There are some people who believe that gravity is repulsive and not attractive: Gravity is a Push. Also see here, with the great quote
Nope. If you set up the differential equations describing “straight” lines (ones which always move in the direction they’re pointing on the curved surface, the solutions form spirals on that surface.
Think about two great circles on a globe, the straightest lines you can get on the surface of a sphere. Now draw those curves on a Mercator projection map. They appear to move towards each other and then away cyclically. If you imagine one as a fixed reference point, you can see the other as accelerating towards it, overshooting, and coming back. If you tried describing particles that moved like that without knowing that the Earth was really curved, you’d probably posit some sort of attractive force between the two.
The rubber sheet is dented because:
Mass-energy density (which are the same thing according to Special Relativity) needs to be broken into 16 pieces to get something which looks the same to all observers in spacetime. This is similar to the way a velocity vector in three-dimensional space needs three components, even though one observer may see motion only in his x-direction.
This 4x4 matrix is “conserved”, meaning that the amount flowing into any given region of spacetime equals the amount flowing out.
General Relativity posits that gravitational attraction is explained by the way initially parallel geodesics (straight lines) move with respect to each other in a curved spacetime. This curvature shows up in the fact that the “dot product” formula for two vectors at a point varies as the point varies. This formula is encoded in the “metric” g.
The only 4x4 matrices made up of the components of g and their derivatives (up to second order) which look the same to all observers and are conserved in the same sense that mass-energy density should be are linear combinations of g itself and the “Einstein tensor” G.
Thus the Einstein equation, T = aG + bg, stating that mass-energy is the same thing as some description of curvature, results.
Gravitrons, in theory. I don’t know anything more about them, except for the fact that they’re not universally recognized by all scientists, and that they were fleetingly mentioned by my physics teacher 3 years ago.
Yes, I understand the geometry of curved surfaces (well, some of it anyway). And I’ll accept that the geodesic curve for particles on that particular surface, when they’re “launched” like the coins are usually launched, would be a spiral of doom into the center hole. I think this is what you’re saying.
However, that can’t be the whole story for the actual physical systems — the actual coins rolling on plastic funnels you see in the mall. For one thing, that would imply that if the Hand of God were to come down and instantly reverse a coin’s velocity vector at some point in its doom spiral, that the coin would switch to spiraling upward and outward, away from the center. Were there no friction, perhaps it would do that, at least up to the height and radius where the coin’s kinetic energy was completely converted into potential energy, where its speed would approach zero and it wouldn’t move upward any further.
Surely though, on the real systems, sustained upward motion isn’t going to happen for any appreciable length of time (where I’m defining “appreciable” in such a way as to favor my case the best, naturally). I don’t have any measurements to back up that claim, admittedly; it just seems physically unrealistic.
Yes, and that in this system the energy lost to friction is negligible. Yeah, it’s not exactly a geodesic, but it’s close enough.
In the rubber sheet model, you can try drawing the geodesics and see that the dent makes a line coming in along one direction go out along another. Remember that my real point in bringing up the coin vortex was to back up the real sense of the rubber sheet model.
Extended nitpick on the coin-vortex tangent:
It’s possible that there are inspiraling geodesics on those surfaces (I haven’t worked it out, so I don’t know either way), but that’s not precisely relevant to this problem. (It is approximately relevant, as I explain below.) A particle, sliding without friction on one of these surfaces [I’m ignoring the angular momentum of the rolling coin here] and constrained to stay on the surface, but acted on by no other force, will always follow one of these “intrinsic surface geodesics.” (By this I just mean the geodesics which are defined by the spatial metric induced by the embedding of the 2-manifold within Euclidean 3-space. I’m using this term to differentiate between these geodesics and a different geodesic I’m talking about below. Note that if there are inspiraling geodesics, then there are also outspiraling ones (since the surface is rotationally symmetric) as well as closed orbits.)
But on Earth the particle or coin is being acted on not only by the surface’s normal force, but also by gravity (with a force, acting along the inward tangent, of mg sin(theta) when the surface is at an angle theta from the horizontal). This will cause the trajectories to curve inward relative to the intrinsic geodesics of the surface. How much curvature occurs depends on the particle’s speed: a rapidly-moving particle will follow the intrinsic geodesic fairly closely, while a slowly-moving particle will curve inward more.
It is possible – and this is what is done in general relativity – to define spacetime geodesics for this system (here in two spatial dimensions and one time dimension), such that a particle sliding without friction on this surface under gravity will follow a spacetime geodesic. These geodesics are somewhat more complicated than the intrinsic geodesics, having two parameters instead of one. (That is, an intrinsic geodesic containing a point P is fully defined by the direction it takes leaving P. A spacetime geodesic is fully defined by the direction and speed it takes leaving P. This is not too surprising; since spacetime geodesics exist in three dimensions rather than two, it takes one further parameter to constrain the geodesic.) In particular, this is enough free parameters that there always exist closed circular orbits, at any radius (ignoring relativistic considerations :)).
So a rapidly-moving particle will approximately follow the intrinsic geodesics. You can test how closely the coins follow geodesics by trying to launch them with different speeds (though IIRC they usually come with a launch ramp, to try to impose somewhat-uniform initial conditions).
Say, science can never totally “explain” anything. Science describes as best it can. Deeper and deeper, with more and more elaborate aspects, and more and more specificity, and replete with arcane formulae, it none the less ultimately describes what we believe about the way nature is. Consequently, the question “Why?” can result in only two responses: 1) a description of our understanding of the nature of the universe, and 2) the larger and ultimate one: That’s the way it is.
xo, C.
IANAP, but my understanding is that the inability to figure out how and/or why gravity works is one the greatest frustrations of physicists. Richard Feynman, for example, used to hate going to gravity conferences, because there were always lots of words but no progress.
As to gravitrons, IIRC, that was merely a proposal by analogy to photons, but has never made headway as an explanation, much less developed into a full-fledged theory.
Gravity attracts for the same reason 2 +2 = 4. “It” just does.