I’m afraid I’m to blame for introducing the word “parallel” into this conversation (albeit in scare quotes, let it be noted), which has unfortunately caused all manner of confusion. Perhaps it would be better, considering the various different definitions people have for “parallel” which diverge or become incoherent as one leaves the Euclidean context, and the problems this ambiguity seems to be causing, to refrain from using it and speak only of distances, directions, intersections, etc., making more explicit what is meant in any particular employment of the word.
Not worrying about general relativity in its entirety yet, just working on small things at a time instead, what exactly is it that you think can’t exist?
A universe in which non-Euclidian actually describes something real. A universe in which time and space are two faces of the same coin. Also, let’s say a curve in space is because it’s being carved out of a four dimension entity: space/time. That’s what I assume you mean by embedding in this case. But, again, while I believe that time is a forth dimension, as I understand it, it is both one way and purely linear, so I don’t understand how embedding applies.
I’m for the most part considered quite bright for three reasons. I have a good memory for ridiculous things even though I can’t remember yesterday’s dinner, I am very good at if-then-else type logic - syllogisms and so-forth -, and I can create or understand most analogies very quickly.
But I don’t understand any of the analogies presented here. I mean, I could under some circumstances, but operating under my notion of space - effectively limitless and mostly empty, I can’t understang why gravity simply attracting objects doesn’t work as well as postulating that the space it’s in itself curved. It seems so overly complicated, so non-sensical. Doesn’t mean that it doesn’t fit the facts; my problem is a lack of a gut level sense, not an intellectual belief.
FWIW, I don’t really believe in dark matter either. The idea of Something that suggests that 80-90% of all the mass in the universe is stuff we have no way whatsover of sensing using any tools whatsoever seems to me to be much less likely than that somewhere along the way, there’s been a snowballing error in calculations
Indistinguishable, for starters, the simple fact that space doesn’t act like a grativity free (from the outside, such as in space), sealed off swimming pool and everything sucked out of it until it’s a vacuum, then plopping in (at high and erratic speeds and directions) a model “Sun” and “planets”, comets, and so-forth.
In fact, I logged in just now to ask this very question.
Above, Mathochist said in passing:
Do calculations come out "more accurate"ly specifically as a result of treating gravitation as the curvature of spacetime rather than as an attractive force? Or rather is it simply more computationally efficient to do so or something? Or what?
-FrL-
The Einstein equation covers more phenomena in a way. One, in particular, is called “frame dragging”. It describes how a rotating gravitating body tends to actually wrap spacetime around with itself as it turns.
The classic example of this is the precession of Mercury. Mercury’s orbit is actually an ellipse, and so it has a “perihelion” where it’s closest to the Sun, and an “aphelion” where it’s farthest (we just past our own aphelion, actually). But the ellipse doesn’t quite close up on itself. If you could sit directly above the Sun and watch Mercury’s ellipse, it would slowly turn around the sun. The perihelion point creeps forward about 56 seconds of arc per year (60 seconds in a minute, 60 minutes in a degree, 360 degrees in a full circle) or 5600 per century.
We knew for a long time ways that this sort of thing could happen. But all the ways classical mechanics talked about – from the “precession of the equinoxes”, to the gravitational effects of all the other planets, to the tiny amount by which the Sun isn’t quite a perfect sphere – all added up to 5557 seconds per century. There’s a shortfall of 43 seconds. People guessed all sorts of things, like another planet inside the orbit of Mercury, but nothing really panned out.
But when you do the calculations in GR, you realize that the Sun is massive. It’s so huge and heavy that as it turns, it actually drags spacetime itself around with it, and since Mercury is in spacetime (you see where I’m going with this), Mercury gets dragged around the Sun by 43 arcseconds every century. This is in addition to the amount by which Mercury’s orbit moves through spacetime due to Newtonian effects.
So that’s just one concretely measurable way that GR’s model of gravitation-as-geometry supersedes the Newtonian model of “action at a distance”.
Curvatures in the sense that GR uses them consist of matrices with ten independent components, but most of them are almost always incredibly tiny in situations like those we see within the solar system. This is Important because these ten quantities are the variables in a system of highly nonlinear (and thus highly complicated) dfferential equations. If you just say that the small quantities are exactly zero, it simplifies the equations a lot, and the real solutions to the real equations won’t differ very much from the solutions to the simplified equations. And the simplified equations give back exactly Newtonian gravitation.
So under normal circumstances the difference between GR and Newtonian mechanics is vanishingly small. When you have an object as heavy and spinning as fast as the Sun, and you’re probing its gravitational field with an object as light and as close as Mercury, you still only get a discrepancy of 43 arcseconds per century. That’s how small the difference is within our solar system.
We may as well just use Newtonian mechanics to do our calculations, and in fact we do. All of our interplanetary satellites wing their way from moon to moon using nothing more than classical mechanics. But when we want to understand the life cycle of a star, or how stars zoom around in the core of a globular cluster, or how they form into galaxies… then we need the additional terms of GR, which become just as important as the contributions from classical mechanics in these situations.
Ya know, maybe they should start teaching relatavistic physics to little kids rather than Newtonian. I mean, when you get down to it, why should it be any more plausible that two objects are attracted to one another than that space is curved? They’re both counter-intuitive ideas. But because I’ve known about gravity all my life, I accept it without question. Perhaps if I’d been told that it happened because of magic space curvature instead of magic object attraction, I’d be able to accept it better.