Geodesic equation Definition and 68 Threads

I understand why the geodesic equation works in flat space. It just basically gives a set of differential equations to solve for a path as a function of a single variable s where the output is the coordinates of whichever parameterization of the space you are using. But the derivation I know and...

The geodesic equation follows from vanishing variation ##\delta S = 0## with ##S[C] = \int_C ds = \int_a^b dt \sqrt{g_{ab}\,\dot{x}^a\,\dot{x}^b}## In many cases one uses the energy functional with ##\delta E = 0## instead: ##E[C] = \int_a^b dt \, {g_{ab}\,\dot{x}^a\,\dot{x}^b}## Can...

I'm trying to derive the Geodesic equation, \ddot{x}^{α} + {Γ}^{α}_{βγ} \dot{x}^{β} \dot{x}^{γ} = 0. However, when I take the Lagrangian to be {L} = {g}_{γβ} \dot{x}^{γ} \dot{x}^{β}, and I'm taking \frac{\partial {L}}{\partial \dot{x}^{α}}, I don't understand why the partial derivative of...

Hello Everybody, Carroll introduces in page 106 of his book "Spacetime and Geometry" the variational method to derive the geodesic equation. I have a couple of questions regarding his derivation. First, he writes:" it makes things easier to specify the parameter to be the proper time τ...

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...

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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...

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} -...

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...

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

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...

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}} +...

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...

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 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...

I don't understand the equation of the geodesic y=y(x) for the surface given by z=f(x,y) : a(x)y''(x)=b(x)y'(x)^3+c(x)y'(x)^2+d(x)dxdy-e(x) the functions a,b,c,d,e are here not very important, what I don't understand, is that there is terms in \frac{dy}{dx} and dxdy...What does this mean ?