Light Cone Distribuitions

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:

[itex]\delta_{\pm}\left(\sigma\right)=\theta_{\pm}\left(x,\Sigma\right)\delta\left(\sigma\right)[/itex],

so they define the light cone Dirac Functionals [itex]\delta_{\pm}[/itex] with the functional [itex]\delta (\sigma)[/itex]. But they don't define [itex]\delta (\sigma)[/itex]. 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 [itex]\delta (\sigma)[/itex] 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.

 

Oh, yes, that makes more sense.

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

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

Sorry if i'm being annoying. Thank you.