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The meaning of classical relativistic fields
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To discuss the operational meaning of classical relativistic fields and
their arguments, we consider a special case of classical
electrodynamics.
In a region Omega without charges or currents, such as in a vacuum,
Maxwell's equations read
\nabla \cdot E (t,x)= 0,
\nabla \cdot B(t,x)= 0,
\nabla \times E (t,x)= - \partial_t B(t,x),
\nabla \times B(t,x) = c^{-2} \partial_t E(t,x).
Experimentally, one can check the values of the field by measuring the
Lorentz force
F(t,x)=q(E(t,x)+v \times B(t,x))
that the electromagnetic field exerts at time t on a particle with
charge q and velocity v at position x. Assuming that the fields don't
vary too much in space and time, such tests allow one to determined
the field everywhere in Omega to a certain accuracy. High frequency
behavior cannot be tested in that way, but needs the methods of
quantum optics. In that case, averages of products of the fluctuating
part of the fields, collected into coherence matrices (that may depend
on one or two position arguments), can be measured using optical tools.
We conclude that both the components of the fields and their arguments
have an immediate, observer-independent physical meaning.
The Maxwell equations in vacuum are Poincare covariant, having the
standard transformation behavior under translations, rotations, and
Lorentz boosts. But things may look different when considered by
different observers in their particular decomposition into space and
time. The Lorentz transformations that mediate between observers in
different Lorentz frames cause the standard effects of relativity when
different observers look at the same objective situation.
There is an observer-dependent decomposition of space-time position x
into a time component t=c^{-1} u dot x -- where c is the speed of
light, u=(u_0,\u) is the 4-velocity of the observer, a future-pointing
unit vector in the Minkowski inner product, and u^2 = u_0^2-\u^2 --,
and its instantaneous space slice, a 3-dimensional hyperplane orthogonal
to u and passing through the observer's 4-position. In particular, in
coordinates where the observer is at rest, \u=0, u_0=1, the time
coordinate is t=x_0/c, and x=(ct,\x) with \x labeling the space
coordinates.
This decomposition is convenient for our non-relativistic intuition.
But this decomposition has no operational meaning at all, not even for
the observer itself.
Indeed, an observer at rest in the origin of its rest frame cannot
know at time t anything about the world in the present, the
instantaneous space slice consisting of all x=(x_0,\x) with fixed
x_0=ct -- except what happens at the origin itself. The reason is that
relativity forbids the communication of information at a speed >c, so
that whatever information an observer may have recorded at time t in
its memory must be due to events happening in the past causal cone,
consisting of all x=(x_0,\x) with x_0<=ct-|\x|. (For example, we cannot
see now what happens at the sun now, only what happened 8 minutes ago.)
Similarly, an observer at rest in the origin of its rest frame cannot
influence at time t anything about the world in the present -- except
what happens at the origin itself. The reason is that relativity
forbids the propagation of influences at a speed >c, so that whatever
an observer does at time t can affect only events in the future
causal cone, consisting of all x=(x_0,\x) with x_0>=ct+|\x|.
Thus the present is completely inaccessible and completely uncapable
of being influenced by the observer, except at the origin.
In particular, measurements can be made only at the origin itself,
and learning about measurements made by others now is possible only in
the future. Moreover, since space is unbounded but the intersection of
any space slice with the past causal cone, one never can learn
_everything_ about any particular time in the past.
We conclude that the local observer frames, and the associated
splitting of space-time into slices of 3-spaces at fixed time, have no
observational relevance and are spurious remnants of a nonrelativistic
view of a relativistic theory. They become relevant only in as far one
wants to make a nonrelativistic approximation and a corresponding
expansion in c^{-1}.