I am not sure whether this is a physics question, a maths question or even a linguistics question; please forgive me if I have chosen the wrong platform. I am trying to understand what it really means for spacetime to be curved, but I cannot figure out how to combine in my mind the notions of space, time and curvature in such a way that it makes sense. That is surely because I am using the wrong notions for one or more of those concepts, but I am neither a physicist nor a mathematician (and not even a native English speaker, for that matter) so I am forced to resort to those notions as I understand them from my education and my day to day experience. Please allow me to describe those concepts as I view them and the problems I find at combining them, in the hope that someone can point out my mistakes and help me understand the alluded question.
I can understand the word "space" in different ways depending on the context in which it is used, but when I view itself as the context in which things exist, I understand it as an infinite set of three-dimensional locations, each described as a triplet of distances from a chosen reference point along three orthogonal axes. Macroscopically, different regions (subsets of locations) may be "occupied" or "filled" with different material objects. For example, my body occupies a specific volume of space underneath which there is a volume occupied by my chair, and so on. Microscopically, however, it appears that most of those seemingly filled volumes are actually empty space, since most of the matter in each atom making up those things is confined to a tiny volume in its nucleus, and the rest is spanned (rather than occupied) by an ordered cloud of increasingly complex, negatively charged electron orbitals. However, if I am not mistaken, electrostatic repulsion between electrons in orbitals of adjacent atoms prevents those atoms from overlapping and thus they behave as if their mostly empty volumes were indeed full of matter, hence yielding the macroscopic effects that I can directly observe. But the point to be made is how I may be able to understand space as the context in which objects can exist; an infinite set of three-dimensional locations which may be, in effect, empty or full of matter.
Space can also provide a context for abstract things that do not physically occupy the locations where they appear; for example, distances as differences between locations. If I have a location A and a separate location B, both with respect to a common reference origin, then I also have, in my mind, a line segment joining them, describing not only the distance between both locations but also the direction from one to the other, expressed as the pair of angles the line forms in relation to a set of chosen reference axes.
I may also be able to understand the notion of time as the appreciation, in my mind, that things outside my head (as well as inside my body) are constantly changing. The relative distances and orientations between the volumes occupied by most material structures change from one instant to the next, as those objects move with respect to one another. The rate at which they move often changes too, as the moving objects accelerate and decelerate affected by different forces exerted between them. And that underlying motion conveys further changes in multiple parameters in my environment, all contributing to my overall perception of time passing. But I can trick my brain into seeing time stopped if I look at a scene that does not change, for example that depicted by a still photograph. When things don't seem to move, they don't seem to age.
Change is driven by thermodynamic factors and thus my perception of time is driven by thermodynamic considerations, subtle as they may be. In general, if my brain observes a scene during which local entropy increases, for example a glass smashing into hundreds of pieces moving outwards in all directions, then I perceive time to be progressing forward. The opposite is trickier to see in a universe where global entropy is constantly increasing, but I have had experiences in which local entropy decreased (for example when I helped restore, in the early nineties of the previous century, an old pick-up truck from the late fifties) and it did seem like time went backwards for the restored objects; the truck did end up looking and behaving like it had thirty years before. But of course, we had to put a lot of effort into the restoration.
Since processes occur over time and different events may occur at different instants separated by lapses of different durations, I can understand time as a context in which processes and events happen, very much analogous to the way objects may occupy different locations of space separated by different distances, and I can combine both notions into a complex spacetime in which objects exist and events occur, so I can refer to any location at any instant as a quadruplet where three coordinates describe distances from a reference location along three orthogonal axes, and the fourth describes the lapse between the instant under consideration and a chosen reference time. But I cannot fully consider each coordinate as a single spatiotemporal dimension, if that is a word, because spatial dimensions behave differently from the temporal dimension.
Consider a distance along a specific spatial axis; X for the sake of the argument. If I want to transform it into a distance along a different spatial axis, for example Y, I can perform a vector multiplication between my known distance X and a unit distance along the remaining Z axis and obtain the desired value. In general, I can transform any spatial dimension into any other spatial dimension by multiplying it times the remaining spatial dimension. But in order to transform a distance along any spatial axis into an interval along the temporal axis, I must first introduce another piece of information representing, in the simplest case, a rate of linear motion. Otherwise, how long does it take to cover some distance at an undefined speed? So even though I may be able to understand spacetime as a complex context in which objects exist and events happen, the separate notions of space and time still lurk underneath.
And then we get to the notion of curvature. I may be able to understand the notion of a curve when it is applied to a surface or when it is applied to a trajectory and therefore a line, but in both cases it implies a continuous and contiguous set of infinitesimally small changes in direction; be it in the location of multiple objects at the same time (when considering a curved surface) or in the location of a single object at multiple times (when considering a curved trajectory). No change in direction yields a plane or a straight line and abrupt changes in direction yield angles, but continuous, infinitesimally small changes in direction yield curves.
I think I can understand what it means for surfaces and trajectories to be curved in space, although I find it difficult to extend the notion to things curving in time, since time provides only one dimension without any orthogonal alternatives, making it difficult for me to apply the same notion of "change in direction". But what I cannot understand in any way is what it means for spacetime itself to be curved. As far as I know, time is not located anywhere in space so it does not make sense for me to think of it as curved in space, and space does not have internal constituents that may be ascribed a location, therefore those relative locations cannot be said to be arranged in any specific way; the notion of directionality is not there to evaluate if there is an infinitesimal change in it.
In other words, if I have understood correctly what has often been told to me, the presence or absence of a material structure can change the state of curvature of the surrounding spacetime; very massive objects produce large curvatures whereas lighter objects produce little curvature. It then follows that the curvature spacetime must change as the distribution of objects in it changes; presumably between a planar or linear state (in the absence of material structures) and a tightly curved state (in the presence of the most massive objects). But as the curvature of spacetime changes, what exactly changes its location, and with respect to what, so that we can talk about flatness or tight curvature? What does it mean for spacetime to be curved?
Jim Garrison kindly suggests to consider what would one of Edwin Abbott's Flatlanders experience if his world were a large sphere instead of an infinite plane. But if that were the case, in what sense would the Flatlander be a Flatlander?
I can understand the notion of a planar substrate on which a Flatlander might live. I can understand the notion of a spherical substrate on which a Spacelander might live. And I can even understand how, when only a small angle of the spherical substrate is considered, especially if the radius of the substrate is comparatively large, the degree of curvature on the surface can be so small as to make it behave, to all practical purposes, as a plane. I am a kind of Spacelander living on a kind of sphere and I can still be amazed at the apparent flatness of a pond on a windless day. I have no trouble understanding those notions because there is a substrate made of countless tiny things each of which can be ascribed a location with respect to some common reference point. Once I've got a set of things each located somewhere, I can analyse the angles formed by adjacent things and conclude whether the substrate is planar, spherical, conical, cylindrical or whatever.
My problem starts when I remove the substrate in order to consider the curvature of space(time) itself, because then I am left without things that can be ascribed a location and therefore I cannot evaluate whether they form a plane, a sphere or a hypercube. I realise how I can, in my mind, subdivide space into infinite sizeless points each of which might be ascribed a location (what does a sizeless point denote if not a location?) but, are we saying that those abstract points in my mind are physically located outside my head? Are those the things that actually move towards a more curved or towards a more planar arrangement as material structures come in and out of their vicinity?