Read about geodesic affine parameter | 4 Discussions | Page 1

  1. R

    Alternative form of geodesic equation

    Homework Statement We are asked to show that: ## \frac{d^2x_\mu}{d\tau^2}= \frac{1}{2} \frac{dx^\nu}{d\tau} \frac{dx^{\rho}}{d\tau} \frac{\partial g_{\rho \nu}}{\partial x^{\mu}} ## ( please ignore the image in this section i cannot remove it for some reason ) Homework Equations The...
  2. V

    A Geodesic defined for a non affine parameter

    The geodesic general condition, i.e. for a non affine parameter, is that the directional covariant derivative is an operator which scales the tangent vector: $$\zeta^{\mu}\nabla_{\mu}\zeta_{\nu}=\eta(\alpha)\zeta_{\nu}$$ I have three related questions. When $$\alpha$$ is an affine parameter...
  3. F

    I Example of computing geodesics with 2D polar coordinates

    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...
  4. K

    Affine parameterization of a light ray

    Hello, Is this parameterization correct? - ##r(t) = R = \mbox{const}## ##\theta(t) = 0 = \mbox{const}## ##z(t) = ct## ##t = t## This is supposed to be the null geodesic curve in the case of a light ray, emitted at point {##r=R,\theta=0,z=0,t=0##} parallel to the ##z-##axis in flat spacetime...
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