My question is about how to properly compute the projection of a tensor in a given direction using inner product product in a pseudo-riemannian manifold, since inner product isn't defined positive.
For simplicity, suppose we have a 2-dimensional curved manifold with signature $ (-, +) $ and a generic metric tensor $ g_ {\mu \nu} $. I want to project a time-like vector $ t ^ \mu = (t ^ 0, t ^ 1) $ on another time-like vector $ n ^ \mu = (n ^ 0, n ^ 1) $ and suppose to know that these two vector form an acute angle, so we expect that the projected component $ t ^ \mu_p $ will be something like
$ t ^ \mu_p = An ^ \mu $ with $A> 0$
because the projected component of $ t ^ \mu $ has the same direction of $ n ^ \mu $. To determine $ A $ I suppose I have to use the inner product $ t_ \mu n ^ \mu $, but this of course is negative, since both vectors are time-like. How do I justify mathematically the use of the minus sign that I want in the projection in order to obtain the correct result?
Then, if I want to do the same with a p-rank tensor $ T $, do I have to consider any minus sign somewhere? For example, for a 2-rank tensor the projected component
$ T ^ {\mu \nu} _p = (T _ {\alpha \beta} n ^ \alpha n ^ \beta) n ^ \mu n ^ \nu $
or do I need a minus sign?