the Minkowski metric
/ min-KOFF-skee /
In ordinary space, Pythagoras tells you distance from coordinates: ds^2 = dx^2 + dy^2 + dz^2. Spacetime needs a modified rule because time enters with an opposite sign; the Minkowski metric is that rule — the machine that turns the coordinate differences between two events into an invariant 'spacetime distance' that all observers agree on.
The Minkowski metric is a symmetric rank-2 tensor eta_munu, a 4x4 array of numbers that in inertial Cartesian coordinates (ct, x, y, z) is diagonal: eta_munu = diag(+1, -1, -1, -1) (the 'mostly minus' convention; many texts use diag(-1,+1,+1,+1)). It defines the invariant interval ds^2 = eta_munu dx^mu dx^nu = c^2 dt^2 - dx^2 - dy^2 - dz^2, using the Einstein summation convention (a repeated upper-lower index pair is summed over 0..3). It also raises and lowers indices — V_mu = eta_munu V^nu — and supplies the Lorentz-invariant inner product of two four-vectors, A·B = eta_munu A^mu B^nu.
That minus sign is the whole of special relativity compressed into one symbol: it makes time geometrically different from space, produces the light cone, and singles out the Lorentz transformations as exactly those linear maps that leave eta unchanged. A caveat: unlike the Euclidean metric it is indefinite — ds^2 can be positive, negative, or zero — so 'interval' is not a distance in the everyday sense, and a nonzero separation can still have zero interval (light-like). In curved spacetime (general relativity) eta is replaced by a position-dependent metric g_munu(x), and Minkowski is its flat, gravity-free limit.
With signature (+,-,-,-), two events separated by dt = 1 s and dx = 4x10^8 m have ds^2 = c^2(1)^2 - (4x10^8)^2 = (9x10^16) - (1.6x10^17) < 0, a space-like interval — no signal can connect them, and some observers see them as simultaneous.
The sign of ds^2 classifies the causal relationship between two events.
The overall sign of the signature is a convention with no physical content, but you must fix one and keep it: (+,-,-,-) makes time-like intervals positive with c^2 dtau^2 = ds^2, while (-,+,+,+) flips every sign. Mixing conventions mid-calculation is the classic source of stray minus signs.