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...
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...
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...
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...