Wledig said:I ran across a statement
Hi, sorry for not providing the reference. I read it on Wald (page 287).Where? Please give a specific reference.
Can we be more precise about this statement? If I take the derivative with respect to time of ##E = -P_a \xi^a## it should yield zero, right? But how exactly do I calculate this derivative?the projection of the 4-momentum on the direction of symmetry is conserved. If the Killing vector is timelike, we call this conserved quantity the "energy (of the particle)
"With respect to time" is ambiguous. "With respect to proper time" is more precise. You compute the derivative as you usually would with ##P## being the 4-momentum of the observer and the Killing field ##\xi## is evaluated and differentiated along the world-line of the observer.Wledig said:Hi, sorry for not providing the reference. I read it on Wald (page 287).
Can we be more precise about this statement? If I take the derivative with respect to time of ##E = -P_a \xi^a## it should yield zero, right? But how exactly do I calculate this derivative?
No. There is nothing being imposed in terms of momentum conservation (in fact, momentum conservation is a result from this very type of argumentation for the cases where it applies). You need to put some more ideas into what ##dV^a/d\tau## actually means for a vector field ##V## on a manifold.Wledig said:Sorry, I'm still confused. If I do something like: $$\frac{d}{dt}(-P_a \xi^a) = \frac{1}{c}\frac{d}{d\tau}(-P_a\xi^a)=-\frac{1}{c}\left(\frac{dP_a}{d\tau}\xi^a + P_a\frac{d\xi^a}{d\tau}\right)$$
It vanishes because the derivative of the killing vector is zero by definition and we are imposing momentum conservation? Am I thinking this through correctly?
A Killing vector field is a mathematical concept used in differential geometry and physics. It is a type of vector field that preserves the metric of a manifold, meaning that it does not change the distances between points on the manifold. In simpler terms, it is a vector field that represents a symmetry of the manifold.
Killing vector fields are closely related to killing vectors, which are the infinitesimal generators of isometries (symmetries that preserve the metric) on a manifold. In other words, a killing vector field is a vector field that generates killing vectors. However, not all killing vectors can be expressed as killing vector fields.
Killing vector fields have many important applications in physics, particularly in the study of spacetime and general relativity. They are also used in other areas of mathematics, such as differential equations and Lie groups. Understanding killing vector fields can provide insight into the symmetries and properties of a manifold.
There are various methods for calculating killing vector fields, depending on the specific manifold and metric being studied. In general, the process involves solving differential equations or using geometric methods to find vector fields that satisfy certain conditions, such as preserving the metric or being parallel to the geodesic curves on the manifold.
Yes, it is possible to visualize Killing vector fields using vector field plots or other graphical representations. This can help in understanding the symmetries and properties of a manifold and how the vector field acts on it. However, since killing vector fields can have a high number of dimensions, it may be difficult to visualize them in certain cases.