PeterDonis said:
There are upper and lower indices, which are distinct. Both of the papers you linked to carefully use only upper indices for the spatial metric and only lower indices for the temporal one. That indicates that the two operate on distinct domains. The Trautman paper uses the term "form" for something with lower indices, and "vector" for something with upper indices.
Also, on p. 417, Trautman says that the spatial metric "may be used to define the square of any form and of any spacelike vector, but not of timelike vectors". That indicates to me that there is also no way of forming an inner product between a spacelike and a timelike vector.
I'll have to take a look at the other references you give in post #17.
The nonzero form is defined on Trautman-p416 t_{a}=(\partial_a t).
Then, a vector v^a such that v^at_a=0 is spacelike [and so must be tangent to the hypersurfaces of constant-t... since v^a doesn't pierce the hypersurface (a la MTW's "bongs of a bell")];
if v^at_a \neq 0, then v^a is timelike.
In short, if a vector has a nonzero t-component, it's timelike...otherwise it's spacelike.
[Application.. a future-directed unit-timelike vector (of the form \left( \begin{array}{c} 1 \\ V \end{array}\right) can be 4-velocity of an observer traveling with velocity V... this vector, of course, is tangent to the worldline of the observer. In Galilean physics, V can take any finite value. We excluded infinite values... because that would correspond to being tangent to the constant-t hypersurface.. Let's not allow our observers to do that. (The unit magnitude is determined by the temporal metric.) Note the Galilean time-dilation factor between two observers is 1:
\left( \begin{array}{cc} 1 & V \end{array}\right)\left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right)\left( \begin{array}{c} 1 \\ W\end{array}\right)=1]
A temporal metric can be defined as t_{ab}=t_at_b=(\partial_a t)(\partial_b t) (see Ehlers-p-A121-near Eq 3a)
Then for spacelike vector X^a and timelike vector U^a,
X^at_{ab}U^b=(X^at_{a})t_{b}U^b=0 can be interpreted by saying that they are orthogonal with respect to this temporal metric...
i.e. their inner product via this temporal metric is zero.
(Of course, it's also zero if U^a is spacelike.
In other words, the temporal metric annihilates spacelike vectors.)
For me, the temporal metric is primary
(since it shows up as a limiting case of the Minkowski spacetime metric, in my formulation.
My unified metric is M = \left( \begin{array}{cc} 1 & 0 \\ 0 & -\epsilon^2 \end{array}\right) ).
