The d'Alembertian operator, "□²", is the four-dimensional equivalent of del-squared ("∇²"). It is useful in physics. Here is the definition:
□2
= ∇2 -
(1/c2)(∂2/∂t2)
Maxwell's equations can be rewritten in terms of the d'Alembertian operator as applied to two potentials. This reduces the four equations you are used to down to two:
□2 V = - ρ/ε0
□2 A = - μ0j
Where V is the scalar electric potential, A is the vector potential, ρ is the charge density, j is the current density, and ε0 & μ0 are well-loved constants. To find out what the electric field (E) and the magnetic field (B) are, use these equations:
B = ∇ × A
E = (1/c) ∂/∂t A - ∇V
The next logical step would be to combine A and V into a single four-dimensional vector potential. Then E&M reduces down to one equation. I'll leave that as an exercise for the reader. (Bonus points for anyone who discovers special relativity in the process.)
It should be noted that equations that involve □² tend to have wavelike solutions. Consequently, it is called the wave-equation operator.
Aside
Note that unicode glyphs are often unreadable on some browsers. IE5 under win2k can not display the ∇.
I have also noticed that under IE, the white rectangle (□,□) looks very similar to the symbol for
“no glyph available.” This may cause some confusion
D'Alembertian operator E2 Writeup, Copyright 2002 Frank Grimes.
This writeup is dedicated to the public domain. Do with it what you will. (For details, see http://creativecommons.org/licenses/publicdomain/ )
--Frank Grimes, 2007
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