Exercises on General Relativity and Cosmology

Physikalisches Institut Exercise 08

Universit¨at Bonn 1 June 2011

Theoretische Physik SS 2011

Exercises on General Relativity and Cosmology

Priv. Doz. Dr. S. F¨orste

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Due 8 June 2011

Exercise 8.1: Noether’s theorem (14 credit s)

(a) Consider the action:

S = Z

d^dxL(Φ, ∂_µΦ)

with the following large “active” transformation of, both, the position and of the field:

x → x⁰ ; Φ(x) → Φ⁰(x⁰) = F(Φ(x)). Verify the following form of the transformed

action: (2 credit s)

S⁰ ≡ Z

d^dx L(Φ⁰(x), ∂_µΦ⁰(x))^verify= Z

d^dx

∂x⁰

∂x

L

F(Φ(x)), ∂x^ν

∂x^0µ∂_νF(Φ(x))

(b) Consider a (active) large Lorentz transformation: x^0µ= Λ^µ_νx^ν and Φ⁰(Λx) = L_ΛΦ(x).

(where, Λ is the Lorentz transformation matrix andL_Λ is an appropriate representa- tion of the Lorentz group on the field). Show that the transformed action of 8.1(a)

takes the following form: (2 credit s)

S⁰ = Z

d^dx(L_ΛΦ,(Λ⁻¹ ·∂(L_ΛΦ))_µ) (c) Now consider an infinitesimal (small) active transformation:

x^0µ =x^µ+ω_aδx^µ

δω_a ; Φ⁰(x⁰) = Φ(x) +ω_aδF δω_a(x)

where, {ω_a} is a set of infinitesimal parameters. If the generator G_a is defined as follows:

δ_ωΦ(x)≡Φ⁰(x)−Φ(x)≡ −iω_aG_aΦ(x)

Show that it is explicitly given by: (2 credit s)

iG_aΦ = δx^µ δωa

∂_µΦ− δF δωa

(d) Consider an (active) infinitesimal Lorentz transformation:

x^0µ=x^µ+ω^µ_νx^ν ; F(Φ) ≡L_ΛΦ≈

1−1

2iω_ρνS^ρν

Φ

Show that the generators of the Lorentz transformations are: (2 credit s) L^ρν =i(x^ρ∂^ν−x^ν∂^ρ) +S^ρν

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(e) Plug in the infinitesimal transformations of 8.1(c), with varyingω_a, in 8.1(a) and ex- pand Lagrangian to obtain the followingclassically conserved current: (4 credit s)

j_a^µ=

∂L

∂(∂_µΦ)∂_νΦ−δ_ν^µL δx^ν

δω_a − ∂L

∂(∂_µΦ) δF δω_a

Hint: Use det(1+E) ≈ 1 + Tr(E) for small E. Also, assume that the action is invariant under rigid (ie. constantω_a) transformations

(f) Show that the above current for the Lorentz transformation of 8.1(d) is: (2 credit s) j^µνρ =T_c^µνx^ρ−T_c^µρx^ν + 1

2i ∂L

∂(∂_µΦ)S^νρΦ

where, T_c, is the canonical energy-momentum tensor derived in class.

Exercise 8.2: Various definitions of energy-momentum tensor (6 credit s) In general, the canonical tensor,T_c^µν, is not symmetric. But, since the following modifi- cation does not change the classical conservation property:

T_B^µν =T_c^µν+∂ρB^ρµν ; B^ρµν =−B^µρν

we can hope to get aclassically (ie. only for field configurations obeying e.o.m) symmetric T_B^µν, called the Belinfante tensor. The tensorB^ρµν is, by no means, unique.

(a) Show that one possible choice of B^ρµν is: (2 credit s) B^ρµν = 1

4i

∂L

∂(∂_µΦ)S^νρΦ + ∂L

∂(∂_ρΦ)S^µνΦ + ∂L

∂(∂_νΦ)S^µρΦ

Hint: Use the fact thatS^µν =−S^νµ and use ∂_µj^µνρ = 0 of ex-8.1(f) above.

(b) Consider the following Lagrangian for a massive vector fieldA_µ (in Euclidian space-

time): (2 credit s)

L = 1

4F^αβF_αβ+ 1

2m²A^αA_α

where, F_αβ =∂_αA_β −∂A_βA_α. Write down its Belinfante tensor forB^αµν =F^αµA^ν (c) As was said above, the tensor of 8.2(b) will be symmetric only classically. But, there

exists another common (cf. exercise 0.3(c) for m = 0) expression for an identically (ie. classical and otherwise) symmetric energy-momentum tensor of a vector field:

Tˆ_B^µν =F^µαF^ν_α− 1

4η^µνF^αβF_αβ+m²

A^µA^ν− 1

2η^µνA^αA_α

Show that it coincides with the Belinfante tensor for classical configurations. (2 credit s) Remark: TheHilbert tensor, derived in class, is an identically symmetric energy-momentum tensor. Note that these are all non-gravitational (ie. matter) energy-momentum tensors.

The list of gravitational tensor candidates is longer and contentious.

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