The "Pisa experiment" tells that the motion of a particle in a
gravitational field is independent of the particle's properties. On the
basis of this result a hypothesis is formulated, called the "equivalence
principle". The general relativity is based on this hypothesis.
In free fall the effects of gravity disappear in all possible local
experiments and general relativity reduces locally to special relativity.
An accelerated frame is locally equivalent to a frame in a
gravitation field.
Gravitation field is locally equivalent to a non-inertial frame.
Gravitation forces are equivalent to inertial forces.
Non-inertial frames are curved. Example: constantly
accelerated elevator (Rindler space). (t'Hooft, Chapter 3)
The Rindler space is a (sort of) space of constantly accelerated
observers. The accelerations are chosen such that if two of the
observers are connected with a solid rod, no stresses appear in the rod
in the process of motion. The Rindler space is an easiest model of a
constant uniform gravitational field. It is often used as a test-ground
for all sorts of theories.
We shall see on the example of the Rindler space, that non-inertial
frames have some peculiar features compared to inertial frames: they
are curved, they have horizons, the time goes differently at different
places.
Exercises.
(Kind of difficult for a first exercise, but it actually counts for two
exercises. Again we will have (almost) solved it at the lecture).
Consider the motion of a charge e with mass m in a constant
uniform electric field E which is, say, in X direction. Calculate
x(t), y(t) and x(y).
You might need the equation of motion of a charge in an electro-magnetic
field E, H:
dp/dt = e(E + 1/cv×H),
where the (relativistic) momentum p and the velocity v are connected as