlp.edu.ar/bitstream/handle/10915/125010/Documento_completo.pdf-PDFA.pdf?sequence=1 Eq. 2.24 suggests that the formula remains the same, but I don't see why.
Maybe I'm wrong, but isn't this simply because the Maxwell equations in general covariant form are the same as the equations in flat spacetime because the torsion vanishes? If they retain the same form, the expressions for B and E in terms of the gauge field don't change either.

vanhees71

## 1. What is the relationship between B and A in curved space-time?

The relationship between B and A in curved space-time is described by the equation \nabla \times A = B. This equation is known as the Maxwell's equations in curved space-time and it relates the magnetic field (B) to the vector potential (A).

## 2. Can B and A be measured separately in curved space-time?

Yes, B and A can be measured separately in curved space-time. B can be measured using a magnetic field sensor while A can be measured using a vector potential sensor. However, since they are related by the equation \nabla \times A = B, their measurements will be interdependent.

## 3. How does the curvature of space-time affect the relationship between B and A?

The curvature of space-time affects the relationship between B and A by changing the value of the vector potential A. In curved space-time, the vector potential is not a simple function of the magnetic field B, as it is in flat space-time. Instead, it is influenced by the curvature of space-time, which can change the direction and strength of the magnetic field.

## 4. Is the equation \nabla \times A = B valid in all types of curved space-time?

Yes, the equation \nabla \times A = B is valid in all types of curved space-time. This equation is a fundamental law of electromagnetism and it holds true in both flat and curved space-time. However, in curved space-time, the vector potential A may have a more complicated form due to the curvature of space-time.

## 5. How does the equation \nabla \times A = B relate to Einstein's theory of general relativity?

The equation \nabla \times A = B is a part of the Maxwell's equations, which describe the behavior of electromagnetic fields. These equations are consistent with Einstein's theory of general relativity, which describes the curvature of space-time and its effect on the behavior of matter and energy. In fact, Einstein's theory predicts the existence of gravitational waves, which can be described using the same mathematical framework as electromagnetic waves, further linking the two theories.

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