What is Null geodesics: Definition and 21 Discussions
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.
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...
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...
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...
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...
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...
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##...
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##...
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...
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...
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...
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...
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...
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 =...
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...
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...
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...
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...