= Minkowski metric
{c}
{title2=$\eta_{\mu\nu}=\operatorname{diag}(-1,1,1,1)$}
With the mostly-plus <metric signature> $(-,+,+,+)$ and <coordinates> $X^\mu=(ct,x,y,z)$, the <Minkowski metric> is the <nondegenerate bilinear form>
$$
\eta_{\mu\nu}=\operatorname{diag}(-1,1,1,1),\qquad
U\cdot V=-U^0V^0+U^1V^1+U^2V^2+U^3V^3.
$$
It is a <symmetric bilinear form>, but not a <positive-definite bilinear form>. For example, $U=(1,1,0,0)$ is nonzero yet $U\cdot U=0$, making it a <null vector>. This does not make the <Minkowski metric> degenerate: some other <vectors> have nonzero pairing with $U$.
Applied to infinitesimal displacements, the <Minkowski metric> gives
$$
ds^2=-c^2dt^2+dx^2+dy^2+dz^2.
$$
For a <timelike curve>, a clock records $d\tau=\sqrt{-ds^2}/c$. For a <null curve>, $ds^2=0$. For <spacelike separation>, the positive <Minkowski interval> measures a different kind of separation. A <Lorentz transformation> preserves this form, expressed in <matrix> notation by $\Lambda^T\eta\Lambda=\eta$, although it need not preserve ordinary <Euclidean distance> in the four-component coordinate list.
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