# 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### A Exists ? : Invariant geodesic equation

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

14. ### 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 $$T^{\mu\nu}(y^\sigma)=m\int d \tau \frac{\delta^{(4) }(y^\sigma-x^\sigma(\tau ))}{\sqrt{-g}}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}.\tag{2}$$ The covariant...
15. ### 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. ### 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. ### 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. ### 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...

34. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...