# Plotting a 3d Delta function with a norm coefficient

I am trying to plot the following in Mathematica. I have a 3 dimensional delta function, that is given by

$$|t-x|\delta(-t^2 + x^2+ z^2)$$

I am trying to make a 3d plot where I could see where the delta function is kicking in but also the strength of the function due to the $$|x^-|$$ in front. Using Plot3D, I could easily show where the delta function is activated, but I am struggling to make like a shaded thing that would show that for exemple, when $$t=x$$, then this thing has to be zero.

To find where the delta function is activated, I am using

p1 = Plot3D[{Sqrt[x^2 + z^2], -Sqrt[x^2 + z^2]}, {z, 0, 5}, {x, -5, 5}, BoxRatios -> {1, 1, 1}, ColorFunction -> "RustTones"]

Any help would be much appreciated.

You cannot "plot" the DiracDelta function because it's a distribution.

To get an idea where DiracDelta is "aktiv" try to plot a numerical approximation of DiracDelta:

eps = .001; (* small*)
dirac = Function[x, Exp[-(x^2/(2 eps))]/Sqrt[2 Pi eps]] ;


DensityPlot3D shows the cone t^2-x^2-z^2==0

DensityPlot3D[dirac[x^2 + z^2 - t^2] RealAbs[x - t], {z, -1, 1}, {x, -1, 1}, {t, 0,1}, AxesLabel -> Automatic] • Your dirac is not "a numerical approximation of DiracDelta", but its approximation in the weak topology. Exactly speaking, dirac defines the certain distribution. Dec 10, 2020 at 13:40
• "Exactly" speaking No! The limit of dirac is one possibility to define the DiracDelta-distribution. Dec 10, 2020 at 17:43