Light Cone Distribuitions

[PLuz]

Hello,

I have been reading the excellent review by Eric Poisson, Ian Vega and Adam Pound:http://relativity.livingreviews.org/Articles/lrr-2011-7/fulltext.html [Broken]

In section 12, Eq.12.15, there's something that I don't quite understand. They write:

$\delta_{\pm}\left(\sigma\right)=\theta_{\pm}\left(x,\Sigma\right)\delta\left(\sigma\right)$,

so they define the light cone Dirac Functionals $\delta_{\pm}$ with the functional $\delta (\sigma)$. But they don't define $\delta (\sigma)$. I suppose they intend to define a Dirac distribution along the unique geodesic that links two points in space time, but the Synge world function is defined for space,time and null geodesics, how is $\delta (\sigma)$ only restricted to the light cone?

(You might also want to look in section 13.2 where they generalize for curved spacetime)

Thank you and sorry if it's a silly question.

 

[George Jones]

how is $\delta (\sigma)$ only restricted to the light cone?

$\delta (\sigma)$ is not restricted to the light-cone, the support of $\delta (\sigma)$ is restricted to the light-cone. The support of a function is the closure of the set on which the function is non-zero.

 

[PLuz]

Oh, yes, that makes more sense.

Altough, to clarify, if I have a base point $x_1$ and another point, say, $x_2 \in I^+(x_1)$, how would one see $\delta(\sigma)$ as a function of $x_2$?

I don't understand the argument of $\delta$, what should I visualize? $x_3:=\sigma(x_1,x_2)$ it's a scalar, what does it mean $\delta(x_3)$?

Sorry if I'm being annoying. Thank you.