Minkowski Metric
I wonder what it means?
covariant? Tensor? Metric?
I never did know. I probably never will.
Will it get me to alpha centuri or make me immortal?
I don't think so.
"Jack Sarfatti" wrote in message
...
"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
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