What is the domain of definition of time $t$ in physics? Time $t$ is often used as variable of physical quantity, like $E (r, t)$.
In most EM, acoustic and other wave tutorials, equations (maxwell’s equations, acoustic equations, …) and wave function (EM wave $\cos (\omega t-kz)$ and $\exp(i\omega t)$, harmonic motion $\cos(\omega t)$, pulse wave $f(t)$, etc) have variable $t$, but the range of $t$ is usually not shown explicitly.
For the cases initial conditions are given, or the diagrams of harmonic motion that set $t \geq 0$, the time $t$ might not be allowed to be negative.
However, when performing fourier transform, $t$ is usually set to be from $-\infty$ to $+\infty$.
Then what is the domain of definition of time $t$ in physics (expecially when no explicit definition is given, which is a common case)?
Is the domain of $t$ from $-\infty$ to $+\infty$ or just nonnegative number?
(Please discuss in SI units, I am not sure whether different unit systems will have different conclusions)
 A: t runs between $- \infty$ and $ \infty$. When necessary, especially with integral transforms, one uses Heavyside's function to limit the domain.
A: In physics there is no consensus on the "domain" of time. There really can be no consensus since all theories must be valid for any choice of coordinates (including choice of a parameter we call time).
Writing this a little mathematically all physics must be invariant to the translation of time
$$
t \rightarrow t + a,
$$
where (for example!) $a \in \mathbb{R}$.
Also we usually require that the physics be invariant to scaling
$$
t \rightarrow \lambda t.
$$
Notice the emphasis on physics, this is because the math itself can be different and even have some local problems which usually need further consideration to extract the physics of it all.
To answer your question more directly a less correctly: t is always allowed to be negative, but sometimes this might result in mathematical consequences which should be treated with care.
Note that there are examples where we can take time to be $t \in \mathbb{C}$ so we should really treat is just as an parameter.
A: Physics is not mathematics. We choose our concept of time according to the problem we're addressing. The SI definition of the second changes as we learn to build better clocks.
Mathematically, the domain is chosen to match the problem we're solving. In an "initial value" problem, time starts at the boundary and continues for as long as we choose. If we're analyzing the dynamics of a system that has existed for a long time and will continue to exist for a long time, we might choose a domain of $-\infty$ to $\infty$ for convenience. Depending on the system, a "long time" might only be a nanosecond. Choice of the domain does not imply a commitment to the idea that real physical time matches the domain.
