# 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. ### I Macroscopic objects in free-fall

Hi, very basic question. Take an object like a rock or the Earth itself. If we consider their internal constituents, there will be electromagnetic forces acting between them (Newton's 3th law pairs). From a global perspective if the rock is free from external non-gravitational forces, then it...
2. ### Geodesic on a sphere and on a plane in 2D

I start with the 2D plane. Suppose y(x) is the curve that connects these two points. Its length is given by: $$S=\int_1^2 \, ds=\int_1^2 (1+y'^2)^{\frac {1}{2}} \, dx$$ Applying Euler's equation we get:$$\frac {\partial f} {\partial y'}=A$$$$\dfrac {y'}{(1+y'^2)^{\frac {1}{2}}}=A$$...
3. ### I Is momentum conserved as a body falls through a gravitational field?

If one stands on a large planetary body, like the moon, and throws a large object, like a rock straight up, the object will leave with some velocity, slow down to a stop, and then come back down with the same velocity once it returns to its origin. In Newtonian mechanics, the understanding is...

33. ### I What is the Strange Solution to the EFE with a Born Frame and Rotating System?

The Born frame field (see ref below) describes a rotating system and the proper acceleration ##\vec{a}=\nabla _{{{\vec {p}}_{0}}}\,{\vec {p}}_{0}={\frac {-\omega ^{2}\,r}{1-\omega ^{2}\,r^{2}}}\,{\vec {p}}_{2}##. If ##\omega## depends on coordinate ##r## then...
34. ### I Geodesics in 4D Spacetime: An Overview

The geodesic for 2-D, 3-D are straight lines. For a 4-D spacetime (x1,x2,x3,t) what would be it's geodesic.?? The tangent vector components are ##V^0=\frac{∂t}{∂λ} , V^i=\frac{∂x^i}{∂λ},i=1,2,3## & ##(\nabla_V V)^\mu=(V^\nu \nabla_\nu V)^\mu=0,(\nu,\mu=0,1,2,3)##
35. ### I Covariant derivative of tangent vector for geodesic

For the simple case of a 2-D curve in polar coordinated (r,θ) parametrised by λ (length along the curve). At any λ the tangent vector components are V1=dr(λ)/dλ along ##\hat r## and V2=dθ(λ)/dλ along ##\hat θ##. The non-zero christoffel symbol are Γ122 and Γ212. From covariant derivative...
36. ### I Calculating Acceleration of Gravity w/ Geodesic Deviation: Troubleshooting

I have tried twice now to calculate acceleration of gravity using the general relativistic equation of geodesic deviation and both times my solution is twice the correct answer. What am I doing wrong? As an example here is one problem: Calculate the acceleration of gravity g at the earth’s...
37. ### 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...
38. ### 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.
39. ### Find the curve with the shortest path on a surface (geodesic)

Homework Statement Let ##U## be a plane given by ##\frac{x^2}{2}-z=0## Find the curve with the shortest path on ##U## between the points ##A(-1,0,\frac{1}{2})## and ##B(1,1,\frac{1}{2})## I have a question regarding the answer we got in class. Homework Equations Euler-Lagrange ##L(y)=\int...
40. ### B Geodesic dome parametric formula

I've been researching for the calculus behind geodesic domes, and specifically calculus related to parametric surfaces. I've found http://teachers.yale.edu/curriculum/viewer/new_haven_06.04.05_u#f, but unfortunately, it comes short of providing me the most needed information, and so far I...
41. ### 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...
42. ### I Geodesic Deviation: Definition of Connecting Vector

Hello! I read a derivation for the geodesic deviation: you have 2 nearby geodesics and define a vector connecting points of equal proper time and calculate the second covariant derivative of this vector. I understand the derivation but I am a bit confused about the actual definition of this...
43. ### I Force Experienced on a Curved Geodesic Path

Can a person inside a spaceship falling freely on a geodesic path, experience the same just like a person inside a car experience a force on a turn on Earth i.e when the geodesic path is no more straight near a huge planet. Thanks.
44. ### I Coordinate and proper time, null geodesic

I have a question which asks show that a null geodesic to get to r> R , r some constant, given the space time metric etc, takes infinite coordinate time but finite proper time. ( It may be vice versa ). I just want to confirm that, ofc there is no affine parameter for a null geodesic and so you...
45. ### General Relativity - geodesic - affine parameter

Homework Statement Question attached: Homework Equations see below The Attempt at a Solution [/B] my main question really is 1) what is meant by 'abstract tensors' as I have this for my definition: to part a) ##V^u\nabla_uV^a=0## but you do say that ##V^u=/dot{x^u}## ; x^u is a...
46. ### I Geodesic Maximality: Answers to Relativists' Questions

Hello, i know that relativists like to extend solutions of einstein equations so that they are geodesicly maximal (i.e. geodesics end only in singularity or infinite value of affine parameter). But why only geodesicly? Thus this geodesic maximality imply, that if i take any timelike or...
47. ### A Prove EL Geodesic and Covariant Geodesic Defs are Same via Riemmanian Geometry

... via plugging in the Fundamental theorem of Riemmanian Geometry : ##\Gamma^u_{ab}=\frac{1}{2}g^{uc}(\partial_ag_{bc}+\partial_bg_{ca}-\partial_cg_{ab})## Expanding out the covariant definition gives the geodesic equation as: (1) ##\ddot{x^u}+\Gamma^u_{ab} x^a x^b =0 ## (2) Lagrangian is...
48. ### A Constant along a geodesic vs covariantly constant

some questions I have seen tend to word as show that some quantity/tensor/scalar (e.g let this be ##K##) is constant along an affinely parameterised geodesic, others ask show covariantly constant. the definiton of covariantly constant/ parallel transport is: ## V^a\nabla_u K = 0 ##for the...
49. ### 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...
50. ### 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...