GRAVITATION F10
S. G. RAJEEV
Lecture 12
1.
Maxwell’s Equations in Curved SpaceTime
1.1.
Recall that Maxwell equations in Lorentz covariant form are.
∂
μ
F
μ
ν
=
j
ν
,
F
μ
ν
=
∂
μ
A
ν
−
∂
ν
A
μ
.
1.2.
They follow from the variational principle.
S
=
1
4
ˆ
F
μ
ν
F
μ
ν
dx
+
ˆ
j
μ
A
μ
dx
First,
δ
S
=
ˆ
F
μ
ν
∂
μ
δ
A
ν
dx
+
ˆ
j
ν
δ
A
ν
dx
Now integrate by parts the first term.
1.3.
This leads to a wave equation with source for the electromagnetic
potential.
∂
μ
∂
μ
A
ν
−
∂
ν
[
∂
μ
A
μ
] =
j
ν
It is common to impose the condition
∂
μ
A
μ
= 0
,(the Lorentz gauge) taking
advantage of the gauge invariance
A
μ
→
A
μ
+
∂
μ
Λ
. Then each component of
A
μ
satisfies the wave equation
∂
μ
∂
μ
A
ν
=
j
ν
1.4.
The generally covariant form of Maxwel’s equations is.
D
μ
F
μ
ν
=
j
ν
,
F
μ
ν
=
∂
μ
A
ν
−
∂
ν
A
μ
Recall that the Christo
ff
el symbols cancel out in the antisymmetric derivative of
a covariant vector.
1.5.
In terms of potentials.
D
μ
D
μ
A
ν
−
D
μ
D
ν
A
μ
=
j
μ
We cannot interchange the derivatives in the second term without introducing
some terms involving curvature.
1.6.
An equivalent form of the curved space Maxwell’s equations is.
1
√
−
det
g
∂
μ
−
det
gg
μ
ρ
g
νσ
F
ρσ
=
j
ν
1
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GRAVITATION F10
2
1.7.
This follows from the covariant variational principle.
S
=
1
4
ˆ
F
μ
ν
F
ρσ
g
μ
ρ
g
νσ
−
det
gdx
+
ˆ
j
μ
A
μ
−
det
gdx
1.8.
These equations tell us how the gravitational field a
ff
ects the propa
gation of light.
For example it can tell us how light is di
ff
racted and refracted by
a gravitational field.
Spectacular phenomena such as gravitational lensing follow
from this. More later.
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 Spring '11
 Fred
 Equations, General Relativity, Fundamental physics concepts, Energy density, Aµ, det gdx

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