Time and space in general relativity

[brotherbobby]

(1) I remember reading somewhere that in general relativity, "space" and "time" lose their metrical meanings. Is that true? And yet, we continue talking of space and time in general relativity as spacetime.
(2) Moreover, as someone mentioned in this thread, what happens to the speed of light? In Einstein's 1912 papers (a and b), he abandoned lorentz invariance and made time spatially dependent, $c(x, y, z)$. So the speed of light is no longer an invariant in (the later) general relativity for coordinate systems. But worse, if space and time have lost their metrical meanings, assuming (1) above is true, how can an observer even measure the speed of light? Locally? Does space and time continue to have their meanings locally secure?
(3) Philosophically, focus has now shifted to matter and fields - see Einstein's fifth appendix in his famous expository book on the subject : Relativity - the special and the general theory. I did not understand it and am even baffled by it. I understand that space and time have been linked to matter, energy, pressure and even electromagnetic energy and momentum (via the two-index tensor). Does it mean however that space and time are mere illusions of something more physically fundamental (like the examples above)? But surely we can see space and time. How about an empty universe, with no matter and energy? Is general relativity saying that if matter and energy would be absent, so would space and time, implying there would also not be a universe at all?

 

[Nugatory]

I remember reading somewhere that in general relativity, "space" and "time" lose their metrical meanings. Is that true?

Not really. It is true that we no longer can construct global inertial reference frames in which an observer at rest in a given has a natural universe-wide "time" and "space" that works out to infinity. However, the SR notions of "time" and "space" are special cases (as is everything in special relativity - that's why we call it that) of the more general notion of timelike and spacelike intervals, which work just fine in GR. The concept you're looking for is "foliation".
So no, your premise is not true, at least not in a way that would lead to the your later concerns and conclusions.

see Einstein's fifth appendix in his famous expository book on the subject : Relativity - the special and the general theory.

A cautionary note: You should not be trying to learn relativity from that book, for about the same reasons that we don't teach classical mechanics from Newton's Principiae.

 

[silverrahul]

A cautionary note: You should not be trying to learn relativity from that book, for about the same reasons that we don't teach classical mechanics from Newton's Principiae.

Why is this ? Is this because, it is written in too much complicated, complex, old fashioned way or are some of what it says were actually found out to be wrong later?

 

[jbriggs444]

Why is this ? Is this because, it is written in too much complicated, complex, old fashioned way or are some of what it says were actually found out to be wrong later?

Yes, the former pretty much. It is written in an old fashioned way and for a different audience. With decades and even centuries of experience, it often turns out that there are different ways to motivate the same resulting theory. More fruitful intuitions to create, more helpful symmetries to explore.

Discovering and teaching are different things.