I'd like to do a little bit of classical mechanics, but the particular thing I want to do is a little hefty for one post. We'll split it in two. Today I'll set the stage and tomorrow we'll use it to solve an interesting problem.
The problem involves the orbit of a planet in a gravitational field, and fortunately the initial approach to the problem is not complicated. First we write everything in terms of energy. The total energy of a system is the sum of its kinetic and potential energy. Kinetic energy is the energy associated with motion, and potential energy is (in the case of orbits) the energy associated with its particular position in the gravitational field. We'll write the kinetic energy in terms of polar coordinates, and then write that in terms of the angular momentum L. I'm skipping the details, since while interesting they're not vital at the moment. If you're curious they're in any mechanics textbook (I first learned mechanics from Thornton and Marion).
Some explanation: r is the distance from the center to the oribiting body (say, from the sun to the earth), r-dot is the rate of change of r, L is the angular momentum (roughly, a measure of the mass of the earth and the speed of its orbit), and U is the gravitational potential energy. We can solve this for r-dot if we want:
Now why are we doing this? The orbital dynamics of stars and planets are well-known, why should we start from the very beginning? The reason I'm starting here is that this basic equation doesn't care about the form of the potential energy. We know that in classical gravity it's proportional to 1/r, but for all we know it might have been something else. And if it were, what might the universe have been like? Perhaps orbits wouldn't be stable, or perhaps they wouldn't have been possible in the first place. Maybe there wouldn't have been any such thing as escape velocity, or maybe the relationship between distance and orbital speed would have been different.
While it's an interesting philosophical bull-session question, there is some relevance. Looking at what laws produce what results can tell us something, even if we think it's purely speculative. Small differences between the predictions of Newton's 1/r potential and the actual orbit of Mercury helped spur the development of relativity, and today there's some (admittedly dubious) speculation that the laws may need to be further modified to account for the lack of success of dark matter searches.
I don't know how that will turn out, but I do know that mathematical physics has got the tools to make predictions from it. Counterfactual "what if" scenarios don't always stay counterfactual forever.
In looking at things this way, in terms of energy, you're still living within the traditional paradigm of modern (i.e. late 19th century) classical mechanics. As with elementary particles, it amounts to looking for symmetry, and (implicitly) that the symmetry is simple enough that we can guess it.
An alternative is that it is the equations of motion that are simple. If so, then we need to look at how to modify the equations of motion of general relativity in a simple way. This was the topic of my paper that got an honorable mention at the gravity essay contest and is likely to be my first published paper.
If I may, a general physics question, one I've had for a while:
I know that, as Einstein says, time slows for an object near a gravitational field. Meanwhile, its motion through space becomes more rapid as the body falls, or to put it another way, a falling body gives up some of its time-ward motion in exchange for more space-ward motion. It seems to me, therefore, that gravitational potential energy can be described as the energy that a body has on account of moving forward in time (as we all do at one second per second from our own reference frames).
However, I've never heard anyone actually describe gravitational potential energy in those terms, so I'm wondering if there's some reason why it's not a good or accurate way to think about it.
I shall observe that Marion's textbook was excellent, and Thornton only made it better.
(It is not always the case that the substitute author improves the product by editing it.)
Adrian: I'm not enough of an expert at GR to rule out your idea, but my instincts tell me there are (at least) a couple of potential trouble spots for your idea. One is that the effect of an artificial acceleration has to be indistinguishable from the effect of gravitational acceleration to the object undergoing the acceleration, and it is not obvious that your idea will satisfy this criterion. The other is that from the point of view of the observer, time does not become infinitely slow. If you're the one falling through the event horizon (and you somehow survive the trip), you will see that you get there in finite time. Your friends who are watching will never actually see you get there, since the time as measured in their frame increases without bound. Suppose you send out signals at fixed intervals in your frame: due to gravitational redshift, those signals are detected at ever longer intervals in the frame your friends are in.
Better than theory succeeding is good theory failing. All the fun is in the footnotes.
Teleparallelism defaults to General Relativity if the Equivalence Principle is true. Only the disjoint non-overlap is interesting. Grant funding supports observing zero-risk overlaps but excludes observing falsifications lest the anomaly not be observed - or be observed. The Church is winning not by impressing its dogma but by inculcating its methods of "revealed authority."
Do local left and right shoes vacuum free fall identically? Somebody should look. It does not occur naturally as astronomic bodies. Earth's L-configuration chiral protein amino acids (meat) are cancelled by D-configuration chiral sugars (wood) re a parity Nordtvedt effect.
I think describing modifications to general relativity being necessary to 'account for the lack of success in dark matter searches' is a little a backwards. Modifications to our current understanding of gravity based on general relativity are (possibly) required because this current understanding utterly and epically fails to describe the large-scale structure of the universe. Dark matter is just the most plausible bit of hand waving to account for this failure. Throw in the pioneer and flyby anamolies, and it seems notions about modifying the current 'laws' deserve better than dismissal as dubious speculation.