# What does a 4+1D wave look like at the light cone?

For ease of reference in this post equations are numbered as in ref. 1.

The equation $$(36)$$ cited in the OP reads

$$G_4(r,t)=\frac1{4\pi^2c^3}\left(\frac{\delta(t-r/c)}{r(t^2-r^2/c^2)^{1/2}}-\frac{\eta(t-r/c)}{c(t^2-r^2/c^2)^{3/2}}\right).\tag{36}$$

Here $$\delta$$ is the Dirac delta, and $$\eta$$ is the Heaviside step function.

By itself, this expression doesn't really help understand what's going on: the Dirac delta in the first term is multiplied by a function that is singular at the delta's distributional singularity, and the second term isn't integrable. So, even if there is some way to evaluate this Green's function as a distribution like $$\langle G_4,h\rangle$$ for a smooth test function $$h$$, I suppose the expression would have to be transformed somehow.

Let's go another way. Namely, start from how equation $$(36)$$ was derived. The authors in ref. 1 derived it by integrating the Green's function for (5+1)-dimensional wave equation,

$$G_5=\frac1{8\pi^2c^2}\left(\frac{\delta(\tau)}{r^3}+\frac{\delta'(\tau)}{cr^2}\right),\tag{32}$$

where $$\tau=t-r/c$$, along the line of uniformly distributed sources in 5-dimensional space, using the integral

$$G_{n-1}(r,t)=2\int_r^\infty s(s^2-r^2)^{-1/2}G_n(s,t)ds,\tag{25}$$

where $$r=r_{n-1}$$ is the radial coordinate in $$(n-1)$$-dimensional space.

Remember that a Green's function for a wave equation is the impulse response of the equation, i.e. the wave that appears after the action of the unit impulse of infinitesimal size and duration, $$f(r,t)=\delta(r)\delta(t)$$. For visualization purposes it's useful to replace this impulse with one that is finite at least in one variable, e.g. time. So let's choose another force function $$f(r,t)=\delta(r)F(t)$$, where $$F$$ is defined as the Gaussian function

$$F(t)=\frac1{\sqrt{2\pi}\sigma}\exp\left(-\frac{x^2}{2\sigma^2}\right).$$

Then, following equation $$(34)$$, we'll have the displacement response of the (5+1)-dimensional equation given by

$$\phi_5(r,t)=\frac1{8\pi^2c^2}\left(\frac{F(\tau)}{r^3}+\frac{F'(\tau)}{cr^2}\right).\tag{34}$$

Now, to find the displacement response $$\phi_4(r,t)$$ of the (4+1)-dimensional equation, we can use $$\phi_5$$ instead of $$G_5$$ in $$(25)$$. The resulting integral doesn't seem to have a closed form, so the following animations of $$\phi_4(r,t)$$ were done using numeric quadrature.

In the animations below we use $$c=1$$.

Let's first see how the $$\phi_5(r,t)$$ looks. This will enable us to compare it with $$\phi_4(r,t)$$. The following animation uses $$\sigma=0.005.$$

We can see that it basically has two spikes: one positive and one negative. Let's increase $$\sigma$$ to $$\sigma=0.04$$ to resolve them better:

Now we can finally look at $$\phi_4(r,t)$$. The following animation uses $$\sigma=0.005$$, as the first one above.

Notice the difference of this wave from the $$\phi_5(r,t)$$: the former has a "tail" inside the light cone, i.e. on the LHS of the positive spike here. This tail corresponds (in the limit $$\sigma\to0$$) to the second, negative, term in the equation $$(36)$$, while the positive spike corresponds to the first term with the Dirac delta.

What does it mean that the Green’s function for the 4+1D wave equation has an opposite sign delta-function exactly on the light cone?

is exactly that: there is a negative wake inside the light cone and the opposite sign, i.e. positive, Dirac delta, i.e. infinitely high and narrow spike, on the light cone.

References:

1: H. Soodak, M. S. Tiersten, Wakes and waves in N dimensions, Am. J. Phys. 61, 395 (1993)