The D'Alembertian is a generalization of the Laplacian operator to a space of arbitrary dimension and metric.

Where does the D'Alembertian symbol $\Box$ come from?

According to Wikipedia it has to do with 4-dimensional spaces, but the authors don't cite sources and I don't see why the relation.

  • $\begingroup$ The laplacian in 3d is sometimes denoted by a triangle (sitting on its base: see en.m.wikipedia.org/wiki/Laplace_operator). In 4d, it becomes a square. $\endgroup$ Apr 27 '18 at 0:28
  • $\begingroup$ Why don't a diamond then? Maybe if you know where was this notation used for the first time I can find the context. $\endgroup$ Apr 27 '18 at 0:33

The symbol seems due to Poincaré (1901, p. 456):

For brevity we will introduce the following symbol, by setting, $$ \Delta\mathrm U -\mathrm K_0\frac{d^2\mathrm U}{dt^2}=\Box\mathrm U $$

Here $\mathrm K_0=1\,/\,c^2$ (p. 336). It didn’t immediately catch on: it appears without comment in Poincaré’s (1906, p. 132) (cited by Wikipedia), but not in the book where Lorentz named it (1909, p. 17):

the operation $\Delta-\frac1{c^2}\frac{\partial^2}{\partial t^2}$ may be given the name of the Dalembertian (...) in commemoration of the fact that the mathematician d’Alembert was the first to solve a partial differential equation (...) which contains this operation, or rather the operation $\frac{\partial^2}{\partial x^2}-\frac1{c^2}\frac{\partial^2}{\partial t^2}$

In (1910, p. 175) Lewis (of photon fame) was still suggesting the notation $$ \diamondsuit^2= \frac{\partial^2}{\partial x_1^2} + \frac{\partial^2}{\partial x_2^2} + \frac{\partial^2}{\partial x_3^2} + \frac{\partial^2}{\partial x_4^2}\qquad(x_4=ict), $$ but Sommerfeld (1910, p. 661) and von Laue (1911, p. 71) chose again $\Box$ (pronounced “Delta”).

Note: Sommerfeld (ibid.) says the symbol was already introduced by Cauchy. This is repeated by Silberstein (1912, p. 805; 1914, p. 113), Bateman (1915, p. 6) and Pauli (1921, p. 607), but neither gives a citation. (Perhaps they mean this?)


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