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Unformatted text preview: 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 µν = j ν , F µ ν = ∂ µ Aν − ∂ ν Aµ . 1.2. They follow from the variational principle.
ˆ
ˆ
1
S=
F µν Fµν dx + j µ Aµ dx
4
First,
ˆ
ˆ
µν
δ S = F ∂µ δ Aν dx + j ν δ Aν dx
Now integrate by parts the ﬁrst term.
1.3. This leads to a wave equation with source for the electromagnetic
potential.
∂ µ ∂ µ Aν − ∂ ν [ ∂ µ Aµ ] = j ν It is common to impose the condition ∂µ Aµ = 0,(the Lorentz gauge) taking
advantage of the gauge invariance Aµ → Aµ + ∂µ Λ. Then each component of Aµ
satisﬁes the wave equation
∂ µ ∂ µ Aν = j ν
1.4. The generally covariant form of Maxwel’s equations is.
D µ F µν = j ν , F µ ν = ∂ µ Aν − ∂ ν Aµ Recall that the Christoﬀel symbols cancel out in the antisymmetric derivative of
a covariant vector.
1.5. In terms of potentials.
D µ D µ Aν − D µ D ν Aµ = 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
√
∂µ
− det gg µρ g νσ Fρσ = j ν
− det g
1 GRAVITATION F10 2 1.7. This follows from the covariant variational principle.
ˆ
ˆ
1
S=
Fµν Fρσ g µρ g νσ − det gdx + j µ Aµ − det gdx
4 1.8. These equations tell us how the gravitational ﬁeld aﬀects the propagation of light. For example it can tell us how light is diﬀracted and refracted by
a gravitational ﬁeld. Spectacular phenomena such as gravitational lensing follow
from this. More later.
2. Conservation Laws
2.1. The electric current is a Lorentz vector ﬁeld jµ satisfying.
Dν j ν = 0
2.1.1. This follows from Maxwel’s equations.
D µ F µν = j ν
First,
D ν D µ F µν = D ν j ν
By antisymmetry, l.h.s.
µ
ν
[Dν , Dµ ]F µν = Rν µρ F ρν + Rν µρ F µρ = Rνρ F ρν + Rµρ F µρ The contraction
µ
Rνρ = Rν µρ is called the Ricci tensor. It plays an important role in GR. For now, we just
need that it is symmetric (consequence of the symmetries of the Riemann tensor).
2.2. An equivalent form of the divergence of a vector ﬁeld is.
Dµ j µ = ∂µ
− det gj µ
This follows as for the wave equation 2.3. The conservation of electric charge follows. Consider a region in spacetime bounded by two spacelike surfaces x0 = T1 , x0 = T2 , with T2 > T1 . The
integral of Dµ j µ in this region can (using Gauss’s theorem)
ˆ T1 j 0 1 2 3 − det gdx dx dx = ˆ T2 j0 − det gdx1 dx2 dx3 (We are assuming that the electric current vanishes at spatial inﬁnity.) Thus j 0
integrated with respect to the invariant volume measure is a conserved quantity.
Such conservation laws are very important and we should understand them in
several diﬀerent ways. GRAVITATION F10 3 2.4. Charge conservation follows from gauge invariance. Under the gauge
transformation
Aµ → Aµ + ∂µ Λ
the change in the Maxwell action is
ˆ
ˆ
j µ − det g ∂µ Λdx = −
∂µ
− det gj µ Λdx When the equation of motion is satisﬁed the action should be unchanged under
all inﬁnitesimal transformations. Gauge transformations produce a change that
does not involve the electromagnetic ﬁeld, only its source. Hence the source must
satisfy the identity
∂µ
− det gj µ = 0
if the action is to be extremal. 2.4.1. The electric current can be deﬁned as the variation of the ‘source’ action
w.r.t. the potential.
ˆ
δ S1
, S1 = Aµ j µ − det gdx
j µ − det g =
δ Aµ 2.5. The electric charge is the integral of electric current over a spacelike
surface. A surface is spacelike if its normal vector is timelike. The integral
ˆ
Q = j ν − det gdSν is a scalar quantity equal to the total electric charge in that region of space. The
integral of the electric current on a surface whose normal is spacelike has another
meaning: it is the ﬂux of electric charge through that surface.
3. The Stress Tensor
3.1. There is a tensor ﬁeld T µν whose integral over a spacelike surface
is energymomentum.
ˆ
P µ = T µν − det gdSν Thus T 00 is energy density and T i0 is momentum density. Of course, energy
density includes mass as well as all other forms of energy. What is the meaning of
the remaining components?
3.2. The integral over a surface with spacelike normal is the total ﬂux
of energy or momentum across that surface. If an electromagnetic wave is
propagating through a region, it carries some energy and momentum across such
a surface. The component T ij is the amount of the ith component of momentum
carried across a small surface whose normal is pointed in the j th direction. This
can also be thought of the force felt on that surface per unit area (stress) . Also
T 0j is the energy carried across this surface, or the work being done by moving
that surface an inﬁnitesimal amount in the j th direction.
Thus the tensor T µν combines the mechanical notions of energy density, momentum density, stress into a single entity. GRAVITATION F10 4 3.3. The stress tensor is the variation of matter action w.r.t. the metric
tensor. The energy and momentum denisty of matter, including the e.m. ﬁeld is
the source of gravity. So the variation w.r.t. gµν will give the stress tensor.
T µν
It follows that − det g = δ Sm
δ g µν 3.4. The tensor ﬁeld T µν is symmetric.
3.5. The stress tensor of a massless scalar ﬁeld follows from its action.
ˆ
1
Sφ =
g µν − det g ∂µ φ∂ν φdx
2
Recall that δ log det g = δ tr log g = g µν δ gµν
It is actually a bit more convenient to vary w.r.t. the inverse matrix g µν
δ log det g = −gµν δ g µν , =⇒ δ
Thus
1
δ Sφ =
2
and ˆ δg µν
1
− det g = −
− det ggµν δ g µν
2
1
ρσ
∂µ φ∂ν φ − gµν g ∂ρ φ∂σ φ
− det gdx
2 1
Tµν = ∂µ φ∂ν φ − gµν g ρσ ∂ρ φ∂σ φ
2
3.5.1. The energy density is positive in Minkowski space.
1
2
2
T00 =
[∂0 φ] + [ ∂i φ]
2
3.6. The ﬁeld equations imply that the stress tensor is conserved.
D µ T µν = 0
1
1
Dµ ∂µ φ∂ν φ − gµν g ρσ ∂ρ φ∂σ φ = ∂µ φDµ ∂ν φ − gµν g ρσ Dµ [∂ρ φ∂σ φ]
2
2
But
g ρσ Dν [∂ρ φ∂σ φ] = 2g ρσ [Dρ Dν φ] ∂σ φ = 2 [Dµ ∂ν φ] ∂µ φ
so that the two terms cancel. GRAVITATION F10 5 3.7. The stresstensor of the electromagnetic ﬁeld also can be computed
from its action. Ignoring sources for now,
ˆ
1
S=
Fµν Fρσ g µρ g νσ − det gdx
4
ˆ
ˆ
1
1
µρ νσ
δS =
Fµν Fρσ g δ g
− det gdx −
Fµν Fρσ g µρ g νσ − det ggαβ δ g αβ dx
2
8
so that
1
1
Fµρ Fνσ g ρσ − gµν g ρσ g β Fρα Fσβ
2
8
3.7.1. The energy density T00 and momentum density T0i in ﬂat space are familiar
special caes.
ˆ
ˆ
3
1 2
2
E + B d x,
E × Bd 3 x
2
Tµν = 3.8. Conservation of the stress tensor follows from Maxwell’s equations.
D µ T µν = 0
3.8.1. The stress tensor of the electromagnetic ﬁeld is also traceless.
g µν Tµν = 0 ...
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