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Talk:Counterexamples to Relativity (view source)
Revision as of 04:03, August 5, 2010
, 04:03, August 5, 2010→Curl of the gravitational field
:So, if I understand you properly, gravity both (a) curves spacetime and (b) creates a conservative field. (I derived (a) from your point: "You get into complicated issues of curved coordinate systems (you're in one now; it's what you perceive as "gravity"!)", and (b) from your point: "the gravitational field, under either Newtonian or relativistic mechanics, is a conservative field. Its curl is zero.") However, you're saying that if you are in an accelerating reference frame (such as a quickly-spinning merry-go-round) only (a) occurs; there is no field. Is this correct? If so, why is there this discrepancy between the two? [[User:PhyllisS|PhyllisS]] 22:54, 4 August 2010 (EDT)
:So, if I understand you properly, gravity both (a) curves spacetime and (b) creates a conservative field. (I derived (a) from your point: "You get into complicated issues of curved coordinate systems (you're in one now; it's what you perceive as "gravity"!)", and (b) from your point: "the gravitational field, under either Newtonian or relativistic mechanics, is a conservative field. Its curl is zero.") However, you're saying that if you are in an accelerating reference frame (such as a quickly-spinning merry-go-round) only (a) occurs; there is no field. Is this correct? If so, why is there this discrepancy between the two? [[User:PhyllisS|PhyllisS]] 22:54, 4 August 2010 (EDT)
Yes, gravity curves spacetime. It looks as though you're ready to go to the next level. It has to do with curved ''coordinate systems'' vs. curved ''spacetime''. This is why GR is so complicated. But here goes.
First, we have to recognize that, at some level, we could say that energy is *not* conserved. Spacecraft use "slingshot maneuvers" around one planet to gain extra energy on their way to another planet. I assume you've heard of this. Cassini used three such maneuvers, twice around Venus and once around Earth. So what was going on? If you look at a coordinate system centered on Venus, you would see Cassini come in and go out again, with complete conservation of energy. But, in a coordinate system fixed around the solar system, Venus was moving, so Cassini came in at low speed and went out at high speed. We "stole" some energy out of Venus's orbit. So, to be ''really'' correct, we have to say that gravity is a conservative field in the absence of '''moving''' gravitating bodies.
But, in the larger sense, energy is conserved. Always. Newtonian or relativistic. (But in relativity, the mass figures into the equation. Let's not worry about that just now.)
Now we get to the really cool stuff. Your comment above suggests that you are ready for it.
You will feel a "fictitious force" whenever you are in a "curved spacetime coordinate system". A curved coordinate system would include things like polar coordinates on the plane, or spherical coordinates in 3 dimensions. But this is 4-dimensional spacetime. So I'd like you to take my word for this. On a rotating merry-go-round, your 4-dimensional spacetime coordinate system is curved. (This doesn't require relativity, special or general, to formulate this.) The 3-dimensional slice of it is flat, but, when you bring in time, and the spatial coordinates are accelerating, the overall coordinate system is curved. This creates fictitious forces. In the case of the merry-go-round, the forces are the centrifugal force and the Coriolis force. There are also the fictitious "acceleration G forces" in a rocket. But '''the spacetime itself isn't curved'''. It's like polar coordinates on the plane. Yes, the coordinate system is curved, but the plane isn't. There are Cartesian coordinates on the plane also. Similarly, an observer on the ground is in a flat coordinate system, and doesn't see any fictitious forces. He just sees the mechanism of the ride pushing you inward as you go around. Your recoil against that acceleration is what you perceive as the centrifugal force.
So the moral of the story would seem to be: you can choose a flat coordinate system that exposes the fictitious force for what it is. It's a perception from the curved spacetime coordinate system that the observer is operating in.
But how about gravity? In the case of gravity, '''spacetime itself is curved'''. If spacetime (properly called a "manifold") is curved, every coordinate system is curved, and the fictitious force which is gravity is inescapable.
So your statement "gravity curves spacetime", is exactly correct, but you have to distinguish "curving spacetime" and "curving some particular coordinate system". The actual curvature involves things called "Riemann's tensor" and "Ricci's tensor".
The number of people that understand this is way more than the 3 that Arthur Eddington claimed (it was more than 3 but probably less than 10 at the time), but it's still a pretty complicated subject.
[[User:Simeon|Simeon]] 00:03, 5 August 2010 (EDT)