Pretty essential
Hi, Michel,
I can only guess what you are trying to get at here. My guess is that you are trying to establish something like the following claims:
1. The "Rindler metric" can be obtained from the Minkowski metric by considering a sequence of (parallel) boosts applied to an observer in flat spacetime,
2. The "uniform gravitational field" in gtr is given by the Rindler metric, and this captures the content of "the" equivalence principle of gtr,
3. Therefore (?), assuming "the equivalence principle", the Schwarzschild metric can be obtained by elementary considerations in special relativity alone, without any reference to the vacuum field equations of gtr.
I don't know if this guess is accurate, I can offer some comments about these claims, irrespective of whether you were really trying to make them.
1. Indeed, the so-called "Rindler metric" is simply the usual Minkowski metric represented in another coordinate chart (note that changing coordinates is analogous to changing bases in linear algebra), and this Rindler coordinate chart (and the transformation from the Cartesian coordinate chart) can indeed be worked out from considering an observer in Minkowski vacuum (aka flat spacetime) who is accelerating with constant magnitude and direction of acceleration. This is done, for example, in the classic textbook by Misner, Thorne, & Wheeler, Gravitation (see section 6.6). This involves CONTINUOUS acceleration, but that can be approximated by a sequence of boosts. All you need know here is that acceleration corresponds to the path curvature of the world line of the accelerating test particle (observer), which is given by taking the exterior derivative of the tangent vector X to the curve, namely D_X X.
2. It is not VERY wrong to say this as a first approximation, but deeper examination shows that in some ways this claim is misleading. For one thing, there is more than one "equivalence principle" used in gtr (most textbooks discuss both strong and weak versions). More important, one important and valuable way to think about "the" EP is in terms of how the gravitational field varies over a larger and larger region of spacetime. In a sufficiently small region, a curved spacetime can always be approximated by a flat spacetime. This is simply the Lorentzian analog of a key insight of surface theory: curved two dimensional manifolds immersed in euclidean spaces have a tangent plane at each point. If you look at a larger region (or demand more accuracy), however, you must take account of quadratic curvature effects.
The real question here should be: why do quadratic effects suffice? Why does one not need to consider cubic or even more complicated forms? Why can Riemann and his heirs get away with the simplest thing beyond "bundled" linear forms, namely "bundled" quadratic forms, in order to define a suitable notion of "distance" along a curve on a smooth manifold? Answers, some of them good, have been offered in the literature. Be this as it may, in Riemannian (or Lorentzian) geometry, these quadratic deviations are precisely what is captured by the Riemann curvature tensor. In a vacuum solution in gtr, a kind of tensorial "average" of the Riemann curvature tensor (the Ricci curvature tensor) is identifiable as a relativistic analog of the tidal force tensor known from vacuum solutions in Newtonian gravitation. Thus slogans such as this: "the gravitational field is only observable as second order effects in the geometry". C.f. the Jacobi geodesic deviation formula, for example.
In both Newtonian gravitation nor gtr, in a sense, "all gravitational fields appear uniform on sufficiently small scales"; what is interesting is how they vary on larger scales. In particular, curved spacetimes such as the Schwarzschild vacuum are easily distinguished mathematically (e.g. using curvature invariants) from a Rindler scenario.
3. Like Pervect, I have, from time to time, encountered claims to the effect that the Schwarzschild geometry can be derived from "physical first principles", without appealing to the vacuum field equations of general relativity. Some of these are not entirely incorrect. For example, there are various other classical relativistic theories of gravitation which admit the Schwarzschild spacetime as a solution, even as a vacuum solution, so in that sense you don't need gtr to "derive" the Schwarzschild geometry (as a Lorentzian spacetime, i.e. a smooth manifold endowed with a specific metric tensor). But this probably isn't what you had in mind, and more ambitious claims of this nature tend to be far more dubious. Regarding the debunking eprint cited by Pervect, I note the Richard Price is well known for his work on how black holes respond (according to gtr) to perturbations (roughly speaking, outside the horizon, they radiate away imperfections in their preferred Kerr geometry as gravitational radiation) and is a coauthor of the problem book I mentioned in my first PF post (this being my second post), so he is certainly a leading expert on how gravitational fields behave (according to general relativity) under small perturbations from some reference spacetime, such as the Schwarzschild or Minkowski vacuum solutions of gtr.
Chris Hillman
