What's the physical meaning of a “sphere”, and a pair (or even a series) of “concentric spheres”, in a region with Schwarzschild geometry?

1. Enclos[e] the origin of our Schwarzschild spacetime in a series of concentric spheres [...]

In a spacetime region of Schwarzschild geometry (including "its origin", if necessary), what is the explicit physical meaning* of a set of sufficiently many distinct participants (presumably at least four) being geometrically related to each other as (a subset of) a unique "sphere";
and what is the explicit physical meaning of two distinct such "spheres" being "concentric" to each other?

(*: In the given context of the notion of "physical meaning" is surely understood in the sense spelt out by Einstein, that "All our well-substantiated space-time propositions amount to the determination of space-time coincidences {such as} encounters between two or more material points.".)

• To provide a token of my own attempts to work on a solution to the OP question, and an estimate of its difficulty: I've learned (a while ago) about Karlhede's invariant (see arxiv.org/abs/1404.1845 and references there) whose value apparently allows to identify at least some spheres, or some pairs of concentric spheres, in a region of Schwarzschild geometry. (However, I have not yet fully understood either how Karlhede's invariant might be defined, and consequently how its values ought to be determined, strictly in terms of coincidence determinations, as advertised by Einstein.) – user12262 Sep 4 '15 at 5:59

One of the lessons of relativity is that we are free to choose any coordinates to describe the geometry of spacetime, though obviously some coordinates make more physical sense than others. In the case of a Schwarzschild black hole we choose a set of polar coordinates $(t, r, \theta, \phi)$ called the Schwarzschild coordinates. The singularity at the centre of the black hole is at $r = 0$.
When we say a sphere we simply mean the spacial surface with constant $r$ in the Schwarzschild coordinates. All such spheres are automatically concentric because they are constructed around the same centre i.e. the singularity at $r = 0$.
The Eddington-Finkelstein coordinates use the same $r$ coordinates as the Schwarzschild coordinates, so the meaning of sphere is the same in both.
• John Rennie: "One of the lessons of relativity is that we are free to choose any coordinates to describe the geometry of spacetime" -- The lesson is much stronger: we can describe the geometry of spacetime without referring to coordinates at all, but (merely) to determinations of physical coincidence. "some coordinates make more physical sense than others" -- The more thorough technical statement is that coordinates may or may not be affine to a physically given suitably generalized metric space. So you should address how coordinate values $r$ are to be assigned to any suitable worldlines. – user12262 Sep 2 '15 at 19:19
• @user12262: I'm not sure what you're asking. Are you asking how we define the $r$ coordinate? You seem to know enough about GR that surely you know this. – John Rennie Sep 3 '15 at 5:50
• John Rennie: "Are you asking how we define the $r$ coordinate?" -- Well, since you seem to insist on involving coordinates, and specifically "the $r$ coordinate": In your answer, as it stands, you have already given some relevant part of "how we define the $r$ coordinate", namely, AFAIU: to all elements of the same one "sphere" shall be assigned the same value $r>0$. But what's not yet addressed (and that's my actual question, besides any coordinate assignments): How do we define which events (or which wordlines) contitute or belong to the same one "sphere" in the first place?, – user12262 Sep 3 '15 at 16:36
• [contd.] ... in a region of Schwarzschild geometry. (Also not yet addressed is: which particular value $r \gt 0$ then ought to be assigned to which particular "sphere"; but this part "how we define the $r$ coordinate" is not my interest/question at all.) "You seem to know enough about GR that surely you know this." -- I'm certainly not aware that I'd know how "we" determine which events/worldlines belong to the same sphere, and which don't. If you do indeed know how to answer this (do you?) but you think it wouldn't benefit me personally, you should still provide it to benefit others. – user12262 Sep 3 '15 at 16:38