Hermeneutics of Classical General Relativity

In the Newtonian mechanistic worldview, space and time are distinct and absolute. In Einstein's special theory of relativity (1905), the distinction between space and time dissolves: there is only a new unity, four-dimensional space-time, and the observer's perception of ``space'' and ``time'' depends on her state of motion. In Hermann Minkowski's famous words (1908):

Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

Nevertheless, the underlying geometry of Minkowskian space-time remains absolute.

It is in Einstein's general theory of relativity (1915) that the radical conceptual break occurs: the space-time geometry becomes contingent and dynamical, encoding in itself the gravitational field. Mathematically, Einstein breaks with the tradition dating back to Euclid (and which is inflicted on high-school students even today!), and employs instead the non-Euclidean geometry developed by Riemann. Einstein's equations are highly nonlinear, which is why traditionally-trained mathematicians find them so difficult to solve. Newton's gravitational theory corresponds to the crude (and conceptually misleading) truncation of Einstein's equations in which the nonlinearity is simply ignored. Einstein's general relativity therefore subsumes all the putative successes of Newton's theory, while going beyond Newton to predict radically new phenomena that arise directly from the nonlinearity: the bending of starlight by the sun, the precession of the perihelion of Mercury, and the gravitational collapse of stars into black holes.

General relativity is so weird that some of its consequences -- deduced by impeccable mathematics, and increasingly confirmed by astrophysical observation -- read like science fiction. Black holes are by now well known, and wormholes are beginning to make the charts. Perhaps less familiar is Gödel's construction of an Einstein space-time admitting closed timelike curves: that is, a universe in which it is possible to travel into one's own past!

Thus, general relativity forces upon us radically new and counterintuitive notions of space, time and causality ; so it is not surprising that it has had a profound impact not only on the natural sciences but also on philosophy, literary criticism, and the human sciences. For example, in a celebrated symposium three decades ago on Les Langages Critiques et les Sciences de l'Homme, Jean Hyppolite raised an incisive question about Jacques Derrida's theory of structure and sign in scientific discourse:

When I take, for example, the structure of certain algebraic constructions [ensembles], where is the center? Is the center the knowledge of general rules which, after a fashion, allow us to understand the interplay of the elements? Or is the center certain elements which enjoy a particular privilege within the ensemble? ...With Einstein, for example, we see the end of a kind of privilege of empiric evidence. And in that connection we see a constant appear, a constant which is a combination of space-time, which does not belong to any of the experimenters who live the experience, but which, in a way, dominates the whole construct; and this notion of the constant -- is this the center?

Derrida's perceptive reply went to the heart of classical general relativity:

The Einsteinian constant is not a constant, is not a center.

It is the very concept of variability -- it is, finally, the concept of the game. In other words, it is not the concept of something -- of a center starting from which an observer could master the field -- but the very concept of the game ...

In mathematical terms, Derrida's observation relates to the invariance of the Einstein field equation under nonlinear space-time diffeomorphisms (self-mappings of the space-time manifold which are infinitely differentiable but not necessarily analytic). The key point is that this invariance group ``acts transitively'': this means that any space-time point, if it exists at all, can be transformed into any other. In this way the infinite-dimensional invariance group erodes the distinction between observer and observed; the of Euclid and the G of Newton, formerly thought to be constant and universal, are now perceived in their ineluctable historicity; and the putative observer becomes fatally de-centered, disconnected from any epistemic link to a space-time point that can no longer be defined by geometry alone.