the causal Green's function
For problems that evolve in time, there is a non-negotiable physical rule: an effect cannot precede its cause. If you strike a system at time s, nothing about that strike can be felt at any earlier time. The causal Green's function is the time-dependent point-source response with this rule built in — it is identically zero before the source acts and carries the system's reply forward in time only.
Write G(t, s) (suppressing space for a moment) for the response at time t to a unit impulse delivered at time s. Causality demands G(t, s) = 0 for t < s — no response before the kick. The defining property is L_t G(t, s) = delta(t - s) together with this vanishing for t < s, which singles out the retarded Green's function from among the mathematically valid solutions (the other being the advanced one, supported for t > s reversed, which would let the future influence the past). The payoff is a clean solution by superposition over time: for a forcing f(t) switched on, u(t) = integral from -infinity to t of G(t, s) f(s) ds — note the upper limit t, the signature of causality: only sources from the past contribute. For PDEs this becomes the space-time propagator, e.g. the retarded fundamental solution of the wave equation that spreads a flash outward on the light cone.
Why it is the physically right choice: in electromagnetism it gives the retarded potentials (fields determined by past, not future, charge motion); in signal processing it is the impulse response of a realizable filter; in quantum field theory the retarded versus Feynman propagators encode different boundary conditions in time. The honest point is that mathematics admits the advanced solution equally — the equation alone does not know about the arrow of time — so causality is an extra condition you impose, motivated by physics, not derived from the differential equation. And because the causal Green's function is not time-symmetric, it does not satisfy ordinary reciprocity; instead G of the operator and G of its time-reversed adjoint are related.
A damped oscillator m x'' + c x' + k x = F(t) struck by a unit impulse at time s responds with G(t, s) = (1/(m omega)) e^(-gamma (t - s)) sin(omega (t - s)) for t > s and zero before. The driven motion is x(t) = integral from -infinity to t of G(t, s) F(s) ds.
Only the past forces the present — the upper limit is t.
The equation alone admits both retarded and advanced solutions equally; causality (vanishing before the source) is an extra physical condition you impose, and the causal Green's function is not time-symmetric, so it does not obey ordinary reciprocity.