lightarrow said:
So, if twin B (the one who accelerates) doesn't look out of the window, how can he establish that he accelerates because of a grav. field and not because of his rockets and so, that he have to use another metric instead of the Minkowsky one?
An observer in a flat Minkowski space-time does not HAVE to use a Minkowski metric, it's just the easiest and simplest choice. An accelerating observer in a Minkowski space-time might chose to use Rindler coordinates, for instance.
Take a look at
http://www.eftaylor.com/pub/chapter2.pdf
This has some material from one of Taylor's book I'll quote a brief excerpt:
Nothing is more distressing on first contact with the idea of curved spacetime
than the fear that every simple means of measurement has lost its
power in this unfamiliar context. One thinks of oneself as confronted with
the task of measuring the shape of a gigantic and fantastically sculptured
iceberg as one stands with a meterstick in a tossing rowboat on the surface
of a heaving ocean.
Were it the rowboat itself whose shape were to be measured, the procedure
would be simple enough (Figure 1). Draw it up on shore, turn it
upside down, and lightly drive in nails at strategic points here and there
on the surface. The measurement of distances from nail to nail would
record and reveal the shape of the surface. Using only the table of these
distances between each nail and other nearby nails, someone else can
reconstruct the shape of the rowboat. The precision of reproduction can be
made arbitrarily great by making the number of nails arbitrarily large.
It takes more daring to think of driving into the towering iceberg a large
number of pitons, the spikes used for rope climbing on ice. Yet here too the
geometry of the iceberg is described—and its shape made reproducible—
by measuring the distance between each piton and its neighbors.But with all the daring in the world, how is one to drive a nail into spacetime
to mark a point? Happily, Nature provides its own way to localize a
point in spacetime, as Einstein was the first to emphasize. Characterize the
point by what happens there: firecracker, spark, or collision! Give a point
in spacetime the name event
So a metric is defined by setting up a number of events in space-time, and assigning them coordinates.
If the events are close enough to each other, the distance (in GR, the Lorentz interval) will be determined by some quadratic form written in terms of the coordinates. The coefficients of this quadratic form are the metric.
It might or might not be obvious, but one can recover the notion of distances and times given the Lorentz intervals between all pairs of points. I don't want to digress on how this is done right at the moment, but if you want more info, ask.
So you can have many metrics that describe the same physical situation, by chosing different coordinates, i.e. assigning different numbers for the coordinates of some physically defined set of events.
If you know the metric coefficients in some particular neighborhood, there is a way to determine if the underlying space-time is flat, but it's a bit technical. One basically computes the curvature tensor - if the space-time is flat, there will be no curvature, i.e. all the components of the curvature tensor will vanish.
By this procedure, an accelerating observer using Rindler coordinates and an inertial observer using some Minkowski coordinates will have different coordinates for the same points, and also will have different metrics.
Both will, however, compute a zero curvature tensor.
I hope this helps some.
