Parallel Transport and Covariant Differentiation

Question 1

Classical Gravitation

Answer 1

∇²φ = 4πGρ

Question 2

d’Alembertian

Answer 2

-there is a 4D equivalent for classical gravitation: square = η_μν ∂^ν ∂^μ = (-1/c² ∂²/∂t² + ∇²) -in addition to Newtonian gravity this has an instant response to the presence of a mass, violating relativity

Question 3

Equivalence Principle Weak Version

Answer 3

-the weak version just says that: ma = mg -i.e. can’t distinguish gravity form any other form of acceleration -BUT g can’t be the same everywhere, this leads to the idea of a local inertial frame, a frame which is not accelerating

Question 4

Equivalence Principle Einstein’s Variant

Answer 4

-all laws of physics will be the same in all inertial frames -this is expressed using tensor relationships

Question 5

Local Inertial Frame Definition

Answer 5

-frame which is not accelerating -at a point there is no acceleration -in local coordinates, ε^α, we therefore have: d²ε^α/dτ² = 0 AND ds² = η_αβ dε^α dε^β

Question 6

Local Inertial Frame General Metric

Answer 6

g_μν(x) = η_αβ dε^α/dx^μ dε^β/dx^ν -symmetric

Question 7

Local Inertial Frame Generic Line Element

Answer 7

ds² = g_αβ dx^α dx^β

Question 8

The Geodesic Equation

Answer 8

Γ^ν_αβ = ∂x^ν/∂ε^μ ∂²ε^μ/∂x^α∂x^β -where Γ^ν_αβ is a connection coefficient

Question 9

What properites do we need in a metric for GR?

Answer 9

-LIF must exist: gμν -> ημν locally by coordinate transformation (where g is general metric and η is the LIF metric) -g_μν must exist, a Riemannian manifold -the line element ds²=g_μν dx^μd x^ν must exist -the connection coefficients, Γ, must exist -if a tensor relation in the LIF is true, it must be true everywhere

Question 10

Transformation

Answer 10

V^’μ = ∂x^’μ/∂x^ν V^ν

• ∂x^’μ/∂x^ν is NOT a Lorentz transform since ∂²x^’μ/∂x^α∂x^β does not equal zero
• ∂x^’μ/∂x^ν is a function of position
• as long as ∂x^’μ/∂x^ν is not a singular transformation (inverse exists), 4-vectors and 1-forms behcave as normal

Question 11

Derivative of Scalars

Answer 11

φ_,μ = ∂φ/∂x^μ

-well-behaved because φ has no direction

Question 12

Derivatives of A^μ

Answer 12

A^μ_, is not well-behaved since ∂²x^’μ/∂x^α∂x^β is not zero

Question 13

Derivative of A_

Answer 13

∂A_/∂x^β = ∂/∂x^β [A^α e_α]

-use product rule:

= ∂A_/∂x^β e_α + A^α ∂e_α/∂x^β

-define: ∂e_α/∂x^β = Γ^μ_αβ e_μ

=>

∂A_/∂x^β = [A^α_,β + A^μ Γ^α_μβ] e_α

-since e_α is the basis the stuff in the brackets must e

Question 14

Covariant Derivative

Answer 14

A^α_;β = A^α_,β + A^μ Γ^α_μβ

Question 15

Covariant Derivative of Scalar

Answer 15

-we make the CHOICE that:

φ_;μ = φ_,μ

-since this makes physical sense (φ has no direction)

Question 16

Covariant Derivative of Covariant Vectors

Answer 16

A_μ;β = A_μ,β - A_α Γ^α_μβ

Question 17

Covariant Derivative of General Tensors

Answer 17

• treat as a combination of covariant and contravariant vectors
• 1Γ for each index of a general tensor
• the +/- signs on the Γ correspond to whether th index is contravariant or covariant
• e.g.

A^μ_ν;β = A^μ_ν,β + Γ^μ_αβ A^α_ν - Γ^α_νβ A^μ_α

Question 18

Properties of Γ and the Metric

Answer 18

1. metric compatibility
2. torsion free
3. Γ^γ_βμ = 1/2 g^αγ [g_αβ,μ + g_αμ,β - g_βμ,α]
4. Γ does not transform as a 1/2 tensor so is not a tensor itself
5. Γ depends on g_μν,λ terms

Question 19

Properties of Γ and the Metric

Metric Compatibility

Answer 19

• a metric in a LIF has no derivatives (it is locally flat) so should map onto general coordinates
• requires η_αβ;γ = 0 & g_αβ;γ=0
• free choice but not all possible versions of differential geometry satisfy it

Question 20

Properties of Γ and the Metric

Torsion Free

Answer 20

-partial derivative commute in the LIF:

φ_,β,α = φ_,α,β

-we want this to hold in general as a tensor relation, i.e.

φ_,β;α = φ_,α;β

-this is only true if we choose Γ^μ_αβ = Γ^μ_βα (no twisting)

Question 21

Properties of Γ and the Metric

Γ^γ_βμ = 1/2 g^αγ [g_αβ,μ + g_αμ,β - g_βμ,α]

Answer 21

• given g_αβ;μ=0 and g_αβ=g_βα
• expand terms and cancel to get formula
• in the LIF, g_αβ,μ = 0 => Γ=0

Question 22

Properties of Γ and the Metric

Γ Depends On g_μν,λ

Answer 22

if g_μν,λσ = 0 (no 2nd derivatives) then g_μν,λ is constant in x

• but the geodesic equation is like F=ma and Γ~forces
• these forces are only real if g_μν,λσ isn’t 0 otherwise they are fake and can be transformed away

Question 23

Parallel Transport

Definition

Answer 23

• vectors have constant magnitude
• but, move them so that they remain tangent to the surface, magnitude remains constant but the direction can change
• do this so that the vector remains ‘parallel’ with itself in an infinitesimal sense
• this is the basis of paralell transport

Question 24

Parallel Transport

What is the equation of motion for a particle (vector V_) along a path in which it remains ‘parallel’?

Answer 24

∂V_/∂x^β = V^α_,β e_α + V^α Γ^μ_αβ e_μ

= [V^α_,β + V^μ Γ^α_μβ] e_α

= V^α_;β e_α