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. B

    Would a graviton follow the geodesic?

    A graviton, if massless, is generally expected to travel at c. If so, we would not expect it to follow the geodesic, which is the path a hypothetical particle with infinite speed. Therefore I would think for example that a massless graviton that was gravitationally lensed around a galaxy would...
  2. P

    Strange geodesic in Schwartzschild metric

    The following curve is geodesic in Schwardschild metric: \tau \mapsto [(1-2m/r_0)^{-1/2}\tau,r_0,0,0]. The tangent vector is: [(1-2m/r_0)^{-1/2},0,0,0] , its length is 1 and its product with killing vector \partial_t is equal: (1-2m/r_0)^{1/2} = \textrm{const}. So the body lays at rest...
  3. J

    Does an electron moving along a geodesic radiate?

    layperson here, so please correct any misconceptions i have on this. an electron will emit photons if it is accelerated (including changes to either velocity and/or direction of travel). acceleration occurs if the electron experiences a force. since GR indicates that gravity is not a...
  4. E

    Geodesic and the shortest path

    it comes from the calculus of variation that the shortest path between two points on a surface must be geodesic. then must the geodesic connected two points be the shortest path? if not, what about the example? Thanks for any reply!
  5. Fredrik

    Geodesic implies the well-known identity 0=0

    I'm trying to do excercise 4.8 in "Riemannian manifolds" by John Lee. (It's about showing that the geodesics of \mathbb R^n are straight lines). The result I'm getting is that the definition of a geodesic implies the well-known identity 0=0, which isn't very useful. I must have made a mistake...
  6. A

    How Do Geodesics Behave in a 2D Metric with Signature (-,+)?

    Homework Statement Consider the 2-dim metric {{\it ds}}^{2}=-{\frac {{a}^{2}{{\it dr}}^{2}}{ \left( {r}^{2}-{a}^{2}\right) ^{2}}}+{\frac {{r}^{2}{d\theta }^{2}}{{r}^{2}-{a}^{2}}}, where r > a. What is its signature? Show that its geodesics satisfy {\frac {{a}^{2}{{\it dr}}^{2}}{{d\theta...
  7. W

    Gravitational vs geodesic proper time

    I've been trying to learn GR and I've been back and forth through Schutz's first course book. I think I understand the basic principals, but one thing still eludes me: a traveler in free fall travels along the geodesic, the path of longest proper time. If the path between two points passes...
  8. R

    Principle of Least Action - Straight Worldline on a Geodesic

    What does it mean to say that something moves on a straight wordline in terms of the principle of least action? I know it generally means that action is minimum or stationary but since I only really know some physics from a conceptual standpoint and not a mathematical one I don't really know...
  9. M

    Geodesic on a cylinder - have I done this correctly?

    Geodesic on a cylinder - have I done this correctly?? Homework Statement ds^{2} = a^{2}d\theta^{2} + dz^{2} ds = \sqrt{a^{2}d\theta^{2} + dz^{2}} \int\sqrt{a^{2} + dz'^{2}} d\theta = Min E-L equation df/dz - d/d\theta(df/dz') = 0 df/dz = 0, d/d\theta[\frac{z'}{\sqrt{a^{2}...
  10. M

    Help with the Euler-Lagrange formula for a geodesic

    Homework Statement The metric is: ds^{2} = y^{2}(dx^{2} + dy^{2}) I have to find the equation relating x and y along a geodesic.The Attempt at a Solution ds = \sqrt{ydx^{2} + ydy^{2}} - is this right? ds = \sqrt{y + yy'^{2}} dx F = \sqrt{y + yy'^{2}} So then I apply the Euler-Lagrange...
  11. F

    Sphere geodesic and Christoffel Symbols

    Homework Statement I'm trying (on my own) to derive the geodesic for a sphere of radius a using the geodesic equation \ddot{u}^i + \Gamma^i_{jk}\dot{u}^j\dot{u}^k, where \Gamma^i_{jk} are the Christoffel symbols of the second kind, \dot{u} and \ddot{u} are the the first and second...
  12. S

    Geodesic deviation & Jacobi Equation

    http://en.wikipedia.org/wiki/Jacobi_field also see http://iopscience.iop.org/0305-4470/14/9/029/?ejredirect=.iopscience What's the difference between the jacobi equation and the geodesic deviation equation?
  13. W

    Is the Pen on My Desk a Geodesic and Is My Room an Inertial Frame?

    Hi, I was wondering: I'm sitting at my desk and on my desk lies a pen. Does the pen describe a geodesic? And is the room I'm sitting in an inertial frame? I think the pen doesn't describe a geodesic because it's not in free fall and i think my room is a good approximation of an inertial...
  14. R

    Geodesic in plane when metric depends on single variable?

    Hello, I have a p+1 dimensional manifold describing the parameter space of a family of probability densities. The p+1 dimensions are (beta, t1, t2, ..., tp), all reals, and beta restricted to the positive reals. The (fisher) metric on this manifold is a function of beta only, hence the...
  15. H

    What is the geodesic on a parabolic surface defined by z = A / (x^2 + y^2)?

    I'm trying to find the geodesic between any two points in 3D space, where the geodesic is constrained to a surface defined by z = A / (x^2 + y^2), where A is a constant. I've tried all sorts of variations on the Euler-Lagrange equations (after changing to cylindrical coordinates), but am...
  16. H

    How Do You Calculate a 2D Null Geodesic in the Presence of a Gravitational Mass?

    I am interested in solving the null geodesic between two points in the presence of a gravitational mass, assuming that everything takes place in 2 dimensions (i.e., no Z coordinate). The following is known: -x and y coordinates of first point -x and y coordinates of second point -x and y...
  17. T

    Christoffel symbols and Geodesic equations.

    Homework Statement (a) Consider a 2-dimensional manifold M with the following line element ds2=dy2+(1/z2)dz2 For which values of z is this line element well defined. (b) Find the non-vanishing Christoffel symbols (c) Obtain the geodesic equations parameterised by l. (d) Solve...
  18. J

    Solution help for the Geodesic Equation

    Let’s say an asteroid is about to enter earth’s atmosphere(it will burn up of course). The initially sitting is at: point = [x_{0},y_{0},z_{0},t_{0}] = [r_{0},\\theta_{0},\\phi_{0},t_{0}] With a 4-velocity: V = [v_{1},v_{2},v_{3},v_{4}] The momentum at 0+ p_{0+} = mass*V =...
  19. R

    Geodesic Sphere Homework: Prove Great Circle Path is Shortest

    Homework Statement L = R \int_{\theta_1}^{\theta_2} \sqrt{1 + sin^2(\theta ) \phi ' ^ 2} d\theta Use the result to prove that the geodesic (shortest path) between two given points on a sphere is a great circle. [Hint: The integrand f(phi,phi_prime,theta) in the result is independent of phi so...
  20. M

    Proving Geodesic Length Unchanged by Small Curve Changes

    Homework Statement Prove that the proper length of geodesic between two points is unchanged to first order by small changes in the curve that do not change its endpoints. Homework Equations Length of curve = \int...
  21. M

    Solving Geodesic Equations on Surfaces of Revolution

    Homework Statement I'm given the surface of revolution parametrized by \psi (t, \theta ) = (x(t), y(t)cos \theta, y(t)sin \theta ) where the curve \alpha (t) = (x(t),y(t)) has unit speed. Also given is that \gamma (s) = \psi (t(s), \theta (s)) is a geodesic which implies the following equations...
  22. D

    Geodesic Triangle: Calculating Area on a Sphere

    An equilateral geodesic triangle is right angled. The area of a geodesic triangle on a sphere of radius r is (1/2)\pir^2. But how is that obtained?
  23. D

    What is the proof that great circles are the only geodesics on a sphere?

    I've been reading a few proofs showing that a great circle is geodesic. Most of these proofs start with a parametrization and then show that it satisfies the differential equations of geodesics. The book that I have doesn't even give a proof. It just tells me that the great circles on the sphere...
  24. Philosophaie

    Hard time grasping the concept of the Geodesic

    Have a hard time grasping the concept of the Geodesic. If you are given a velocity and position of say a small astroid entering the gravitational field of the Earth. How do you find the trajectory or path it takes as it falls to Earth? Not afraid of tensors and advanced calculus.
  25. M

    Geodesic Curves Covering Surfaces

    Are there surfaces that have a geodesic curve which completely covers the surface, or (if that's not possible) is dense in the surface? In other words, if you were standing on the surface and started walking in a straight line, eventually you would walk over (or arbitrarily close to) every...
  26. H

    Proper Time For Photon on Null Geodesic

    I can understand the logic from some arguments as to why proper time in a photon's "frame of reference" is zero. I cannot understand how this follows from the argument that (SPACE)2 - (TIME) 2 = 0. This to me says that the SPACE-TIME interval for the photon is zero (null interval) and SPACE =...
  27. M

    What are the 'closed geodesic' ?

    i have been reading about 'Selberg Trace formula' i know what a Laplacian is but i do not know what is the author referring to when he talks about 'Closed Geodesic' i know what the Geodesic of a surface is l(\gamma)=\int_\gamma \sqrt{ g(\dot\gamma(t),\dot\gamma(t)) }\,dt\ , but i do...
  28. T

    Tidal forces and geodesic deviation: a principle question

    hi all, i kind of have a black hole regarding my understanding of the consistensy of tidal forces and geodesic deviation analysis. one can determine some coefficients of the riemman tensor from the tidal forces equations, by getting to a form that is like the form of the geodesic deviation...
  29. M

    Exploring Geodesic Surfaces: Minimizing Curves and Area Theory

    if we have or can have geodesic curves minimizing the integral \sqrt (g_{ab}\dot x_a \dot x_b ) is there a theory of 'minimizing surfaces or Geodesic surfaces' that minimize the Area or a surface ?,
  30. B

    Solving Friedmann's Dust-Filled Equation for Radial Geodesics

    I understand what geodesics are and how to calculate them from Christoffel symbols and all that. But I've just come across a question I have no idea about. I've been given the dust filled Friedmann solution: ds^2 = -dt^2 + a(t)^2 (dX^2 + X^2 dO^2) (O=omega) And been told to show that...
  31. C

    Proving Geodesic Pushed by Isometric Diffeomorphism

    Hello. Suppose that \sigma: (M, g) \to (N, h) is an isometric diffeomorphism between two Riemannian manifolds M and N and let \gamma: [0, 1] \to M be a geodesic on M. Because \sigma preserves distances, and geodesics are locally length minimizing, it is intuitively clear that \sigma_*...
  32. P

    Solving the Geodesic Equation: Raising Contravariant Indices

    Homework Statement I would like to manipulate the geodesic equation. Homework Equations The geodesic equation is usually written as k^{a}{}_{;b} k^{b}=\kappa k^{a} (it is important for my purpose to keep it in non-affine form). It is clear that by contracting with the metric we may...
  33. T

    Radial geodesic distance with Schwarzschild's solution

    Given we have a spherically symmetric gravitational field around a spherically symmetric body of mass M. How can I calculate the actual (geodesic) distance between two points with the same angle but at different distances from the center of the body (and field). I came immediately to think of...
  34. haushofer

    Geodesic coordinates and tensor identities

    Hi, I have a question about deriving tensor identities using geodesic coordinates ( coordinates in which one can put the connection to zero ). For the reference, I'm a master student physics and followed courses on general relativity and geometry. I'm busy with an article of Wald and Lee called...
  35. Loren Booda

    Geodesic applied to twins paradox

    Regarding the twins paradox: do all closed trajectories require nonzero acceleration at some point, or can a closed geodesic fulfill overall the special relativistic requirement of constant velocity?
  36. L

    Geodesic deviation equation

    Hi...does anyone have a good description (or a link to it) on geodesic deviation equation?Most of the references i have are in a setting of relativity, which make me all at sea. Please help me if you know a mathematical characterization of how geodesics from one point deviate (which just...
  37. L

    Geodesics in Modified Euclidean Spaces: Is There a General Statement?

    Hi there: i have a question on geodesics in a Eculidean space equipped with a metric tensor \lambda(x)*I, where I is the identity matrix. Is any general statement that can be made towards the geodesic between two points in this modified space? My feel is that this space is quite special...
  38. M

    Sun forms different geodesic

    Hi, Earth follws a straight path in 4-d space time.ok.now the Earth moves over the geodesic formed by the sun's gravity.now we also have other 7 planets.So does it mean that the sun forms different geodesic for differnt planets. if my question does make some logic than please explain me.
  39. D

    Incoming spacelike radial geodesic

    We need to show that using the Schwarzschild metric, an incoming radial spacelike geodesic satisfies r>= 1m/(1 + E^2) I know that E = (1-2m/r) is constant, and I think that ds squared should be negative for a spacelike geodesic. I try substituting E into the metric and setting ds squared <=0...
  40. D

    Geodesic & Asymptotic Curves: Proving Segment of Straight Line

    Homework Statement Show that a regular curve on a smooth surface is a geodesic and an asymptotic curve if and only if it is a segment of a straight line. The Attempt at a Solution I did the <= implication, which is quite easy. I can't get the other one.
  41. T

    Geodesic Deviation: Deriving Formula with Schtuz's Book

    Can someone explain how to derive the formula for geodesic deviation? i didn't quite understand it. I'm using schtuz's book.
  42. quasar987

    How do we find the null geodesic for a 2D spacetime with a positive constant?

    We consider a 2 dimensional spacetime with coordinates (t,y) and metric g=dt\otimes dt - (t^2+a^2)^2dy\otimes dy where a is a positive constant. In previous subquestions, I have calculated the Christoffel symbols for the metric-compatible connexion and now I am asked to find the null geodesic...
  43. K

    A person travelling through a geodesic.

    A person traveling through a geodesic. Does experiment some kind of acceleration?, since the geodesic equation analogue to Newton one is: \Delta _{u} u =0 and in an Euclidean space there's no acceleration for a particle line X(u)=au+b
  44. K

    Do the L4 and L5 points travel in a geodesic?

    Given a two body system like the sun and the earth: Do the L4 and L5 points travel in a geodesic, or is it just the sun and the Earth which do so?
  45. R

    Geodesic Curvature (Curvature of a curve)

    Can anyone point me to good reference that fully develops the geometry of geodesic curvature? Most of the ones I have manage to derive it, then show it's the normal to the curve, then never mention it again. I want to know how it relates to the metric, first second or third. Thanks.
  46. P

    Geodesics on a Circular Cylinder: Solving Ch6 Q6.4

    Hi, I'm working on marion&thornton ch6 question 6.4. "Show that the geodesic on the surface of a straight circular cylinder is a (partial) helix" I used the example of the geodesic on a sphere in the book, but when i calculate the angle phi i get something like phi=b*z+c, where b and c are...
  47. dextercioby

    How can we simplify calculating geodesic distances on a deformed Earth?

    This is just a mere suggestion for doing such computations without resorting to complicated series expansions of elliptic integrals. Up to the irregular continental shape & sea floor, we could picture our planet as a revolution ellipsoid. This means that, except for points on the same...
  48. rcgldr

    Geodesic sphere made of straws

    First starting making these back in 1973, but those are long gone. Decided to make a 4v icosahedron recently and this is what I ended up with. Web page shows the progress, and the final sphere: 4f.htm
  49. K

    What does the geodesic equation for a surface involve?

    I don't understand the equation of the geodesic y=y(x) for the surface given by z=f(x,y) : a(x)y''(x)=b(x)y'(x)^3+c(x)y'(x)^2+d(x)dxdy-e(x) the functions a,b,c,d,e are here not very important, what I don't understand, is that there is terms in \frac{dy}{dx} and dxdy...What does this mean ?
  50. M

    Light Geodesic Path: Troubleshooting Post Deletion

    can't delete post ? w/e. i'm trying to follow the path numerically. in ase of light geodesic, given gij, xi, and very small deltas dxi on step k, what would be dxi on step k+1?
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