Null geodesics Definition and 22 Threads

In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational forces is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a geodesic.
In general relativity, gravity can be regarded as not a force but a consequence of a curved spacetime geometry where the source of curvature is the stress–energy tensor (representing matter, for instance). Thus, for example, the path of a planet orbiting a star is the projection of a geodesic of the curved four-dimensional (4-D) spacetime geometry around the star onto three-dimensional (3-D) space.

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  1. J

    I Affine parameter and non-geodesic null curves

    Consider the curve (thanks to SE) in flat spacetime, given in Cartesian coordinates by$$x^μ(λ)=\left(λ , R\cos\frac{\lambda⁡}{R} , R\sin\frac{\lambda}{⁡R} ,0\right)$$where ##~R~## is a positive constant. At each point$$\dot x^\mu \dot x_\mu=0$$so it is a null curve but not a geodesic (not a...
  2. Onyx

    B Four Velocity Sign of Time: \dot t>0?

    Is it generally the case even with light like paths that ##\dot t>0##?
  3. cianfa72

    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...
  4. stevendaryl

    A Computing Null Geodesics in Schwarzschild Geometry

    Computing timelike geodesics in the Schwarzschild geometry is pretty straightforward using conserved quantities. You can treat the problem as a variational problem with an effective Lagrangian of ##\mathcal{L} = \frac{1}{2} (Q \frac{dt}{d\tau}^2 - \frac{1}{Q} \frac{dr}{d\tau}^2 - r^2...
  5. E

    I Conformally related metrics have the same null geodesics

    Homework Statement:: i) If ##\bar{g} = \Omega^2 g## for some positive function ##\Omega##, show that ##\bar{g}## and ##g## have the same null geodesics. ii) Let ##\psi## solve ##g^{ab} \nabla_a \nabla_b \psi + \xi R \psi = 0##. Determine ##\xi## such that ##\bar{\psi} = \Omega^p \psi## for...
  6. LightAintSoFast

    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...
  7. C

    How Do You Calculate Null Geodesics for the Given Schwarzschild 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...
  8. P

    A Interpretation of covariant derivative of a vector field

    On Riemannian manifolds ##\mathcal{M}## the covariant derivative can be used for parallel transport by using the Levi-Civita connection. That is Let ##\gamma(s)## be a smooth curve, and ##l_0 \in T_p\mathcal{M}## the tangent vector at ##\gamma(s_0)=p##. Then we can parallel transport ##l_0##...
  9. SonnetsAndMath

    A Radial, exterior, outgoing, null geodesics in Schwarzschild

    I'm a little confused about the proper way to find these null geodesics. From the line element, $$c^2 d{\tau}^2=\left(1-\frac{r_s}{r}\right) c^2 dt^2-\left(1-\frac{r_s}{r}\right)^{-1}dr^2-r^2(d{\theta}^2+\sin^2\theta d\phi^2),$$ I think we can set ##d\tau##, ##d\theta## and ##d\phi## to ##0##...
  10. P

    Non-radial null geodesics in Eddington-Finkelstein coordinates

    Homework Statement My end goal is to plot null geodesics around a black hole with realistic representations within the horizon (r<2GM, with c=1) using Mathematica. I've done this for outside the horizon using normal Schwarzschild coordinates and gained equation (1) below, and then used this...
  11. Ibix

    I Null geodesics in Schwarzschild spacetime

    I was looking at null geodesics in Schwarzschild spacetime. Carroll's lecture notes cover them here: https://preposterousuniverse.com/wp-content/uploads/grnotes-seven.pdf He notes (and justifies) that orbits lie in a plane and chooses coordinates so it's the equatorial plane, then uses Killing...
  12. myra2016

    Find Null Geodesics with affine parameter

    Homework Statement The metric is given by https://dl.dropboxusercontent.com/u/86990331/metric12334.jpg H is constant; s is an affine parameter, and so r(0)=t(0)=0. Apologise in advance because I'm not very good with LaTex. So I used Word for equations, and upload handwritten attempt at...
  13. B

    Null geodesics and null curves

    What is the exact difference between null geodesics and null curves? Please explain both qualitatively and quantitatively.
  14. T

    How should I think about null geodesics?

    I am kinda being thrown into pretty intense physics and this really doesn't have too much to do with what I'm doing but I was wondering if null geodesics have zero length, what are the other dimensions or parameters that accounts for the apparent movement of particles? I am a visual learner and...
  15. C

    Null geodesics of the FRW metric

    When working with light-propagation in the FRW metric $$ds^2 = - dt^2 + a^2 ( d\chi^2 + S_k(\chi) d\Omega^2)$$ most texts just set $$ds^2 = 0$$ and obtain the equation $$\frac{d\chi}{dt} = - \frac{1}{a}$$ for a light-ray moving from the emitter to the observer. Question1: Do we not strictly...
  16. C

    Existence of affine parameters of null geodesics

    We have a general spacetime interval ##ds^2 = g_{\mu \nu} dx^\mu dx^\nu##. One way to define an affine parameter is to define it to be any parameter ##u## which is related to the path length ##s## by ##u = as + b## for two constants ##a,b##. One can show that for the tangent vector ##u^\alpha =...
  17. WannabeNewton

    How Does Fermi-Walker Transport Behave in Rotating Space-Times?

    Hi all. It is well known that in Schwarzschild space-time, a torque-free gyroscope in circular orbit at any permissible angular velocity at the photon radius (also known as the photon sphere i.e. ##r = 3M##) will, if initially tangent to the circle, remain tangent to the circle everywhere along...
  18. P

    Chapter 21 Ray D'Inverno Scalar Optics, congruence of null geodesics

    First of all this is my first thread, so I apologize for any mistake. Perhaps this is a stupid question, but i need some help in exercise 21.10 of D'Inverno, to write down geodesic equation for l^a, which is a vector tangent to a congruence of null geodesics and then by a rescaling of l^a...
  19. D

    Null geodesics of a Kerr black hole

    Homework Statement Hi, From the Kerr metric, in geometrized units, \left(1 - \frac{2M}{r}\right) \left(\frac{dt}{d\lambda}\right)^2 + \frac{4Ma}{r} \frac{dt}{d\lambda}\frac{d\phi}{d\lambda} - \frac{r^2}{\Delta} \left(\frac{dr}{d\lambda}\right)^2 - R_a^2...
  20. P

    Conformal invariance of null geodesics

    Hi, folks. I hope this is the right forum for this question. I'm not actually taking any classes, but I am doing self-study using D'Inverno's Introducing Einstein's Relativity. I have a solution, and I want someone to check it for me. Homework Statement Prove that the null geodesics of two...
  21. Z

    Can someone please explain to me why massless particles follow null geodesics?

    Anything that is massless must follow null geodesics. Why is this?
  22. F

    Null geodesics of light from a black hole accretion disk

    Sorry I don't know latex so this may look a little messy. Homework Statement I'm trying to solve the equation for null geodesics of light traveling from a rotating black hole accretion disk to an observer at r = infinity. The point of emission for each photon is given by co-ordinates r, phi...
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