Wolfram Alpha:
Search by keyword:
Astronomy
Chemistry
Classical Physics
Climate Change
Cosmology
Finance and Accounting
General Relativity
Lagrangian and Hamiltonian Mechanics
Macroeconomics
Mathematics
Microeconomics
Particle Physics
Probability and Statistics
Programming and Computer Science
Quantum Field Theory
Quantum Mechanics
Semiconductor Reliability
Solid State Electronics
Special Relativity
Statistical Mechanics
String Theory
Superconductivity
Supersymmetry (SUSY) and Grand Unified Theory (GUT)
test
The Standard Model
Topology
Units, Constants and Useful Formulas
Electromagnetic 4 - Potential
-----------------------------
A^{μ} = (φ/c, A) where φ is the electric potential and A
is the magnetic vector potential. Therefore,
A_{μ} = η_{μν}A^{ν} = (φ/c, -A) η is the Minkowski metric of
the form +---
The corresponding 4 - gradient operators are,
∂_{μ} = ∂/∂x^{μ} = {(1/c)∂/∂t, ∇}
and
∂^{ν} = η^{μν}∂_{μ} = {(1/c)∂/∂t, -∇}
Now,
E = -∂A/∂t - ∇φ
Consider only x dimension,
E_{x} = -∂A_{1}/∂t - ∂φ/∂x
Now, from before,
∂_{0}A_{1} = (1/c)∂A_{1}/∂t + ∂(φ/c)/∂x
Therefore, in order to get the RHS to equal E_{x}, we
have to multiply by -c.
E_{x} = -c{∂A_{1}/∂t - ∂φ/∂x}
Similarly, for the B field we can find the B_{x} component
of the curl.
B_{x} = ∇ x A_{1} = ∂_{2}A_{3} - ∂_{3}A_{2}
So E and B have the same form. We can now define an
asymmetric tensor, F_{μν} such that,
F_{μν} = ∂_{μ}A_{ν} - ∂_{ν}A_{μ}
where,
F_{00} = ∂_{0}A_{0} - ∂_{0}A_{0} = 0
F_{01} = ∂_{0}A_{1} - ∂_{1}A_{0} = -E_{x}/c
F_{02} = ∂_{0}A_{2} - ∂_{2}A_{0} = -E_{y}/c
F_{03} = ∂_{0}A_{3} - ∂_{3}A_{0} = -E_{z}/c
F_{10} = ∂_{1}A_{0} - ∂_{0}A_{1} = E_{x}/c
F_{11} = ∂_{1}A_{1} - ∂_{1}A_{1} = 0
F_{12} = ∂_{1}A_{2} - ∂_{2}A_{1} = B_{z}
F_{13} = ∂_{1}A_{3} - ∂_{3}A_{1} = -B_{y}
F_{20} = ∂_{2}A_{0} - ∂_{0}A_{2} = E_{y}/c
F_{21} = ∂_{2}A_{1} - ∂_{1}A_{2} = -B_{z}/c
F_{22} = ∂_{2}A_{2} - ∂_{2}A_{2} = 0
F_{23} = ∂_{2}A_{3} - ∂_{3}A_{2} = B_{x}
F_{30} = ∂_{3}A_{0} - ∂_{0}A_{3} = E_{z}/c
F_{31} = ∂_{3}A_{1} - ∂_{1}A_{3} = B_{y}/c
F_{32} = ∂_{3}A_{2} - ∂_{2}A_{3} = -B_{x}
F_{33} = ∂_{3}A_{3} - ∂_{3}A_{3} = 0
so we can write (note by convention the B components
are written as the -ve of the actual values).
- -
| 0 E_{x}/c E_{y}/c E_{z}/c |
|-E_{x}/c 0 -B_{z} B_{y} | = F_{μν} ... The COVARIANT
|-E_{y}/c B_{z} 0 -B_{x} | ELECTROMAGNETIC
|-E_{z}/c -B_{y} B_{x} 0 | TENSOR
- -
- -
| 0 -E_{x}/c -E_{y}/c -E_{z}/c |
| E_{x}/c 0 -B_{z} B_{y} | = F^{μν} ... The CONTRAVARIANT
| E_{y}/c B_{z} 0 -B_{x} | ELECTROMAGNETIC
| E_{z}/c -B_{y} B_{x} 0 | TENSOR
- -
Note: F_{μν} = η_{μα}η_{νβ}F^{αβ} and F^{μν} = η^{μα}η^{νβ}F_{αβ}
Properties:
F^{μν} = -F^{νμ} ... antisymmetric
F^{μν} is gauge invariant but not Lorentz invariant (see note
on Gauge Theory).
Inner product:
F_{μν}F^{μν} = 2(B^{2} - E^{2}/c^{2}) which is invariant under a Lorentz
transformation.
Maxwell's Equation in Flat Spacetime
------------------------------------
∇.E = ρ/ε and ∇ x B = μ_{0}J + (1/c^{2})∂E/∂t reduce to,
∂F^{μν}/∂x^{μ} = ∂_{μ}F^{μν} = μ_{0}j^{ν}
where j^{ν} = ρ∂x^{ν}/∂t = (cρ, j) ... the 4-current
∇.B = 0 and ∇ x E = -∂B/∂t reduce to,
∂F_{μν}/∂x^{γ} + ∂F_{νγ}/∂x^{μ} + ∂F_{γμ}/∂x^{μ} = ∂_{γ}F_{μν} + ∂_{μ}F_{νγ} + ∂_{ν}F_{γμ} = 0
Maxwell's Equation in Curved Spacetime
--------------------------------------
Replace the partial derivative with the covariant
derivative to get,
∇_{μ}F^{μν} = μ_{0}J^{ν}
∇_{γ}F_{μν} + ∇_{μ}F_{νγ} + ∇_{ν}F_{γμ} = 0
This is equal to,
(∂_{γ}F_{μν} - Γ^{σ}_{γμ}F_{σν} - Γ^{σ}_{γν}F_{μσ}) + (∂_{μ}F_{νγ} - Γ^{σ}_{μν}F_{σγ} - Γ^{σ}_{μγ}F_{νσ})
+ (∂_{ν}F_{γμ} - Γ^{σ}_{νγ}F_{σμ} - Γ^{σ}_{νμ}F_{γσ}) = 0
We can use the antisymmetric properties of F_{ab} = -F_{ba} to get,
(∂_{γ}F_{μν} - Γ^{σ}_{γμ}F_{σν} - Γ^{σ}_{γν}F_{μσ}) + (∂_{μ}F_{νγ} - Γ^{σ}_{μν}F_{σγ} + Γ^{σ}_{μγ}F_{σν}) +
+ (∂_{ν}F_{γμ} + Γ^{σ}_{νγ}F_{μσ} Γ^{σ}_{νμ}F_{σγ}) = 0
We can use the symmetry of the Christofell symbols
Γ^{σ}_{αβ} = Γ^{σ}_{βα} to get,
∂_{γ}F_{μν} + ∂_{μ}F_{νγ} + ∂_{ν}F_{γμ} - Γ^{σ}_{γμ}F_{σν} + Γ^{σ}_{γμ}F_{σν} - Γ^{σ}_{γν}F_{μσ}
+ Γ^{σ}_{γν}F_{μσ} - Γ^{σ}_{μν}F_{σγ} + Γ^{σ}_{μν}F_{σγ} = 0
Which simplifies to,
∂_{γ}F_{μν} + ∂_{μ}F_{νγ} + ∂_{ν}F_{γμ} = 0