Minkowski Metric

"The Minkowski metric, unlike the Riemann
metric, by definition is NOT a generally covariant quantity. This is
now conventional wisdom in the history and foundations of
gravitational physics." wrote Z


The above remark is wrong. inkowskiIn fact the Minkowski metric is a
covariant second rank tensor under GCTs in globally flat spacetime and
it is invariant under Lorentz boosts connecting geodesic observers.

Even when you make the split in intrinsically curved space-time

guv(curved) = (Minkowski)uv + huv(curved)

each term on RHS is generally covariant separately.

Consider 1+1 space-time for computational simplicity.
The O(1,1) Lorentz boosts between geodesic inertial observers are

xi' = Li'^ixi

i = 0,1

x0' = #(x0 - @x1)

x1' = #(-@x0 + x1)

c = 1

# = (1 - @^2)^-1/2

In ordinary notation # = gamma, @ = v = relative speed between geodesic
inertial observers

L0'^0 = #

L0'^1 = - #@

L1'^0 = - #@ = L0'^1

L1'^1 = # = L0'^0

(M)i'j' = L^ii'L^jj'(M)ij

where "Minkowski" = M

M00 = 1

M11 = -1

M01 = M10 = 0

M0'0' = L0'^iL0'^jMij

= L0'^0L0'^0M00 + L0'^1L0'^1M11

= L0'^0L0'^0 - L0'^1L0'^1M11

= #^2 - #^2@^2 = +1 = M00

M0'0' = M00

M1'1' = L1'^iL1'^jMij

= L1'^0 L1'^0M00 + L1'^1L1'^1M11

= L1'^0 L1'^0 - L1'^1L1'^1

= #^2@^2 - #^2 = -1 = M11

M1'1' = M11

M0'1' = L0'^iL1'^jMij

= L0'^0L1'^0M00 + L0'^1L1'^1M11

= L0'^0L1'^0 - L0'^1L1'^1

= 0 = M1'0' = M10 = M01

Therefore the canonical form for the Minkowski metric is invariant under
the linear Lorentz boosts connecting geodesic observers.

Furthermore, the Minkowski metric

(Minkowski)uv

under GCT general nonlinear local coordinate transformations Xu'^u is a
covariant 2nd rank symmetric tensor

(Minkowski)u'v' = Xu'^u Xv'^v (Minkowski)uv