Question

How to derive the relation satisfied by “gravitational magnetic field” from an equation of the Weyl tensor?

Answer 1

Let us call the spacetime MM with a metric gabgab. There is a unit spacelike vector field ηaηa orthogonal to a hypersurface. So that we can define the so-called gravitational electric and magnetic fields using Weyl tensor:

Eab=CambnηmηnEab=Cambnηmηn, Bab=∗CambnηmηnBab=∗Cambnηmηn

where ∗Cambn=ϵampqCpqbn∗Cambn=ϵampqCddbnpq with ϵampqϵampq the alternating tensor. This definition is not the standard one, but it does not matter in my question. Plus, I should not call Eab,BabEab,Babgravitational electric, magnetic fields because ηaηais not timelike. But let me use these names to represent the two fields. This is just nomenclature.

The Weyl tensor satisfies the following relation:

[Math Processing Error]∇[mCab]cd=2gc[mCab]pqηp+2gd[mCab]cpηp+η[mCab]cd

I want to show that

D[aEb]c=0,D[aBb]c=0D[aEb]c=0,D[aBb]c=0

Here, DaDa is the covariant derivative compatible with the induced metric hab=gab−ηaηbhab=gab−ηaηb.

It is easy to prove D[aEb]c=0D[aEb]c=0, by simply contracting ηqηcηqηc with both sides of the equation of Weyl tensor and obtaining:

∇mEbd−∇bEmd=ηmEbd−ηbEmd∇mEbd−∇bEmd=ηmEbd−ηbEmd

and then projecting this to the hypersurface to get the result.

But it is not so easy to get the 2nd result: D[aBb]c=0D[aBb]c=0. One possible method is to contract ηfηbϵcdefηfηbϵcdef, then you will get the following complicated expression:

Hope it helps you

Answer 2

Let us call the spacetime M with a metric gab. There is a unit spacelike vector field ηa orthogonal to a hypersurface. So that we can define the so-called gravitational electric and magnetic fields using Weyl tensor:

Eab=Cambnηmηn, Bab=∗Cambnηmηn

where ∗Cambn=ϵampqCpqbn with ϵampq the alternating tensor. This definition is not the standard one, but it does not matter in my question. Plus, I should not call Eab,Bab gravitational electric, magnetic fields because ηa is not timelike. But let me use these names to represent the two fields. This is just nomenclature.

The Weyl tensor satisfies the following relation:

∇[mCab]cd=2gc[mCab]pqηp+2gd[mCab]cpηp+η[mCab]cd

I want to show that

D[aEb]c=0,D[aBb]c=0

Here, Da is the covariant derivative compatible with the induced metric hab=gab−ηaηb.

It is easy to prove D[aEb]c=0, by simply contracting ηqηc with both sides of the equation of Weyl tensor and obtaining:

∇mEbd−∇bEmd=ηmEbd−ηbEmd

and then projecting this to the hypersurface to get the result.

But it is not so easy to get the 2nd result: D[aBb]c=0. One possible method is to contract ηfηbϵcdef, then you will get the following complicated expression:

∇mBae−∇aBme=4ηb(ϵefabEmd−ϵefmbEad)−2ηb∗Ceamb−(ηmBae−ηaBme)