The question doesn't make sense. Can you give a reference that explains what you are trying to do?bobrubino said:
I am trying to project a black and white hole onto a field of Schwarzschild spacetime and I don't know what to do. Do you have any advice?PeterDonis said:
Your question still doesn't make sense. Again, where are you getting this from? What makes you think this is even a meaningful thing to do? Why do you want to do it? A reference would be helpful.bobrubino said:
Schwarzschild spacetime is a black hole and a white hole. Your question makes no sense.bobrubino said:
The maximally extended Schwarzschild is both a black hole and a white hole. If you are just trying to visualize that, my preference is Kruskal coordinatesbobrubino said:
A Killing Vector is a vector field on a spacetime manifold that represents symmetries of the spacetime. Specifically, it is a solution to Killing's equation, which states that the Lie derivative of the metric tensor with respect to this vector field is zero. This implies that the spacetime has a symmetry along the direction of the Killing Vector, such as time translation or rotational symmetry.
The Schwarzschild field refers to the spacetime geometry described by the Schwarzschild solution to Einstein's field equations. This solution represents the gravitational field outside a spherically symmetric, non-rotating, uncharged massive object. It is characterized by the Schwarzschild metric, which depends only on the radial coordinate and describes how spacetime is curved by the presence of the mass.
The Schwarzschild metric has two obvious Killing Vectors. One corresponds to time translation symmetry, given by the vector ∂/∂t, and the other corresponds to rotational symmetry around the azimuthal angle, given by the vector ∂/∂φ. These vectors satisfy Killing's equation in the Schwarzschild spacetime, indicating the presence of these symmetries.
Projecting a Killing Vector onto a Schwarzschild field involves expressing the Killing Vector in the coordinate system used for the Schwarzschild metric. This typically means writing the components of the Killing Vector in terms of the Schwarzschild coordinates (t, r, θ, φ). For instance, the time translation Killing Vector is simply (1, 0, 0, 0) in these coordinates, while the rotational Killing Vector is (0, 0, 0, 1).
Projecting Killing Vectors onto the Schwarzschild field is crucial for understanding the symmetries and conserved quantities in this spacetime. For example, the time translation Killing Vector is associated with the conservation of energy, and the rotational Killing Vector is associated with the conservation of angular momentum. These projections help in solving geodesic equations and in the analysis of particle orbits and other physical phenomena in the Schwarzschild spacetime.