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. H

    A Geodesic Distance & Maximally Symmetric Spacetimes: Why Does it Matter?

    Any physical quantity ##K(t,x,x')## on a maximally symmetric spacetime only depends on the geodesic distance between the points ##x## and ##x'##. Why is this so? N.B.: This statement is different from the statement that The geodesic distance on any spacetime is invariant under an arbitrary...
  2. 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...
  3. nomadreid

    I Survival Time in Black Hole: Myth Debunked

    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...
  4. binbagsss

    Solving Geodesic Equations with Euler-Lagrange and Noether's Theorem

    Homework Statement Homework Equations There are 5 equations we can use. We have the fact that Lagrangian is a constant for an affinely parameterised geodesic- 0 in this case for a light ray : ##L=0## And then the Euler-Lagrange equation for each of the 4 variables. The Attempt at a Solution...
  5. T

    I Geodesic Deviation in Static Spacetime: Flat Geometry

    Regarding Einstein's static universe John Baez explains in The Meaning of Einstein's Equation To see this, consider a small ball of test particles, initially at rest relative to each other, that is moving with respect to the matter in the universe. In the 13 local rest frame of such a ball, the...
  6. 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}...
  7. tom.stoer

    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...
  8. 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
  9. P

    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}...
  10. 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
  11. Lunct

    B Does an object moving in a geodesic accelerate?

    So in GR, objects orbiting the sun, for example, move in a geodesic - a straight line something curved. Without GR (using Newtonian Gravity), I can easily say because planets orbiting the sun are doing so in a ellipse, they are accelerating. However, would they still be accelerating when you add...
  12. 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...
  13. M

    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...
  14. shihab-kol

    What Is the Nature of Black in the Visible Spectrum VIBGYOR?

    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?
  15. binbagsss

    General relativity, geodesic, KVF, chain rule covariant derivatives

    Homework Statement To show that ##K=V^uK_u## is conserved along an affinely parameterised geodesic with ##V^u## the tangent vector to some affinely parameterised geodesic and ##K_u## a killing vector field satisfying ##\nabla_a K_b+\nabla_b K_a=0## Homework Equations see above The Attempt at...
  16. binbagsss

    General relativity, geodesic question

    Homework Statement question attached Homework EquationsThe Attempt at a Solution Attempt : Check if ##V^{\alpha}\nabla_{\alpha}V^u=0## Since Minkowski space, connection tensors/christoffel symbols are zero so this reduces to: ##V^{\alpha}\partial_{\alpha}V^u=0## where...
  17. davidge

    Solving Geodesic Equations for ##\phi(r)##

    The system of DE arises when using the Schwarzschild metric of General Relativity on the geodesics equation. I was trying to solve these equations for ##\phi## as a function of ##r##. I followed Weinberg (Weinberg's book S&G relativity) who uses ##t' = 1 / B(r)## in his book. So now we have...
  18. M

    A EoM in Schwarzschild geometry: geodesic v Hamilton formalism

    Hi there guys, Currently writing and comparing two separate Mathematica scripts which can be found here and also here. The first one I've slightly modified to suit my needs and the second one is meant to reproduce the same results. Both scripts are attempting to simulate the trajectory of a...
  19. binbagsss

    GR conditions conserved quantities AdS s-t; t-l geodesic

    Homework Statement Question attached Homework Equations The Attempt at a Solution part a) ##ds^2=\frac{R^2}{z^2}(-dt^2+dy^2+dx^2+dz^2)## part b) it is clear there is a conserved quantity associated with ##t,y,x## From Euler-Lagrange equations ## \dot{t}=k ## , k a constant ; similar for...
  20. P

    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...
  21. joneall

    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...
  22. V

    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?
  23. P

    I Understanding Schutz's Geodesic Deviation Eq. 6.84

    I have some problem understanding the section on "Geodesic deviation" in schutz, more specifically I'm confused by eq. 6.84: Eq 6.84 reads (ξ is the 'connecting vector' from one geodesic to Another, V is the tangent vector): We can use (6.48) to obtain ∇V∇Vξα = ∇V(∇Vξα) = (d/dλ)(∇Vξα) =...
  24. M

    A How to Integrate the geodesic equations numerically?

    Hello there, I've been considering the geodesic equations of motion for a test particle in Schwarzschild geometry for some time now. Similar to what we can do with the Kepler problem I would like to be able to numerically integrate the equations of motion. I'm quite interested to see how...
  25. binbagsss

    I Solve Null Geodesic: Affine Parameter & Coordinate Time - Q1,Q2,Q3

    I am asked a question about how far a light ray travels, the question is to be solved by solving for the null goedesic. I am given the initial data: the light ray is fired in the ##x## direction at ##t=0##. The relvant coordinates in the question are ##t,x,y,z##, let ##s## be the affine...
  26. binbagsss

    Solve Timelike Geodesic: Find A & B for Curve

    Homework Statement The question is to find ##A## and ##B## such that the specified curve (we are given a certain parameterisation , see below) is a timelike geodesic , where we have ##|s| < 1 ## I am just stuck on the last bit really. So since the geodesic is affinely paramterised...
  27. L

    I Meaning of the sign of the geodesic curvature

    My question is : what is the meaning of the geodesic curvature sign for a coordinate patch? for a surface? Thank you.
  28. V

    A Geodesic defined for a non affine parameter

    The geodesic general condition, i.e. for a non affine parameter, is that the directional covariant derivative is an operator which scales the tangent vector: $$\zeta^{\mu}\nabla_{\mu}\zeta_{\nu}=\eta(\alpha)\zeta_{\nu}$$ I have three related questions. When $$\alpha$$ is an affine parameter...
  29. 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!
  30. 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}...
  31. m4r35n357

    I Explore Black Hole Orbits with My Kerr Orbit Simulator on YouTube

    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...
  32. V

    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 The question is why are geodesics...
  33. Elnur Hajiyev

    A Can geodesic deviation be zero while curvature tensor is not

    I understand(or assume understand) that geodesic deviation describes how much parallel geodesics diverge/converge on manifolds while moving along these geodesic. But is not it a definition for intrinsic curvature? If it is same as Riemann curvature tensor in terms of describing curvature, why...
  34. 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} =...
  35. A

    Intersection of Hyperboloid & 2-Plane=Ellipse

    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...
  36. A

    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...
  37. T

    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...
  38. 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...
  39. bcrowell

    Alternative definitions of geodesic

    I teach both physics and math at a community college, and I've volunteered to give a short talk for students at our weekly math colloquium that has to do with curvature and non-curvature singularities in relativity. This is a tall order, given that I can't even assume that all the students will...
  40. D

    Geodesic Transport of Small 2D Surface on 3D Manifold

    Hello, I've just read and I think I have understood the following result : If we were to geodesically transport all points of a small 2D surface, so small that it would be flat for all purposes, in a direction vertically above it, and if this surface belongs in an arbitrary 3D manifold, then in...
  41. 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?
  42. 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...
  43. S

    Nature of Geodesic: Determine Without Knowing Metric?

    Hello! Please help: A world line is given to us. It is known that it is a geodesic. The metric, however, is not known. Since we don't know the metric, it should not be possible to tell whether the geodesic is spacelike/timelike/null (Right?) But since the geodesic is known (x,y,z,t), we can find...
  44. Andre' Quanta

    Geodesic deviation equation

    My teacher of General Relativity has proposed a demonstration of the geodesic deviation equation based on normal coordinates, the problem is that for me the procedure is wrong, could you help me to find the problem? Suppose to have a differentiable manifold M of dimension 4, and two geodesics x...
  45. U

    What is the geodesic in this case?

    Homework Statement Using the geodesic equation, find the conditions on christoffel symbols for ##x^\mu(\tau)## geodesics where ##x^0 = c\tau, x^i = constant##. Show the metric is of the form ##ds^2 = -c^2 d\tau^2 + g_{ij}dx^i dx^j##. Homework EquationsThe Attempt at a Solution The geodesic...
  46. 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...
  47. 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...
  48. A

    Geodesic Deviation in 2D: Is There Directional Dependence?

    In 2 dimensions, is the geodesic deviation equation governed by a single scalar, independent of the direction of the geodesics? That is, if ξ is the separation of two nearby geodesics, do we have d^2 \xi/ds^2 + R\xi = 0 where R is a scalar that is completely independent of the direction of the...
  49. U

    Light-like Geodesic - What are the limits of integration?

    Homework Statement Consider the following geodesic of a massless particle where ##\alpha## is a constant: \dot r = \frac{\alpha}{a(t)^2} c^2 \dot t^2 = \frac{\alpha^2}{a^2(t)} Homework EquationsThe Attempt at a Solution Part (a) c \frac{dt}{d\lambda} = \frac{\alpha}{a} a dt =...
  50. U

    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}...
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