Geodesics Definition and 181 Threads

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...

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?

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...

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.

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}...

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...

I was told by someone that for computer vision AI, a photo of say an apple and an orange exists on some high dimensional manifold, and the goal is to learn a geodesic between the two objects. What does this mean? Does this mean that the photo of one of the images is just a tuple of coordinates...

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...

To find the geodesics of a space(time), what we need to do is extremizing the functional ##\displaystyle \int_{\lambda_1}^{\lambda_2}\sqrt{g_{\mu \nu} \frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}} d\lambda ##. But sometimes the presence of the square root makes the equation of motion too...

I need to find the geodesics of AdS3 spacetime. But my searches have given me nothing. Can anyone give a reference where they're calculated? Thanks

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...

Consider the metric of ##S^{2}##: $$ds^{2}=d\theta^{2}+\sin^{2}(\theta)d\phi^{2}$$ Then in order to determine the geodesics on this surface one can minimise the integral $$s=\int_{l_{1}}^{l_{2}}\sqrt{\left(\frac{d\theta}{dl}\right)^{2}+\sin^{2}(\theta)\left(\frac{d\phi}{dl}\right)^{2}}dl$$ where...

As I understand it, a curve ##x^{\mu}(\lambda)## (parametrised by some parameter ##\lambda##) connecting two spacetime events is a geodesic if it is locally the shortest path between the two events. It can be found by minimising the spacetime distance between these two events...

In particular how does matter "clump" together to form stars and planets, and how do Galaxy/star systems form? For the latter question is the answer simply that near massive enough bodies, the spacetime curvature is significant enough that the geodesics within its vicinity are closed curves...

I was trying to solve this excercise: Now I was able to find the eq. of geodetics (or directly by Christoffel formulas calculation or by the Lagrangian for a point particle). And I verified that such space constant coordinate point is a geodetic. Now, for the second point I...

Hello I am little bit confused about lagrange approximation to geodesic equation: So we have lagrange equal to L=gμνd/dxμd/dxν And we have Euler-Lagrange equation:∂L/∂xμ-d/dt ∂/∂x(dot)μ=0 And x(dot)μ=dxμ/dτ. How do I find the value of x(dot)μ?

I am trying to find and solve the geodesics equation for polar coordinates. If I start by the definition of Christoffel symbols with the following expressions : $$de_{i}=w_{i}^{j}\,de_{j}=\Gamma_{ik}^{j}du^{k}\,de_{j}$$ with $$u^{k}$$ is the k-th component of polar coordinates ($$1$$ is for...

I have these questions: 1) Why must light always move along a geodesic line? What is the principle behind that? 2) A second question about spacetime: We mostly depict or imagine spacetime as a net of flexible fiber that extends everywhere as a plane as we see it.. As we are looking it, what...

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...

Hi, I recently tried to derive the equations for a geodesic path on a sphere of radius 1 (which are supposed to come out to be a great circle) using the formula \dfrac{d^2 x^a}{dt^2}+\Gamma^a_{bc} \dfrac{dx^b}{dt}\dfrac{dx^c}{dt}=0 for the geodesic equation, with the metric...

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...

[Moderator's note: this post has been spun off into its own thread.] I'm a retired engineer trying to get my head around GR, its effects in our everyday non-relativistic world, and its reduction to Newtonian gravity. I hope this is not too much of a digression from the current string. As I...

Not a formal course - just a question I decided to try to tackle with what I've gleaned from Stanford's lectures on Youtube, but still putting this here on account of this. So, I've been watching the Stanford GR series, and I have two motivations for messing around with this type of metric; 1...

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...

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...

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

I would be interested in knowing if others think I have the correct analysis of whether length is stationary and/or extremal in the cases of geodesics that are timelike, null, and spacelike. Timelike In Minkowski space, the proper time ##\tau=\int \sqrt{g_{ij}dx^i dx^j}## (+--- metric) is both...

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...

Assuming that the expectation that all matter and energy are quantized is correct, I'm making a further assumption that "random" means something like, "hypothetically predictable, but only by means at least possibly impractical on any permanent or general basis whatsoever, such as enumeration...

Suppose to have a killing vector that its norm is null, so at the same time is also a null geodesic. Does the metric have special propierty? What can i say about the Killing vector and its proprierties?

Suppose to have a Lie group that is at the same time also a Riemannian manifold: is there a relation between Christoffel symbols and structure constants? What can i say about the geodesics in a Lie group? Do they have special properties?

Looks like my main pet GR project is about to enter something akin to maintenance mode, since it now does all I currently need it to. It's nothing earth-shattering at first glance, but is very concise (e.g. ~100 lines of Python for the simulator script) and should be easier to understand than...

I am trying to get my head around curvature, geodesics and acceleration in GR. I've put together the following paragraph that attempts to describe qualitatively how I think these things play together. In Newtonian mechanics, a freely falling object accelerating towards the Earth implies a force...

In Dirac's book on GRT, top of page 17, he has this: (I'll use letters instead of Greeks) gcdgac(dva/ds) becomes (dvd/ds) I seems to me that that only works if the metric matrix is diagonal. (1) Is that correct? (2) If so, that doesn't seem to be a legitimate limitation on the property of...

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...

Homework Statement Using the Reissner Nordstrom line element, which I've given in the relevant equations section, I'm looking to show that the time like Geodesics obey the equation again show below. Homework Equations Line Element[/B] ##ds^2= - U(r)c^2dt^2 +\frac{dr^2}{U(r)} +r^2(d\theta^2 +...

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 =...

Homework Statement I'm working on a problem from my gravitation book. The question is the following: Given \begin{equation} \frac{D}{Ds} T^\mu = 0 \end{equation}, where \begin{equation} T^\mu \left(s,a\right) = \frac{\partial z^\mu}{\partial s} \end{equation} is the tangent vector to a...

We've all seen an image similar to this one: This is displaying the projection of GR Geodesics onto 3-D space (well, 2D in the picture). I'm still working my way through the General Relativity texts, so I'm not yet able to do the calculation on my own. Can anyone give me a formula that I can...

I've found the equations of motion for a charged test particle in the Kerr-Newman geometry from a number of sources. However, they aren't very reliable and disagree on small details, so I'm trying to derive it myself. I'm completely stuck at the last step though, where you "use" the...

I have an object (A) at some altitude above the Earth ellipsoid, and a point (B) on the surface of the Earth. Since you're not confined to the surface of the Earth as you travel from A (at altitude), to B, I'm getting confused. If I were to create a (Cartesian) vector pointing from object...

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 everyone, :) This is a question that one of my friends sent me. It is kind of open ended and I don't have any clue about the particular area of research he is undertaking. Therefore I am posting the question here with the hope that anybody knowledgeable in this area would be able to help...

Hello, I was wondering the following. Suppose you start with a Riemannian manifold M. Say you know one geodesic. Pushing this geodesic forward through an isometry M -> M gives again a geodesic. Can this procedure give you all geodesics? Thinking of the plane or the sphere it seems...

I'm taking an undergraduate level GR course, and from my text (Lambourne), the author describes a geodesic as a curve that "always goes in the same direction", and says that the tangent vector to the curve at some point u+du (where u is the parameter variable from which all the vector components...

"The geodesics in R n are the straight lines parametrized by constant velocity". This can be proved with the geodesic equation: \ddot{x^a} + \Gamma^a_{bc} \dot{x^b} \dot{x^c}=0 Locally we can find a coordinate system such that \Gamma =0 , and thus: \ddot{x^a} =0 So along a geodesic at some...

I have a related question which may broaden the image for conceptual clarity. Imagine an object (me) moving through "flat" outer space, far from any gravitational bodies. We can say that the geodesic I am traveling along is essentially straight or flat, as is its worldline, correct? So me...

Suppose there is a radially free falling object starting at r(t=0) = r0 > rS with some initial velocity v. And suppose there is a radial light ray starting at R(t=0) = R0 > r0. Suppose that both the object and the light ray reach the singularity at the same time. Question: is there a simple...