Martin Perl, the author mentioned in the paper, was a Nobel prize winner and one of the first people I met when I got to SLAC. His office was across the hall from mine. Unfortunately, Martin passed away this year. He was a very nice, smart, and humble man.
This cleared up that question for me. Noether showed that every physical symmetry implies a conservation law and vice versa, the symmetry of time flow and the conservation of energy are one such pair. Due to the fact that time is not flowing uniformly (Relativity) energy is not conserved.
The converse of Noether's theorem does not turn out to be true in general. Noether showed that if there is a symmetry in the system then there is a corresponding conservation law. But the existence of a conserved quantity does not (in general) imply the existence of some symmetry in the physical system.
(I should note that there are more careful ways to get something that looks like the converse of Noether's theorem, but the literature is fairly complicated -- you have to distinguish between trivial conservation laws and non-trivial conservation laws. Here's one answer online: http://physics.stackexchange.com/questions/24596/is-the-conv...)
When the Universe is taken as a whole do the changes of gravitational energy in spacetime exactly cancel out the changes of matter energy (including dark matter) due to the expansion of spacetime? How do we know (is there some mathematical reason to believe this is the case)?
When the author says that spacetime absorbs matter energy or gives matter energy, what does that really mean? Does it mean it can be used as a convenient sink or well in regards to changes of potential energy? What are the rules governing that transfer or do we just make things balance and decide that spacetime did it?
I am trying to shake this uncomfortable feeling that it is too convenient. I'd love to know the ground those kinds of operations stand on.
If energy is not conserved across distortions in space time, then neither is momentum. That em-drive people have been testing is forcing a distortion in a resonating wave that should just be varying the flux density from one end to the other, it it looks the epitome of something that should just sit there and do nothing. It's a lopsided microwave oven. But that could make a pinch in spacetime that isn't completely symmetrical at the neck, so you end up with a sort of spacetime ramjet.
Probably not, but it would be nice.
edit - this nonsensical waffle was brought to you today by too much coffee, not enough sleep, and the number 7.
Well, we know that momentum - and energy - are only conserved across a long enough timeframe. I'm working from 30 year memories year (special relativity class during my undergrad) so bear with me....
Consider a charged object moving at high speed (nearing c) toward an object with the same charge, but on a path to pass "beside it" (it doesn't really matter by how much, so long as the one will not hit the other); treat the second object as a stationary reference frame, to make this conceptually easier.
At any given time, there is force of repulsion between the two objects. Normally (relative speed much, much less than c), this force can be approximated as an r-squared force acting along the line separating the two objects.
So far so good. No problems with conservation of anything here, all the vectors in the various force diagrams add up just fine.
But the objects are moving relative to each other at speed nearing the speed of light, and the repulsive force is transmitted at the speed of light...
...which means that at any given instant, the force exerted on one particle by the other is parallel not to the line drawn from one to the other at the moment the force is exerted (at the moment it arrives) but parallel to that line from a few moments ago.
At any given time, when the objects are close enough to one another, the forces acting on each object are an angle to the line drawn from one the other.
In order to have conservation of anything, one has to make a vector integration over a sufficiently long time frame ("long enough" before and "long enough" after the collision, that is, the moment the incoming object passes by or begins moving away from the other).
Otherwise, things just don't add up.
Like I said, I am casting my mind back to a brain-popping moment from 30 years ago in which we sat there slack jawed, eyes agape, and a little aghast, starting at the blackboard while the prof stood there with a Cheshire Cat grin tossing his chalk up and down, waiting for us to accept - or at least merely acknowledge - that we weren't in Kansas anymore, that the universe was rabbit holes all the way down, and that we'd been lied to since junior high.
> momentum - and energy - are only conserved across a long enough timeframe
> In order to have conservation of anything, one has to make a vector integration over a sufficiently long time frame
My understanding was that the time frame was strictly a matter of the Heisenberg Uncertainty Principle -- if the time is less than specified by HUP for the situation, then conservation generally does not hold, but if time is greater than that, then conservation is strictly held.
The simple model of virtual particles involved in Feynman diagrams is that they are precisely the ones that take time less than HUP.
Unless you're talking about relativistic effects where adjustments are needed, but after those adjustments, it's back to the above again.
To some extent, but mostly this is long-explored territory with few dragons.
It's not often put this way, but you could say that the virtual particles < HUP establish the wavefunctions that have effects fully evaluated by the time of > HUP, such as the phase cancellations that cause light to take the path of least time, which is otherwise puzzling.
I could be wrong, as I'm a bit rusty on this topic as well, but I was under the impression that the electric field was only distorted by relativistic effects if one of the electrons is actually accelerating. If it is just zipping along constantly at its high-fraction-of-c then I believe its corresponding field appears to be doing the same thing _for all observers within the radius of the radiated EM field that was created by the acceleration of the electron[1]_
Or have I made a relativity-noob mistake and what I said is only true in the frame of reference of the moving electron?
[1] Sorry for not having a better way to describe this concept. If it has a name (maybe light cone? Not sure) I'd love to know.
This is more or less the idea behind the Alcubierre drive [1]. If you have some source of exotic matter that has negative energy density it is possible to construct solutions to Einstein's field equations that permit faster-than-light travel by essentially bending the surrounding spacetime around the spaceship.
That is actually not a drive, since it is not (and, given the known laws of physics, cannot be) mounted on the ship it propels.
So lotsofmangos's idea is not quite the same, as it can be on-ship (although trying to use it as a FTL drive would either require infinite energy or rip the ship apart due to tidal forces (or, more likely, both).
This is interesting, and I would like to find out more about it. That equation with the Del & T is meaningless to me, since I don't know what the letters stand for. A search for "energy-momentum conservation" turns up lots of college physics labs about energy conservation and momentum conservation.
Can anyone help me find out more?
Also, I have read that Newton's three laws of motion, if suitably formulated so that they are meaningful in a non-euclidean space, and being careful to say "derivative of momentum", NOT "mass times derivative of velocity", continue to be exactly right, as far as modern physics is aware. But since Newton's laws essentially say, "momentum is conserved", that would not be true if this article is correct (right)?
