Geodesic Definition and 256 Threads

I have a question. Is it true that any curve in 2-dimensional manifold which tangent vector is null at each point is null geodesic? (In 2-dimensional manifold there are only 2 null direcitions at each point).

Hello Anybody know how we can solve the equations of geodesic deviation in a given spacetime whether approximately or exactly? Thanks in advance

A graviton, if massless, is generally expected to travel at c. If so, we would not expect it to follow the geodesic, which is the path a hypothetical particle with infinite speed. Therefore I would think for example that a massless graviton that was gravitationally lensed around a galaxy would...

The following curve is geodesic in Schwardschild metric: \tau \mapsto [(1-2m/r_0)^{-1/2}\tau,r_0,0,0]. The tangent vector is: [(1-2m/r_0)^{-1/2},0,0,0] , its length is 1 and its product with killing vector \partial_t is equal: (1-2m/r_0)^{1/2} = \textrm{const}. So the body lays at rest...

layperson here, so please correct any misconceptions i have on this. an electron will emit photons if it is accelerated (including changes to either velocity and/or direction of travel). acceleration occurs if the electron experiences a force. since GR indicates that gravity is not a...

it comes from the calculus of variation that the shortest path between two points on a surface must be geodesic. then must the geodesic connected two points be the shortest path? if not, what about the example? Thanks for any reply!

I'm trying to do excercise 4.8 in "Riemannian manifolds" by John Lee. (It's about showing that the geodesics of \mathbb R^n are straight lines). The result I'm getting is that the definition of a geodesic implies the well-known identity 0=0, which isn't very useful. I must have made a mistake...

Homework Statement Consider the 2-dim metric {{\it ds}}^{2}=-{\frac {{a}^{2}{{\it dr}}^{2}}{ \left( {r}^{2}-{a}^{2}\right) ^{2}}}+{\frac {{r}^{2}{d\theta }^{2}}{{r}^{2}-{a}^{2}}}, where r > a. What is its signature? Show that its geodesics satisfy {\frac {{a}^{2}{{\it dr}}^{2}}{{d\theta...

I've been trying to learn GR and I've been back and forth through Schutz's first course book. I think I understand the basic principals, but one thing still eludes me: a traveler in free fall travels along the geodesic, the path of longest proper time. If the path between two points passes...

What does it mean to say that something moves on a straight wordline in terms of the principle of least action? I know it generally means that action is minimum or stationary but since I only really know some physics from a conceptual standpoint and not a mathematical one I don't really know...

Geodesic on a cylinder - have I done this correctly?? Homework Statement ds^{2} = a^{2}d\theta^{2} + dz^{2} ds = \sqrt{a^{2}d\theta^{2} + dz^{2}} \int\sqrt{a^{2} + dz'^{2}} d\theta = Min E-L equation df/dz - d/d\theta(df/dz') = 0 df/dz = 0, d/d\theta[\frac{z'}{\sqrt{a^{2}...

Homework Statement The metric is: ds^{2} = y^{2}(dx^{2} + dy^{2}) I have to find the equation relating x and y along a geodesic.The Attempt at a Solution ds = \sqrt{ydx^{2} + ydy^{2}} - is this right? ds = \sqrt{y + yy'^{2}} dx F = \sqrt{y + yy'^{2}} So then I apply the Euler-Lagrange...

Homework Statement I'm trying (on my own) to derive the geodesic for a sphere of radius a using the geodesic equation \ddot{u}^i + \Gamma^i_{jk}\dot{u}^j\dot{u}^k, where \Gamma^i_{jk} are the Christoffel symbols of the second kind, \dot{u} and \ddot{u} are the the first and second...

http://en.wikipedia.org/wiki/Jacobi_field also see http://iopscience.iop.org/0305-4470/14/9/029/?ejredirect=.iopscience What's the difference between the jacobi equation and the geodesic deviation equation?

Hi, I was wondering: I'm sitting at my desk and on my desk lies a pen. Does the pen describe a geodesic? And is the room I'm sitting in an inertial frame? I think the pen doesn't describe a geodesic because it's not in free fall and i think my room is a good approximation of an inertial...

Hello, I have a p+1 dimensional manifold describing the parameter space of a family of probability densities. The p+1 dimensions are (beta, t1, t2, ..., tp), all reals, and beta restricted to the positive reals. The (fisher) metric on this manifold is a function of beta only, hence the...

I'm trying to find the geodesic between any two points in 3D space, where the geodesic is constrained to a surface defined by z = A / (x^2 + y^2), where A is a constant. I've tried all sorts of variations on the Euler-Lagrange equations (after changing to cylindrical coordinates), but am...

I am interested in solving the null geodesic between two points in the presence of a gravitational mass, assuming that everything takes place in 2 dimensions (i.e., no Z coordinate). The following is known: -x and y coordinates of first point -x and y coordinates of second point -x and y...

Homework Statement (a) Consider a 2-dimensional manifold M with the following line element ds2=dy2+(1/z2)dz2 For which values of z is this line element well defined. (b) Find the non-vanishing Christoffel symbols (c) Obtain the geodesic equations parameterised by l. (d) Solve...

Let’s say an asteroid is about to enter earth’s atmosphere(it will burn up of course). The initially sitting is at: point = [x_{0},y_{0},z_{0},t_{0}] = [r_{0},\\theta_{0},\\phi_{0},t_{0}] With a 4-velocity: V = [v_{1},v_{2},v_{3},v_{4}] The momentum at 0+ p_{0+} = mass*V =...

Homework Statement L = R \int_{\theta_1}^{\theta_2} \sqrt{1 + sin^2(\theta ) \phi ' ^ 2} d\theta Use the result to prove that the geodesic (shortest path) between two given points on a sphere is a great circle. [Hint: The integrand f(phi,phi_prime,theta) in the result is independent of phi so...

Homework Statement Prove that the proper length of geodesic between two points is unchanged to first order by small changes in the curve that do not change its endpoints. Homework Equations Length of curve = \int...

Homework Statement I'm given the surface of revolution parametrized by \psi (t, \theta ) = (x(t), y(t)cos \theta, y(t)sin \theta ) where the curve \alpha (t) = (x(t),y(t)) has unit speed. Also given is that \gamma (s) = \psi (t(s), \theta (s)) is a geodesic which implies the following equations...

An equilateral geodesic triangle is right angled. The area of a geodesic triangle on a sphere of radius r is (1/2)\pir^2. But how is that obtained?

I've been reading a few proofs showing that a great circle is geodesic. Most of these proofs start with a parametrization and then show that it satisfies the differential equations of geodesics. The book that I have doesn't even give a proof. It just tells me that the great circles on the sphere...

Have a hard time grasping the concept of the Geodesic. If you are given a velocity and position of say a small astroid entering the gravitational field of the Earth. How do you find the trajectory or path it takes as it falls to Earth? Not afraid of tensors and advanced calculus.

Are there surfaces that have a geodesic curve which completely covers the surface, or (if that's not possible) is dense in the surface? In other words, if you were standing on the surface and started walking in a straight line, eventually you would walk over (or arbitrarily close to) every...

I can understand the logic from some arguments as to why proper time in a photon's "frame of reference" is zero. I cannot understand how this follows from the argument that (SPACE)2 - (TIME) 2 = 0. This to me says that the SPACE-TIME interval for the photon is zero (null interval) and SPACE =...

i have been reading about 'Selberg Trace formula' i know what a Laplacian is but i do not know what is the author referring to when he talks about 'Closed Geodesic' i know what the Geodesic of a surface is l(\gamma)=\int_\gamma \sqrt{ g(\dot\gamma(t),\dot\gamma(t)) }\,dt\ , but i do...

hi all, i kind of have a black hole regarding my understanding of the consistensy of tidal forces and geodesic deviation analysis. one can determine some coefficients of the riemman tensor from the tidal forces equations, by getting to a form that is like the form of the geodesic deviation...

if we have or can have geodesic curves minimizing the integral \sqrt (g_{ab}\dot x_a \dot x_b ) is there a theory of 'minimizing surfaces or Geodesic surfaces' that minimize the Area or a surface ?,

I understand what geodesics are and how to calculate them from Christoffel symbols and all that. But I've just come across a question I have no idea about. I've been given the dust filled Friedmann solution: ds^2 = -dt^2 + a(t)^2 (dX^2 + X^2 dO^2) (O=omega) And been told to show that...

Hello. Suppose that \sigma: (M, g) \to (N, h) is an isometric diffeomorphism between two Riemannian manifolds M and N and let \gamma: [0, 1] \to M be a geodesic on M. Because \sigma preserves distances, and geodesics are locally length minimizing, it is intuitively clear that \sigma_*...

Homework Statement I would like to manipulate the geodesic equation. Homework Equations The geodesic equation is usually written as k^{a}{}_{;b} k^{b}=\kappa k^{a} (it is important for my purpose to keep it in non-affine form). It is clear that by contracting with the metric we may...

Given we have a spherically symmetric gravitational field around a spherically symmetric body of mass M. How can I calculate the actual (geodesic) distance between two points with the same angle but at different distances from the center of the body (and field). I came immediately to think of...

Hi, I have a question about deriving tensor identities using geodesic coordinates ( coordinates in which one can put the connection to zero ). For the reference, I'm a master student physics and followed courses on general relativity and geometry. I'm busy with an article of Wald and Lee called...

Regarding the twins paradox: do all closed trajectories require nonzero acceleration at some point, or can a closed geodesic fulfill overall the special relativistic requirement of constant velocity?

Hi...does anyone have a good description (or a link to it) on geodesic deviation equation?Most of the references i have are in a setting of relativity, which make me all at sea. Please help me if you know a mathematical characterization of how geodesics from one point deviate (which just...

Hi there: i have a question on geodesics in a Eculidean space equipped with a metric tensor \lambda(x)*I, where I is the identity matrix. Is any general statement that can be made towards the geodesic between two points in this modified space? My feel is that this space is quite special...

Hi, Earth follws a straight path in 4-d space time.ok.now the Earth moves over the geodesic formed by the sun's gravity.now we also have other 7 planets.So does it mean that the sun forms different geodesic for differnt planets. if my question does make some logic than please explain me.

We need to show that using the Schwarzschild metric, an incoming radial spacelike geodesic satisfies r>= 1m/(1 + E^2) I know that E = (1-2m/r) is constant, and I think that ds squared should be negative for a spacelike geodesic. I try substituting E into the metric and setting ds squared <=0...

Homework Statement Show that a regular curve on a smooth surface is a geodesic and an asymptotic curve if and only if it is a segment of a straight line. The Attempt at a Solution I did the <= implication, which is quite easy. I can't get the other one.

Can someone explain how to derive the formula for geodesic deviation? i didn't quite understand it. I'm using schtuz's book.

We consider a 2 dimensional spacetime with coordinates (t,y) and metric g=dt\otimes dt - (t^2+a^2)^2dy\otimes dy where a is a positive constant. In previous subquestions, I have calculated the Christoffel symbols for the metric-compatible connexion and now I am asked to find the null geodesic...

A person traveling through a geodesic. Does experiment some kind of acceleration?, since the geodesic equation analogue to Newton one is: \Delta _{u} u =0 and in an Euclidean space there's no acceleration for a particle line X(u)=au+b

Given a two body system like the sun and the earth: Do the L4 and L5 points travel in a geodesic, or is it just the sun and the Earth which do so?

Can anyone point me to good reference that fully develops the geometry of geodesic curvature? Most of the ones I have manage to derive it, then show it's the normal to the curve, then never mention it again. I want to know how it relates to the metric, first second or third. Thanks.

Hi, I'm working on marion&thornton ch6 question 6.4. "Show that the geodesic on the surface of a straight circular cylinder is a (partial) helix" I used the example of the geodesic on a sphere in the book, but when i calculate the angle phi i get something like phi=b*z+c, where b and c are...

This is just a mere suggestion for doing such computations without resorting to complicated series expansions of elliptic integrals. Up to the irregular continental shape & sea floor, we could picture our planet as a revolution ellipsoid. This means that, except for points on the same...

First starting making these back in 1973, but those are long gone. Decided to make a 4v icosahedron recently and this is what I ended up with. Web page shows the progress, and the final sphere: 4f.htm