Light Cone Distributions: Review by Eric Poisson, Ian Vega and Adam Pound

[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

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.