What is Poincare's "Fourth Geometry"?

Question

In Science and Hypothesis, Poincare cryptically describes a "Fourth Geometry." Can anyone clarify what he is talking about? Is there a standard name for this geometry?

The Fourth Geometry.—Among these explicit axioms there is one which seems to me to deserve some attention, because when we abandon it we can construct a fourth geometry as coherent as those of Euclid, Lobatschewsky, and Riemann. To prove that we can always draw a per- pendicular at a point A to a straight line AB, we consider a straight line AC movable about the point A, and initially identical with the fixed straight line AB. We then can make it turn about the point A until it lies in AB produced. Thus we assume two propositions—first, that such a rotation is possible, and then that it may continue until the two lines lie the one in the other produced. If the first point is conceded and the second rejected, we are led to a series of theorems even stranger than those of Lobatschewsky and Riemann, but equally free from contradiction. I shall give only one of these theorems, and I shall not choose the least remarkable of them. A real straight line may be perpendicular to itself.

Answer 1

It is the Minkowski plane, the lightlike lines are “perpendicular to themselves”, see e.g. Stachel, Poincaré and the Origins of Special Relativity. To get this geometry, one needs to expand what “geometry” means. The classical geometries of Euclid, Lobachevsky, and Riemann have what is now called sign-definite metrics, which define real distances. In contrast, the Minkowski plane has an indefinite metric with signature $1,1$, so the “distances” can be imaginary (along the timelike lines). It does have constant curvature, like the classical geometries, but is only homogeneous, not isotropic, spacelike and timelike directions can not be interchanged by Lorentz transformations (which preserve the metric). It can also be realized as the Lorentz geometry on the hyperboloid of one sheet, and is sometimes denoted $H^{1,1}$.

The Minkowski plane first appeared on a long list of geometries from Klein's Erlangen program (1872), where he classified them based on a choice of “absolute”, a conic in the projective plane preserved by a subgroup of projective transformations. But it remained largely unexplored until Minkowski connected a four-dimensional version of it to the spacetime of special relativity in 1907. Poincare originally singled it out in Les Géométries non Euclidiennes (1891) as deserving more attention, and, ironically, used it in Science and Hypothesis (1902) to illustrate his views on the conventional nature of choosing a physical geometry, see Einstein's Pathway to the Special Theory of Relativity by Weinstein, pp.517-18. Only the pair physics + geometry is empirically determined, according to him, not each piece separately, and we are free to keep Euclidean geometry, as the simplest, come what may, by modifying the physics. This was the reason he favored Lorentz's interpretation of special relativity, with its Euclidean background space, over Einstein's to the end of his life.