What does spacetime curvature means?
That's maybe a bit too general of question to answer here. You'll get better answers if you Google around for "gravity curvature" and see what you find, read through it, then try some more specific questions here.
And while you're Googling, make sure you understand geodesics.
The concept of space-time was introduced due to the way the Lorentz transform works. One need to consider space and time together, as a unit, rather than as separate individual entities. Specifically in relativity, simultaneity is relative and depends on the spatial motion of the observer.
Space-time can be visualized through the space-time diagram of relativity, with the understanding that the coordinates used to describe space and time on the diagram transform via the Lorentz transform and not the way one is used to.
It's hard to visualize curved space-time, but it's not that hard to envision a curved surface. The mathematical details needed to handle space-time curvature properly for general relativity can get advanced, for instance there are multiple sorts of curvature. But hopefully one has an intuitive idea that the surface of a globe has a different geometry than that of a plane, the former being curved, the later being flat. IT's also worth noting that some types of curvature affect the accuracy of flat maps, for instance flat maps can represent only small portions of the globe "to scale", any large flat map will not accurately represent angles and distances.
Combining space-time and curvature, into "space-time curvature", the most simple intuitive idea is drawing your space-time diagrams on curved surfaces. While this could work in principle, it would be impractical, so instead the theory focuses on the mathematics of correcting for the way distances and angles are distorted by using flat maps. The mathematical construct that is used to find the actual distances in the presence of the distortion of curved geometry (it can be regarded as the fundamental way of changing coordinates into distances that always works, curvature or no curvature) is known as the metric.
Spacetime curvature is the way to model tidal gravitational effects.
I would agree that that's the most important effect, but I wouldn't use it as a definition because there are other effects of curvature that don't fall into that description.
Specifically, one can decompose the Riemann tensor which describes curvature into the Electrogravitic part, which is what you describe, the Magnetogravitic part, which represents frame-dragging effects (which don't, as far as I know, have an interpretation as tidal effects), and the Topogravitic part, which has a reasonable interpreation as "curved space", but no interpretation in terms of tidal effects that I'm aware of.
Besides not being complete, I also think it's also not terribly clear to those who aren't already familiar with the theory why we call tidal forces space-time curvature rather than tidal forces.
The result does follow logically from the idea of drawing space-time diagrams on curved surfaces, as I'm sure you already know, I suppose that my idea is that the best way to present the topic is to talk about representing space-time by space-time diagrams, then applying the notion of curvature to talk about drawing space-time diagrams on curved surfaces, and finally bringing in the idea of how the result looks like tidal forces last, as a consequence of geodesic deviation. But this full presentation seems to long for presentation in a single post to me :(. Perhaps there is some less-longwinded way to explain why "tidal forces" arise naturally from a consideration of curved space-time, but I don't see any shortcuts.
Another issue, I suspect, is explaning why explaining tidal forces is sufficient to explain gravity (not only sufficient, but better, because tidal forces are a part of a tensor).
For instance, I've noticed resistance and a lack of satisfaction when I try to explain the "gravity" of a moving body by explaining just its tidal forces.
It depends on how you interpret the word "tidal". MTW, for example, gives a very general interpretation: tidal gravity is present whenever initially parallel geodesics don't remain parallel. This, of course, is precisely the definition of curvature. (Kip Thorne also takes this point of view in Black Holes And Time Warps.)
They do under the general definition given above: frame dragging is just a way in which initially parallel geodesics don't remain parallel.
Under the general interpretation above, it does. But this part does raise an issue: the curvature described is purely space curvature (i.e., it refers to spacelike geodesics, not timelike or null geodesics, which is what the electrogravitic and magnetogravitic parts refer to). I can see why "tidal effects" does not intuitively seem like a good description of pure space curvature, since "tidal" implies some sort of change with time.
I think this is because "gravity" is a more general term than "tidal gravity"; it's only tidal gravity that directly corresponds to spacetime curvature. It's true that for the other effects of "gravity" to be present, tidal gravity has to be present, but I can see why that would not be obvious to someone who isn't familiar with GR.
Space can be described by euclidean geometry, but the einstienian geometry of space and time is something that does not provide any physical picture like euclidean does, i think it is imagination of albert einstein where thought is more important thus lacking pureness of geometry and mathematics.
General Relativity is geometry described mathematically.
Can You explain how?
Ok. That would take some time to understand but did einstein proposed spacetime curvature or it is an outcome discovered later.
the spacetime curvature was not born from Einstein. It's born from a manifold's treatment. As such it was developed way before Einstein (the works of Gauss for example, or an even stronger name for curvature: Riemmann). If spacetime was a manifold (since Lorentz's times), I don't see why people wouldn't try to generalize it (allow the metric to depend on spacetime points) and try to find what is there... Einstein was able to connect gravity with it in a physical way, while Hilbert achieved the same thing in a more delicate way some days later...
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