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. A

    Alternate form of geodesic equation

    Homework Statement We're asked to show that the geodesic equation \frac{du^{a}}{dt} +\Gamma^{a}_{bc}u^{b}u^{c}=0 can be written in the form \frac{du_{a}}{dt}=\frac{1}{2}(\partial_{a}g_{cd})u^{c}u^{d} Homework Equations...
  2. andrewkirk

    Need help reproducing geodesic DE from a paper

    I am trying to understand the paper 'Spectral shifts in General Relativity' by Narlikar. The paper considers a light ray emanating from the origin of a FLRW coordinate system in a universe whose hypersurfaces of constant time (in that coordinate system) are homogeneous and isotropic. The...
  3. andrewkirk

    Lightlike radial null geodesic - how do we know it has constant theta and phi?

    Consider a light ray emanating from the origin of a FLRW coordinate system in a homogeneous, isotropic universe. The initial velocity of that ray will have only x0 (t) and x1 (r) components. In papers I have seen it is assumed that its velocity will continue to have zero circumferential...
  4. M

    Finding the geodesic function for scalar * function= scalar

    In this expression the junk on the left is a scalar. The stuff before the integral is another scalar. The integral is a time-like curve between x1 and x2 and at imagine fgf(x1) is a lower left corner of the rectangle and fgf(x2) is the upper right corner and x2-x1 is the length of the base of...
  5. C

    Calculus of Var, Euclidean geodesic

    Homework Statement Calculate the geodesic for euclidean polar coordinates given ds^{2}=dr^{2}+r^{2}dθ^{2} Homework Equations standard euler-lagrange equation The Attempt at a Solution I was able to reduce the euler-lagrange equation to \frac{d^{2}r}{dθ^{2}}-rλ=0 where...
  6. S

    How to get components of Riemann by measuring geodesic deviation?

    Hi all, I'm now reading Chap 11 of Gravitation by Wheeler, etc. In exercise 11.7, by introducing Jacobi curvature tensor, which contains exactly the same information content as Riemann curvature tensor, we are asked to show that we can actually measure ALL components of Jacobi curvature tensor...
  7. Telemachus

    Geodesic on a cone, calculus of variations

    I have to find the geodesics over a cone. I've used cylindrical coordinates. So, I've defined: x=r \cos\theta y=r \sin \theta z=Ar Then I've defined the arc lenght: ds^2=dr^2+r^2d\theta^2+A^2dr^2 So, the arclenght: ds=\int_{r_1}^{r_2}\sqrt { 1+A^2+r^2 \left ( \frac{d\theta}{dr}\right )^2...
  8. M

    Geodesic Upper Half Plane help

    The metric is ds^2=\frac{dx^2+dy^2}{y^2}. I have used the Euler-Lagrange equations to find the geodesics, and my equations are \dot{x}=Ay^2, \ddot{y}+\frac{\dot{x}^2-\dot{y}^2}{y}=0. I cannot seem to find the first integral for the second equation. I know it is \dot{y}=y\sqrt{1-Ay^2}, but I...
  9. E

    Geodesic Equation - Physics Explained

    http://mykomica.org/boards/shieiuping/physics/src/1335180831965.jpg http://mykomica.org/boards/shieiuping/physics/src/1335180965708.jpg
  10. R

    Geodesic Equation for Straight Lines in Euclidean Space

    Homework Statement If a general parameter ##t=f(s)## is used to parameterize a straight line in Euclidean space, then the geodesic equation takes the form ##\frac{d^2u^i}{dt^2}+\Gamma^i_{jk}\frac{du^j}{dt}\frac{du^k}{dt}=h(s)\frac{du^i}{dt}##, where...
  11. R

    Geodesic Deviation on an Infinite Cylinder

    Homework Statement Ants follow geodesics on a surface which is an infinite cylinder. Do the geodesics deviate? By considering only the paths of itself and its neighbors, can an ant decide whether it is on a cylinder of a plane? The Attempt at a Solution My answer to the first question is...
  12. A

    Metric Connection from Geodesic Equation

    For the following two-dimensional metric ds^2 = a^2(d\theta^2 + \sin^2{\theta}d\phi^2) using the Euler-Lagrange equations reveal the following equations of motion \ddot{\phi} + 2\frac{\cos{\theta}}{\sin{\theta}}\dot{\theta}\dot{\phi} = 0 \ddot{\theta} -...
  13. N

    Geodesic and MiSaTaQuWa equation of motion

    I am new to General Relativity and confused by the geodesic equation and MiSaTaQuWa equation. Most of the book saying that the geodesic equation is the motion of a particle in curved-spacetime. However, I read somewhere about this MiSaTaQuWa equation of motion. What is the difference between...
  14. Q

    Solve Geodesic Problem: Show ku^β = u^α∇αu^β

    Homework Statement I'm working my way through Wald's GR book and doing this geodesic problem: Show that any curve whose tangent satisfies u^\alpha \nabla_\alpha u^\beta = k u^\beta , where k is a constant, can be reparameterized so that \tilde{u}^\alpha \nabla_\alpha \tilde{u}^\beta =...
  15. S

    If Geodesic are light cone, how can be test particle trajectiories

    I had this doubt studing GR, but let's consider SR for semplicity, where g_{\mu\nu}=\eta_{\mu\nu}the geodesics are 0=ds^2=dt^2-dr^2 we obtain the constraint we obtain the constraint r=(+/-)t So it is a well known light cone, but in SR we have that a (test?)particle can always move in a...
  16. L

    Solving the Geodesic equations for a space

    I'm in an intro course and my shaky ability to solve differential equations is apparent. How would you go about solving \ddot{r}-r\ddot{\theta}=0 \ddot{\theta}+\frac{1}{r}\dot{r}\dot{\theta}=0 It might be obvious. They're the geodesic equations for a 2d polar coordinate system (if...
  17. L

    What flows on a surface can be geodesic flows?

    The question is what flows on a surface can be geodesic flows. Specifically, starting with a smooth vector field on a surface - perhaps with isolated singularities - when is there a Riemannian metric so that the vector field has constant length and is tangent to geodesics on the surface? Here...
  18. C

    Coordinate change to remove asymptotic geodesic?

    Through my mathematical fumblings, I think I have found a metric which gives a solution of the geodesic equation of motion that is asymptotic. It is a diagonal metric, with g00 = (x_1)^(-3) and g11 = 1. I am largely self-taught with SR so I may be miles off, but I think this gives a G.E. of M...
  19. L

    Geodesic flows on compact surfaces

    Does a geodesic flow on a compact surface - compact 2 dimensional Riemannian manifold without boundary - always have a geodesic that is orthogonal to the flow?
  20. U

    Mathematica Geodesic expansion in Mathematica

    Does anybody knows the package that can, given metric and equation of hypersurface (spacelike or null )calculate induced metric, external curvature and expansion (Raychaudhuri equation) in Mathematica. Thanks
  21. Philosophaie

    The three vectors in the Null Geodesic equation

    From a metric maybe the Schwarzschild, you can find g in co and contra varient forms. From that you can calculate Affinity. My question is from the Null Geodesic equation (ds=0) what do the three contravarient vectors represent? Do they represent the path of a planet around the sun or the...
  22. L

    Derivation of geodesic equation from hamiltonian (lagrangian) equations

    Homework Statement Hello, I would like to derive geodesics equations from hamiltonian H=\frac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu} using hamiltonian equations. A similar case are lagrangian equations. With the definition L=g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu I tried to solve the...
  23. T

    Fermi-Walker transport geodesic

    Hi guys, here's my question: An accelerated observer (both in curved or non-curved space) who Fermi Walker transports his own basis vectors set along his world line will have the metric in the minkowsky form \eta_{\mu\nu} at each point of the world line? AND if the observer follows a...
  24. Y

    If already known the Action and unkown the Metric, how to get geodesic equation?

    If already known the form of Action and unkown the Metric, how to derive the geodesic equation?
  25. TrickyDicky

    Geodesic motion from the EFE

    In order to clarify what the EFE tells us about geodesic motion, it is important to remember that by the local flatness theorem, we can at any point p introduce a coordinate system (Riemann normal coordinates) so that the first derivatives of the metric at that point vanish. We can choose to...
  26. P

    Deriving the Geodesic formulae

    [b]1. A straight line in flat space may be defined by the equation: (when I use the ^ symbol in this case it means like upper subscript not to the power) (U^v)(d/dx^v)(U^u)=0 (U^u=dx^u/ds) derive the geodesic equation. Please help I'm completely clueless all I can really see to do...
  27. jfy4

    Geodesic Motion for the Gödel Space-time

    (Hopefully, Part 1 of 2) This is one of my favorite metrics, and I decided that while tedious, and old-fashioned, I would practice for my GR studies by finding the Christoffell symbols and write out the equations for geodesics using the Gödel metric, then attempting to solve them. First...
  28. V

    Newtonian limit to schwarschild radial geodesic

    Hello Everyone, While trying to find the Newtonian limit to radial geodesic I was able to find that \frac{d^2r}{d\tau^2}=\frac{GM}{r^2} In the weak field limit we can naively replace \tau by "t" and recover Newtons Law, this though does not sound very rigorous. Can some-one suggest a...
  29. C

    Connection transformation from geodesic equations

    I don't know if the tex is displaying properly. On my computer all I see is the geodesic equations in every tex field. In the past when this has happened, it has been fine for others viewing it, but if it doesn't make sense, I will upload a pdf or something. Thanks. Homework Statement The...
  30. M

    Proving Geodesic Curvature of Curve in Surface is Equal to Projection

    Homework Statement Show that the geodesic curvature of an oriented curve C in S at a point p in C is equal to the curvature of the plane curve obtained by projecting C onto the tangent plane along the normal to the surface at p. Homework Equations Meusnier's theorem, and k^2 = (k_g)^2 +...
  31. M

    Constant Normal Curvature on Curves Lying on a Sphere?

    Homework Statement What curves lying on a sphere have constant geodesic curvature? Homework Equations k^2 = (k_g)^2 + (K_n)^2 The Attempt at a Solution I'm trying to understand the solution given in the back of the book. It says, a curve on a sphere will have constant curvature...
  32. M

    Prove that a great circle is a geodesic

    Homework Statement L = R \int \sqrt{1+ sin^2 \theta \phi ' ^ 2} d\theta from theta 1 to theta 2 Using this result, prove that the geodesic (shortest path) between two given points on a sphere is a great circle. [Hint: The integrand f(\phi,\phi',\theta) in the result is independent of...
  33. W

    Solve Geodesic Problem for f:[a,b] to R

    If f:[a,b] \to R is a positive real function and\gamma(u,v) = ( f(u)\cos (v), f(u) \sin (v), u) then show that \gamma(t) = \sigma(u(t), c) is a geodesic in Mwhere c is a constant between 0 and2\pi and M=\sigma(U) where U= \{ (u,v)| a<u<b and 0<v< 2\pi \} Actually , I tried to calculate the...
  34. WannabeNewton

    Solving geodesic equations on the surface of a sphere

    Homework Statement Find the geodesics on the surface of a sphere of radius a by: (a) writing the geodesic equations for the spherical coordinates given by: x = rsinTcosP y = rsinTsinP z = rcosT for T and P(the r - equation can be ignored as a = constant); (b) exhibit a particular...
  35. D

    Timelike geodesic equations for the Schwarzschild metric

    I'm following a slightly confusing set of notes in which I can't tell what exactly the timelike geodesic equations for the Schwarzschild metric are (seems to have about 3 different equations for them). How are these derived, or alternatively, does anyone have a link to a site in which they...
  36. haushofer

    Geodesic equation via conserved stress tensor

    Hi, I have a question which was raised after reading the article "Derivation of the string equation of motion in general relativity" by Gürses and Gürsey. The geodesic equation for point particles can apparently be obtained as follows. First one takes the stress tensor of a point particle...
  37. K

    Geodesic Dome Angles: Help for Math Wizards!

    Hi, Any math wizards here willing to assist me in determining cut angles for parts to construct geodesic dome greenhouse. I am building a 13' diameter geodesic dome. I am using the info from geodesic dome calculators on line for a 2V version. It is composed of 6 pentagonal frames with...
  38. C

    BRS: Using geodesic congruences and frame fields to compute optical experience

    I. Overview Another SA asked me to elaborate on a remark I made to the effect that frequency shift phenomena always (even in Minkowski vacuum) involve at least the following ingredients: two (proper time parameterized) timelike curves C, C' an event A on C ("emission event") an...
  39. S

    Geodesic Curvature of a curve on a flat surface

    Sorry if this ends up being a naive question, but I have just a little conundrum. I'm dealing with curves in R2 and the Gauss-Bonnet theorem is a very useful result with what I'm currently doing, what with Gaussian curvature of a flat surface being zero, which is all fine...
  40. W

    Timelike Geodesic and Christoffel Symbols

    Homework Statement How do I show the following metric have time-like geodesics, if \theta and R are constants ds^{2} = R^{2} (-dt^{2} + (cosh(t))^{2} d\theta^{2}) Homework Equations v^{a}v_{a} = -1 for time-like geodesic, where v^{a} is the tangent vector along the curve The Attempt at a...
  41. W

    Show that Geodesic is space-like everywhere

    Homework Statement If the geodesic is space-like somewhere, show that the geodesic is space-like everywhere. Homework Equations Geodesic equation: \ddot{X}^{\mu}+\Gamma^{\mu}_{\alpha \beta}\dot{X}^{\alpha}\dot{X}^{\beta} = 0 The Attempt at a Solution I looked at the metric...
  42. S

    Finding Null Geodesic Equations for Einstein's Metric in Curved Space

    Homework Statement I'm given the metric for Einstein's universe, ds2 = c2dt2 - dr2/(1 - kr2) - r2d(theta)2 - r2sin2(theta)d(phi)2 and asked to find the null geodesic equations and show that in the plane theta=\pi/2, the curves satisfy the equation: (dr/d(phi))2 = r2(1-kr2)(mr2-1) where m is a...
  43. T

    Geodesic curves for an ellipsoid

    Homework Statement The problem asks to find the shortest distance between two points on Earth, assuming different equatorial and polar radii i.e. the coordinates are represented as: x = a*cos(theta)*sin(phi) y = a*sin(theta)*sin(phi) z = b*cos(phi) Homework Equations The Attempt at a...
  44. I

    Principal, Asymptotic, and Geodesic Curves

    Homework Statement Is there a curve on a regular surface M that is asymptotic but not principal or geodesic? Homework Equations The given definitions of asymptotic, principal, and geodesic: A principal curve is a curve that is always in a principal direction. An asymptotic curve is a...
  45. I

    Deriving Geodesic Deviation - Help Appreciated

    Hi there, I'm trying to understand the derivation of geodesic deviation given here: http://wps.aw.com/wps/media/objects/500/512494/supplements/Ch21.pdf but I can't figure out why x(t)+\chi(t) obeys the geodesic equation (eq.(7)). Of course x(t) does, since it is per definition a...
  46. M

    CD: What Makes a Spacetime Geodesically Complete?

    I'm reading an article (http://arxiv.org/abs/gr-qc/0403075) which proves that a certain spacetime is geodesically complete. It does this by proving that the first derivatives fo all coordinates have finite bounds. My question is why this is enough. Is it just a simple ODE result? We know...
  47. P

    Geodesic equation in new coordinates question

    Homework Statement Suppose \bar{x}^{\mu} is another set of coordinates with connection components \bar{\Gamma}^{\mu}_{\alpha\beta}. Write down the geodesic equation in new coordinates. Homework Equations Using the geodesic equation: 0 = \frac{d^{2}x^{\mu}}{ds^{2}} +...
  48. B

    Timelike Geodesic: Proving c^2 from $\ddot x^{\mu}$

    My lecturer has written: \ddot x^{\mu} + \Gamma^{\mu}{}_{\alpha \beta} \dot x^{\alpha} \dot x^{\beta} = 0 where differentiation is with respect to some path parameter \lambda. If we choose \lambda equal to proper time \tau then it can be readily proved that c^2 = g_{\mu \nu}(x)...
  49. P

    Null geodesic in 2 dimensional manifold

    I have a question. Is it true that any curve in 2-dimensional manifold which tangent vector is null at each point is null geodesic? (In 2-dimensional manifold there are only 2 null direcitions at each point).
  50. A

    Solving the Equation of Geodesic Deviation

    Hello Anybody know how we can solve the equations of geodesic deviation in a given spacetime whether approximately or exactly? Thanks in advance
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