What is Geodesic equation: Definition and 68 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. Kostik

    A Dirac "GTR" Eq. 27.11 -- how to show that a boundary term vanishes?

    In Dirac's "General Theory of Relativity", p. 53, eq. (27.11), Dirac is deriving Einstein's field equations and the geodesic equation from the variation ##\delta(I_g+I_m)=0## of the actions for gravity and matter. Here ##p^\mu=\rho v^\mu \sqrt{-g}## is the momentum of an element of matter. He...
  2. cianfa72

    I Lie dragging vs Fermi-Walker transport along a given vector field

    We had a thread long time ago concerning the Lie dragging of a vector field ##X## along a given vector field ##V## compared to the Fermi-Walker transport of ##X## along a curve ##C## through a point ##P## that is the integral curve of the vector field ##V## passing through that point. We said...
  3. cianfa72

    I Wald synchronous reference frame proof

    Hi, on Wald's book on GR there is a claim at pag. 43 about the construction of synchronous reference frame (i.e. Gaussian coordinate chart) in a finite region of any spacetime. In particular he says: $$n^b\nabla_b (n_aX^a)=n_aX^b\nabla_b \, n^a$$Then he claims from Leibnitz rule the above equals...
  4. E

    General relativity - Using Ricc and Weyl tensor to find the area

    I have the following question to solve:Use the metric:$$ds^2 = -dt^2 +dx^2 +2a^2(t)dxdy + dy^2 +dz^2$$ Test bodies are arranged in a circle on the metric at rest at ##t=0##. The circle define as $$x^2 +y^2 \leq R^2$$ The bodies start to move on geodesic when we have $$a(0)=0$$ a. we have to...
  5. E

    A Solving Geodesics with Metric $$ds^2$$

    I have the following question to solve:Use the metric: $$ds^2 = -dt^2 +dx^2 +2a^2(t)dxdy + dy^2 +dz^2$$ Test bodies are arranged in a circle on the metric at rest at $$t=0$$. The circle define as $$x^2 +y^2 \leq R^2$$ The bodies start to move on geodesic when we have $$a(0)=0$$ a. we have to...
  6. H

    I Carroll GR: Geodesic Eq from Var Principles

    On pages 106-107 of Spacetime & Geometry, Carroll derives the geodesic equation by extremizing the proper time functional. He writes: What I am unclear on is the step in 3.47. I understand that the four velocity is normalized to -1 for timelike paths, but if the value of f is fixed, how can we...
  7. Hamiltonian

    I Help Deriving Geodesic Equation from David Tong Notes

    I was following David tongs notes on GR, right after deriving the Euler Lagrange equation, he jumps into writing the Lagrangian of a free particle and then applying the EL equation to it, he mentions curved spaces by specifying the infinitesimal distance between any two points, ##x^i##and ##x^i...
  8. I

    A Geodesic Eq Derived from Einstein Field Equations?

    Since the EFE describes the shape of spacetime, it describes the way black holes, for example, evolve. Can one derive the geodesic equation from it in some limit ?
  9. jk22

    A Exists ? : Invariant geodesic equation

    Does there exist a form of the geodesic equation which is invariant under coordinates change ?
  10. jk22

    A How to Read Geodesic Equation: Vector, 3-D & EFE Solutions

    In the formula : ##\frac{d^2 x^\mu}{d\tau^2}=-\Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau}## How is the ##x^\mu## understood : a 4-vector or the ##\mu##-st component simply ? If it is a vector, how to write it in spherical coordinate with extra time dimension ? Btw...
  11. George Keeling

    I Solving Geodesic Eq.: Mysterious Conservation Eq. (Sec. 5.4 Carroll)

    I'm still on section 5.4 of Carroll's book on Schwarzschild geodesics Carroll says "In addition, we always have another constant of the motion for geodesics: the geodesic equation (together with metric compatibility) implies that the quantity $$...
  12. George Keeling

    I Question about geodesics on a sphere

    I am working from Sean Carroll's Spacetime and Geometry : An Introduction to General Relativity and have got to the geodesic equation. I wanted to test it on the surface of a sphere where I know that great circles are geodesics and is about the simplest non-trivial case I can think of. Carroll...
  13. Jason Bennett

    I Alternative form of geodesic equation for calculating Christoffels

    From Thomas Moore A General Relativity Workbook I have the geodesic equation as, $$ 0=\frac{d}{d \tau} (g_{\alpha \beta} \frac{dx^\beta}{d \tau}) - \frac{1}{2} \partial_\alpha g_{\mu\nu} \frac{dx^\mu}{d \tau} \frac{dx^\nu}{d \tau} $$ as well as $$ 0= \frac{d^2x^\gamma}{d \tau^2} +...
  14. sergiokapone

    I Derivation of Geodesics Eq from EM Tensor of Point Particle

    The energy-momentum tensor of a free particle with mass ##m## moving along its worldline ##x^\mu (\tau )## is \begin{equation} T^{\mu\nu}(y^\sigma)=m\int d \tau \frac{\delta^{(4) }(y^\sigma-x^\sigma(\tau ))}{\sqrt{-g}}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}.\tag{2} \end{equation} The covariant...
  15. R

    Alternative form of geodesic equation

    Homework Statement We are asked to show that: ## \frac{d^2x_\mu}{d\tau^2}= \frac{1}{2} \frac{dx^\nu}{d\tau} \frac{dx^{\rho}}{d\tau} \frac{\partial g_{\rho \nu}}{\partial x^{\mu}} ## ( please ignore the image in this section i cannot remove it for some reason ) Homework Equations The...
  16. A

    I Why is √(gμνdxμdxν) the Lagrangian for Geodesic Eq?

    From the invariance of space time interval the metric dΓ2=dt2-dx2-dy2-dz2 dΓ2=gμνdxμdxν dΓ=√(gμνvμvμ)dt dΓ=proper time. Can someone please help me in sort out why the term √(gμνdxμdxν) is taken as the Lagrangian,as geodesic equation is solved by taking this to be the Lagrangian.
  17. A

    I Deriving Geodesic Equation from Lagrangian

    Hi, If I have a massive particle constrained to the surface of a Riemannian manifold (the metric tensor is positive definite) with kinetic energy $$T=\dfrac 12mg_{\mu\nu} \dfrac{\text dx^{\mu}}{\text dt} \dfrac{\text dx^{\nu}}{\text dt}$$ then I believe I should be able to derive the geodesic...
  18. Alex Petrosyan

    I Origin of the half factor in Euler-Lagrange for geodesics

    I was wondering where does the 1/2 factor come from in the Euler-Lagrange equation, that is: L = \sqrt{g_{\mu \nu} \dot{x}^\mu \dot{x}^\nu} implies that \partial_\mu L = \pm \frac{1}{2} (\partial_\mu g_{\mu \nu} \dot{x}^\mu \dot{x}^\nu ) I'm not sure I entirely understand where it comes...
  19. P

    A Lense-Thirring effect - General Relativity

    Let us assume a "toy-metric" of the form $$ g=-c^2 \mathrm{d}t^2+\mathrm{d}x^2+\mathrm{d}y^2+\mathrm{d}z^2-\frac{4GJ}{c^3 r^3} (c \mathrm{d}t) \left( \frac{x\mathrm{d}y-y\mathrm{d}x}{r} \right)$$ where ##J## is the angular-momentum vector of the source. Consider the curve $$ \gamma(\tau)=(x^\mu...
  20. S

    B Geodesic equation in Minkowski space clarification

    Hi, So the geodesic equation is saying in my frame of reference I may see acceleration and then in your frame of reference you may see gravity? So by just changing coordinates you can create a "force" ? And also is this relevant to the Minkowski space or do I need to be in GR to be able to get...
  21. F

    I Deduce Geodesics equation from Euler equations

    I am using from the following Euler equations : $$\dfrac{\partial f}{\partial u^{i}}-\dfrac{\text{d}}{\text{d}s}\bigg(\dfrac{\partial f}{\partial u'^{i}}\bigg) =0$$ with function ##f## is equal to : $$f=g_{ij}\dfrac{\text{d}u^{i}}{\text{d}s}\dfrac{\text{d}u^{j}}{\text{d}s}$$ and we have...
  22. S

    I Geodesic Equation: Understanding Proper Time & x^α

    Hello! I am a bit confused about the geodesic equation. So for a massive particle it is given by: ##\frac{d}{d\tau}\frac{dx^\alpha}{d\tau}+\Gamma^\alpha_{\mu\beta}\frac{dx^\mu}{d\tau}\frac{dx^\beta}{d\tau}=0##, where ##\tau## is the proper time, but in general can be any affine parameter. I am...
  23. W

    Given the metric, find the geodesic equation

    Homework Statement Given that ##ds^2 = r^2 d\theta ^2 + dr^2## find the geodesic equations. Homework Equations The Attempt at a Solution I think the ##g_{\mu\nu} = \left( \begin{array}{ccc} 1& 0 \\ 0 & r^2 \end{array} \right)## Then I tried to use the equation ##\tau = \int_{t_1}^{t_2}...
  24. binbagsss

    I Derivation of geodesic equation from the action - quick question

    Hi, I am following this : https://en.wikipedia.org/wiki/Geodesics_in_general_relativity and all is good except how do we get ## \delta g_{uv}=\partial_{\alpha}g_{uv}\delta x^{\alpha}## Many thanks
  25. A

    I Why are the equations for dt/du and Dt[a]/Du equal in the geodesic equation?

    In the geodesic equation why is dt/du=λ(u)t ,where t is the tangent vector along the curve and why Dt[a]/Du=λ(u)dx[a]/du equated same,as given in hobson
  26. 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...
  27. M

    A Deriving Equations of Motion in GR

    Question Background: I'm considering the Eddington-Robertson-Schiff line element which is given by (ds)^2 = \left( 1 - 2 \left(\frac{\mu}{r}\right) + 2 \left(\frac{\mu^2}{r^2}\right) \right) dt^2 - \left( 1 + 2 \left( \frac{\mu}{r} \right) \right) (dr^2 + r^2 d\theta^2 + r^2 \sin^2{\theta}...
  28. A

    I Geodesic Equation: Lagrange Approximation Solution for Schwarzschild Metric

    Hello so if we have geodesic equation lagrange approximation solution: d/ds(mgμνdxν/ds)=m∂gμν∂xλdxμ/ds dxν/ds. So if we have schwarzschild metric (wich could be used to describe example sun) which is:ds2=(1-rs/r)dt2-(1-rs/r)-1dr2-r2[/SUP]-sin22. But that means that ∂gμν/∂xλ=0. So that means that...
  29. 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?
  30. A

    I Solving Geodesic Equations with Killing Vectors: Is There a General Solution?

    Hello I am concered about way of solving geodesic equation. Is there a general solution to geodesic equation? How to calculate the Cristoffel symbol at the right side of the equation? Thanks for helping me out!
  31. 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...
  32. C

    Finding the geodesic equation from a given line element

    Homework Statement We've got a line element ds^2 = f(x) du^2 + dx^2 From that we should find the geodesic equation Homework Equations Line Element: ds^2 = dq^j g_{jk} dq^k Geodesic Equation: \ddot{q}^j = -\Gamma_{km}^j \dot{q}^k \dot{q}^m Christoffel Symbol: \Gamma_{km}^j = \frac{g^{jl}}{2}...
  33. D

    Deriving geodesic equation using variational principle

    I am trying to derive the geodesic equation using variational principle. My Lagrangian is $$ L = \sqrt{g_{jk}(x(t)) \frac{dx^j}{dt} \frac{dx^k}{dt}}$$ Using the Euler-Lagrange equation, I have got this. $$ \frac{d^2 x^u}{dt^2} + \Gamma^u_{mk} \frac{dx^m}{dt} \frac{dx^k}{dt} =...
  34. NihalRi

    How do I use the geodesic equation for locations on earth

    So I've gone through the process of deriving the geodesic equation, I thought I understood it. I hoped that once the equation was obtained I'd be able to do simple replacements and find the shortest path between two locations on earth. I'm really stuck right now though so does anyone know how...
  35. P

    Geodesic Eq: Deriving 2nd Term on RHS

    As the geodesic equation in a form of is quite familiar for me. But I still cannot derive it in terms of time coordinate parameter; I can't get the second term on the right hand side what I can get is ½{d[lngαβ(dxα/dt)(dxβ/dt)]/dt}dxμ/dt How can I obtain that term?
  36. S

    What is the Purpose of the Geodesic Equation in General Relativity?

    I started studying the geodesic equation: ∂2xμ/∂s2 = - Γμab(∂xa/∂s)(∂xb/∂s) where the term s is proper time according to the wiki(https://en.wikipedia.org/wiki/Geodesics_in_general_relativity). The 2nd derivative on the left side of the equation is the acceleration in the xμ direction. Now my...
  37. D

    Deriving geodesic equation from energy-momentum conservation

    Hi all, I am trying to follow the calculation by samalkhaiat in this thread: https://www.physicsforums.com/threads/finding-equations-of-motion-from-the-stress-energy-tensor.547502/page-2 (post number 36). I am having some difficulty getting the equation above equation (11) (it was an unnumbered...
  38. G

    Black hole electron: How can we drop the geodesic equation?

    Hi, Einstein once showed that if we assume elementary particles to be singularities in spacetime (e.g. black hole electrons), then it is unnecessary to postulate geodesic motion, which in standard GR has to be introduced somewhat inelegantly by the geodesic equation. I don't have access to...
  39. U

    Solving this space-time Metric

    Homework Statement (a) Find ##\dot \phi##. (b) Find the geodesic equation in ##r##. (c) Find functions g,f,h. (d) Comment on the significance of the results. Homework Equations The metric components are: ##g_{00} = -c^2## ##g_{11} = \frac{r^2 + \alpha^2 cos^2 \theta}{r^2 +\alpha^2}##...
  40. 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...
  41. 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...
  42. U

    Energy-Momentum Tensor Algebra

    Homework Statement (a) Show acceleration is perpendicular to velocity (b)Show the following relations (c) Show the continuity equation (d) Show if P = 0 geodesics obey: Homework EquationsThe Attempt at a SolutionPart (a) U_{\mu}A^{\mu} = U_{\mu}U^v \left[ \partial_v U^{\mu} +...
  43. 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...
  44. U

    Index Notation: Understanding LHS = RHS

    I was reading my lecturer's notes on GR where I came across the geodesic equation for four-velocity. There is a line which read: Summing them up, \partial_i g_{aj} u^i u^j - \frac{1}{2} \partial_a g_{ij} u^i u^j = \frac{1}{2} u^i u^j \partial_a g_{ij} I'm trying to understand how LHS = RHS...
  45. 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...
  46. FreeThinking

    MTW "Gravitation" page 263, equation 10.27?

    Homework Statement (This is self-study.) In the equation just above 10.27 on page 263 of "Gravitation" by Misner, Thorne, and Wheeler, the first term is: \frac{\partial}{\partial x^{\beta}} (\frac{dx^{\alpha}}{d\lambda}) \frac{dx^{\beta}}{d\lambda} which becomes the first term in (10.27)...
  47. 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...
  48. 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|...
  49. 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.
  50. 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...