# 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. ### I Curved spacetime bulging in time direction

Let us start with a simple two-dimensional space that is asymptotically flat, like e.g. sheet of paper on a desk. Geodesics in this space are straight lines in the traditional sense. Now add a bulge in the center of this space. without affecting the asymptotic flatness. Say, we simply distort...
2. ### I Incomplete geodesics in a singularity, do they warrant quantum concerns?

Question: The idea of a continuum breaks down for a singularity when a geodesic become incomplete (the breaking of the idea that there was a continuous succession, where no part could be distinguished from neighboring parts, except by arbitrary division), and so with that does this indicate a...
3. ### A What Is the Gauss Embedding Theorem Mentioned in the Video Talk?

Does anybody know to which "Gauss embedding theorem" the speaker in this video talk at minute 14 (point 5. in the displayed notes) is refering too? Sounds to be a standard result in differential geometry, but after detailed googling I found nothing to which the speaker may refering too in the...
4. ### I About global inertial frame in GR - revisited

Hi, reading this old thread I'd like a clarification about the following: Fermi Normal hypersurface at an event on a comoving FLRW worldline is defined by the collection of spacetime orthogonal geodesics. Such geodesics should be spacelike since they are orthogonal to the timelike comoving...
5. ### I A question about geodesics and notion of parallelism

I think I have a slight misconception maybe, but I was wondering about this question. Usually when we say that the vectors are parallel, we say that it means that there's an equation ##k = \alpha l##, for the vectors ##k## and ##l## and some scalar ##\alpha##. In the context of differential...
6. ### I How can a square-based lattice turn into a circular-based one?

Hi All, I was just watching the video on the Veritassium channel about scientific dissemination of general relativity. And I could see once again that when you try to use animations and/or computer graphics to show the transition from free space geodesics to geodesics around a planet, what you...
7. ### I Inertial Gravity in Relativistic Geodesics - Forum Discussion

Hello friends of the Forum. I want to ask you why the inertial acceleration in free fall in the relativistic geodesic equations is assumed equal to zero in free fall and equal to 9.8 m/s at rest on the earth's surface. On the other hand, assuming that zero acceleration in free fall, what would...
8. ### 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...
9. ### B Solve General Geodesics in FLRW Metric w/ Conformal Coordinates

Once having converted the FLRW metric from comoving coordinates ##ds^2=-dt^2+a^2(t)(dr^2+r^2d\phi^2)## to "conformal" coordinates ##ds^2=a^2(n)(-dn^2+dr^2+r^2d\phi^2)##, is there a way to facilitate solving for general geodesics that would otherwise be difficult, such as cases with motion in...
10. ### I Connection & Affine Geodesics: Q&A

1 Does the connection vanish along a affine geodesic? 2 In《Introducing Einstein's Relativity》Ed2 on page 96"It can be shown that the result(the connection vanishes at P) can be extended to obtain a coordinate system in which the connection vanishes along a curve,but not in general to a...
11. ### I Geodesics with arbitrary parametrization

Let ##x=(x^1,\ldots,x^m)## be local coordinates in a manifold ##M##; and let ##\{\Gamma^i_{jk}(x)\}## be a connection. Assume that we have a curve ##x=x(t),\quad \dot x\ne 0##. Is this curve geodesic or not? My guess is that the answer is "yes" iff for all ##k,n## the function ##x(t)##...
12. ### What is the function to extremise for finding geodesics on a Helicoid?

I've got that length of a curve on the surface is: $$L=\int_{-\infty}^{\infty}\sqrt{1+\frac{4\pi^{2}}{h^{2}}\rho^{2}+\left(\frac{d\rho}{dz}\right)^{2}}dx$$ So the function to extremise is: $$f(\rho,\rho')=\sqrt{1+\frac{4\pi^{2}}{h^{2}}\rho^{2}+\left(\frac{d\rho}{dz}\right)^{2}}$$ Where...
13. ### I Do all geodesics begin and end at the big bang?

So. It was late night, the limpid sky a near cloudless darkness, somewhat lightened by the waxing moon. I being somewhat stoned and looking at the stars and constellations as I sometimes do, and it came to seem to me, in my imagination, that In the spaces between the stars, I was observing (in...
14. ### A Geodesics on three dimensional graph

Suppose there is a three dimensional graph (such as z=x^2+y^2). Suppose there is a point on the surface of the 3 dimensional graph, for example at (x,y,z)=(1,1,2). Suppose the point is moving along the surface (along a geodesic) according to a unit vector, such as <0,1,0>. Is there a...
15. ### I Detecting Gravitational Waves w/ Interferometers: Explained

Hi, I would like to ask for some clarification about the physics involved in the gravitational waves detection using interferometers. Starting from this thread Light speed and the LIGO experiment I'm aware of the two ends of an arm of the interferometer (e.g. LIGO) can be taken as the...
16. ### B Twin Paradox: Orbital Geodesics, Gravity, & Time Dilation

~ Shower Thoughts ~ Twin A is in a spaceship, Twin B is in a spaceship. Both in 'deep space'. B follows a highly elliptical geodesic which goes around a planet (or black hole) with strong gravity, very far away. When they meet again, who is younger and why? I genuinely don't know what this...
17. ### I Time Dilation in Gravitational Fields vs Acceleration on Earth

If I'm standing on Earth, is my time dilation actually greater than if I was in a rocket accelerating at 9.8m/s^2 in deep space due to me being in a gravitational field on top of the acceleration? Geodesics experience time dilation in gravitational fields, so it seems like there is an additive...

27. ### I Parallel Transport & Geodesics: Explained

I am currently reading Foster and Nightingale and when it comes to the concept of parallel transport, the authors don't go very deep in explaining it except just stating that if a vector is subject to parallel transport along a parameterized curve, there is no change in its length or direction...
28. ### A Geodesics of the 2-sphere in terms of the arc length

I'm trying to evaluate the arc length between two points on a 2-sphere. The geodesic equation of a 2-sphere is: $$\cot(\theta)=\sqrt{\frac{1-K^2}{K^2}}\cdot \sin(\phi-\phi_{0})$$ According to this article:http://vixra.org/pdf/1404.0016v1.pdfthe arc length parameterization of the 2-sphere...
29. ### I Geodesics subject to a restriction

Hi, I'm trying to solve a differential geometry problem, and maybe someone can give me a hand, at least with the set up of it. There is a particle in a 3-dimensional manifold, and the problem is to find the trajectory with the smallest distance for a time interval ##\Delta t=t_{1}-t_{0}##...
30. ### 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...
31. ### 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...

33. ### I Geodesics under coordinate transformation

I will start with an example. Consider components of metric tensor g' in a coordinate system $$g'= \begin{pmatrix} xy & 1 \\ 1 & xy \\ \end{pmatrix}$$ We can find a transformation rule which brings g' to euclidean metric g=\begin{pmatrix} 1 & 0 \\ 0 & 1\\ \end{pmatrix}, namely...

35. ### Geodesics and Motion in an EM Field

I've also attached my attempt as a pdf file. My main issue seems to be I only get one A partial term. Any help would be appreciated.
36. ### A Equations for computing null geodesics in Schwarzschild spacetime

My project for obtaining my master's degree in computer science involved ray tracing in Schwarzschild spacetime in order to render images of black holes. These light rays had to be computed numerically using the geodesic equation. However, I ran into a problem. The geodesic equation is given as...
37. ### I Free Particles & Geodesics: Do They Go Hand-in-Hand?

Is it correct to say that free particles always follow geodesics?
38. ### I Proving Geodesics in Hyperbolic Geometry

Given ##ds^2 = y^{-2}(dx^2 + dy^2)##, I am trying to prove that a demicircle centred on the x-axis, written parametrically as ##x=r\cos\theta + x_0 ## and ##y= r \sin \theta ## are geodesics. Where ##r## is constant and ##\theta \in (0,\pi)## I have already found the general form of the...
39. ### Null geodesics given line element

Hi, I'm the given the following line element: ds^2=\Big(1-\frac{2m}{r}\large)d\tau ^2+\Big(1-\frac{2m}{r}\large)^{-1}dr^2+r^2(d\theta ^2+\sin ^2 (\theta)d\phi ^2) And I'm asked to calculate the null geodesics. I know that in order to do that I have to solve the Euler-Lagrange equations. For...
40. ### 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...
41. ### B Understanding Orbital Geodesics in General Relativity

In the following link, https://en.wikipedia.org/wiki/Geodesics_in_general_relativity they write:"the path of a planet orbiting around a star is the projection of a geodesic of the curved 4-D spacetime geometry around the star onto 3-D space." Is there anything wrong with the following circular...
42. ### I Number of Geodesics Between Two Points: Topology vs. Homeomorphism

In general, is the number of distinct geodesics between two fixed points purely a feature of the topology of the manifold? i.e. Can there ever exist two topologically equivalent (I can’t remember the proper word right now- is homeomorphic the right one?) manifolds, ##\mathcal M## and ##\mathcal...
43. ### Cosmological model, geodesics question

Homework Statement I am unsure of Q3 but have posted my solutions to other parts Homework EquationsThe Attempt at a Solution 3) ok so it is clear that because the metric components are independent of ##x^i## each ##x^i## has an associated conserved quantity ##d/ds (\dot{x^i})=0##. (1) The...
44. ### A Compute Commutator of Covariant Derivative & D/ds on Vector Fields

Hi, let ##\gamma (\lambda, s)## be a family of geodesics, where ##s## is the parameter and ##\lambda## distinguishes between geodesics. Let furthermore ##Z^\nu = \partial_\lambda \gamma^\nu ## be a vector field and ##\nabla_\alpha Z^\mu := \partial_\alpha Z^\mu + \Gamma^\mu_{\:\: \nu \gamma}...
45. ### I Demonstration of relation between geodesics and FLRW metric

I am reading a book of General Relativity and I am stuck on a demonstration. If I consider the FLRW metric as : ##\text{d}\tau^2=\text{d}t^2-a(t)^2\bigg[\dfrac{\text{d}r^2}{1-kr^2}+r^2(\text{d}\theta^2+\text{sin}^2\theta\text{d}\phi^2)\bigg]## with ##g_{tt}=1##, ##\quad...
46. ### 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...
47. T

### I Incompleteness of geodesics as ##t \longrightarrow 0##

The ##t## in the title states the time evolution of our universe. How does one show that Relativity doesn't require geodesics extending indefinetely in the past? That is, how does one show that the theory allows for geodesics to have a starting point in the past? Is it hard to show?
48. ### Explain why all geodesics on a sphere are arcs of great circ

Homework Statement Show that any geodesic with constant ##\theta## lies on the equator of a sphere, with the north pole being on the ##\theta = 0## line. Hence explain why all geodesics on a sphere will be arcs of a great circle. Homework EquationsThe Attempt at a Solution I've had a go at it...
49. ### I Non-radial geodesics in Schwarzschild spacetime

Consider a non-radial timelike geodesic outside the event horizon. Will it nevertheless cross the horizon radially or are non-radial geodesics also possible inside? I couldn't find any reference regarding a possible angle dependence in this respect.
50. ### Euler-Lagrange Equations for geodesics

Homework Statement The Lagrange Function corresponding to a geodesic is $$\mathcal{L}(x^\mu,\dot{x}^\nu)=\frac{1}{2}g_{\alpha \beta}(x^\mu)\dot{x}^\alpha \dot{x}^\beta$$ Calculate the Euler-Lagrange equations Homework Equations The Euler Lagrange equations are \frac{\mathrm{d}}{\mathrm{d}s}...