Parallel Transport and Covariant Differentiation Flashcards
Classical Gravitation
∇²φ = 4πGρ
d’Alembertian
-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
Equivalence Principle Weak Version
-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
Equivalence Principle Einstein’s Variant
-all laws of physics will be the same in all inertial frames -this is expressed using tensor relationships
Local Inertial Frame Definition
-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
Local Inertial Frame General Metric
g_μν(x) = η_αβ dε^α/dx^μ dε^β/dx^ν -symmetric
Local Inertial Frame Generic Line Element
ds² = g_αβ dx^α dx^β
The Geodesic Equation
Γ^ν_αβ = ∂x^ν/∂ε^μ ∂²ε^μ/∂x^α∂x^β -where Γ^ν_αβ is a connection coefficient
What properites do we need in a metric for GR?
-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
Transformation
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
Derivative of Scalars
φ_,μ = ∂φ/∂x^μ
-well-behaved because φ has no direction
Derivatives of A^μ
A^μ_, is not well-behaved since ∂²x^’μ/∂x^α∂x^β is not zero
Derivative of A_
∂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
Covariant Derivative
A^α_;β = A^α_,β + A^μ Γ^α_μβ
Covariant Derivative of Scalar
-we make the CHOICE that:
φ_;μ = φ_,μ
-since this makes physical sense (φ has no direction)
Covariant Derivative of Covariant Vectors
A_μ;β = A_μ,β - A_α Γ^α_μβ
Covariant Derivative of General Tensors
- 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
Properties of Γ and the Metric
- metric compatibility
- torsion free
- Γ^γ_βμ = 1/2 g^αγ [g_αβ,μ + g_αμ,β - g_βμ,α]
- Γ does not transform as a 1/2 tensor so is not a tensor itself
- Γ depends on g_μν,λ terms
Properties of Γ and the Metric
Metric Compatibility
- 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
Properties of Γ and the Metric
Torsion Free
-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)
Properties of Γ and the Metric
Γ^γ_βμ = 1/2 g^αγ [g_αβ,μ + g_αμ,β - g_βμ,α]
- 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
Properties of Γ and the Metric
Γ Depends On g_μν,λ
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
Parallel Transport
Definition
- 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
Parallel Transport
What is the equation of motion for a particle (vector V_) along a path in which it remains ‘parallel’?
∂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
