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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?

UPDATE:

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?

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closed as too broad by AccidentalFourierTransform, StephenG, Alfred Centauri, Chris, WillO Feb 24 '18 at 1:58

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ We live on a curved surface. Try to draw a square 1 mile on a side using laser surveying equipment. Start at one corner, draw a 1 mile line due north. Turn left exactly 90° and draw a 1 mile line. Repeat twice more and you will not be exactly at your starting point, and the line going south won’t be pointing 180° either. The error is small, but gets bigger the larger the square. Or consider that you can draw a “right triangle” starting at 0°N 0°W (on the equator), up to the North Pole, then down to 0°N 90°W and back along the equator... three right angles. Extend to 3 dimensions... $\endgroup$ – Jim Garrison Feb 12 '18 at 4:41
  • $\begingroup$ @JimGarrison that seems to be a decent start to an answer... $\endgroup$ – Kyle Kanos Feb 21 '18 at 11:15
  • $\begingroup$ Thank you @JimGarrison, although I am not sure what you mean by "extend to 3 dimensions..." since both your examples are three-dimensional to begin with. Any curved surface implies three dimensions (using only two dimensions you can get a curved line or a planar surface, but not a curved surface) and your mention of using the equator as one of the sides and the north pole as the opposite vertex of a "right triangle" suggests to me a sphere, which is clearly three-dimensional. So what do you mean by "extend to 3 dimensions"? Thank you $\endgroup$ – Carvo Loco Feb 21 '18 at 20:09
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    $\begingroup$ Have you read Edwin Abbott’s book “Flatland”? If not, I strongly suggest you read it. Then consider what a Flatlander would experience if his world was the surface of a large sphere (i.e. the Earth’s surface) instead of an infinite plane. Locally it would appear flat but over large distances curious effects would be observed, such as that right triangle with 3 90° angles. $\endgroup$ – Jim Garrison Feb 22 '18 at 1:41
  • $\begingroup$ Thanks again @JimGarrison. I have updated the question with a reference to your comment and a request for further clarification, if that is not an abuse of your time. $\endgroup$ – Carvo Loco Feb 24 '18 at 0:37
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Let me just make something clear: there is a perfectly well defined mathematical notion of what it means for a space to be curved. My answer is not a substitute for a book on differential geometry and/or general relativity, just a quick overview.

Mathematically, curvature means that you can model space(time) as a (pseudo-)Riemannian manifold with nonzero Riemann tensor. There is no important difference between space and spacetime being curved; spacetime is essentially a four-dimensional space, with a few quirks that are not very relevant here. Spatial dimensions are different from the time dimension but not as much as you'd expect, so it turns out that spacetime is the best model to describe reality.

In more accessible terms, curvature means that you can draw a triangle and find that the sum of its interior angles is not 180º. It means that you can draw a circle and find that its circumference is not $2\pi$ times its radius. It means that if you try to draw a square starting from one corner, after drawing the four sides you will not return to the same place as where you started. It means that if you grab an arrow and transport it along a closed loop keeping it parallel to itself, you will not end up with the same arrow that you started with.

It doesn't really make sense to say that a dimension such as time is curved, but if you want to understand what it means, the simplest aspect is that time passes differently in different places, corresponding to the gravitational field. Since objects move in such a way as to maximize the time they measure between two events, this "position-dependent time" affects their trajectories. This is not the whole story, but in many situations it's the most important effect.

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  • $\begingroup$ Thank you Javier, I appreciate your answer. Although my question was not as much geared at obtaining a detailed but highly abstract mathematical description as it was at understanding how those abstract ideas fit in the reality I find outside my head. For example, you say that I can draw a circle and find that its circumference is not 2π times its radius, and I am almost sure I can do that in my mind by resorting to non-euclidean geometries. But how would I do that in reality? How would I draw a circle with a circumference different from 2π times its radius? $\endgroup$ – Carvo Loco Feb 24 '18 at 0:32
  • $\begingroup$ As for the curvature of time, what do you mean when you say that "objects move in such a way as to maximize the time they measure between two events"? If I drop a hammer onto the surface of the moon, what two events will the hammer take into consideration and, especially, how will the hammer measure the time between those events in order to move in such way that the measurement will be maximum? I honestly do not understand what you are trying to say. $\endgroup$ – Carvo Loco Feb 24 '18 at 0:32
  • $\begingroup$ @Carvo well, it's not like the hammer measures time. Or rather, the laws of physics don't comment on it; they just say how things are. Given that you drop the hammer from a given height at some given time and that it arrives to the ground some given time later, the path taken will make the hammer's time the greatest. It could go slow, it could go fast, it could go up and then down, it depends on the specific numbers. $\endgroup$ – Javier Feb 24 '18 at 0:59
  • $\begingroup$ In response to your first question, every circle you've ever drawn has had a circumference-diameter ratio different from pi (don't know if bigger or smaller), since you live in curved spacetime. It's just that the difference is two small to notice. $\endgroup$ – Javier Feb 24 '18 at 1:00
  • $\begingroup$ ... and since I don't know an exact value for π, and since the tools that assist me in my circle making are far from high-precision... Ok, I think I may be following you, but this raises new questions in my mind. Is that deviation from the "perfect circleness", if we can call it that way, dependant upon the orientation of the circle with respect to the structure causing the curvature? Does it matter if I draw my circle on the floor or on a wall? $\endgroup$ – Carvo Loco Feb 24 '18 at 1:27
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Carvo, these are deep questions and it is obvious you have spent a lot of time thinking about them. I offer you two paths forward.

First, rest assured that from the standpoint of mathematics, much of what you mention has been solved in the sense that the meaning of "curved space" is a settled question, and Einstein's equations of general relativity describe it to a high degree of accuracy. The experts on this site know how to work with those equations to solve real-life questions about the structure of the universe, but their labors will not be comprehensible to someone who does not grasp the mathematics. I invite them to weigh in and offer their perspectives on this here.

Second- the closest I can come to describing the meaning of "curved space" in nonmathematical terms is to say that (and this is an approximation to the underlying truth) the presence of mass or energy causes spacetime to get "bunched up" in its vicinity. That "bunching up" process is responsible for bending light beams that pass near massive objects like the Sun and for making clocks slow down in the vicinity of massive objects.

Please note that I am reasoning by analogy here, in the interests of giving you some (perhaps superficial) insight into the deeper stuff, which I cannot claim to fully understand myself, but about which I encourage others here to furnish to you their inputs.

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  • $\begingroup$ Thank you @nielsnielsen. The problem I stumble with in your argument is the "bunching up" of spacetime (and I also wonder what those double quotes mean). I can understand how things may bunch up within a region because things are located somewhere with respect to other things; if the average distance between things in a set is small, those things are bunched up. But where exactly is spacetime? Is it to my left or to my right? Is it above my head or below my feet? I do not see how spacetime can be ascribed a precise location and thus I cannot see what "bunching up" may mean when applied to it. $\endgroup$ – Carvo Loco Feb 24 '18 at 0:33
  • $\begingroup$ @carvo loco, by "bunching up" I mean in a metaphorical sense that spacetime gets denser in the vicinity of mass, and the denser it gets, the slower time runs and the more light gets bent when passing through such a region. So if we imagine spacetime to be represented by a uniform 3-dimensional grid of lines, the spacing between the gridlines gets progressively squashed the closer you get to the central mass. This is the best I can do without getting into the math, which is really horrendous (and too much for me in fact!). $\endgroup$ – niels nielsen Feb 24 '18 at 4:45
  • $\begingroup$ In order for me to understand the notion of gridline density, I need to think of gridlines as things separated by distances of non-gridline; only then I can look at those "empty" distances and judge whether the gridlines are squashed (short distances separating adjacent gridlines) or not squashed (large distances separating adjacent gridlines). But if I had infinite gridlines each immediately adjacent to the next leaving no gaps inbetween, then I wouldn't be able to determine any variations in the gridline density because the grid would be infinitely dense at every point. (Cont'd) $\endgroup$ – Carvo Loco Mar 1 '18 at 1:10
  • $\begingroup$ (Cont.) Are you suggesting that spacetime conforms some sort of units separated by distances of non-spacetime, so that those distances of non-spacetime between the units of spacetime can become shorter or larger in the presence or absence of a nearby mass? How can we talk about the spacing between things, when the things considered are themselves space? $\endgroup$ – Carvo Loco Mar 1 '18 at 1:10
  • $\begingroup$ no, that is not what I am suggesting. have you consulted any of the standard references on this topic? $\endgroup$ – niels nielsen Mar 1 '18 at 2:41

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