The Field Equation, Euler-Lagrange and Metrics

Question 1

Covariant Derivative - Parallel Transport Interpretation

Answer 1

• the covariant derivative of a vector measures how much it changes relative to what it would have been if parallel transported
• remember covariant derivative in the direction along parallel transport vanishes

Question 2

Riemann Tensor

Definition

Answer 2

• the most common way to express the curvature of Riemannian manifolds
• it assigns a tensor to each point of a Riemann manifold that measures the extent to which the metric tensor is not locally isometric to that of Euclidian space

Question 3

Riemann Tensor in Flat Space

Answer 3

-in completely flat space:
g_μν -> η_μν
-and Γ^α_βγ and its derivatives vanish
=>
R^ρ_σμν = 0
-there is no real gravity

Question 4

How many values of the Riemann tensor are there?

Answer 4

-in N dimensions there are:
1/12 N² (N² - 1)
-independent components

Question 5

Symmetry of the Riemann Tensor

Answer 5

-many of the R^ρ_σμν are identical
-partly due to torsion free metric compatibility
R_ασμν = - R_σαμν
AND
R^ρ_σμν = - R^ρ_σνμ

Question 6

Riemann Tensor in 1D

Answer 6

-in 2D there are:
1/12 1² (1² - 1) = 0
-independent componenets of the Riemann tensor
=>
-cannot have curvature in 1D

Question 7

Ricci Tensor

Definition

Answer 7

R_μν = R^λ_μλν

Question 8

Curvature Scalar

Definition

Answer 8

R = g^αβ R_αβ

Question 9

Einstein Tensor

Definition

Answer 9

G^αβ = R^αβ - 1/2 g^αβ R =

Question 10

Einstein Tensor

Symmetry

Answer 10

G^αβ = G^βα

-symmetric since both R^αβ and g^αβ are symmetric

Question 11

Einstein Tensor

Covariant Derivative

Answer 11

G^αβ_;β = 0

Question 12

The Field Equation

Equation

Answer 12

R_μν - 1/2 g_μν R = k T_μν

Question 13

The Field Equation

Interpretation

Answer 13

R_μν - 1/2 g_μν R = k T_μν

• LHS is stuff, RHS is curvature
• no stuff <=> no curvature

Question 14

Action for General Relativity

Answer 14

S = ∫ dS
= ∫√ [gαβ dx^α dx^β]
= ∫√ [gαβ dx^α/dλ dx^β/dλ] dλ

Question 15

Euler-Lagrange Equation

Answer 15

d/dλ (∂L/∂x^α’) - ∂L/∂x^α = 0
-with the Lagrangian:
L = √ [gαβ dx^α/dλ dx^β/dλ] = ds/dλ

Question 16

Euler-Lagrange Equation

L’ = L²

Answer 16

-replacing L with L²:
L² = gαβ dx^α/dλ dx^β/dλ
-recovers the same Euler Lagrange equation
-so use this instead as it simplifies things by removing the square root

Question 17

How to get the Lagrangian from the metric?

Answer 17

ds² = u(r) dr² +u(θ) dθ²
=>
L = u(r) (dr/dλ)² + u(θ) (dθ/dλ)²

Question 18

How to apply the Euler Lagrange Equation

Answer 18

d/dλ (∂L/∂x^α’) - ∂L/∂x^α = 0

• find L
• then work through α for each index i.e. α=r, α=θ etc.

Question 19

Euler-Lagrange Equation

Geodesic Equation

Answer 19

-when you sub into the Euler Lagrange equation from the metric you recover the geodesic equation

Question 20

Euler-Lagrange Equation

Connection Coeffcients

Answer 20

-the geodesic equation from the Euler Lagrange equation must match this form of the geodesic equation:
d²x^α/dλ² + Γ^α_μβ dx^μ/dλ dx^β/dλ = 0

Question 21

Euler-Lagrange Equation

Constants of Motion

Answer 21

-no explicit dependence on a given variable, then:
∂L/∂x^α = 0
=>
-can identify constants of motion:
∂L/∂x^α’ = const.
-but with L = gαβ x^α’ x^β’ this just means:
gγα x^α’ = const.
-the specific nature of the constant depends on the problem

Question 22

What if gμν is diagonal?

Answer 22

• reflects underlying symmetry

- many of the Γ -> 0 and finding g^μν is trivial

Question 23

Requirements of a Metric for GR

Answer 23

• gμν can’t be singular everywhere or it has no inverse
• but it is ok to have e.g. a single point where det(gμν) is singular
• a ‘good’ metric will be generally well behaved and ideally reflect any underlying symmetry

Question 24

Metrics

k & Rs

Answer 24

-Rs is the radius of curvature and k=1/Rs²
-so:
k = 0 - flat space
k > 0 - positively curved, spherical three-space
k < 0 - negatively curved, hyperbolic