Geodesic Definition and 256 Threads

Hi all, I would like to know if somebody know the cases when we have in the space time manifold (and in general in any manifold) geodesic incompleteness. I know that a case can be a singularity in the curvature scalar (or in general, a singularity in any component of the Riemann tensor)...

Hi everyone, While reading http://relativity.livingreviews.org/Articles/lrr-2011-7/fulltext.html reference I bumped into a result. Can anyone get from Eq.19.1 to Eq.19.3? I've also been struggling to get from that equation to the one before 19.4 (which isn't numbered)...anyone? Thank...

Homework Statement We're asked to show that the geodesic equation \frac{du^{a}}{dt} +\Gamma^{a}_{bc}u^{b}u^{c}=0 can be written in the form \frac{du_{a}}{dt}=\frac{1}{2}(\partial_{a}g_{cd})u^{c}u^{d} Homework Equations...

I am trying to understand the paper 'Spectral shifts in General Relativity' by Narlikar. The paper considers a light ray emanating from the origin of a FLRW coordinate system in a universe whose hypersurfaces of constant time (in that coordinate system) are homogeneous and isotropic. The...

Consider a light ray emanating from the origin of a FLRW coordinate system in a homogeneous, isotropic universe. The initial velocity of that ray will have only x0 (t) and x1 (r) components. In papers I have seen it is assumed that its velocity will continue to have zero circumferential...

In this expression the junk on the left is a scalar. The stuff before the integral is another scalar. The integral is a time-like curve between x1 and x2 and at imagine fgf(x1) is a lower left corner of the rectangle and fgf(x2) is the upper right corner and x2-x1 is the length of the base of...

Homework Statement Calculate the geodesic for euclidean polar coordinates given ds^{2}=dr^{2}+r^{2}dθ^{2} Homework Equations standard euler-lagrange equation The Attempt at a Solution I was able to reduce the euler-lagrange equation to \frac{d^{2}r}{dθ^{2}}-rλ=0 where...

Hi all, I'm now reading Chap 11 of Gravitation by Wheeler, etc. In exercise 11.7, by introducing Jacobi curvature tensor, which contains exactly the same information content as Riemann curvature tensor, we are asked to show that we can actually measure ALL components of Jacobi curvature tensor...

I have to find the geodesics over a cone. I've used cylindrical coordinates. So, I've defined: x=r \cos\theta y=r \sin \theta z=Ar Then I've defined the arc lenght: ds^2=dr^2+r^2d\theta^2+A^2dr^2 So, the arclenght: ds=\int_{r_1}^{r_2}\sqrt { 1+A^2+r^2 \left ( \frac{d\theta}{dr}\right )^2...

The metric is ds^2=\frac{dx^2+dy^2}{y^2}. I have used the Euler-Lagrange equations to find the geodesics, and my equations are \dot{x}=Ay^2, \ddot{y}+\frac{\dot{x}^2-\dot{y}^2}{y}=0. I cannot seem to find the first integral for the second equation. I know it is \dot{y}=y\sqrt{1-Ay^2}, but I...

http://mykomica.org/boards/shieiuping/physics/src/1335180831965.jpg http://mykomica.org/boards/shieiuping/physics/src/1335180965708.jpg

Homework Statement If a general parameter ##t=f(s)## is used to parameterize a straight line in Euclidean space, then the geodesic equation takes the form ##\frac{d^2u^i}{dt^2}+\Gamma^i_{jk}\frac{du^j}{dt}\frac{du^k}{dt}=h(s)\frac{du^i}{dt}##, where...

Homework Statement Ants follow geodesics on a surface which is an infinite cylinder. Do the geodesics deviate? By considering only the paths of itself and its neighbors, can an ant decide whether it is on a cylinder of a plane? The Attempt at a Solution My answer to the first question is...

For the following two-dimensional metric ds^2 = a^2(d\theta^2 + \sin^2{\theta}d\phi^2) using the Euler-Lagrange equations reveal the following equations of motion \ddot{\phi} + 2\frac{\cos{\theta}}{\sin{\theta}}\dot{\theta}\dot{\phi} = 0 \ddot{\theta} -...

I am new to General Relativity and confused by the geodesic equation and MiSaTaQuWa equation. Most of the book saying that the geodesic equation is the motion of a particle in curved-spacetime. However, I read somewhere about this MiSaTaQuWa equation of motion. What is the difference between...

Homework Statement I'm working my way through Wald's GR book and doing this geodesic problem: Show that any curve whose tangent satisfies u^\alpha \nabla_\alpha u^\beta = k u^\beta , where k is a constant, can be reparameterized so that \tilde{u}^\alpha \nabla_\alpha \tilde{u}^\beta =...

I had this doubt studing GR, but let's consider SR for semplicity, where g_{\mu\nu}=\eta_{\mu\nu}the geodesics are 0=ds^2=dt^2-dr^2 we obtain the constraint we obtain the constraint r=(+/-)t So it is a well known light cone, but in SR we have that a (test?)particle can always move in a...

I'm in an intro course and my shaky ability to solve differential equations is apparent. How would you go about solving \ddot{r}-r\ddot{\theta}=0 \ddot{\theta}+\frac{1}{r}\dot{r}\dot{\theta}=0 It might be obvious. They're the geodesic equations for a 2d polar coordinate system (if...

The question is what flows on a surface can be geodesic flows. Specifically, starting with a smooth vector field on a surface - perhaps with isolated singularities - when is there a Riemannian metric so that the vector field has constant length and is tangent to geodesics on the surface? Here...

Through my mathematical fumblings, I think I have found a metric which gives a solution of the geodesic equation of motion that is asymptotic. It is a diagonal metric, with g00 = (x_1)^(-3) and g11 = 1. I am largely self-taught with SR so I may be miles off, but I think this gives a G.E. of M...

Does a geodesic flow on a compact surface - compact 2 dimensional Riemannian manifold without boundary - always have a geodesic that is orthogonal to the flow?

Does anybody knows the package that can, given metric and equation of hypersurface (spacelike or null )calculate induced metric, external curvature and expansion (Raychaudhuri equation) in Mathematica. Thanks

From a metric maybe the Schwarzschild, you can find g in co and contra varient forms. From that you can calculate Affinity. My question is from the Null Geodesic equation (ds=0) what do the three contravarient vectors represent? Do they represent the path of a planet around the sun or the...

Homework Statement Hello, I would like to derive geodesics equations from hamiltonian H=\frac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu} using hamiltonian equations. A similar case are lagrangian equations. With the definition L=g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu I tried to solve the...

Hi guys, here's my question: An accelerated observer (both in curved or non-curved space) who Fermi Walker transports his own basis vectors set along his world line will have the metric in the minkowsky form \eta_{\mu\nu} at each point of the world line? AND if the observer follows a...

If already known the form of Action and unkown the Metric, how to derive the geodesic equation?

In order to clarify what the EFE tells us about geodesic motion, it is important to remember that by the local flatness theorem, we can at any point p introduce a coordinate system (Riemann normal coordinates) so that the first derivatives of the metric at that point vanish. We can choose to...

[b]1. A straight line in flat space may be defined by the equation: (when I use the ^ symbol in this case it means like upper subscript not to the power) (U^v)(d/dx^v)(U^u)=0 (U^u=dx^u/ds) derive the geodesic equation. Please help I'm completely clueless all I can really see to do...

(Hopefully, Part 1 of 2) This is one of my favorite metrics, and I decided that while tedious, and old-fashioned, I would practice for my GR studies by finding the Christoffell symbols and write out the equations for geodesics using the Gödel metric, then attempting to solve them. First...

Hello Everyone, While trying to find the Newtonian limit to radial geodesic I was able to find that \frac{d^2r}{d\tau^2}=\frac{GM}{r^2} In the weak field limit we can naively replace \tau by "t" and recover Newtons Law, this though does not sound very rigorous. Can some-one suggest a...

I don't know if the tex is displaying properly. On my computer all I see is the geodesic equations in every tex field. In the past when this has happened, it has been fine for others viewing it, but if it doesn't make sense, I will upload a pdf or something. Thanks. Homework Statement The...

Homework Statement Show that the geodesic curvature of an oriented curve C in S at a point p in C is equal to the curvature of the plane curve obtained by projecting C onto the tangent plane along the normal to the surface at p. Homework Equations Meusnier's theorem, and k^2 = (k_g)^2 +...

Homework Statement What curves lying on a sphere have constant geodesic curvature? Homework Equations k^2 = (k_g)^2 + (K_n)^2 The Attempt at a Solution I'm trying to understand the solution given in the back of the book. It says, a curve on a sphere will have constant curvature...

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

If f:[a,b] \to R is a positive real function and\gamma(u,v) = ( f(u)\cos (v), f(u) \sin (v), u) then show that \gamma(t) = \sigma(u(t), c) is a geodesic in Mwhere c is a constant between 0 and2\pi and M=\sigma(U) where U= \{ (u,v)| a<u<b and 0<v< 2\pi \} Actually , I tried to calculate the...

Homework Statement Find the geodesics on the surface of a sphere of radius a by: (a) writing the geodesic equations for the spherical coordinates given by: x = rsinTcosP y = rsinTsinP z = rcosT for T and P(the r - equation can be ignored as a = constant); (b) exhibit a particular...

I'm following a slightly confusing set of notes in which I can't tell what exactly the timelike geodesic equations for the Schwarzschild metric are (seems to have about 3 different equations for them). How are these derived, or alternatively, does anyone have a link to a site in which they...

Hi, I have a question which was raised after reading the article "Derivation of the string equation of motion in general relativity" by Gürses and Gürsey. The geodesic equation for point particles can apparently be obtained as follows. First one takes the stress tensor of a point particle...

Hi, Any math wizards here willing to assist me in determining cut angles for parts to construct geodesic dome greenhouse. I am building a 13' diameter geodesic dome. I am using the info from geodesic dome calculators on line for a 2V version. It is composed of 6 pentagonal frames with...

I. Overview Another SA asked me to elaborate on a remark I made to the effect that frequency shift phenomena always (even in Minkowski vacuum) involve at least the following ingredients: two (proper time parameterized) timelike curves C, C' an event A on C ("emission event") an...

Sorry if this ends up being a naive question, but I have just a little conundrum. I'm dealing with curves in R2 and the Gauss-Bonnet theorem is a very useful result with what I'm currently doing, what with Gaussian curvature of a flat surface being zero, which is all fine...

Homework Statement How do I show the following metric have time-like geodesics, if \theta and R are constants ds^{2} = R^{2} (-dt^{2} + (cosh(t))^{2} d\theta^{2}) Homework Equations v^{a}v_{a} = -1 for time-like geodesic, where v^{a} is the tangent vector along the curve The Attempt at a...

Homework Statement If the geodesic is space-like somewhere, show that the geodesic is space-like everywhere. Homework Equations Geodesic equation: \ddot{X}^{\mu}+\Gamma^{\mu}_{\alpha \beta}\dot{X}^{\alpha}\dot{X}^{\beta} = 0 The Attempt at a Solution I looked at the metric...

Homework Statement I'm given the metric for Einstein's universe, ds2 = c2dt2 - dr2/(1 - kr2) - r2d(theta)2 - r2sin2(theta)d(phi)2 and asked to find the null geodesic equations and show that in the plane theta=\pi/2, the curves satisfy the equation: (dr/d(phi))2 = r2(1-kr2)(mr2-1) where m is a...

Homework Statement The problem asks to find the shortest distance between two points on Earth, assuming different equatorial and polar radii i.e. the coordinates are represented as: x = a*cos(theta)*sin(phi) y = a*sin(theta)*sin(phi) z = b*cos(phi) Homework Equations The Attempt at a...

Homework Statement Is there a curve on a regular surface M that is asymptotic but not principal or geodesic? Homework Equations The given definitions of asymptotic, principal, and geodesic: A principal curve is a curve that is always in a principal direction. An asymptotic curve is a...

Hi there, I'm trying to understand the derivation of geodesic deviation given here: http://wps.aw.com/wps/media/objects/500/512494/supplements/Ch21.pdf but I can't figure out why x(t)+\chi(t) obeys the geodesic equation (eq.(7)). Of course x(t) does, since it is per definition a...

I'm reading an article (http://arxiv.org/abs/gr-qc/0403075) which proves that a certain spacetime is geodesically complete. It does this by proving that the first derivatives fo all coordinates have finite bounds. My question is why this is enough. Is it just a simple ODE result? We know...

Homework Statement Suppose \bar{x}^{\mu} is another set of coordinates with connection components \bar{\Gamma}^{\mu}_{\alpha\beta}. Write down the geodesic equation in new coordinates. Homework Equations Using the geodesic equation: 0 = \frac{d^{2}x^{\mu}}{ds^{2}} +...

My lecturer has written: \ddot x^{\mu} + \Gamma^{\mu}{}_{\alpha \beta} \dot x^{\alpha} \dot x^{\beta} = 0 where differentiation is with respect to some path parameter \lambda. If we choose \lambda equal to proper time \tau then it can be readily proved that c^2 = g_{\mu \nu}(x)...