What is a 'spacelike surface' in relativity? 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
$$\int_{w_1}d\sigma_{\mu} J^{\mu}_a=\int_{w_2}d\sigma_{\mu} J^{\mu}_a.$$

In my understanding, spacelike-surfaces are surfaces of constant $t$ (surfaces perpendicular to the time axis in the figure below, like the which passes through the origin), but the text above states to take a spacelike-surface limited in time, which means that my definition of spacelike-surfaces isn't the correct one.
I would appreciate some clarification.

 A: The definition of a space-like surface is a little bit more general than your understanding. A space-like surface is any surface for which (all of) the tangent vectors to that surface are everywhere space-like. Surfaces of constant t for some given time-coordinate t satisfy this requirement, but not all space-like surfaces have to be surfaces of constant t for some coordinate t.
Imagine working in (1+1) dimensions (so that we can work with 2-D space-time diagrams). A space-like surface is any curve which always has tangent vectors whose angle (absolute value) w.r.t. the horizonal (on a standard space-time diagram) is $<45^\circ$. This is because vectors which are $<45^\circ$ to the horizontal are space-like in a space-time diagram. Surfaces of constant $t$ in this case would correspond to horizontal lines which is obviously a subset of such space-like surfaces. Surfaces of constant $t'$ given some arbitrary $t'$ (but still a "time coordinate") would be straight lines with angle $<45^\circ$ to the horizontal.
A: Your definition of the surface is correct.  I think you misread the text.  They are talking about a volume that is bounded by two such surfaces "in the time direction" and unbounded "in the space directions".
