Measurement in General Relativity
"If we knew what it was we were doing, it would not be called research, would it?"
- Albert Einstein
L&L do not contradict Einstein at all. They agree that locally
you cannot distinguish a real gravity field from a "fake" one.
However, the meaning of "real gravity field" in L&L's usage
is not the same the way Einstein uses the same words.
Einstein only means "g-force" by "gravity field" - he does not
mean "curvature." If by "real gravity field" one means curvature
then of course one must correlate separated local measurements
in some way in order to measure the "geodesic deviation."
Note for a clean curvature geodesic deviation measurement you need
2 geodesic test particles that do not themselves feel any g-forces or any non-gravity forces
even though the non-geodesic LNIF observer doing the measurement
will feel g-forces! That is (LC|LNIF) =/= 0. Note that the connection field, emergent from locally gauging the Poincare group,
is a property not of the observed test particle but of the observing detector!
Using Bohr's method of the "total experimental arrangement" and Einstein's method
of the gedankenexperiment is also useful in classical measurement theory in
GR and will help in quantum gravity issues as well.
The geodesic equation on the observed test particle's point center of mass only is
d^2x^u(Observed)/dt^2 + (LC|Observer)^uvw(dx^v(Observed)/ds)(dx^w(Observed)/ds) = 0
Where in the rest LNIF frame of the observer
(LC)00^i=1,2,3 ~ (non-gravity force on the observer-detector)^i
One dramatic illustration of why we need Einstein's curved space-time is the fact that if Alice and Bob stand still at North Pole and South Pole respectively they radially accelerate away from each other each with
g ~ c^2rs/r^2
in opposite directions at fixed distance from each other!
This is a paradox in Euclidean geometry, but not so in non-Euclidean geometries!
This nice example from Stephen Hawking shows the intimate intuitive connection of curvature to the equivalence principle together with the need to look globally in a topological way. This is the method of Roger Penrose, which showed why there must be light-like surrounding surfaces or event horizons that you will not find using only local algebra! This is similar to superconductivity in which you will not find the ground state ODLRO coherence using only local perturbation theory to finite order. You have to make an infinite sum and when you do that you have an emergent qualitatively new phenomenon such as "consciousness" in living matter - "More is different" (P.W. Anderson).
On Jan 7, 2007, at 12:16 AM, Z wrote:
But most of this is taken almost directly from Einstein, Pauli, and
Landau & Lifshitz.
Pauli wrote explicitly in his Encyclopedia article on relativity that
in Einstein's model the pseudo-field produced by frame acceleration is
"no more or less real" than a homogeneous gravitational field produced
Landau and Lifshitz reintroduced the physical distinction between the
two fields, and refer to the "actual" gravitational field, as opposed
to the effects of frame acceleration. Other authors, such as Eddington
and Tolman, referred to the "permanent" gravitational field. This is
what I'm calling the "true" gravitational field.