Geodesic

Two geodetic lines, in red, on a curved surface, those geodesic coincide with the trajectories of two particles in the spherical gravitational field of a central mass according to the general theory of relativity.

Geodetic triangle on a sphere. The geodesic line would be any of the arches that form the triangles.

In geometry, the geodesic line is defined as the line of minimum length that joins two points on a given surface, and is contained in this surface. The osculating plane of the geodesic is perpendicular at any point to the tangent plane to the surface. The geodesics of a surface are the "straightest" possible (with less curvature) fixed a point and a given direction on said surface.

More generally, one can speak of geodesics in "curved spaces" of higher dimension called Riemannian manifolds where, if space contains a natural metric, then geodesics are (locally) the shortest distance between two points in space. A physical example of the semi-Riemannian variety is that which appears in the theory of general relativity, which states that material particles move along temporary geodesics of curved space-time.

The term "geodetic" It comes from the word geodesy, the science of measuring the size and shape of the planet Earth; in the original sense, it was the shortest path between two points on the Earth's surface, specifically, the segment of a great circle.

Introduction

A Riemanianna manifold (M, g) is a differentiable manifold endowed with an additional structure of metric space that allows generalizing concepts of Euclidean geometry to more general spaces. Specifically, the tangent space at each point is endowed with a dot product, which determines the metric on the manifold, given by:

${displaystyle g_{ij}{big Δ}_{p}={bigg langle }{frac {partial }{partial x_{i}}}}}{bigg Δ},{frac {partial }{partial x_{j}}}}{bigg Δ}{bigg}{bigg}{bigg}{bigg}}{bigg}{bigg}{bigg}}{bigg}{bigg}{bigg}}{bigg}}{bigg}{bigg}{bigg}{bigg}}}{bigg}}}{$

where <,> is the previously defined scalar product and p is any point of the manifold M.

Based on this metric, the length L_C along a curve contained in it is evaluated thanks to the g_{ij components} of the metric tensor g as follows:

${displaystyle L_{C}=int _{C}{sqrt {sum _{i,j}g_{ij}x'_{i}(t)x'_{j}(t)}}}}}$

Where x_i(t) is the parametric expression of the points on the curve parameterized by the parameter t. All geodesic lines are extremals of the previous integral. One of the ways to obtain the equations of the geodesics is to minimize the previous functional. In that case the Euler-Lagrange equations provide the geodesic curves. In the next section we will follow a different approach.

Geodesics as zero acceleration curves

${displaystyle mathbb {R} ^{n}$It has an affin structure that allows 'connect' tangent spaces of different points in a natural way. Given a ${displaystyle pin mathbb {R} ^{n}$, we can identify ${displaystyle T_{p}mathbb {R} ^{n}$ with his own ${displaystyle mathbb {R} ^{n}$, which allows to differentiate vector fields in a simple and intuitive way. This is not the case for more general Riemannian varieties, in which tangent spaces at each point are abstract vectorial spaces, so it makes no sense to perform operations directly among them. To illustrate this, be it ${displaystyle Xin chi (M)}$ a vector field in M and ${displaystyle gamma (t)in M}$ a curve ${displaystyle gamma:(-varepsilonvarepsilon)rightarrow M;quad tmapsto gamma (t)}$. Intuitively, applying calculation techniques in ${displaystyle mathbb {R} ^{n}$If we would like to calculate the field derivative ${displaystyle X}$ in a ${displaystyle t_{0}in (-varepsilonvarepsilon)}$ along the curve ${displaystyle gamma }$, we would write:

${displaystyle X'VAIL_{t_{0}}}=lim _{hto 0}{frac {x1}{gamma (t_{0}+h)}-X_{gamma (t_{0}}}}{h}}}}{h}}}}$

However, ${displaystyle XATA_{gamma }(t_{0}+h)}$ and ${displaystyle XATA_{gamma }(t_{0})}$ belong to different vector spaces and the previous operation makes no sense. To do this it is necessary to 'connect' the vector spaces of the different points of the variety so that the previous operation makes sense. To do this, the concept of affin connection arises.

Kind Connection

An aphin connection is an app ${displaystyle nabla:chi (M)times chi (M)rightarrow chi (M),quad (X,Y)mapsto nabla _{Y}X}$ that fulfills: ${displaystyle i);quad nabla _{fcdot Y_{1+gcdot Y_{2}}X=fnabla _{Y_{1}}}X+gnabla _{Y_{2}X}$

${displaystyle ii);quad nabla _{Y}(X_{1}+X_{2})=nabla _{Y}X_{1}+nabla _{Y}X_{2}}$

${displaystyle iii)quad nabla _{Y}(fcdot X)=fcdot nabla _{y}X+Xcdot Y(f)}$

functions ${displaystyle f,gin C^{infty }(M)}$ and fields ${displaystyle X_{1},X_{2},Y_{1},Y_{2}in chi (M)}$.

If we choose a card around a p-point of variety: ${displaystyle (Uni p,varphi =(x_{1},dotsx_{n})}$, then we can express the connection according to the local coordinates. In fact, be it:

${displaystyle X=sum _{i=1}{n}f_{i}(x_{1},dotsx_{n}){frac {partial }{partial x_{i}}}}}}}}}}$ e ${displaystyle Y=sum _{j=1}{n}g_{j}(x_{1},dotsx_{n}){frac {partial }{partial x_{j}}}}}}}}}}}$. Let's express the connection. ${displaystyle nabla _{Y}X}$ based on local coordinates:

${displaystyle nabla _{Y}X=nabla _{sum _{j=1}{n}{n}{j}(x_{1},dotsx_{n}{frac}{partial }{partial x_{j}}}}}{bigg(}{i={n}{n}{n}{n}{s}{x}{$${displaystyle =sum _{i=1}{n}f_{i}sum _{j=1}{n{bigg (}{frac {partial g_{j}}{partial x_{j}}}}{{f}}{cd}{cd}{cd}{cd}{cd}{cd}{f}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{$${displaystyle =sum _{k=1}{n}{bigg (}sum _{i,j}f_{ig_{j}Gamma _{i,j}^{k}+Y(f_{k}){bigg)}{frac {partial }{partial x_{k}}}}}}}}{bigg$Where ${displaystyle Gamma _{i,j}^{k}(x_{1},dotsx_{n}}})$ are the symbols of Christoffel of the connection and are given by the following expression:

${displaystyle nabla _{frac {partial }{partial x_{i}}{frac {partial }{partial x_{j}}}}=sum _{k=1}{n}Gamma _{i,j}{frac {partial }{partial }{partial x_{k}}}}}}}}}}}}}}}}$.

Covariant derivative and parallel transport

Now. ${displaystyle Yin chi (M)}$ a field in M and ${displaystyle c:(-varepsilonvarepsilon)rightarrow M}$ an integral curve of that field, so ${displaystyle c'(t)=YUD_{c(t)}}$ and be ${displaystyle Xin chi (M)}$ another field. Sea ${displaystyle {overline {c}(t)=(varphi circ c)(t)}$ the local expression of the curve. In this situation, the field ${displaystyle nabla _{Y}XATA_{c(t)}}$ only depends on ${displaystyle XATA_{c(t)}$ and ${displaystyle c(t)}$. This motivates the following notation: ${displaystyle nabla _{c'(t)}v(t):=nabla _{Y}XATA_{c(t)}}}$Where ${displaystyle v(t)=XUD_{c(t)}}$ is the restriction of field X to curve c. The previous expression particularizes the concept of connection to the restrictions of fields on a whole curve. We define covariant derivative along a curve as:

${displaystyle {frac {Dv}{dt}}=nabla _{c'(t)v}(t)}$.

The covariant derivative is a generalization of acceleration on Riemannian manifolds.

A vector field ${displaystyle v(t)}$ is said parallel along a curve ${displaystyle c(t)}$ if your covariant derivative is overturned. Given a curve ${displaystyle c(t)}$ and a vector ${displaystyle v_{0}in TpM}$ for some ${displaystyle pin M}$, the existence of such a parallel field is guaranteed by the theorem of existence and uniqueness of ordinary differential equations. In fact, if we take the expression of the covariant derivative and equal 0 we get a linear system of ordinary differential equations:

${displaystyle {frac {Dv(t)}{dt}}=sum _{k=1^}{n}{bigg (}v'_{k}(t)+sum _{i,j}v_{i}(t){overline {c}_{j}(t) Gamma _{j,i}{k}({overline {c}}_{1},dots{overline {c}_{n}){bigg)}{frac {partial }{partial x_{k}}}}}}=0}$, ${displaystyle v(0)=v_{0}. !$

Geodesics

Continuing with the concept that the covariant derivative somehow represents acceleration, it is natural to conceive geodesic as curves of null covariant derivation. Since acceleration is not a magnitude that possesses the curves but the vectors, then it seems natural that if ${displaystyle gamma (t)}$ It's a geodetic to fulfill:

${displaystyle nabla _{gamma '}gamma '=0. !$

That is, the geodesics are curves whose tangent vector is a parallel field along itself. Here, as you can see, ${displaystyle v(t)=gamma '(t)}$. Expanding the previous equation we get the second order equation system:

${displaystyle gamma ''_{k}(t)+sum _{i,j}Gamma _{i,j}{k}gamma '_{i}gamma '_{j}=0,quad k=1,2,...,n}$,

which is resolved with the initial conditions ${displaystyle gamma (0)=p;quad gamma '(0)=v}$. Note that if ${displaystyle gamma }$ It's a geodetic, then ${displaystyle Δgamma '(t)ѕ=cte}$. This implies, in fact, that geodesics have null intrinsic acceleration. To understand this better, imagine that our variety is the sphere unity ${displaystyle M=mathbb {S} ^{2}subset mathbb {R} ^{3}$. So, the geodesics are curves that run along with constant velocity in the sphere, which means they have, in physical terms, tangential null acceleration, but seen as curves of ${displaystyle mathbb {R} ^{3}$They would still have normal acceleration different from 0. That normal acceleration vector, however, does not belong to the tangent space at a point in the sphere, so that, intrinsically, for an observer living in the sphere, that curve would indeed go without acceleration.

Examples

• Geodesics in Euclide Space ${displaystyle mathbb {E} ^{n}$ It's the straight lines.
• The geodesics in ${displaystyle mathbb {H} ^{2}}$, hyperbolic space, are circle arches.
• The geodesics in ${displaystyle mathbb {S} ^{2}}$ are maximum circumference arches. If the sphere is considered to be embedded in the three-dimensional euclid space then the maximum circles are obtained as intersection of the sphere with a plane passing through its center. In particular, the meridians of a sphere and the equator are geodetic lines. Using coordinates ${displaystyle scriptstyle {(r,thetaphi)}}$spherical for a radio dial R, geodetic equations are simply:

(♪)${displaystyle {ddot {theta }-sin(theta)cos(theta){dot {dot {phi }}{2}=0,qquad {ddot {phi }}{frac {2cos(theta)}{sin(theta}{dot {theta }{$

In particular, a meridian that crosses the north and south poles, responds to the parametric equations:

(**)${displaystyle theta (s)={frac {s}{R}}},qquad phi (s)=phi _{0}={mbox{cte. !$

Which satisfies the equations (*) trivially.

Integral curves on the tangent bundle

The geodetic equations system obtained before does not depend on ${displaystyle gamma }$ explicitly, only of ${displaystyle gamma'}$ and ${displaystyle gamma '}$. This implies that such second order and size equations system n, can easily be transformed into a first order and size 2n equation system. Indeed, calling ${displaystyle delta _{k}(t)=gamma '_{k}(t)}$ We get the system:

${displaystyle delta _{k}(t)=gamma '_{k}(t)}$; ${displaystyle delta '_{k}=-sum _{i,j}^{k}{i}delta _{j}}$

with initial conditions ${displaystyle gamma _{k}(0)=gamma _{k}^{0}$ and ${displaystyle delta _{k}(0)=gamma _{k}'^{0}$. Looking well, this system can be interpreted as a first order EDOs system on the tangent fiberd variety. Indeed, our new variables ${displaystyle (gamma _{1}dotsgamma _{n},delta _{1}dotsdelta _{n})}$ are a set of N coordinates and N speeds, such that ${displaystyle phi ^{-1}:(gamma _{1}dotsgamma _{n},delta _{1}dotsdelta _{n})mapsto sum _{i}delta _{1}{frac {partial }{partial x}}{bigg ̄}{,{(gamma}$ is the reverse of the homeomorphism of a letter in the tangent fiberdo. Thus, geodesics can be interpreted as integral curves of a field on the tangent fiberd, called the geodesic field.