为何Weyl张量的磁分量在球对称条件下会消失？

Let's break this down step by step, starting with what the electric and magnetic Weyl components actually represent, then tying it to spherical symmetry's strict constraints.

First, a quick recap: When we split the Weyl tensor into electric and magnetic parts, we're separating its tidal effects relative to a set of observers moving along timelike curves:

• The electric Weyl component describes the "stretching-squeezing" tidal forces, analogous to how an electric field acts on charges.
• The magnetic Weyl component describes a "twisting" tidal effect, similar to a magnetic field's rotational influence on charged particles.

Now, spherical symmetry is an extremely rigid condition—it means the spacetime looks identical no matter which direction you face or how you rotate around the central object. Here's why this eliminates the magnetic component:

1. No preferred direction = no room for a "twist"
   The magnetic Weyl component acts as a 3D antisymmetric tensor (or equivalently, an axial vector). For such a tensor to be non-zero, there has to be a clear preferred direction associated with it—the axis around which the twist occurs. But spherical symmetry explicitly forbids any preferred direction: if you had a non-zero magnetic component pointing along some axis, rotating the spacetime 90 degrees around another axis would shift that component's direction, which directly contradicts the spacetime being spherically symmetric. The only way to satisfy this symmetry rule is for the magnetic component to be zero everywhere.

2. Invariance under SO(3) rotations
   Mathematically, the symmetry group of a spherically symmetric spacetime is the 3D rotation group SO(3). Any non-zero magnetic Weyl component would transform in a non-trivial way under these rotations, breaking the symmetry. The only tensor that stays invariant under all SO(3) rotations (other than the metric itself) is the zero tensor—so the magnetic Weyl component has to vanish.

As a quick contrast, cylindrical symmetry has a clear preferred axis (the central axis of the cylinder). A non-zero magnetic Weyl component can align with this axis, and rotations around the axis won't alter it—so it can exist without breaking the symmetry.

内容的提问来源于stack exchange，提问作者user2129579