What is Geodesics: Definition and 182 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. P

    A The Connection Between Geodesics and the Lagrangian | Explained in Textbook

    I've recently read in a textbook that a geodesic can be defined as the stationary point of the action \begin{align} I(\gamma)=\frac{1}{2}\int_a^b \underbrace{g(\dot{\gamma},\dot{\gamma})(s)}_{=:\mathcal{L}(\gamma,\dot{\gamma})} \mathrm{d}s \text{,} \end{align} where ##\gamma:[a,b]\rightarrow...
  2. FallenApple

    What is the concept of geodesics between images in computer vision AI?

    I was told by someone that for computer vision AI, a photo of say an apple and an orange exists on some high dimensional manifold, and the goal is to learn a geodesic between the two objects. What does this mean? Does this mean that the photo of one of the images is just a tuple of coordinates...
  3. binbagsss

    General Relativity geodesics, killing vector, conserved quantities

    Homework Statement Homework EquationsThe Attempt at a Solution [/B] Let ##k^u## denote the KVF. We have that along a geodesic ##K=k^uV_u## is constant , where ##V^u ## is the tangent vector to some affinely parameterised geodesic. ##k^u=\delta^u_i## , ##V^u=(\dot{t},\vec{\dot{x}})## so...
  4. SonnetsAndMath

    A Radial, exterior, outgoing, null geodesics in Schwarzschild

    I'm a little confused about the proper way to find these null geodesics. From the line element, $$c^2 d{\tau}^2=\left(1-\frac{r_s}{r}\right) c^2 dt^2-\left(1-\frac{r_s}{r}\right)^{-1}dr^2-r^2(d{\theta}^2+\sin^2\theta d\phi^2),$$ I think we can set ##d\tau##, ##d\theta## and ##d\phi## to ##0##...
  5. P

    Non-radial null geodesics in Eddington-Finkelstein coordinates

    Homework Statement My end goal is to plot null geodesics around a black hole with realistic representations within the horizon (r<2GM, with c=1) using Mathematica. I've done this for outside the horizon using normal Schwarzschild coordinates and gained equation (1) below, and then used this...
  6. ShayanJ

    A Finding geodesics using the squared Lagrangian

    To find the geodesics of a space(time), what we need to do is extremizing the functional ##\displaystyle \int_{\lambda_1}^{\lambda_2}\sqrt{g_{\mu \nu} \frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}} d\lambda ##. But sometimes the presence of the square root makes the equation of motion too...
  7. ShayanJ

    A Geodesics of AdS3: Find a Reference

    I need to find the geodesics of AdS3 spacetime. But my searches have given me nothing. Can anyone give a reference where they're calculated? Thanks
  8. Ibix

    I Null geodesics in Schwarzschild spacetime

    I was looking at null geodesics in Schwarzschild spacetime. Carroll's lecture notes cover them here: https://preposterousuniverse.com/wp-content/uploads/grnotes-seven.pdf He notes (and justifies) that orbits lie in a plane and chooses coordinates so it's the equatorial plane, then uses Killing...
  9. F

    I Geodesics on S2 Surface: Arc-Length Parametrisation

    Consider the metric of ##S^{2}##: $$ds^{2}=d\theta^{2}+\sin^{2}(\theta)d\phi^{2}$$ Then in order to determine the geodesics on this surface one can minimise the integral $$s=\int_{l_{1}}^{l_{2}}\sqrt{\left(\frac{d\theta}{dl}\right)^{2}+\sin^{2}(\theta)\left(\frac{d\phi}{dl}\right)^{2}}dl$$ where...
  10. F

    I Geodesics and affine parameterisation

    As I understand it, a curve ##x^{\mu}(\lambda)## (parametrised by some parameter ##\lambda##) connecting two spacetime events is a geodesic if it is locally the shortest path between the two events. It can be found by minimising the spacetime distance between these two events...
  11. F

    I Formation of Bound Systems, Stars & Galaxies in General Relativity

    In particular how does matter "clump" together to form stars and planets, and how do Galaxy/star systems form? For the latter question is the answer simply that near massive enough bodies, the spacetime curvature is significant enough that the geodesics within its vicinity are closed curves...
  12. S

    A Solve Gravitational Interferometer & Geodesics

    I was trying to solve this excercise: Now I was able to find the eq. of geodetics (or directly by Christoffel formulas calculation or by the Lagrangian for a point particle). And I verified that such space constant coordinate point is a geodetic. Now, for the second point I...
  13. A

    B Euler-Lagrange equation for calculating geodesics

    Hello I am little bit confused about lagrange approximation to geodesic equation: So we have lagrange equal to L=gμνd/dxμd/dxν And we have Euler-Lagrange equation:∂L/∂xμ-d/dt ∂/∂x(dot)μ=0 And x(dot)μ=dxμ/dτ. How do I find the value of x(dot)μ?
  14. F

    I Example of computing geodesics with 2D polar coordinates

    I am trying to find and solve the geodesics equation for polar coordinates. If I start by the definition of Christoffel symbols with the following expressions : $$de_{i}=w_{i}^{j}\,de_{j}=\Gamma_{ik}^{j}du^{k}\,de_{j}$$ with $$u^{k}$$ is the k-th component of polar coordinates ($$1$$ is for...
  15. C

    B Light follows Geodesics-Spacetime-Big Bang-Time dilation

    I have these questions: 1) Why must light always move along a geodesic line? What is the principle behind that? 2) A second question about spacetime: We mostly depict or imagine spacetime as a net of flexible fiber that extends everywhere as a plane as we see it.. As we are looking it, what...
  16. myra2016

    Find Null Geodesics with affine parameter

    Homework Statement The metric is given by https://dl.dropboxusercontent.com/u/86990331/metric12334.jpg H is constant; s is an affine parameter, and so r(0)=t(0)=0. Apologise in advance because I'm not very good with LaTex. So I used Word for equations, and upload handwritten attempt at...
  17. A

    I Geodesics on a sphere and the Christoffel symbols

    Hi, I recently tried to derive the equations for a geodesic path on a sphere of radius 1 (which are supposed to come out to be a great circle) using the formula \dfrac{d^2 x^a}{dt^2}+\Gamma^a_{bc} \dfrac{dx^b}{dt}\dfrac{dx^c}{dt}=0 for the geodesic equation, with the metric...
  18. V

    I Light deflection and geodesics

    It is known that light beam bends near massive body and the object sendind deflected the beam is seen in shifted position. Now about spacetime curvature. As I undestand the things are like that: http://i11.pixs.ru/storage/3/3/4/pic2png_7037348_21446334.png The question is why are geodesics...
  19. Bob R

    I Geodesics in Curved Spacetime: What is the Significance of the ct Increment?

    [Moderator's note: this post has been spun off into its own thread.] I'm a retired engineer trying to get my head around GR, its effects in our everyday non-relativistic world, and its reduction to Newtonian gravity. I hope this is not too much of a digression from the current string. As I...
  20. MattRob

    Geodesics in a Given, Arbitrary Metric, dt Coefficient Only

    Not a formal course - just a question I decided to try to tackle with what I've gleaned from Stanford's lectures on Youtube, but still putting this here on account of this. So, I've been watching the Stanford GR series, and I have two motivations for messing around with this type of metric; 1...
  21. Elnur Hajiyev

    A Can geodesic deviation be zero while curvature tensor is not

    I understand(or assume understand) that geodesic deviation describes how much parallel geodesics diverge/converge on manifolds while moving along these geodesic. But is not it a definition for intrinsic curvature? If it is same as Riemann curvature tensor in terms of describing curvature, why...
  22. A

    Intersection of Hyperboloid & 2-Plane=Ellipse

    I want to try and see the intersection between the hyperboloid and the 2-plane giving an ellipse. So far I have the following: I'm going to work with ##AdS_3## for simplicity which is the hyperboloid given by the surface (see eqn 10 in above notes for reason) ##X_0^2-X_1^2-X_2^2+X_3^2=L^2## If...
  23. A

    Why Do Infinitely Heavy Particles Travel Like Massless Ones in AdS Space?

    I'm interested in calculating the geodesics of AdS3. I've been following the analysis in this link (http://www.ncp.edu.pk/docs/snwm/Pervez_hoodbhoy_002_AdS_Space_Holog_Thesis.pdf). I actually agree with all of the mathematics in the calculation and just have a query regarding the physics behind...
  24. B

    Null geodesics and null curves

    What is the exact difference between null geodesics and null curves? Please explain both qualitatively and quantitatively.
  25. bcrowell

    Stationary/extremal L for timelike/null/spacelike geodesics

    I would be interested in knowing if others think I have the correct analysis of whether length is stationary and/or extremal in the cases of geodesics that are timelike, null, and spacelike. Timelike In Minkowski space, the proper time ##\tau=\int \sqrt{g_{ij}dx^i dx^j}## (+--- metric) is both...
  26. T

    How should I think about null geodesics?

    I am kinda being thrown into pretty intense physics and this really doesn't have too much to do with what I'm doing but I was wondering if null geodesics have zero length, what are the other dimensions or parameters that accounts for the apparent movement of particles? I am a visual learner and...
  27. slatts

    Do geodesics of measure zero allow past-eternal inflation?

    Assuming that the expectation that all matter and energy are quantized is correct, I'm making a further assumption that "random" means something like, "hypothetically predictable, but only by means at least possibly impractical on any permanent or general basis whatsoever, such as enumeration...
  28. Andre' Quanta

    Killing vector tangents to geodesics

    Suppose to have a killing vector that its norm is null, so at the same time is also a null geodesic. Does the metric have special propierty? What can i say about the Killing vector and its proprierties?
  29. Andre' Quanta

    Are There Special Properties of Geodesics in a Lie Group?

    Suppose to have a Lie group that is at the same time also a Riemannian manifold: is there a relation between Christoffel symbols and structure constants? What can i say about the geodesics in a Lie group? Do they have special properties?
  30. m4r35n357

    "Simple" 4D Kerr Geodesics simulator, source code and docs

    Looks like my main pet GR project is about to enter something akin to maintenance mode, since it now does all I currently need it to. It's nothing earth-shattering at first glance, but is very concise (e.g. ~100 lines of Python for the simulator script) and should be easier to understand than...
  31. N

    Curvature, Geodesics and Acceleration in GR

    I am trying to get my head around curvature, geodesics and acceleration in GR. I've put together the following paragraph that attempts to describe qualitatively how I think these things play together. In Newtonian mechanics, a freely falling object accelerating towards the Earth implies a force...
  32. E

    Geodesics: Stationary Property & Dirac

    In Dirac's book on GRT, top of page 17, he has this: (I'll use letters instead of Greeks) gcdgac(dva/ds) becomes (dvd/ds) I seems to me that that only works if the metric matrix is diagonal. (1) Is that correct? (2) If so, that doesn't seem to be a legitimate limitation on the property of...
  33. C

    Null geodesics of the FRW metric

    When working with light-propagation in the FRW metric $$ds^2 = - dt^2 + a^2 ( d\chi^2 + S_k(\chi) d\Omega^2)$$ most texts just set $$ds^2 = 0$$ and obtain the equation $$\frac{d\chi}{dt} = - \frac{1}{a}$$ for a light-ray moving from the emitter to the observer. Question1: Do we not strictly...
  34. A

    Timelike Geodesics: Solving Reissner Nordstrom Line Element

    Homework Statement Using the Reissner Nordstrom line element, which I've given in the relevant equations section, I'm looking to show that the time like Geodesics obey the equation again show below. Homework Equations Line Element[/B] ##ds^2= - U(r)c^2dt^2 +\frac{dr^2}{U(r)} +r^2(d\theta^2 +...
  35. C

    Existence of affine parameters of null geodesics

    We have a general spacetime interval ##ds^2 = g_{\mu \nu} dx^\mu dx^\nu##. One way to define an affine parameter is to define it to be any parameter ##u## which is related to the path length ##s## by ##u = as + b## for two constants ##a,b##. One can show that for the tangent vector ##u^\alpha =...
  36. J

    Calculus of variations applied to geodesics

    Homework Statement I'm working on a problem from my gravitation book. The question is the following: Given \begin{equation} \frac{D}{Ds} T^\mu = 0 \end{equation}, where \begin{equation} T^\mu \left(s,a\right) = \frac{\partial z^\mu}{\partial s} \end{equation} is the tangent vector to a...
  37. RCopernicus

    Geodesics - Some help, please.

    We've all seen an image similar to this one: This is displaying the projection of GR Geodesics onto 3-D space (well, 2D in the picture). I'm still working my way through the General Relativity texts, so I'm not yet able to do the calculation on my own. Can anyone give me a formula that I can...
  38. michael879

    Can Kerr-Newman Geodesics Be Separated in Hamilton-Jacobi Equation with Charge?

    I've found the equations of motion for a charged test particle in the Kerr-Newman geometry from a number of sources. However, they aren't very reliable and disagree on small details, so I'm trying to derive it myself. I'm completely stuck at the last step though, where you "use" the...
  39. G

    Earth Geodesics - Rhumb Line vs Great Circle

    I have an object (A) at some altitude above the Earth ellipsoid, and a point (B) on the surface of the Earth. Since you're not confined to the surface of the Earth as you travel from A (at altitude), to B, I'm getting confused. If I were to create a (Cartesian) vector pointing from object...
  40. WannabeNewton

    Null geodesics and rotation

    Hi all. It is well known that in Schwarzschild space-time, a torque-free gyroscope in circular orbit at any permissible angular velocity at the photon radius (also known as the photon sphere i.e. ##r = 3M##) will, if initially tangent to the circle, remain tangent to the circle everywhere along...
  41. P

    Chapter 21 Ray D'Inverno Scalar Optics, congruence of null geodesics

    First of all this is my first thread, so I apologize for any mistake. Perhaps this is a stupid question, but i need some help in exercise 21.10 of D'Inverno, to write down geodesic equation for l^a, which is a vector tangent to a congruence of null geodesics and then by a rescaling of l^a...
  42. Sudharaka

    MHB Gaussian Mixture Models and Geodesics

    Hi everyone, :) This is a question that one of my friends sent me. It is kind of open ended and I don't have any clue about the particular area of research he is undertaking. Therefore I am posting the question here with the hope that anybody knowledgeable in this area would be able to help...
  43. N

    Are any two geodesics the same after applying an isometry?

    Hello, I was wondering the following. Suppose you start with a Riemannian manifold M. Say you know one geodesic. Pushing this geodesic forward through an isometry M -> M gives again a geodesic. Can this procedure give you all geodesics? Thinking of the plane or the sphere it seems...
  44. S

    Intuition of Geodesics: Visualizing Tangent & Parallel Transported Vectors

    I'm taking an undergraduate level GR course, and from my text (Lambourne), the author describes a geodesic as a curve that "always goes in the same direction", and says that the tangent vector to the curve at some point u+du (where u is the parameter variable from which all the vector components...
  45. A

    So the freely falling rock will follow a single geodesic on its way to Earth.

    "The geodesics in R n are the straight lines parametrized by constant velocity". This can be proved with the geodesic equation: \ddot{x^a} + \Gamma^a_{bc} \dot{x^b} \dot{x^c}=0 Locally we can find a coordinate system such that \Gamma =0 , and thus: \ddot{x^a} =0 So along a geodesic at some...
  46. DiracPool

    Understanding Acceleration and Geodesics in Curved Spacetime

    I have a related question which may broaden the image for conceptual clarity. Imagine an object (me) moving through "flat" outer space, far from any gravitational bodies. We can say that the geodesic I am traveling along is essentially straight or flat, as is its worldline, correct? So me...
  47. tom.stoer

    Radial Schwarzschild geodesics - again

    Suppose there is a radially free falling object starting at r(t=0) = r0 > rS with some initial velocity v. And suppose there is a radial light ray starting at R(t=0) = R0 > r0. Suppose that both the object and the light ray reach the singularity at the same time. Question: is there a simple...
  48. D

    When can we extend the minimal geodesics indefinitely?

    What sort of structure must a manifold possesses in order to talk about minimal geodesics between two points on it? When can we extend the minimal geodesics indefinitely?
  49. T

    Relative acceleration of geodesics and spacetime curvature

    Mass curves spacetime. The relative acceleration of nearby geodesics of free test particles indicates the sign of the spacetime curvature. Convergent geodesics mean positive, divergent negative curvature. But also the metric expansion of space curves spacetime. The geodesics may be convergent...
  50. L

    Lagrange multipliers for finding geodesics on a sphere

    Homework Statement Find the geodesics on a sphere g(x,y,z)=x^{2}+y^{2}+z^{2}-1=0 arclength element ds=\sqrt{dx^{2}+dy^{2}+dz^{2}} Homework Equations f(x,y,z)=\sqrt{x'^{2}+y'^{2}+z'^{2}} where x'^{2} \text{means} \frac{dx^{2}}{ds^{2}} and not d^{2}x/ds^{2} The Attempt at a...
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