What is Geodesic: Definition and 254 Discussions

In geometry, a geodesic () is commonly a curve representing in some sense the shortest path (arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line" to a more general setting.
The noun "geodesic" and the adjective "geodetic" come from geodesy, the science of measuring the size and shape of Earth, while many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). The term has been generalized to include measurements in much more general mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph.
In a Riemannian manifold or submanifold geodesics are characterised by the property of having vanishing geodesic curvature. More generally, in the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. Applying this to the Levi-Civita connection of a Riemannian metric recovers the previous notion.
Geodesics are of particular importance in general relativity. Timelike geodesics in general relativity describe the motion of free falling test particles.

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  1. U

    Quick expression on geodesic equation

    Taken from Hobson's book: How did they get this form? \dot u^{\mu} = - \Gamma_{v\sigma}^\mu u^v u^\sigma \dot u^{\mu} g_{\mu \beta} \delta_\mu ^\beta = - g_{\mu \beta} \delta_\mu ^\beta \Gamma_{v\sigma}^\mu u^v u^\sigma \dot u_{\mu} = - \frac{1}{2} g_{\mu \beta} \delta_\mu ^\beta...
  2. binbagsss

    Weak Field Approx, algebra geodesic equation

    My book says in the slow motion approx, so ## v << c ##, ##v=\frac{dx^{i}}{dt}=O(\epsilon) ## It then states: i) ##\frac{dx^{i}}{ds}=\frac{dt}{ds}\frac{dx^{i}}{dt}=O(\epsilon) ## ii) ## \frac{dx^{0}}{ds}=\frac{dt}{ds}=1+O(\epsilon) ## The geodesic equation reduces from...
  3. K

    Are all null geodesics also affine?

    Hi according to the text I am reading a curve is geodesic if these conditions are met ##\frac{d}{ds}(2g_{mi} \frac{dx^{i}}{ds})-\frac{\partial g_{jk}}{\partial x^{m}}\frac{dx^{j}}{ds}\frac{dx^{k}}{ds} = 0##, where ##m=1,...,N## a curve is a null geodesic if exactly the same conditions are...
  4. U

    Closed Universe - FRW Equation

    Homework Statement (a) Show that the equations satisfy FRW equations. (b) Show the metric when ##\eta## is taken as time Homework EquationsThe Attempt at a Solution [/B] The FRW equation is: 3 \left( \frac{\dot a}{a} \right)^2 = 8\pi G \rho Using ##\frac{da}{dt} = \frac{da}{d\eta}...
  5. U

    Tensor Contraction: Contracting ##\mu## with ##\alpha##?

    What do they mean by 'Contract ##\mu## with ##\alpha##'? I thought only top-bottom indices that are the same can contract? For example ##A_\mu g^{\mu v} = A^v##.
  6. U

    Geodesic Deviation Equation Solved

    Taken from my lecturer's notes on GR: I'm trying to understand what goes on from 2nd to 3rd line: N^\beta \nabla_\beta (T^\mu \nabla_\mu T^\alpha) - N^\beta \nabla_\beta T^\mu \nabla_\mu T^\alpha = -T^\beta \nabla_\beta N^\mu \nabla_\mu T^\alpha Using commutator relation ## T^v \nabla_v...
  7. U

    Contracting \mu & \alpha - What Does It Mean?

    What do they mean by contracting ##\mu## with ##\alpha## ?
  8. U

    Proper distance, Area and Volume given a Metric

    Homework Statement [/B] (a) Find the proper distance (b) Find the proper area (c) Find the proper volume (d) Find the four-volume Homework EquationsThe Attempt at a Solution Part (a) Letting ##d\theta = dt = d\phi = 0##: \Delta s = \int_0^R \left( 1-Ar^2 \right) dr = R \left(1 -...
  9. U

    Quick question on Geodesic Equation

    Starting with the geodesic equation with non-relativistic approximation: \frac{d^2 x^{\mu}}{d \tau^2} + \Gamma_{00}^{\mu} \left( \frac{dx^0}{d\tau} \right)^2 = 0 I know that ## \Gamma_{\alpha \beta}^{\mu} = \frac{\partial x^{\mu}}{\partial y^{\lambda}} \frac{\partial^2 y^{\lambda}}{\partial...
  10. U

    General Relativity - Deflection of light

    Homework Statement Find the deflection of light given this metric, along null geodesics. Homework EquationsThe Attempt at a Solution [/B] Conserved quantities are: e \equiv -\zeta \cdot u = \left( 1 - \frac{2GM}{c^2r} \right) c \frac{dt}{d\lambda} l \equiv \eta \cdot u = r^2 \left( 1 -...
  11. E

    Understanding Null Geodesics in Relativity: Insights from Dirac's Book

    In Dirac's book on relativity, he begins and ends his section on proving the stationary property of geodesics with references to "null geodesics". His last sentence is: "Thus we may use the stationary condition as the definition of a geodesic, except in the case of a null geodesic." What is a...
  12. binbagsss

    Killing Vectors conserved quantity along geodesic proof

    I am trying to follow a proof that given a Kiling vector ##V^{u}##, the quantity ##V_{u}U^{u} ## is conserved along a geodesic. I am given the Killiing Equation: ## \bigtriangledown_{(v}U_{u)}=0 ## [1] Below ## U^{u} ## is tangent vector ## U^{u} = \frac{dx^{u}}{d\lambda} ## The proof...
  13. Superposed_Cat

    Geodesic equation proof confusing me

    Hi all, I was looking through this proof and have no idea where the "u" comes from., any help apreciated. http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++&bg=eedbbd&fg=000000&s=0 http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++&bg=eedbbd&fg=000000&s=0...
  14. M

    Choice of action to geodesic equations

    Hello, I am quite new to GR and I have a question regarding the construction of the action to find the geodesic equation. In pretty much every book, you'll find: ##S=-m ∫ dS## using: ## dS=dS\frac{d\tau}{d\tau}=\sqrt{g_{\mu \nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}}d\tau## with...
  15. X

    Linearising the Geodesic Deviation Equation

    Homework Statement Write down the Newtonian approximation to the geodesic deviation equation for a family of geodesics. ##V^\mu## is the particle 4-velocity and ##Y^\mu## is the deviation vector. Homework Equations D = V^\mu\nabla_\mu \\ V^\mu\approx(1,0,0,0) \\...
  16. W

    Hyperbolic Manifold With Geodesic Boundary?

    Hi All, I am trying to figure out the details on giving a surface S a hyperbolic metric with geodesic boundary, i.e., a metric of constant sectional curvature -1 so that the (manifold) boundary components, i.e., a collection of disjoint simple-closed curves are geodesics under this metric. So...
  17. S

    Geodesic quation coordinate time

    Hi guys So I am having trouble reparameterizing the geodesic equation in terms of coordinate time. Normally you have: \frac{d^2 x^{\alpha}}{d \tau^2} + \Gamma_{nm}^{\alpha} \frac{d x^{n}}{d \tau}\frac{d x^{m}}{d \tau} = 0 Where \tau is the proper time. I class we were told to express the...
  18. I

    Geodesic exponential map distance

    Homework Statement Hi all. For some reason I have been having a lot of difficulty with this problem in Peter Petersen's text. The problem is Prove: ##d(exp_p(tv), exp_p(tw)) = |t||v-w| + O(t^2 )## Homework Equations The exponential map is the usual geodesic exponential map. And ##d(p,q)## is...
  19. P

    Why no absolute derivative in this example of geodesic deviation?

    On the surface of a unit sphere two cars are on the equator moving north with velocity v. Their initial separation on the equator is d. I've used the equation of geodesic deviation...
  20. C

    Null geodesic definition (by extremisation?)

    Hi, How can null geodesics be defined? Obviously the concept of parallel-transport, of the tangent to the curve, applies equally well to null curves as to time/space-like curves. Technically this is only the definition for an "auto-parallel", not for a "geodesic". For example in...
  21. C

    Geodesic Conjugate Points Explained

    Dear all, I was reading "Nature of space and time" By Penrose and Hawking pg.13, > If $$\rho=\rho_0$$ at $$\nu=\nu_0$$, then the RNP equation > > $$\frac{d\rho}{d\nu} = \rho^2 + \sigma^{ij}\sigma_{ij} + \frac{1}{n} R_{\mu\nu} l^\mu l^\nu$$ implies that the convergence $$\rho$$ will become...
  22. Greg Bernhardt

    What is the Concept and Calculation of Geodesic Deviation in Physics?

    Definition/Summary Where two particles very close together have the same velocities, their two geodesics are parallel, though only instantaneously, and so the gap (a 4-vector, of time and distance) between them has zero rate of increase, but has non-zero acceleration. The acceleration (a...
  23. M

    Geodesic Radius of Curvature Calculation Method

    I am trying to compute the geodesic (or tangent) radius of curvature of the geodesic circle by using the below formula. \frac{1}{\rho_c}=\frac{\partial G/\partial S}{2\sqrt{E} G} where s is the arc length parameter and E, G are the coefficents of the first fundamental form. Can you...
  24. M

    Computing geodesic distances from structural data

    Greetings, I am attempting to compute geodesic distances on manifolds where structural data have been sparsely sampled. First, off I am not well versed in the mathematics of differential geometry but I do have some knowledge (from an introductory differential geometry class in my...
  25. m4r35n357

    Geodesic Deviation in Spacetime: Exploring the Possibilities

    I've been meaning to ask this for some time, and now I've plucked up the courage! It is puzzling to me that many fundamental relationships in GR are explained in terms of euclidean space. Taking for example the geodesic deviation equation, it occurs to me that if defined in 3+1 spacetime there...
  26. Demon117

    Geodesic equations and Christoffel symbols

    I've been thinking about this quite a bit. So it is clear that one can determine the Christoffel symbols from the first fundamental form. Is it possible to derive the geodesics of a surface from the Christoffel symbols?
  27. E

    Integrating Geodesic Equations: Kevin Brown

    Kevin Brown, in his excellent book "Reflections on Relativity" p. 409, "immediately" integrates 2 geodesic equations: \frac{d^{2}t}{ds^{2}}=-\frac{2m}{r(r-2m)}\frac{dr}{ds}\frac{dt}{ds} \frac{d^{2}\phi}{ds^{2}}=-\frac{2}{r}\frac{dr}{ds}\frac{d\phi}{ds} to get...
  28. I

    Radial Null Geodesic: Solving the Equation

    Hi, I've found geodesic equations for the metric: \begin{equation} ds^{2} = -c^{2} \alpha dt^{2} + \frac{1}{ \alpha } dr^{2} + d \omega ^{2} \end{equation} where \begin{equation} \alpha = 1 - \frac{r^{2}}{r_{s}^{2}} \end{equation} I have found that for a light ray: \begin{equation}...
  29. V

    General parameterisation of the geodesic equation

    Hello all, In Carroll's on page 109 it is pointed out that for derivation of the geodesic equation, 3.44, a "hidden" assumption is that we have used an affine parameter. Some few lines below we see that "any other parametrization" could be used, called alpha, but in that case the general...
  30. Demon117

    Geodesic curvature, normal curvature, and geodesic torsion

    I am struggling to make sense out some things. Hopefully someone can help or at least offer some different point of view. Let's examine a differential curve parameterized by arc length that maps some interval into an oriented surface (lets call it N(s)). The surface has a unit normal field...
  31. N

    Why Is the Norm of the Tangent Vector Constant in Geodesic Equations?

    I am trying to derive the geodesic equation by extremising the integral $$ \ell = \int d\tau $$ Now after applying Euler-Lagrange equation, I finally get the following: $$ \ddot{x}^\tau + \Gamma^\tau_{\mu \nu} \dot{x}^\mu \dot{x}^\nu = \frac{1}{2} \dot{x}^\tau \frac{d}{ds} \ln \left|...
  32. O

    Second order geodesic equation.

    Hello all, I have a geodesic equation from extremizing the action which is second order. I am curious as to what the significance is of having 2 independent geodesic equations is. Also I was wondering what the best way to deal with this is.
  33. darida

    How do I use bivectors to find the electric field in a weak magnetic field?

    Geodesic equation: m_{0}\frac{du^{\alpha}}{d\tau}+\Gamma^{\alpha}_{\mu\nu}u^{\mu}u^{\nu}= qF^{\alpha\beta}u_{\beta} Weak-field: ds^{2}= - (1+2\phi)dt^{2}+(1-2\phi)(dx^{2}+dy^{2}+dz^{2}) Magnetic field, B is set to be zero. I want to find electric field, E, but don't know where to start, so...
  34. stevendaryl

    Affine parametrization for null geodesic?

    The geodesic equation for a path X^\mu(s) is: \frac{d}{d s} U^\mu + \Gamma^\mu_{\nu \tau} U^\nu U^\tau = 0 where U^\mu = \frac{d}{ds} X^\mu But this equation is only valid for affine parametrizations of the path. For a timelike path, being affine means that the parameter s must be...
  35. P

    Parameterize a geodesic using one of the coordinates

    I've been working on a problem where I have to find the geodesics for a given Riemannian Manifold. To present my doubt, I tried to find a simpler example that would demonstrate my uncertainty but the one I found, and shall present bellow, has actually a simplification that my problem doesn't, so...
  36. P

    Lightlike geodesic in AdS5xS5, plane wave background

    Homework Statement My question is about a step in the lecture notes [http://arxiv.org/abs/hep-th/0307101] on page 6, and it is probably quite trivial: I want to see why a lightlike particle in AdS_5\times S^5 sees the metric as plane wave background. The metric is ds^2=R^2(-dt^2...
  37. B

    Geodesic Equation in Flat & Curved Spaces

    I understand why the geodesic equation works in flat space. It just basically gives a set of differential equations to solve for a path as a function of a single variable s where the output is the coordinates of whichever parameterization of the space you are using. But the derivation I know and...
  38. D

    Say I have my pen on my desk; does it describe a geodesic?

    Let's say I have my pen on my desk; does it describe a geodesic.? Or not because there is the normalforce working on it.
  39. tom.stoer

    Geodesic Equation: Generalizing for Functions F

    The geodesic equation follows from vanishing variation ##\delta S = 0## with ##S[C] = \int_C ds = \int_a^b dt \sqrt{g_{ab}\,\dot{x}^a\,\dot{x}^b}## In many cases one uses the energy functional with ##\delta E = 0## instead: ##E[C] = \int_a^b dt \, {g_{ab}\,\dot{x}^a\,\dot{x}^b}## Can...
  40. V

    Geodesic in 2D Space: Understanding the Statement

    Homework Statement I am having trouble understanding how the following statement (taken from some old notes) is true: >For a 2 dimensional space such that ds^2=\frac{1}{u^2}(-du^2+dv^2) the timelike geodesics are given by u^2=v^2+av+b where a,b are constants. Homework Equations...
  41. L

    Question about Geodesic Equation Derivation using Lagrangian

    I'm trying to derive the Geodesic equation, \ddot{x}^{α} + {Γ}^{α}_{βγ} \dot{x}^{β} \dot{x}^{γ} = 0. However, when I take the Lagrangian to be {L} = {g}_{γβ} \dot{x}^{γ} \dot{x}^{β}, and I'm taking \frac{\partial {L}}{\partial \dot{x}^{α}}, I don't understand why the partial derivative of...
  42. F

    Geodesic of Sphere in Spherical Polar Coordinates (Taylor's Classical Mechanics)

    Homework Statement "The shortest path between two point on a curved surface, such as the surface of a sphere is called a geodesic. To find a geodesic, one has to first set up an integral that gives the length of a path on the surface in question. This will always be similar to the integral...
  43. sergiokapone

    Homogeneous gravitational field and the geodesic deviation

    In General Relativity (GR), we have the _geodesic deviation equation_ (GDE) $$\tag{1}\frac{D^2\xi^{\alpha}}{d\tau^2}=R^{\alpha}_{\beta\gamma\delta}\frac{dx^{\beta}}{d\tau}\xi^{\gamma}\frac{dx^{\delta}}{d\tau}, $$ see e.g...
  44. S

    Killing vectors and Geodesic equations for the Schwarschild metric.

    Hello Everybody, Instead of solving the geodesic equations for the Schwarzschild metric, in many books (nearly in all books that I consulted), conserved quantities are looked at instead. So take for eg. Carroll, he looks at the killing equation and extracts the equation K_\mu...
  45. S

    Derivation of the Geodesic equation using the variational approach in Carroll

    Hello Everybody, Carroll introduces in page 106 of his book "Spacetime and Geometry" the variational method to derive the geodesic equation. I have a couple of questions regarding his derivation. First, he writes:" it makes things easier to specify the parameter to be the proper time τ...
  46. F

    Multivariable Calculus: Geodesic problem

    Homework Statement Consider the parametrization of a torus: \tau(u,v)=((2+cos(v)cos(u),(2+cosv)sinu,sinv) The distance from the origin to the center of the tube of the torus is 2 and he radius of the tube is 1. Let the coordinates on \mathbb R^3 be (x,y,z) . If p = \tau(u,v) then u is...
  47. bcrowell

    State of the art re the geodesic hypothesis

    Stated loosely, the geodesic hypothesis says that test particles follow geodesics, where "test particle" means it has to be small in some sense (size, mass, ...), and there is an ambiguity in the word "geodesics" because we want to talk about geodesics of the spacetime that would have existed...
  48. B

    Classical Mechanics: Minimization of geodesic on a sphere

    Homework Statement Use the result (6.41) of Problem 6.1 to prove that the geodesic (shortest path) between two given points on a sphere is a great circle. [Hint: The integrand f(ψ,ψ',θ) in (6.41) is independent of ψ, so the Euler-Lagrange equation reduces to ∂f/ψ' = c, a constant. This gives...
  49. D

    What causes geodesic incompleteness in spacetime manifolds?

    Hi all, I would like to know if somebody know the cases when we have in the space time manifold (and in general in any manifold) geodesic incompleteness. I know that a case can be a singularity in the curvature scalar (or in general, a singularity in any component of the Riemann tensor)...
  50. P

    Geodesic Equation from conservation of energy-momentum

    Hi everyone, While reading http://relativity.livingreviews.org/Articles/lrr-2011-7/fulltext.html reference I bumped into a result. Can anyone get from Eq.19.1 to Eq.19.3? I've also been struggling to get from that equation to the one before 19.4 (which isn't numbered)...anyone? Thank...
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