# Geodesic Definition and 34 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 Natural parametrization of a curve

Hello, I need the natural parametrization or a geodesic curve contained in the surface z=x^2+y^2, that goes through the origin, with x(s=0)=0, y(s=0)=0, dx/ds (s=0)=cos(a) and dy/ds(s=0)=sin(a), with "a" constant, expressed as a function of the arc length, i.e., I need r(s)=r(x(s),y(s)). Thank...
2. ### I Parallel transport on flat space

When parallel transporting a vector along a straight line on flat space, does the connection (when calculating the covariant derivative) always equal zero? Do things change at all when using an arbitrary connection, rather than Christoffel symbols?
3. ### I Check for geodesically-followed path in a coordinate-free way

Hi, My question can result a bit odd. Consider flat spacetime. We know that inertial motions are defined by 'zero proper acceleration'. Suppose there exist just one free body in the context of SR flat spacetime (an accelerometer attached to it reads zero). We know that 'zero proper...

11. ### I Survival time in a black hole

In a thread a decade ago https://www.physicsforums.com/threads/how-to-survive-in-a-black-hole-myth-debunked.170829/, there was a discussion about the paper https://arxiv.org/abs/0705.1029v1, in which the authors discuss the way to maximize one's survival (proper) time after passing the event...
12. ### A Maximizing survival time when falling into a black hole

Unfortunately I didn't find a thread discussing this issue. First I will sketch the standard argument that one should not use the rocket engine and try to accelerate away from the singularity. Then I will try to identify the problematic part of this argument and ask for your comments. 1) For...
13. ### 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...
14. ### A Lagrangian of Geodesics

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...
15. ### I Two Conserved Quantities Along Geodesic

Hi Everyone! I have done three years in my undergrad in physics/math and this summer I'm doing a research project in general relativity. I generally use a computer to do my GR computations, but there is a proof that I want to do by hand and I've been having some trouble. I want to show that...
16. ### Black colour and light

In the visible spectrum VIBGYOR, there is no black colour.So, what do we percieve as 'black' ? Another of my queries is that when dispersion takes place there is a change in wavelength but not so in case of frequency. But they are related inversely. So, why does this happen?
17. ### A Stress tensor in 3D Anti-De Sitter Space

I am doing some mathematical exercises with 3D anti-de sitter face using the metric ds2=-(1+r2)dt2+(1+r2)-1+r2dφ2 I found the three geodesics from the Christoffel symbols, and they seem to look correct to me. d2t/dλ2+2(r+1/r)*(dt/dλ)(dr/dλ)=0...
18. ### I Understanding parallel transfer

I've read Collier's book on General Relativity and consulted parts of Schutz, Hartle and Carroll. In the terms they use, i have yet to gain anything resembling an intuitive understanding of parallel transport. In fact, it seems to me it is usually presented backwards, saying that the geodesic...
19. ### I Metric tensor derived from a geodesic

Let we have a 2D manifold. We choose a coordinate system where we can construct all geodesics through any point. Is it enough to derive a metric from geodesic equation? Or do we need to define something else for the manifold?
20. ### 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}...
21. ### I Black hole orbits

Now my Kerr orbit simulator is pretty much feature complete, I have started to look at producing videos . . . I have just started a channel on YouTube to accumulate some of the more interesting examples. Aside from creating the simulation, the most difficult part was to generate useful initial...
22. ### I Light deflection and geodesics

It is known that light beam bends near massive body and the object sendind deflected the beam is seen in shifted position. Now about spacetime curvature. As I undestand the things are like that: http://i11.pixs.ru/storage/3/3/4/pic2png_7037348_21446334.png [Broken] The question is why are...
23. ### 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} =...
24. ### Geodesics by Plane Intersection

I want to try and see the intersection between the hyperboloid and the 2-plane giving an ellipse. So far I have the following: I'm going to work with ##AdS_3## for simplicity which is the hyperboloid given by the surface (see eqn 10 in above notes for reason) ##X_0^2-X_1^2-X_2^2+X_3^2=L^2## If...
25. ### Interesting Effect of Conformal Compactification on Geodesic

I'm trying to understand why timelike geodesics in Anti de-Sitter space are plotted as sinusoidal waves on a Penrose diagram (a nice example of the Penrose diagram for AdS is given in Figure 2.3 of this thesis: http://www.nbi.dk/~obers/MSc_PhD_files/MortenHolm_Christensen_MSc.pdf). Bearing in...
26. ### A Opposite "sides" of a surface - Differential Geometry.

How, if at all, would differential geometry differ between the opposite "sides" of the surface in question. Simplest example: suppose you look at vectors etc on the outside of a sphere as opposed to the inside. Or a flat plane? Wouldn't one of the coordinates be essentially a mirror while...
27. ### 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...
28. ### Comoving/Proper distance, transverse comoving distance

I'm utterly confused by co-moving distance, transverse comoving distance and proper distance. Is comoving distance = proper distance? Then what is transverse comoving distance? Here's what I know so far: The FRW metric can either be expressed as ds^2 = c^2dt^2 - a^2(t) \left[ \frac{dr^2}{1-kr^2}...
29. ### Closed Universe - FRW Equation

Homework Statement (a) Show that the equations satisfy FRW equations. (b) Show the metric when ##\eta## is taken as time Homework Equations The Attempt at a Solution [/B] The FRW equation is: 3 \left( \frac{\dot a}{a} \right)^2 = 8\pi G \rho Using ##\frac{da}{dt} = \frac{da}{d\eta}...
30. ### Quick one-line on Tensor Contraction

What do they mean by 'Contract ##\mu## with ##\alpha##'? I thought only top-bottom indices that are the same can contract? For example ##A_\mu g^{\mu v} = A^v##.
31. ### Contracting a Tensor

What do they mean by contracting ##\mu## with ##\alpha## ?
32. ### Proper distance, Area and Volume given a Metric

Homework Statement [/B] (a) Find the proper distance (b) Find the proper area (c) Find the proper volume (d) Find the four-volume Homework Equations The Attempt at a Solution Part (a) Letting ##d\theta = dt = d\phi = 0##: \Delta s = \int_0^R \left( 1-Ar^2 \right) dr = R \left(1 -...
33. ### General Relativity - Deflection of light

Homework Statement Find the deflection of light given this metric, along null geodesics. Homework Equations The Attempt at a Solution [/B] Conserved quantities are: e \equiv -\zeta \cdot u = \left( 1 - \frac{2GM}{c^2r} \right) c \frac{dt}{d\lambda} l \equiv \eta \cdot u = r^2 \left( 1...
34. ### Linearising the Geodesic Deviation Equation

Homework Statement Write down the Newtonian approximation to the geodesic deviation equation for a family of geodesics. ##V^\mu## is the particle 4-velocity and ##Y^\mu## is the deviation vector. Homework Equations D = V^\mu\nabla_\mu \\ V^\mu\approx(1,0,0,0) \\...