  We show that tensoriality constraints in noncommutative Riemannian geometry
in the 2-dimensional bicrossproduct model quantum spacetime algebra
[x,t]=\lambda x drastically reduce the moduli of possible metrics g up to
normalisation to a single real parameter which we interpret as a time in the
past from which all timelike geodesics emerge and a corresponding time in the
future at which they all converge. Our analysis also implies a reduction of
moduli in n-dimensions and we study the suggested spherically symmetric
classical geometry in n=4 in detail, identifying two 1-parameter subcases where
the Einstein tensor matches that of a perfect fluid for (a) positive pressure,
zero density and (b) negative pressure and positive density with ratio
w_Q=-{1\over 2}. The classical geometry is conformally flat and its geodesics
motivate new coordinates which we extend to the quantum case as a new
description of the quantum spacetime model as a quadratic algebra. The
noncommutative Riemannian geometry is fully solved for $n=2$ and includes the
quantum Levi-Civita connection and a second, nonperturbative, Levi-Civita
connection which blows up as \lambda\to 0. We also propose a `quantum Einstein
tensor' which is identically zero for the main part of the moduli space of
connections (as classically in 2D). However, when the quantum Ricci tensor and
metric are viewed as deformations of their classical counterparts there would
be an O(\lambda^2) correction to the classical Einstein tensor and an
O(\lambda) correction to the classical metric.
