What is Affine parameter: Definition and 13 Discussions
In geometry, a geodesic () is commonly a curve representing in some sense the shortest path (arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line" to a more general setting.
The noun "geodesic" and the adjective "geodetic" come from geodesy, the science of measuring the size and shape of Earth, while many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). The term has been generalized to include measurements in much more general mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph.
In a Riemannian manifold or submanifold geodesics are characterised by the property of having vanishing geodesic curvature. More generally, in the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. Applying this to the Levi-Civita connection of a Riemannian metric recovers the previous notion.
Geodesics are of particular importance in general relativity. Timelike geodesics in general relativity describe the motion of free falling test particles.
Hi, there. I am doing differentiation with respect to an affine parameter ##s##, I am not sure whether my idea is right or wrong.
Let ##C## be a geodesic for light and the path length ##s## on it be the affine parameter. Now I need to calculate ##\frac {\partial f}{\partial s}##, with ##f##...
In the usual Schwarzschild coordinates the Lagrangian can be written: $$\mathcal{L}= \frac{\dot r^2}{1-\frac{2M}{r}} - \left( 1- \frac{2M}{r} \right) \dot t^2 + r^2 \dot \phi^2$$ where all derivatives are with respect to a (affine) parameter ##\lambda##, and where for convenience I have...
Suppose you have a smooth parametrized path through spacetime ##x^\mu(s)##. If the path is always spacelike or always timelike (meaning that ##g_{\mu \nu} \dfrac{dx^\mu}{ds} \dfrac{dx^\nu}{ds}## always has the same sign, and is never zero), then you can define a smooth function of ##s##...
As far as I know, a object will experience time slower when its speed is close to the speed of light.
But photons themselves moves at the speed of light, does that mean that they experience no time?
Homework Statement
Question attached:
Homework Equations
see below
The Attempt at a Solution
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my main question really is
1) what is meant by 'abstract tensors' as I have this for my definition:
to part a)
##V^u\nabla_uV^a=0##
but you do say that ##V^u=/dot{x^u}## ; x^u is a...
I am asked a question about how far a light ray travels, the question is to be solved by solving for the null goedesic.
I am given the initial data: the light ray is fired in the ##x## direction at ##t=0##.
The relvant coordinates in the question are ##t,x,y,z##, let ##s## be the affine...
As I understand it, a curve ##x^{\mu}(\lambda)## (parametrised by some parameter ##\lambda##) connecting two spacetime events is a geodesic if it is locally the shortest path between the two events. It can be found by minimising the spacetime distance between these two events...
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...
Homework Statement
The metric is given by
https://dl.dropboxusercontent.com/u/86990331/metric12334.jpg
H is constant; s is an affine parameter, and so r(0)=t(0)=0.
Apologise in advance because I'm not very good with LaTex. So I used Word for equations, and upload handwritten attempt at...
If ##\sigma## is an affine paramter, then the only freedom of choice we have to specify another affine parameter is ##a\sigma+b##, a,b constants. [1]
For the tangent vector, ##\xi^{a}=dx^{a}/du##, along some curve parameterized by ##u##
My book says that ' if ##\xi^{a}\xi_{a}\neq 0##, then by...
Hi,
I've heard it said that for Schwarzschild spacetimes, then the coordinate 'r' is an affine parameter for radial null geodesics in the region exterior to the horizon. It seems weird to me that one of the coords is an affine parameter.
How can this be seen? I know that the radial null...
Wald (p. 41) defines a geodesic as a curve whose tangent vector satisfies
T^a\nabla_aT^b=0 . . . . . (3.3.1)
Then he says that we could have defined it by requiring
T^a\nabla_aT^b=\alpha T^b . . . . . (3.3.2)
instead, where \alpha is "an arbitrary function on the curve", but we...