Covariant Derivatives to Parallel transport

When does force-free motion occur?

When the geodesic equation is satisfied: $\frac{d^2x^\nu}{d\tau^2}+\Gamma^ν_{αβ}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau}=0$.

(As opposed to $\ddot x=0$ in Newtonian physics.)

When is a trajectory a geodesic?

A curve $C$ given by $x^\mu(τ)$ with affine parameter $τ$ is called a geodesic if it satisfies the geodesic equation.

What do the Christoffel symbols tell us?

How the basis vectors change when going from a tangent plane at one point to the tangent plane at another point in the manifold.

• i.e., from one tangent space to another.

Action principle

The action principle states that the dynamics of a physical system are given in terms of a variational problem of a functional, the action $S$, given in terms of a function, the Lagrangian $L$, which contains all the information of the system.

Proper time interval

For the path of a particle in a given spacetime, the proper time interval is $dτ^2 = −ds^2 = −g_{\mu ν}dx^µdx^ν$ where the signature of the metric is chosen to be $-+++$ here.

Action $S$

$$S=\int d\tau=\int\sqrt{-g_{\mu\nu}dx^\mu dx^\nu}$$

Principles of special relativity

1. The principle of relativity, which states that the physical laws have the same form in all inertial reference frames.

2. The universality of the speed of light c, that is, the speed of light in vacuum is the same in all inertial reference frames.

Weak equivalence principle

The weak equivalence principle states that: uniform gravitational fields are equivalent to frames that accelerate uniformly relative to inertial frames.

This refers to uniform gravitational fields, and so only applies locally.

Strong equivalence principle

The strong equivalence principle states that: the laws of nature in a freely falling local inertial system are identical to the laws in Special Relativity, i.e., to those in the absence of the gravitational field.

The tensor version of this principle states that: all physical laws that can be expressed in tensor notation in flat spacetime have exactly the same form in a local inertial frame in curved spacetime.

Length of a parallel transported vector

Does not change, i.e., the inner product will stay the same, and so the angle between two parallel transported vectors will also stay the same.

What does parallel transport refer to?

Refers to keeping the angle between a vector and its tangent vector the same when transporting the vector along a curve.

In curved spacetime, parallel transport is path dependent, so the initial and final vectors will look different (even though their angles to their tangent vectors and lengths are the same).

Covariant derivative of a contravariant vector

The covariant derivative of a contravariant vector field is $∇_νv^\mu = ∂_νv^\mu +Γ^\mu_{νρ}v_ρ$

Covariant derivative of a covariant vector

The covariant derivative of a covariant vector field is $\nabla_\nu u_\mu=\partial_\nu u_\mu-\Gamma^\rho_{\nu\mu} u_\rho$

Transformation of $∇_νv^\mu$

$∇_νv^\mu$ transform like a (1,1)-tensor under general coordinate transformations, i.e., $∇'_ νv'^\mu = \frac{∂x^σ} {∂x'^ν} \frac{∂x'^\mu}{ ∂x^ρ} ∇_σv^ρ$.

Transformation of $\nabla_\nu u_\mu$

$\nabla_\nu u_\mu$ transforms like a (0,2)-tensor under general coordinate transformations.

Transformation of the covariant derivative applied to (M,N) tensors

$$∇_ν\,T^{\mu_1\mu_2...\mu_M}_{ρ_1ρ_2...ρ_N} = ∂νT^{\mu_1\mu_2...\mu_M}_{ρ_1ρ_2...ρ_N}+\Gamma^{\mu_1}_{\nu\lambda}\,T^{\lambda\mu_2...\mu_M}_{\rho_1ρ_2...ρ_N}+\Gamma^{\mu_2}_{\nu\lambda}\,T^{\mu_1\lambda...\mu_M}_{ρ_1ρ_2...ρ_N}+\,...\,-\Gamma^{\lambda}_{\nu\rho_1}\,T^{\mu_1\mu_2...\mu_M}_{\lambdaρ_2...ρ_N}-\Gamma^{\lambda}_{\nu\rho2}\,T^{\mu_1\mu_2...\mu_M}_{ρ_1\lambda...ρ_N}\,-\,...$$

Result of the application of the covariant derivative operator $∇_ν$

Application of $∇_ν$to an (M,N)-tensor gives an (M,N +1)-tensor.

For a scalar (a (0,0)-tensor) the covariant derivative operator $∇_ν$ becomes the ordinary partial derivative because $∂_νϕ$ is already a covariant vector, i.e., a (0,1)-tensor.

Properties of the covariant derivative operator $∇_ν$

• It is linear

• Follows the Leibniz rule: $\nabla\left(A^{\mu_1\mu_2....\mu_M}_ {ρ_1ρ_2....ρ_N} \,B^{\,α_1α_2....α_M}_{\beta_1\beta_2...\beta_N}\right)=\nabla_\nu\left(A^{\mu_1\mu_2....\mu_M}_ {ρ_1ρ_2....ρ_N} \right) \,B^{\,α_1α_2....α_M}_{\beta_1\beta_2...\beta_N}+A^{\mu_1\mu_2....\mu_M}_ {ρ_1ρ_2....ρ_N}\,\nabla_\nu\left( B^{α_1α_2....α_M}_{\beta_1\beta_2...\beta_N}\right)$

Covariant derivative of the metric tensor

The covariant derivative of the metric tensor vanishes, $∇_νg_{\muλ} = 0$.

As a consequence, raising and lowering indices commutes with the covariant derivative, e.g., $∇_νv_\mu = ∇_ν(g_{\muρ}v^ρ) = g_{\muρ}∇_νv^ρ$ .

Covariant derivative commutation relations

The covariant derivative commutes with contraction, where contraction turns an (M,N) tensor into an (M-1,N-1) tensor by equating two indices.

The covariant derivative commutes with raising and lowering of indices via the metric tensor.

Covariant curl of a covariant vector

$$∇_νv_\mu −∇_\mu v_ν =∂_νv_\mu −∂_\mu v_ν$$

Covariant divergence of a contravariant vector

$$∇_νv^ν=∂_νv^ν+Γ^ν_ {νλ}v^λ= \frac{1}{ \sqrt{|g|}}∂_\mu\left( \sqrt{|g|}v^\mu\right)$$

The d’Alembertian of a scalar

The d’Alembertian of a scalar $\phi$ is given by $\Box\phi:=\nabla^\nu\nabla_\nu\phi=\frac{1}{ \sqrt{|g|}}∂_\mu\left( \sqrt{|g|}∂_\mu\phi\right)$

Operator $D_\tau$

Defined as $D_\tau:=\dot x^\nu\nabla_\nu$.

Geodesics in reference to $D_\tau$

Geodesics (with $τ$ an appropriate affine parameter) are curves which have vanishing covariant acceleration:

$$\text{Geodesic with affine parameter }τ: \,a^\mu=D_τv^\mu=v^\nu∇_\nu v^\mu=0$$

where the velocity $v^\mu=\dot x^\mu$ is the tangent vector to a curve.

Geodesics are autoparallels.

Parallel transport

An (M,N) tensor $T^{α_1α_2...α_M}_{β_1β_2...β_N}$ is said to be parallel transported along a curved $x^\mu(τ)$ if $D_\tau\, T^{α_1α_2...α_M}_{β_1β_2...β_N}=0$.

Application of $D_\tau$ on a metric

As $∇_\mu g_{νλ} = 0$ we can write $D_\tau g_{\mu\nu}=0$.