I am studying Noether’s theorem in field theory and I am not understanding what spacelike-surfaces mean. I will reproduce the bit of the argument below that contains the term “spacelike-sufaces” in the context I am not understanding.
There will be a conserved ccurrent for each group generator $a$. Each will result in a conserved charge (that is, an integral of motion). To see this, take in spacetime a volume unbounded in the space-like direction, but limited in time by two space-like surfaces $w_1$ and $w_2$. Integrating $\partial_{\mu} J^{\mu}_a=0$ over this volume, we get an integral over the boundary surface, composed of $w_1$, $w_2$ and the time-like boundaries supposed to be at infinity. If we now suppose the current to be zero at infinity on these boundaries, we remain with.
$$\int_{w_1}d\sigma_{\mu} J^{\mu}_a=\int_{w_2}d\sigma_{\mu} J^{
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