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. L

    Finding geodesics on a cone of infinite height

    Homework Statement Find the geodesics on a cone of infinite height, x^{2}+y^{2} = \tan{\alpha}^{2}z^{2} using polar coordinates (x,y,z)=(r\cos{\psi},r\sin{\psi},z) with z=r\tan(\alpha) The Attempt at a Solution I am not sure with how should I expres the element dz^{2} ? When it is a...
  2. P

    Solving Radially-Directed Geodesics in AdS Space

    Homework Statement We consider global AdS given by the coordinates (\rho,\tau, \Omega_i), i=1,\ldots,d and the metric ds^2=L^2(-cosh^2\,\rho\,d\tau^2+d\rho^2+sinh^2\,\rho\,d \Omega_i{}^2) Find the trajectory \tau(\rho), radially-directed geodesics, strating from \rho=\rho_0 with proper...
  3. tom.stoer

    Geodesics in a constant gravitational field

    I want to interpret geodesics in a constant gravitational field. As a simple example I start with flat Minkowski spacetime ds^2 = -dT^2 + dX^2 + dY^2 + dZ^2 with a geodesic (in terms of coordinate time T) X^\mu(T) = (T, X=A, 0, vT) where A is an arbitrary constant and v ≤ c...
  4. B

    Why are geodesics parabolae on earth's surface?

    It's a naive question, but I'm pretty sure my professor said that space-time is locally flat (and I'm pretty sure that the volume of my room counts as "locally"). That said, I would expect falling objects to follow straight trajectories, but that's obviously not the general case. I thought...
  5. B

    Re-parametrization of Geodesics: Can You Confirm?

    Hello. If I find a solution of the geodesic equation and I change the parametrization, the new function does not satisfy this equation for a general re-parametrization. But the world line is the same. Can you confirm it: does it come from the fact that we usually choose \nabla_VV=0...
  6. E

    Exploring BTZ Black Hole & Geodesics

    Hi everybody. I am well aware that there is only one black hole in 2+1, i.e., the BTZ one. I also know that for vanishing and positive cosmological constants we get solutions with a conical singularity. My question is more about the interpretation of these last results. Assume that in the BTZ...
  7. M

    Anti-de Sitter spacetime metric and its geodesics

    Hello, everybody. I have some doubts I hope you can answer: I have read that the n+1-dimensional Anti-de Sitter (from now on AdS_{n+1}) line element is given, in some coordinates, by: ds^{2}=\frac{r^{2}}{L^{2}}[-dt^{2}+\sum\limits_{i=1}^{n-1}(dx^{i})^{2}]+\frac{L^{2}}{r^{2}}dr^{2} This can be...
  8. D

    Null geodesics of a Kerr black hole

    Homework Statement Hi, From the Kerr metric, in geometrized units, \left(1 - \frac{2M}{r}\right) \left(\frac{dt}{d\lambda}\right)^2 + \frac{4Ma}{r} \frac{dt}{d\lambda}\frac{d\phi}{d\lambda} - \frac{r^2}{\Delta} \left(\frac{dr}{d\lambda}\right)^2 - R_a^2...
  9. V

    Locally inertial coordinates on geodesics

    It's a standard fact of GR that at a given point in space-time, we can construct a coordinate system such that the metric tensor takes the form of Minkowski spacetime and its first derivatives vanish. Equivalently, we can make the Christoffel symbols vanish at point. Moreover, the fact that, in...
  10. G

    Geodesic Equations: Newtonian vs Einstein

    \dfrac{d^2 x}{dt^2}=-\nabla \Phi \dfrac{d^2 x^\mu}{d\tau^2}= -\Gamma^{\mu}_{\alpha \beta}{}\dfrac{dx^\alpha}{d\tau}\dfrac{dx^\beta}{d\tau} These two equations, to be true, the way they are written should ring a bell. They are similar yet not identical. What is the meaning behind them...
  11. snoopies622

    Geodesics in a rotating coordinate system

    In a uniformly rotating coordinate system the trajectories of freely moving objects are influenced by an apparent centrifugal and Coriolis force. Is there a coordinate system or metric (or both) in which these trajectories are geodesics instead?
  12. PerpStudent

    Geodesics in Rindler Space: How Do They Differ from Minkowski Space?

    How would one determine a geodesic in Rindler space? Why would geodesics not be simply the same as those of Minkowsky space? Is it not analogous to using polar vs. Cartesian coordinates in euclidean space, where a straight line is the same in either case?
  13. G

    Finding the shapes of all timelike geodesics

    Homework Statement Consider the two-dimensional spacetime with the line element dS2 = -X2dT2+dX2. Find the shapes X(T) of all timelike geodesics in this spacetime. 2. The attempt at a solution I have the solution to this problem but I don't understand one step. For timelike worldlines dS2 =...
  14. B

    Geodesics vs Projectile: Exploring the Equivalence in Curved Space-Time

    Yes, I want to make sure that geodesics of a particle moving in curved space time is the same thing of projectile trajectories. I start from assuming that 1-\frac{2GM}{r}\approx1-2gr and then calculate the schwarzschild metric in this form \Sigma_{\mu\nu}=\begin{bmatrix}\sigma & 0\\ 0 &...
  15. T

    Intuitive explanation of parallel transport and geodesics

    Hello, First of all, please excuse me if I posted in the inappropriate place.. While a student few years ago, I used to work a lot with advanced differential geometry concepts, but never got an intuitive view of HOW humanity got to think about parallel transport, why it contained two words...
  16. J

    Photons of different energy follow different geodesics?

    a massive body like a star creates a warped spacetime in its vicinty. this warped geometry of space is reflected by the geodesics appropriate to its mass. a photon passing by this massive object is not, as per GR, "attracted" to the star via some "force", but rather simply follows what it sees...
  17. N

    Timelike geodesics in Schwarzschild Metric.

    Please explain me how to derive the Timelike geodesics in Schwarzschild Metric. Thank you.
  18. N

    Finding Geodesics What I wish to understand, is how to solve

    Finding Geodesics What I wish to understand, is how to solve this one: given this metric: ds^2= \frac {dt^2} {t^2}- \frac{dx^2} {t^2} I have to calculate the geodesics. S=\int{ \frac {d} {d\lambda} \sqrt{\frac {1} {t^2} \frac {dt^2} {d\lambda^2}- \frac{1} {t^2} \frac {dx^2}...
  19. B

    Solving Geodesics on a Cone Using Euler-Lagrange

    Homework Statement We shall find the equation for the shortest path between two points on a cone, using the Euler-Lagrange equation. Homework Equations The Attempt at a Solution x = r sin(β) cos(θ) y = r sin(β) sin(θ) z = r cos(β) dx = dr sin(β) cos(θ) - r sin(β) sin(θ) dθ...
  20. Demon117

    The Poincaré Group and Geodesics in Minkowski Spacetime

    The poincare' group is the group of isometries of Minkowski spacetime, in a nutshell. In terms of an actual physical definition it is the group of all distance preserving maps between metric-spaces in Minkowski spacetime. What is the difference between this and geodesics?
  21. M

    Differential Geometry Surface with planar geodesics is always a sphere or plane

    Homework Statement Show that if M is a surface such that every geodesic is a plane curve, then M is a part of a plane or a sphere. Homework Equations - If a geodesic, \alpha, on M is contained in a plane, then \alpha is also a line of curvature. - Let p be any point on a surface M and...
  22. P

    Conformal invariance of null geodesics

    Hi, folks. I hope this is the right forum for this question. I'm not actually taking any classes, but I am doing self-study using D'Inverno's Introducing Einstein's Relativity. I have a solution, and I want someone to check it for me. Homework Statement Prove that the null geodesics of two...
  23. C

    Geodesics of a Sphere: Minimizing Integral and Solving for Great Circle Equation

    Homework Statement Hi, just want to get a couple of things straight regarding finding the geodesics of a sphere not using polar coordinates, but rather, Lagrange multipliers... I want to minimize I = int (|x-dot|2 dt) subject to the constraint |x|=1 (sphere) which gives an Euler equation...
  24. T

    General Relativity - Killing Vectors and Geodesics

    Hi, I'm stuck on the last bit the attached question where we're given the metric ds^2=-du^2+u^2dv^2 and have to use equation (*) to find the geodesic equations. They tell us to use V^a=\dot{x}^a the tangent vector to the geodesic and presumably we use the three killing vectors they gave us, so...
  25. L

    Geodesics: Constant Velocity & Affine Parameterization

    Affinely parameterised geodesics satisfy \nabla_XX=0 Why does this mean they have constant velocity? Thanks.
  26. WannabeNewton

    Geodesics on a cone in flat space

    So if you take a sphere with coordinates (r, \theta,\phi) and keep \theta constant you get a cone. The geodesic equations reduce to (by virtue of the euler - lagrange equations): \frac{\mathrm{d} ^{2}r}{\mathrm{d} s^{2}} - r\omega ^{2}\frac{\mathrm{d} \phi }{\mathrm{d} s} = 0 where \omega =...
  27. Z

    Can someone please explain to me why massless particles follow null geodesics?

    Anything that is massless must follow null geodesics. Why is this?
  28. G

    Geodesics on Surfaces: Proving the Relationship to Particle Motion

    Homework Statement Prove that a particle constrained to move on a surface f(x,y,z)=0 and subject to no forces, moves along the geodesic of the surface. Homework Equations The Attempt at a Solution OK, we should prove that the path the particle takes and the geodesic are given by...
  29. D

    What Is the Shortest Path on a Sphere If Not a Line of Latitude?

    Simple question about geodesics. I have a question which I guess will be easy to answer for anyone who is familiar with the geometry involved in GR. Firstly, I have a numbered list which shows my (current) understanding of geodesics. If there is any wrong with my understanding please let...
  30. bcrowell

    Hawking singularity theorem - what if not all geodesics incomplete?

    Hawking singularity theorem -- what if not all geodesics incomplete? The Penrose singularity theorem tells us that once you get a trapped surface, at least one geodesic is guaranteed to be incomplete, going forward in time. But this doesn't mean that 100% of the mass of a collapsing star has to...
  31. TrickyDicky

    Exploring Geodesics: Understanding Curved Paths in Outer Space

    Let's imagine a test particle in outer space not being subjected to any significant force, gravitational(far enough from any massive object) or any other. Its path would be describing a geodesic that follows the universe curvature, right? Would that be an euclidean straight path, or would it...
  32. jfy4

    Phase, Geodesics, and Space-Time Curvature

    Please read and critique this argument for me please, any help is appreciated. Imagine a geodesic, and a matter wave that traverses this geodesic. The action of this matter wave determines the motion of the matter wave along this geodesic over a given space-time interval, and is specified...
  33. jfy4

    Phase, Geodesics, and Space-Time Curvature

    Please read and critique this argument for me please, any help is appreciated. Imagine a geodesic, and a matter wave that traverses this geodesic. The action of this matter wave determines the motion of the matter wave along this geodesic over a given space-time interval, and is specified...
  34. TrickyDicky

    What are the properties of Minkowski spacetime geodesics?

    I have some difficulties understanding how Minkowski spacetime is flat and therefore its geodesics should remain parallel, but at the same time I see it described in other sites as hyperbolic and then geodesics should diverge. Any comment on my confusion about this will be welcome. Thanks
  35. S

    Geodesics and straight lines on a surface

    Homework Statement Let \gamma be a stright line in a surface M. Prove \gamma is a geodeisc The Attempt at a Solution In a plane we know a straight line is the shortest distance between two point. I am not sure if this applies to straight lines on a surface. Further more, there...
  36. G

    General relativity question - geodesics.

    I'm doing some revision for a General relativity exam, and came across this question: A Flat Earth space-time has co-ordinates (t, x, y, z), where z > 0, and a metric ds2 = ((1 + gz)2)dt2 − dx2 − dy2 − dz2 where g is a positive constant. Write down the geodesic equations in this space-time...
  37. B

    Relationship btwn Killing Vectors and Geodesics

    In general, what can one say about the relationship between geodesic motion of (massive and massless) particles and the killing vectors associated with the metric?
  38. Q

    Physical intuition behind geodesics and parallel transport

    Hi all, Sorry if this is a dumb question, but what exactly do we mean by the term parallel transport? Is it just the physicist's way of saying isometry? Also, in my class we have just defined geodesics, and we're told that having a geodesic curve cis equivalent to demanding that the unit...
  39. F

    Is Gravity Really a Force? Examining Einstein's Theory of General Relativity

    From what I understand, Einstein basically scrapped the concept of gravity being a force and instead said that energy (and thereby mass) and momentum causes spacetime to curve. Objects still travel on geodesics in spacetime (Newton's first law), but since it is curved, the geodesics near massive...
  40. M

    Particles (not?) following geodesics in GR

    particles (not??) following geodesics in GR In a three-month old thread https://www.physicsforums.com/showthread.php?p=2557522&posted=1#post2557522 one of the tutors ("atyy") said: "And GR in full form does not have particles traveling on geodesics..." What does that mean? How can a free...
  41. F

    Modeling Light Geodesics in FLWR Metric: Trajectory and Convergence Analysis

    hello, I developed an application that models the trajectory of a light geodesic in the FLWR metric leaving from a galaxy and coming to our. I made for the moment the euclidean case (k=0) with a zero cosmological constant . So, the metric can be written ...
  42. F

    Null geodesics of light from a black hole accretion disk

    Sorry I don't know latex so this may look a little messy. Homework Statement I'm trying to solve the equation for null geodesics of light traveling from a rotating black hole accretion disk to an observer at r = infinity. The point of emission for each photon is given by co-ordinates r, phi...
  43. mnb96

    Geodesics in non-smooth manifolds

    Hello, I will expose a simplified version of my problem. Let's consider the following transformation of the x-axis (y=0) excluding the origin (x\neq 0): \begin{cases} \overline{x}=x \\ \overline{y}=1/x \end{cases} Now the x-axis (excluding the origin) has been transformed into an hyperbola...
  44. Spinnor

    Are there geodesics for Calabi–Yau manifold?

    Say I sit at some point P in a Calabi-Yau manifold. Are there geodesics which start from P and return to P? Are there "geodesics" which start from P and return to P but may make a "side trip first"? Is the number of geodesics which start at P and end at P infinite or finite and does that...
  45. R

    Explanation of Wiki regarding Geodesics as Hamiltonian Flows:

    In the article from Wikipedia called: Geodesics as Hamiltonian Flows at: http://en.wikipedia.org/wiki/Geodesics_as_Hamiltonian_flows" It states the following: It is frequently said that geodesics are "straight lines in curved space". By using the Hamilton-Jacobi approach to the...
  46. D

    Effective potential and geodesics in G.R.

    The geodesics around a spherical mass (Schwarzschild solution) in G.R. can be described by \frac{1}{2}\left(\frac{dr}{d\lambda}\right)^2 + V(r) = \mathcal{E} where V(r) is the effective potential \frac{1}{2}\epsilon - \epsilon\frac{GM}{r} + \frac{L^2}{2r^2} - \frac{GML^2}{r^3} and...
  47. D

    Parallel transport and geodesics

    A vector field is parallel transported along a curve if and only if the the corariant derivative of the vector field along the path is 0. That is \frac{d}{d\lambda} V^\mu + \Gamma^\mu_{\sigma \rho} \frac{dx^\sigma}{d\lambda} V^\rho = 0 This is basically what every book says. But what...
  48. A

    Physics Major's Questions on Geodesics

    I am sorry with the bad title and I am physics major with very weak math. So I come to the forum to rescue me. Basically I have one question, what does a "point-like creature" on a one dimensional line "sees" on different geodesics? if the line is flat, then the creature can sees everything on...
  49. F

    What is the relationship between free falling bodies and spacetime geodesics?

    http://www.youtube.com/watch?v=8MWNs7Wfk84&feature=PlayList&p=858478F1EC364A2C&index=2" , Edmund Bertschinger is talking about Einstein's field equations . during the lecture , under the title of : "Gravity as sapcetime curvature (GR viewpoint) " , he wrote : "Freely falling bodies move along...
  50. L

    Variational Calculus : Geodesics w/ Constraints

    Homework Statement Consider the cylinder S in R3 defined by the equation x^2+y^2=a^2 (a). The points A=(a,0,0) \: and \: B = (a \cos{\theta}, a \sin{\theta}, b) both lie on S. Find the geodesics joining them. (b). Find 2 different extremals of the length functional joining A=(a,0,0)...
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