In the setting of general relativity, I came across a source term of the wave equation of the following form:

$$
\frac{1}{\sqrt{q}}\,\delta^{(3)}(p-\gamma(t))
$$

where $p\in M$ is a point in our 4d spacetime and $\gamma(t)$ is a trajectory that the source takes in the 4d spacetime. $\sqrt{q}$ is the 3d metric determinant of a preferred 3d slicing of $M$. $\delta^{(3)}(p-\gamma(t))$ is a 3d Delta distribution which means that we should have

$$
\int_M\,\delta^{(3)}(p-\gamma(t))\,f(p)\,d^3x=f(\gamma(t))\,.
$$

Of course, this is rather the physics short hand notation that $\delta^{(3)}(p-\gamma(t))$ is a map $C^\infty_c(M)\to C^\infty(\mathbb{R})$ that maps a function $f$ to $f\circ \gamma$.

We would like to show that the Lie derivative of the source along a certain vector field $T$ vanishes if $\gamma(t)$ is a Killing trajectory, that is $\gamma$ is an integral curve of $T$.

However, we are very confused about the rigorous treatment of this expression. Our intuition states the following:

The delta distribution should transform as scalar density of weight 1 under changes of the 3d frames and as a scalar under changes of the frame along the forth direction. However, we are not sure how to make this rigorous, nor how to find the Lie derivative of such a combined expression.

The object $1/\sqrt{q}$ should be an inverse object, that is it is a scalar density of weight $-1$ under changes of the 3d frame and a regular scalar along the forth direction.

Combining these two statements, it would make sense that the original object
$$
\frac{1}{\sqrt{q}}\,\delta^{(3)}(p-\gamma(t))
$$
is a regular 4d scalar. This would also make a lot of sense because it serves as source term of a regular 4d scalar wave equation.

Finally, it makes somehow sense that the Delta distribution is invariant under the Killing vector field $T$ iff the trajectory $\gamma$ is an integral curve of $T$. But we are not sure how to prove this and how to deal rigorously with the delta distribution.

This post imported from StackExchange Physics at 2014-06-11 21:30 (UCT), posted by SE-user user30835