4 ms·

Sprinkle in some "ds" in your terms in > (since the distance metric is x²+y²+z²-ct²) and you just have the line-element for flat (Minkowski) spacetime in Mink

Sprinkle in some "ds" in your terms in > (since the distance metric is x²+y²+z²-ct²) and you just have the line-element for flat (Minkowski) spacetime in Minkowski coordinates. Change to polar coordinates, and you have to write the line-element down differently (even in SR). Change from flat spacetime and you have to write the line-element down differently. Change the metric signature for convenience from +++- to ---+, and you have to write the line-element down differently. The line-element depends on a choice of how one is putting down coordinates on a spacetime, and the choice of the spacetime itself. Handily, we can turn to the lecture notes linked at the very top of this discussion. My paragraph above is essentially the content of §2.1.1 (Basic concepts, Manifolds and Metrics) of the lecture notes linked at the top, but I guess the section is just not basic enough to have found its way into this thread. See eq (2.3) for where to stuff in the "d"s and the paragraph after eq (2.1) about why they're ds rather than ∆s (or nothings). Also see equations (5.1) and (5.2), (5.41, footnote 57) for examples of line-elements for non-flat spacetimes; eqns (5.6), (5.35), (6.1) for more exotic line-elements; and (5.37-5.40) for examples of 1+1 line-elements. See footnote 10 for a pithy comment on choice of metric signature. ("Some people." Gee, thanks. cf. footnote 12.) > In SR/GR we also see time as separate from space is, unfortunately, almost entirely incorrect. The closest we can get to repairing this notion is by a procedure such as slicing or threading and introducing operators to knit the resulting hypervolumes together in some orderly way. This is discussed to an extent in §§5.1-5.2, or in most numerical relativity textbooks (like, especially given the content of §5 of the lecture notes linked at the top, Miguel Alcubierre's excellent introductory textbook! ISBN-13: 978-0199656158). I'm sorry to snipe at you, but this comes after reading the discussion here in which you assert and then retract numerous statements about the content of relativity theory. All that sure has confused me, and I'm sure I'm not alone.

tsimionescu 5y ago
I have no problem with the snipe, I was indeed confused on some items (particularly, I stupidly claimed that different observers can disagree on the order of events that are part of some event's light cone). You're right of course about the missing ds or deltas in the Minkowski metric, and there's also a missing square, since that is the square of the 4-distance, not the 4-distance. However, on the basic point that Minkowski spacetime is different from a Cartesian 4-dimensional space, and thus time can be seen as fundamentally different in nature than the space coordinates, in not sure why you claim I am mistaken. Maybe the specific wording I chose, "separate", was the issue? Time and space are of course tied together in SR&GR, whereas they are more firmly separated in Newtonian mechanics. But "separate" can also mean "different", and I think it is corect to say they are still different things even in GR, though they are tied to each other.

tsimionescu 5y ago

I have no problem with the snipe, I was indeed confused on some items (particularly, I stupidly claimed that different observers can disagree on the order of events that are part of some event's light cone). You're right of course about the missing ds or deltas in the Minkowski metric, and there's also a missing square, since that is the square of the 4-distance, not the 4-distance. However, on the basic point that Minkowski spacetime is different from a Cartesian 4-dimensional space, and thus time can be seen as fundamentally different in nature than the space coordinates, in not sure why you claim I am mistaken. Maybe the specific wording I chose, "separate", was the issue? Time and space are of course tied together in SR&GR, whereas they are more firmly separated in Newtonian mechanics. But "separate" can also mean "different", and I think it is corect to say they are still different things even in GR, though they are tied to each other.