Geodesics - An overview

1. Formula
   Observation

   $x^\mu = x^\mu(\lambda)$

2. Audio
   Observation

   The narrator says the curve can be written in terms of a parameter, which we will give the symbol lambda.

$\lambda$

Parameter used to describe a geodesic curve; later specialized to an affine parameter when $f(\lambda)=0$ for all $\lambda$.

Curve parameter; no explicit domain is stated in the video.

1. Formula
   Observation

   $x^\mu = x^\mu(\lambda)$

2. Audio
   Observation

   Every space-time component of the curve can be written as a function of lambda.

$x^\mu$

Space-time components of the curve, each written as a function of the parameter $\lambda$.

Components of a curve in space-time; the video does not specify the index range explicitly.

1. Formula
   Observation

   $\frac{d\tilde{T}}{d\lambda} = f(\lambda)\tilde{T}$

2. Formula
   Observation

   $\tilde{T}\cdot\tilde{T} \equiv g$_{ij}T^iT^$j = 0$

3. Audio
   Observation

   where T~ is the tangent to the geodesic

1. The slide uses $\tilde{T}$ in the derivative equation but writes the components as $T^i$ and $T^j$ in the norm expression; the video does not explain this notation shift.

$\tilde{T}$

Tangent vector to the geodesic.

Tangent 4-vector along the geodesic.

1. Formula
   Observation

   $\frac{d\tilde{T}}{d\lambda} = f(\lambda)\tilde{T}$

2. Formula
   Observation

   $f(\lambda)=0$\ $\forall\lambda$

3. Audio
   Observation

   $f(\lambda)$ is a scalar function

$f(\lambda)$

Scalar proportionality function relating the rate of change of the tangent vector to the tangent vector itself.

Scalar function of the curve parameter $\lambda$.

1. Formula
   Observation

   $\tilde{T}\cdot\tilde{T} \equiv g$_{ij}T^iT^$j = 0$

2. Formula
   Observation

   $L = \int_A^B \sqrt{g_{ij}\dot{x}^i\dot{x}^j}$\,$d\lambda$

3. Formula
   Observation

   $F = \int_A^B g$_{ij}$\dot{x}^i\dot{x}^j$\,$d\lambda$

4. Formula
   Observation

   $\mathscr{L} = g$_{ij}$\dot{x}^i\dot{x}^j$

1. The video does not state symmetry properties or signature conventions for g_{ij}.

g_{ij}

Metric tensor components used to form the squared norm of the tangent vector and the integrands for length and F.

Tensor components with lower indices i,j.

1. Formula
   Observation

   g_{ij}T^iT^j

1. The relation between $T^i$,$T^j$ and $\tilde{T}$ is shown but not verbally explained.

$T^i$,\ $T^j$

Components of the tangent vector appearing in the contraction g_{ij}T^iT^j.

Vector components with upper indices.

1. Formula
   Observation

   $\sqrt{g_{ij}\dot{x}^i\dot{x}^j}$

2. Formula
   Observation

   g_{ij}$\dot{x}^i\dot{x}^j$

3. Audio
   Observation

   where the derivatives are with respect to the parameter lambda

$\dot{x}^i$,\ $\dot{x}^j$

Derivatives of the curve components with respect to the parameter $\lambda$.

Componentwise derivatives of $x^\mu(\lambda)$.

1. Formula
   Observation

   $L = \int_A^B \sqrt{g_{ij}\dot{x}^i\dot{x}^j}$\,$d\lambda$

2. Audio
   Observation

   The shortest distance between two points A and B can be written as ... To find the geodesic curves all we have to do is minimise L.

L

Length functional between points A and B, to be minimized to obtain geodesic curves.

Functional of the curve.

1. Formula
   Observation

   $F = \int_A^B g$_{ij}$\dot{x}^i\dot{x}^j$\,$d\lambda$

2. Audio
   Observation

   minimising F given here is equivalent to minimising L and is much easier

F

Alternative functional whose minimization is equivalent to minimizing L when $\lambda$ is affine.

Functional of the curve.

1. Formula
   Observation

   $\mathscr{L} = g$_{ij}$\dot{x}^i\dot{x}^j$

2. Audio
   Observation

   for the case of the integral F the Lagrangian is given by this

$\mathscr{L}$

Lagrangian used in the Euler-Lagrange equation for the integral F.

Function of $x^k$ and $\dot{x}^k$.

1. Formula
   Observation

   $\Gamma^k_{ij}$

2. Audio
   Observation

   where we have introduced the connection coefficients

1. The video does not define the connection coefficients explicitly.

$\Gamma^k_{ij}$

Connection coefficients appearing in the geodesic equation.

Array of coefficients indexed by i,j,k.

1. Formula
   Observation

   A geodesic between two points is the shortest length curve between those two points.

2. Audio
   Observation

   Let us start by defining a geodesic. A geodesic between two points is the shortest length curve between those two points.

The video defines a geodesic operationally as the curve of shortest length joining two specified points. The narrator also states that other equivalent definitions exist, but this one is the version used in the lesson.

1. Applies to a curve connecting two points.

2. The definition is presented as one usable formulation among equivalent alternatives.

1. Formula
   Observation

   $x^\mu = x^\mu(\lambda)$

2. Audio
   Observation

   Like any curve we can write a geodesic in terms of a parameter ... every space-time component of the curve can be written as a function of lambda.

A geodesic is treated as a curve in space-time and written componentwise as functions of a parameter $\lambda$. This sets up the later tangent-vector equation and the variational formulas.

1. $\lambda$ is the chosen curve parameter.

2. Each space-time component $x^\mu$ is expressed as a function of $\lambda$.

1. Definition of a geodesic

1. Formula
   Observation

   $\frac{d\tilde{T}}{d\lambda} = f(\lambda)\tilde{T}$

2. Formula
   Observation

   $f(\lambda)=0$\ $\forall\lambda$

3. Audio
   Observation

   the rate of change of the tangent of the geodesic is proportional to the tangent itself ... From this we can define an affine parameter as one for which the function of proportionality is zero for all lambda.

For a geodesic, the tangent vector satisfies a proportionality law with a scalar factor $f(\lambda)$. An affine parameter is then defined as the special choice of parameter for which that proportionality function vanishes identically.

1. $\tilde{T}$ is the tangent to the geodesic.

2. $f(\lambda)$ is a scalar function of the parameter.

3. The affine case is the subcase $f(\lambda)=0$ for all $\lambda$.

1. Parameterizing a geodesic

1. Formula
   Observation

   A geodesic is timelike (spacelike, null) if its tangent 4-vector is timelike (spacelike, null).

2. Audio
   Observation

   Geodesics like four vectors can be classified as time-like, space-like or null. The general rule is that a geodesic is time-like if its tangent four vector is time-like, and analogous for space-like and null geodesics.

The video classifies geodesics according to the causal character of their tangent 4-vector: timelike, spacelike, or null. The rule is stated uniformly across the three cases.

1. The classification depends on the tangent 4-vector of the geodesic.

2. The same rule applies to timelike, spacelike, and null cases.

1. Parameterizing a geodesic

1. Formula
   Observation

   $\tilde{T}\cdot\tilde{T} \equiv g$_{ij}T^iT^$j = 0$

2. Audio
   Observation

   In other words, a null geodesic is one for which this dot product is zero.

1. The video does not define the metric signature or explain why this quadratic form classifies null vectors.

A null geodesic is characterized by the vanishing of the tangent-vector norm. The displayed formula writes that norm as the contraction g_{ij}T^iT^j and sets it equal to zero.

1. $\tilde{T}$ is the tangent vector to the geodesic.

2. g_{ij} are metric components.

3. The equality to zero defines the null case.

1. Classification of geodesics by tangent type

1. Formula
   Observation

   $\tilde{T}\cdot\tilde{T} \equiv g$_{ij}T^iT^$j \neq 0$

2. Audio
   Observation

   And a non-null geodesic is one for which it is not zero.

The complement of the null case is called non-null in the video: the tangent-vector norm is nonzero.

1. Same notation as the null case.

2. Nonzero norm distinguishes the non-null class.

1. Null geodesic condition

1. Formula
   Observation

   $L = \int_A^B \sqrt{g_{ij}\dot{x}^i\dot{x}^j}$\,$d\lambda$

2. Audio
   Observation

   The shortest distance between two points A and B can be written as ... where the derivatives are with respect to the parameter lambda. To find the geodesic curves all we have to do is minimise L.

1. The video does not derive this formula from the line element; it presents it directly.

To turn the geometric idea of shortest path into a calculational problem, the video introduces the length functional L between endpoints A and B. Geodesics are then obtained by minimizing this functional.

1. Endpoints are A and B.

2. $\dot{x}^i$ and $\dot{x}^j$ denote derivatives with respect to $\lambda$.

3. Minimizing L gives the geodesic curves.

1. Definition of a geodesic
2. Parameterizing a geodesic

1. Formula
   Observation

   If $\lambda$ is affine minimising F (below) is equivalent to minimising L:

2. Formula
   Observation

   $F = \int_A^B g$_{ij}$\dot{x}^i\dot{x}^j$\,$d\lambda$

3. Audio
   Observation

   However, this is tricky, but in the case that lambda is an affine parameter, minimising F given here is equivalent to minimising L and is much easier.

1. The equivalence is asserted without proof in this clip.

Because minimizing L is described as tricky, the video introduces a simpler functional F. When $\lambda$ is affine, minimizing F is stated to be equivalent to minimizing L.

1. Requires $\lambda$ to be an affine parameter.

2. F is used as an easier surrogate for L under that condition.

1. Affine parameter
2. Length functional L

1. Formula
   Observation

   $\frac{d}{d\lambda}\left(\frac{\partial\mathscr{L}}{\partial\dot{x}^k}\right) = \frac{\partial\mathscr{L}}{\partial x^k}$

2. Formula
   Observation

   $\mathscr{L} = g$_{ij}$\dot{x}^i\dot{x}^j$

3. Audio
   Observation

   To minimise this integral, we can use the Euler-Lagrange equation given here, where for the case of the integral F the Lagrangian is given by this.

1. The video does not prove the Euler-Lagrange equation; it is invoked as a standard tool.

The minimization of the integral F is carried out using the Euler-Lagrange equation. For this specific problem, the Lagrangian is taken to be $\mathscr{L}=g$_{ij}$\dot{x}^i\dot{x}^j$.

1. Applied to the integral F.

2. Uses $\lambda$ as the independent parameter.

3. The component index k labels the Euler-Lagrange equation for each coordinate.

1. Equivalent functional F for affine parameter

1. Formula
   Observation

   $\frac{d^2x^k}{d\lambda^2} + \Gamma^k_{ij}\frac{dx^i}{d\lambda}\frac{dx^j}{d\lambda} = 0$

2. Audio
   Observation

   The general solution to this equation is this, where we have introduced the connection coefficients ... This equation is called the geodesic equation, and it specifies the geodesics.

1. The intermediate derivation from the Euler-Lagrange equation to the displayed result is omitted in the video.

2. The connection coefficients are named but not defined here.

After applying the Euler-Lagrange equation, the video states the resulting differential equation for the curve components. This is identified as the geodesic equation, and it is said to specify the geodesics.

1. $\Gamma^k_{ij}$ are connection coefficients.

2. The equation is presented as the general solution of the Euler-Lagrange step for F.

3. It specifies geodesics.

1. Euler-Lagrange equation applied to F
2. Parameterizing a geodesic

1. Audio
   Observation

   I note here that there are other equivalent definitions, but this is the one we will use.

1. The alternative equivalent definitions are mentioned but not listed.

There are other equivalent definitions of a geodesic, although the video uses the shortest-length-curve definition.

1. The discussion is within the setting of general relativity presented in the video.

Existential claim about other definitions; no further quantifier detail is given.

1. Formula
   Observation

   If $\lambda$ is affine minimising F (below) is equivalent to minimising L:

2. Audio
   Observation

   in the case that lambda is an affine parameter, minimising F given here is equivalent to minimising L and is much easier.

1. No proof of the equivalence is shown in this clip.

If $\lambda$ is an affine parameter, then minimizing $F = \int_A^B g$_{ij}$\dot{x}^i\dot{x}^j$\,$d\lambda$ is equivalent to minimizing $L = \int_A^B \sqrt{g_{ij}\dot{x}^i\dot{x}^j}$\,$d\lambda$.

1. $\lambda$ is an affine parameter.

2. The endpoints are A and B.

3. F and L are the functionals displayed in the video.

Conditional statement for the affine-parameter case.

1. Formula
   Observation

   $\frac{d^2x^k}{d\lambda^2} + \Gamma^k_{ij}\frac{dx^i}{d\lambda}\frac{dx^j}{d\lambda} = 0$

2. Audio
   Observation

   This equation is called the geodesic equation, and it specifies the geodesics.

1. The video does not state boundary conditions or uniqueness results associated with specifying geodesics.

The equation $\frac{d^2x^k}{d\lambda^2} + \Gamma^k_{ij}\frac{dx^i}{d\lambda}\frac{dx^j}{d\lambda} = 0$ is called the geodesic equation and specifies the geodesics.

1. $\Gamma^k_{ij}$ are connection coefficients.

2. $x^k(\lambda)$ are the curve components.

Universal over the component index k as displayed; no further quantifier detail is spoken.

1. Formula
   Observation

   $\frac{d}{d\lambda}\left(\frac{\partial\mathscr{L}}{\partial\dot{x}^k}\right) = \frac{\partial\mathscr{L}}{\partial x^k}$

2. Formula
   Observation

   $\mathscr{L} = g$_{ij}$\dot{x}^i\dot{x}^j$

3. Formula
   Observation

   $\frac{d^2x^k}{d\lambda^2} + \Gamma^k_{ij}\frac{dx^i}{d\lambda}\frac{dx^j}{d\lambda} = 0$

4. Audio
   Observation

   To minimise this integral, we can use the Euler-Lagrange equation ... The general solution to this equation is this ... This equation is called the geodesic equation

1. The actual algebra from $\mathscr{L}=g$_{ij}$\dot{x}^i\dot{x}^j$ to the geodesic equation is skipped.

2. The definition of $\Gamma^k_{ij}$ is not shown.

1. Expression

   $$\frac{d}{d\lambda}\left(\frac{\partial\mathscr{L}}{\partial\dot{x}^k}\right) = \frac{\partial\mathscr{L}}{\partial x^k}$$
   Explanation

   Start from the Euler-Lagrange equation for the integral being minimized.

   Justification

   The narrator explicitly says this equation is used to minimize the integral.

   Shown in the video
2. Expression

   $$\mathscr{L} = g_{ij}\dot{x}^i\dot{x}^j$$
   Explanation

   Substitute the Lagrangian appropriate to the functional F.

   Justification

   The slide and narration identify this as the Lagrangian for the case of the integral F.

   Shown in the video
3. Expression

   $$\frac{d^2x^k}{d\lambda^2} + \Gamma^k_{ij}\frac{dx^i}{d\lambda}\frac{dx^j}{d\lambda} = 0$$
   Explanation

   The resulting general solution is presented as the geodesic equation.

   Justification

   The video states this is the general solution after introducing connection coefficients, but omits the intermediate derivation.

   Shown in the video

The geodesic equation is obtained as the Euler-Lagrange result for the simplified functional F, with connection coefficients appearing in the final form.

1. Formula
   Observation

   $\frac{d\tilde{T}}{d\lambda} = f(\lambda)\tilde{T}$

2. Formula
   Observation

   $f(\lambda)=0$\ $\forall\lambda$

3. Audio
   Observation

   From this we can define an affine parameter as one for which the function of proportionality is zero for all lambda.

1. Expression

   $$\frac{d\tilde{T}}{d\lambda} = f(\lambda)\tilde{T}$$
   Explanation

   The tangent to a geodesic changes proportionally to itself, with scalar factor $f(\lambda)$.

   Justification

   Directly stated in the video.

   Shown in the video
2. Expression

   $$f(\lambda)=0\ \forall\lambda$$
   Explanation

   Choose the parameter so that the proportionality factor vanishes identically.

   Justification

   This is exactly how the video defines an affine parameter.

   Shown in the video

An affine parameter is the parameter choice that removes the proportionality term in the tangent evolution equation.

1. Animation
   Observation

   The video consists of static slides with text and formulas appearing sequentially: title, Definition, Parameterising a Geodesic, Null and non-null Geodesics, Finding the Geodesics, Thanks for Watching, Software used.

2. Caption evidence
   Observation

   Reference text appears at the bottom of several slides, including Hobson et al., Sharan, Infinite-dimensional Lie groups, and a StackExchange citation.

1. Title card

2. Definition slide

3. Parameterising a Geodesic slide

4. Null and non-null Geodesics slide

5. Finding the Geodesics slide

6. Closing slide

7. Software-used slide

1. Formulas appear one after another on each slide.

2. Section headings change from definition to parameterization, classification, and computation.

3. The final slides switch from mathematical content to acknowledgments and software credits.

1. The presentation remains text-and-formula based throughout.

2. No worked numerical example or diagrammatic geometry figure is shown.

The visual structure mirrors the lecture order: define geodesics, parameterize them, classify null versus non-null, then derive the geodesic equation variationally.

1. Audio
   Observation

   However, this is tricky, but in the case that lambda is an affine parameter, minimising F given here is equivalent to minimising L and is much easier.

One might assume the geodesic problem should always be attacked by minimizing the square-root length functional L directly.

The video warns that this is tricky and instead uses the simpler functional F when $\lambda$ is affine, stating that minimizing F is equivalent to minimizing L in that case.

1. Audio
   Observation

   where we have introduced the connection coefficients, something I won't go into detail here.

A viewer may expect the displayed geodesic equation to come with a definition of $\Gamma^k_{ij}$.

The video explicitly says it will not go into detail about the connection coefficients in this clip.

1. Formula
   Observation

   $x^\mu = x^\mu(\lambda)$

2. Audio
   Observation

   Like any curve we can write a geodesic in terms of a parameter

After defining what a geodesic is, the video applies curve parameterization to express it analytically.

1. Formula
   Observation

   $\frac{d\tilde{T}}{d\lambda} = f(\lambda)\tilde{T}$

2. Formula
   Observation

   $f(\lambda)=0$\ $\forall\lambda$

The affine parameter is a special choice of the general curve parameter $\lambda$ for which the proportionality function vanishes.

1. Formula
   Observation

   $\tilde{T}\cdot\tilde{T} \equiv g$_{ij}T^iT^$j = 0$

The null condition is one member of the broader timelike/spacelike/null classification of geodesics.

1. Formula
   Observation

   $\tilde{T}\cdot\tilde{T} \equiv g$_{ij}T^iT^$j \neq 0$

The video contrasts null geodesics, whose tangent norm is zero, with non-null geodesics, whose tangent norm is nonzero.

1. Formula
   Observation

   $L = \int_A^B \sqrt{g_{ij}\dot{x}^i\dot{x}^j}$\,$d\lambda$

2. Audio
   Observation

   To find the geodesic curves all we have to do is minimise L.

The shortest-path definition is turned into a variational problem by introducing the length functional L.

1. Formula
   Observation

   If $\lambda$ is affine minimising F (below) is equivalent to minimising L:

The equivalence between minimizing F and minimizing L is stated only under the assumption that $\lambda$ is affine.

1. Audio
   Observation

   To minimise this integral, we can use the Euler-Lagrange equation given here, where for the case of the integral F the Lagrangian is given by this.

Once F replaces L as the easier functional, the video applies the Euler-Lagrange method to minimize it.

1. Formula
   Observation

   $\frac{d^2x^k}{d\lambda^2} + \Gamma^k_{ij}\frac{dx^i}{d\lambda}\frac{dx^j}{d\lambda} = 0$

2. Audio
   Observation

   The general solution to this equation is this ... This equation is called the geodesic equation

The geodesic equation is presented as the result of carrying out the Euler-Lagrange procedure for the chosen Lagrangian.

1. Formula
   Observation

   A geodesic between two points is the shortest length curve between those two points.

1. Definition of a geodesic

1. Formula
   Observation

   $f(\lambda)=0$\ $\forall\lambda$

1. Affine parameter

1. Formula
   Observation

   $\tilde{T}\cdot\tilde{T} \equiv g$_{ij}T^iT^$j = 0$

2. Formula
   Observation

   $\tilde{T}\cdot\tilde{T} \equiv g$_{ij}T^iT^$j \neq 0$

1. Null geodesic condition
2. Non-null geodesic condition

1. Audio
   Observation

   minimising F given here is equivalent to minimising L and is much easier.

1. Equivalent functional F for affine parameter
2. Length functional L

1. Formula
   Observation

   $\frac{d}{d\lambda}\left(\frac{\partial\mathscr{L}}{\partial\dot{x}^k}\right) = \frac{\partial\mathscr{L}}{\partial x^k}$

2. Formula
   Observation

   $\frac{d^2x^k}{d\lambda^2} + \Gamma^k_{ij}\frac{dx^i}{d\lambda}\frac{dx^j}{d\lambda} = 0$

1. Intermediate algebra is omitted.

1. Euler-Lagrange equation applied to F
2. Geodesic equation

1. Audio
   Observation

   where we have introduced the connection coefficients, something I won't go into detail here.

1. Geodesic equation