1. Electromagnetism in Curved Spacetime
Cho Minseok

2. Electromagnetic Potential and Field
The electromagnetic potential
𝐴 𝑎
The electromagnetic field :
𝐹 𝛼𝛽 = 𝜕 𝛼 𝐴 𝛽 − 𝜕 𝛽 𝐴 𝛼
𝜕 𝛼 =𝜕/𝜕 𝑥 𝛼 : partial derivative

3. Electromagnetic Field
The electromagnetic field
𝐹 𝛼𝛽 = 𝜕 𝛼 𝐴 𝛽 − 𝜕 𝛽 𝐴 𝛼
The electromagnetic field satisfies
𝜕 𝛼 𝐹 𝛽𝛾 + 𝜕 𝛽 𝐹 𝛾𝛼 + 𝜕 𝛾 𝐹 𝛼𝛽 =0
Faraday’s law
Gauss’s law

4. Electromagnetic Displacement Field
The electric displacement 𝐷 and magnetic field 𝐻
𝐷 =∗ 𝐸 + 𝑃
𝐻 =∗ 𝐵 − 𝑀
∗ : Hodge dual
The tensors 𝜀 and 𝜇 −1
𝜀 0 𝑎𝑏 𝑐 = 𝜀 0 𝜖 𝑎𝑏𝑑 𝑔 𝑑𝑐
𝜇 0 −1 𝑎 [𝑏𝑐] = 𝜇 0 −1 𝜖 𝑎𝑑𝑒 𝑔 𝑑𝑏 𝑔 𝑒𝑐

5. Electromagnetic Displacement Field
In natural units, 𝜀 0 = 𝜇 0 −1 =𝑐=1.
Then electromagnetic displacement
𝒟 𝜇𝜈 = 𝑔 𝜈𝛼 𝐹 𝛼𝛽 𝑔 𝛽𝜈 −𝑔 − ℳ 𝜇𝜈
ℳ 𝜇𝜈 : magnetization-polarization tensor

6. Electric Current Densiity
The electric current density is the divergence of 𝒟 𝜇𝜈 In a vacuum,
𝐽 𝜇 = 𝜕 𝜈 𝒟 𝜇𝜈
Ampere’s law
Gauss’s law
𝒟 𝜇𝜈 : antisymmetric →
𝜕 𝜇 𝐽 𝜇 = 𝜕 𝜇 𝜕 𝜈 𝒟 𝜇𝜈 =0
electric current density conservation

7. Lorentz Force Density and Lagrangian
The Lorentz force density
𝑓 𝜇 = 𝐹 𝜇𝜈 𝐽 𝜈
The Lagrangian density of classical electrodynamics
ℒ=− 1 4 𝐹 𝛼𝛽 𝐹 𝛼𝛽 −𝑔 + 𝐴 𝛼 𝐽 𝛼

8. Electromagnetic Stress-Energy Tensor
The stress-energy tensor
𝑇 𝜇𝜈 = 𝐹 𝜇𝛼 𝑔 𝛼𝛽 𝐹 𝛽𝜈 − 1 4 𝑔 𝜇𝜈 𝐹 𝜎𝛼 𝑔 𝛼𝛽 𝐹 𝛽𝜌 𝑔 𝜌𝜎
The stress-energy tensor is traceless
𝑇 𝜇𝜈 𝑔 𝜇𝜈 =0
massless photon

9. Electromagnetic Stress-Energy Tensor
We can define a mixed tensor density
𝔗 𝜇 𝜈 ≡ 𝑇 𝜇𝛾 𝑔 𝛾𝜈 −𝑔
Then we can find
𝛻 𝑏 𝔗 𝑎 𝑏 + 𝑓 𝑏 =0
− 𝜕 𝜈 𝔗 𝜇 𝜈 =− Γ 𝜇𝜈 𝜎 𝔗 𝜎 𝜈 + 𝑓 𝜇
EM energy ↓
Work done by EM field on G
Work done on matter = +

10. Electromagnetic Wave Equation
The wave equation is modified in two ways.
generalization of the Lorenz gauge
&
covariant derivative
𝛻 𝑎 𝐴 𝑎 =0
□ 𝐴 𝑎 =− 𝐽 𝑎 + 𝑅 𝑏 𝑎 𝐴 𝑏
The wave equation
□ 𝐹 𝑎𝑏 =−2 𝑅 𝑎𝑐𝑏𝑑 𝐹 𝑐𝑑 + 𝑅 𝑎𝑒 𝐹 𝑏 𝑒 − 𝑅 𝑏𝑐 𝐹 𝑎 𝑒 + 𝛻 𝑏 𝐽 𝑎 − 𝛻 𝑎 𝐽 𝑏

11. Maxwell’s equation The Einstein field equation
𝐺 𝑎𝑏 =8𝜋𝐺 Λ 𝑔 𝑎𝑏 + 𝐹 𝑎𝑐 𝑔 𝑐𝑑 𝐹 𝑑𝑏 − 1 4 𝑔 𝑎𝑏 𝐹 𝑐𝑑 𝑔 𝑑𝑒 𝐹 𝑒𝑓 𝑔 𝑓𝑐

12. In the differential geometric view
The Maxwell’s equation
𝛻∧𝐹=0
𝛻∧ ∗𝐹=𝐽

13. Reference me
Christos G Tsagas. (2005). Electromagnetic fields in curved spacetimes