Grav12 - GRAVITATION F10 S. G. RAJEEV Lecture 12 1....

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Unformatted text preview: GRAVITATION F10 S. G. RAJEEV Lecture 12 1. Maxwell’s Equations in Curved Space-Time 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 space-like 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 space-like surface. A surface is space-like if its normal vector is time-like. 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 space-like 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 space-like surface is energy-momentum. 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 space-like 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 stress-tensor 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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