Geodesics Definition and 181 Threads

What sort of structure must a manifold possesses in order to talk about minimal geodesics between two points on it? When can we extend the minimal geodesics indefinitely?

Mass curves spacetime. The relative acceleration of nearby geodesics of free test particles indicates the sign of the spacetime curvature. Convergent geodesics mean positive, divergent negative curvature. But also the metric expansion of space curves spacetime. The geodesics may be convergent...

Homework Statement Find the geodesics on a cone of infinite height, x^{2}+y^{2} = \tan{\alpha}^{2}z^{2} using polar coordinates (x,y,z)=(r\cos{\psi},r\sin{\psi},z) with z=r\tan(\alpha) The Attempt at a Solution I am not sure with how should I expres the element dz^{2} ? When it is a...

Homework Statement We consider global AdS given by the coordinates (\rho,\tau, \Omega_i), i=1,\ldots,d and the metric ds^2=L^2(-cosh^2\,\rho\,d\tau^2+d\rho^2+sinh^2\,\rho\,d \Omega_i{}^2) Find the trajectory \tau(\rho), radially-directed geodesics, strating from \rho=\rho_0 with proper...

I want to interpret geodesics in a constant gravitational field. As a simple example I start with flat Minkowski spacetime ds^2 = -dT^2 + dX^2 + dY^2 + dZ^2 with a geodesic (in terms of coordinate time T) X^\mu(T) = (T, X=A, 0, vT) where A is an arbitrary constant and v ≤ c...

It's a naive question, but I'm pretty sure my professor said that space-time is locally flat (and I'm pretty sure that the volume of my room counts as "locally"). That said, I would expect falling objects to follow straight trajectories, but that's obviously not the general case. I thought...

Hello. If I find a solution of the geodesic equation and I change the parametrization, the new function does not satisfy this equation for a general re-parametrization. But the world line is the same. Can you confirm it: does it come from the fact that we usually choose \nabla_VV=0...

Hi everybody. I am well aware that there is only one black hole in 2+1, i.e., the BTZ one. I also know that for vanishing and positive cosmological constants we get solutions with a conical singularity. My question is more about the interpretation of these last results. Assume that in the BTZ...

Hello, everybody. I have some doubts I hope you can answer: I have read that the n+1-dimensional Anti-de Sitter (from now on AdS_{n+1}) line element is given, in some coordinates, by: ds^{2}=\frac{r^{2}}{L^{2}}[-dt^{2}+\sum\limits_{i=1}^{n-1}(dx^{i})^{2}]+\frac{L^{2}}{r^{2}}dr^{2} This can be...

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...

It's a standard fact of GR that at a given point in space-time, we can construct a coordinate system such that the metric tensor takes the form of Minkowski spacetime and its first derivatives vanish. Equivalently, we can make the Christoffel symbols vanish at point. Moreover, the fact that, in...

\dfrac{d^2 x}{dt^2}=-\nabla \Phi \dfrac{d^2 x^\mu}{d\tau^2}= -\Gamma^{\mu}_{\alpha \beta}{}\dfrac{dx^\alpha}{d\tau}\dfrac{dx^\beta}{d\tau} These two equations, to be true, the way they are written should ring a bell. They are similar yet not identical. What is the meaning behind them...

In a uniformly rotating coordinate system the trajectories of freely moving objects are influenced by an apparent centrifugal and Coriolis force. Is there a coordinate system or metric (or both) in which these trajectories are geodesics instead?

How would one determine a geodesic in Rindler space? Why would geodesics not be simply the same as those of Minkowsky space? Is it not analogous to using polar vs. Cartesian coordinates in euclidean space, where a straight line is the same in either case?

Homework Statement Consider the two-dimensional spacetime with the line element dS2 = -X2dT2+dX2. Find the shapes X(T) of all timelike geodesics in this spacetime. 2. The attempt at a solution I have the solution to this problem but I don't understand one step. For timelike worldlines dS2 =...

Yes, I want to make sure that geodesics of a particle moving in curved space time is the same thing of projectile trajectories. I start from assuming that 1-\frac{2GM}{r}\approx1-2gr and then calculate the schwarzschild metric in this form \Sigma_{\mu\nu}=\begin{bmatrix}\sigma & 0\\ 0 &...

Hello, First of all, please excuse me if I posted in the inappropriate place.. While a student few years ago, I used to work a lot with advanced differential geometry concepts, but never got an intuitive view of HOW humanity got to think about parallel transport, why it contained two words...

a massive body like a star creates a warped spacetime in its vicinty. this warped geometry of space is reflected by the geodesics appropriate to its mass. a photon passing by this massive object is not, as per GR, "attracted" to the star via some "force", but rather simply follows what it sees...

Please explain me how to derive the Timelike geodesics in Schwarzschild Metric. Thank you.

Finding Geodesics What I wish to understand, is how to solve this one: given this metric: ds^2= \frac {dt^2} {t^2}- \frac{dx^2} {t^2} I have to calculate the geodesics. S=\int{ \frac {d} {d\lambda} \sqrt{\frac {1} {t^2} \frac {dt^2} {d\lambda^2}- \frac{1} {t^2} \frac {dx^2}...

The poincare' group is the group of isometries of Minkowski spacetime, in a nutshell. In terms of an actual physical definition it is the group of all distance preserving maps between metric-spaces in Minkowski spacetime. What is the difference between this and 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...

Homework Statement Hi, just want to get a couple of things straight regarding finding the geodesics of a sphere not using polar coordinates, but rather, Lagrange multipliers... I want to minimize I = int (|x-dot|2 dt) subject to the constraint |x|=1 (sphere) which gives an Euler equation...

Hi, I'm stuck on the last bit the attached question where we're given the metric ds^2=-du^2+u^2dv^2 and have to use equation (*) to find the geodesic equations. They tell us to use V^a=\dot{x}^a the tangent vector to the geodesic and presumably we use the three killing vectors they gave us, so...

Affinely parameterised geodesics satisfy \nabla_XX=0 Why does this mean they have constant velocity? Thanks.

So if you take a sphere with coordinates (r, \theta,\phi) and keep \theta constant you get a cone. The geodesic equations reduce to (by virtue of the euler - lagrange equations): \frac{\mathrm{d} ^{2}r}{\mathrm{d} s^{2}} - r\omega ^{2}\frac{\mathrm{d} \phi }{\mathrm{d} s} = 0 where \omega =...

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

Homework Statement Prove that a particle constrained to move on a surface f(x,y,z)=0 and subject to no forces, moves along the geodesic of the surface. Homework Equations The Attempt at a Solution OK, we should prove that the path the particle takes and the geodesic are given by...

Simple question about geodesics. I have a question which I guess will be easy to answer for anyone who is familiar with the geometry involved in GR. Firstly, I have a numbered list which shows my (current) understanding of geodesics. If there is any wrong with my understanding please let...

Hawking singularity theorem -- what if not all geodesics incomplete? The Penrose singularity theorem tells us that once you get a trapped surface, at least one geodesic is guaranteed to be incomplete, going forward in time. But this doesn't mean that 100% of the mass of a collapsing star has to...

Let's imagine a test particle in outer space not being subjected to any significant force, gravitational(far enough from any massive object) or any other. Its path would be describing a geodesic that follows the universe curvature, right? Would that be an euclidean straight path, or would it...

Please read and critique this argument for me please, any help is appreciated. Imagine a geodesic, and a matter wave that traverses this geodesic. The action of this matter wave determines the motion of the matter wave along this geodesic over a given space-time interval, and is specified...

Please read and critique this argument for me please, any help is appreciated. Imagine a geodesic, and a matter wave that traverses this geodesic. The action of this matter wave determines the motion of the matter wave along this geodesic over a given space-time interval, and is specified...

I have some difficulties understanding how Minkowski spacetime is flat and therefore its geodesics should remain parallel, but at the same time I see it described in other sites as hyperbolic and then geodesics should diverge. Any comment on my confusion about this will be welcome. Thanks

Homework Statement Let \gamma be a stright line in a surface M. Prove \gamma is a geodeisc The Attempt at a Solution In a plane we know a straight line is the shortest distance between two point. I am not sure if this applies to straight lines on a surface. Further more, there...

I'm doing some revision for a General relativity exam, and came across this question: A Flat Earth space-time has co-ordinates (t, x, y, z), where z > 0, and a metric ds2 = ((1 + gz)2)dt2 − dx2 − dy2 − dz2 where g is a positive constant. Write down the geodesic equations in this space-time...

In general, what can one say about the relationship between geodesic motion of (massive and massless) particles and the killing vectors associated with the metric?

Hi all, Sorry if this is a dumb question, but what exactly do we mean by the term parallel transport? Is it just the physicist's way of saying isometry? Also, in my class we have just defined geodesics, and we're told that having a geodesic curve cis equivalent to demanding that the unit...

From what I understand, Einstein basically scrapped the concept of gravity being a force and instead said that energy (and thereby mass) and momentum causes spacetime to curve. Objects still travel on geodesics in spacetime (Newton's first law), but since it is curved, the geodesics near massive...

particles (not??) following geodesics in GR In a three-month old thread https://www.physicsforums.com/showthread.php?p=2557522&posted=1#post2557522 one of the tutors ("atyy") said: "And GR in full form does not have particles traveling on geodesics..." What does that mean? How can a free...

hello, I developed an application that models the trajectory of a light geodesic in the FLWR metric leaving from a galaxy and coming to our. I made for the moment the euclidean case (k=0) with a zero cosmological constant . So, the metric can be written ...

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...

Hello, I will expose a simplified version of my problem. Let's consider the following transformation of the x-axis (y=0) excluding the origin (x\neq 0): \begin{cases} \overline{x}=x \\ \overline{y}=1/x \end{cases} Now the x-axis (excluding the origin) has been transformed into an hyperbola...

Say I sit at some point P in a Calabi-Yau manifold. Are there geodesics which start from P and return to P? Are there "geodesics" which start from P and return to P but may make a "side trip first"? Is the number of geodesics which start at P and end at P infinite or finite and does that...

In the article from Wikipedia called: Geodesics as Hamiltonian Flows at: http://en.wikipedia.org/wiki/Geodesics_as_Hamiltonian_flows" It states the following: It is frequently said that geodesics are "straight lines in curved space". By using the Hamilton-Jacobi approach to the...

The geodesics around a spherical mass (Schwarzschild solution) in G.R. can be described by \frac{1}{2}\left(\frac{dr}{d\lambda}\right)^2 + V(r) = \mathcal{E} where V(r) is the effective potential \frac{1}{2}\epsilon - \epsilon\frac{GM}{r} + \frac{L^2}{2r^2} - \frac{GML^2}{r^3} and...

A vector field is parallel transported along a curve if and only if the the corariant derivative of the vector field along the path is 0. That is \frac{d}{d\lambda} V^\mu + \Gamma^\mu_{\sigma \rho} \frac{dx^\sigma}{d\lambda} V^\rho = 0 This is basically what every book says. But what...

I am sorry with the bad title and I am physics major with very weak math. So I come to the forum to rescue me. Basically I have one question, what does a "point-like creature" on a one dimensional line "sees" on different geodesics? if the line is flat, then the creature can sees everything on...

http://www.youtube.com/watch?v=8MWNs7Wfk84&feature=PlayList&p=858478F1EC364A2C&index=2" , Edmund Bertschinger is talking about Einstein's field equations . during the lecture , under the title of : "Gravity as sapcetime curvature (GR viewpoint) " , he wrote : "Freely falling bodies move along...

Homework Statement Consider the cylinder S in R3 defined by the equation x^2+y^2=a^2 (a). The points A=(a,0,0) \: and \: B = (a \cos{\theta}, a \sin{\theta}, b) both lie on S. Find the geodesics joining them. (b). Find 2 different extremals of the length functional joining A=(a,0,0)...