If $G(\vec{x}, \vec{\xi}, t)$ is the Green function of wave equation, then solutions for inhomogeneous wave equation will be
$\left( -\Delta + \frac{1}{v^2}\frac{\partial^2}{\partial t^2}\right) u(\vec{x}, t) = f(\vec{x}, t)$
can be formed as
$$u(\vec{x}, t) = \int d\xi d\tau G_I (\vec{x}, \vec{\xi}, t-\tau)f(\vec{\xi}, \tau)$$
where functions $G_I$ are ($\eta$ is Heaviside's function):
$G_R(\vec{x}, \vec{\xi}, t-\tau) = G(\vec{x}, \vec{\xi}, t-\tau)\eta(t-\tau)$ $G_A(\vec{x}, \vec{\xi}, t-\tau) = -G(\vec{x}, \vec{\xi}, t-\tau)\eta(\tau-t)$
Why are there two Green's functions? Are they the only Green's functions? What conditions do they meet?
Where is the "-" sign from?
What initial conditions does solution for each of the functions meet?
What is their difference?
Appreciate it!