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I put in some links and rescued the field equation from the previous version. I think we also need to explain the differences between Newton's and Einstein's theory a bit, for instance as they relate to black holes. --AxelBoldt

The current text mentions 2 things as fundamental in general Relativity:

  1. You need a reference frame to describe motion
  2. Reference frames can only be defined with respect to material objects; the text seems to imply that these are gravitating and therefore G.R. is a theory of gravitation too.

I do not believe the 2nd assumption is true, and esp. not its implication. I have a different perspective: suppose that the principle of General Relativity can be formulated as follows:

"Laws of physics must be formulated in such a way that they are independent of the frame of reference of the observer."

So G.R. is a theory about theories of physics, as much as a theory of the physical universe itself. It is a recipe for making good theories, which may or may not be consistent with the universe as we actually observe it. The popularity of thought experiments in G.R. demonstrates that its focus is on how we should describe the world, rather than how we actually observe it. The theories work only as far as the universe itself is consistent, understandable, and can be described by logic and mathematics - and there is no a priori reason that it is this way. But to the extent that the universe can be described by theories, the principle formulated above gives an important property of a "good" theory.

Implications for the theory of G.R. based on this principle:

  1. Light has a constant velocity. It is well known that Einstein was troubled by the description of induction in a coil in a variable magnetic field: depending if you are in the coil or in the magnetic field, the one generates the other depending from you perspective. Indeed, the Maxwell-Heavyside equations do not require different formulae: exactly the same equations can be used and applied in your local frame of reference, and the results are identical if you go to the other frame. So this is a "good" theory. Now from these equations the velocity of light can be computed, and it is a constant, independent of your frame of reference. Of course an absolute velocity that is not relative to observers is in conflict with Galilean dynamics. Einstein drew the ultimate consequence, and chose to adjust the geometry of space rather than the theory. The Lorentz transformations immediately follow from this choice, and are the only solution consistent with this choice.
  2. Gravity == acceleration. In a thought experiment, you can play billiard in a train that moves with constant velocity, and not notice that it moves. If it accelerates however, you will notice because the inertial mass of the balls will drive them to one end of the table (your local frame of reference). Now exactly the same will happen if the train moves up a slope with constant velocity: they will roll downhill because gravity pulls on their gravitational mass. However, you are unable to distinguish this event from the previous one -- iff inertial mass is identical to gravitational mass. So a "good" theory would describe both events with the same equations.
Now in another thought experiment you are in an elevator that is accelerated, and a beam of light is sent through one of the sides. Because light has a finite velocity, its path in the elevator case (your local frame of reference) is curved. Now from the previous thought experiment, we require that our theory does not distinguish this situation from the situation that the elevator moves with constant velocity in a gravitational field. Therefore a "good" theory of G.R. predicts (requires!) that light is deflected by a gravitational field - and this has been actually observed.

-- Tompeters