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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}.