Lorentzian scalar product

Question

Let $X$ be a future-directed vector and $Y$ a past-directed one in a time-oriented space-time (manifold). We want to compute $g(X,Y)$. I choose a coordinate in which $X=X^0\partial_0$ with $X^0>0$ since $X$ is future-directed and the space-time is time-oriented. One then gets in this coordinate $$ g(X,Y)=\underbrace{X^0}_{>0} \underbrace{Y^0}_{<0} \underbrace{ g(\partial_0,\partial_0)}_{<0} > 0\,, $$ where I assumed that $g_{00}<0$. Is the scalar product of future-directed vector with past-directed vector in Lorentzian geometry always positive? I think this is not a good answer. Is there a general chart independent solution?

Answer 1

In your metric, diag(-,+,+,+) $\textit{space-like}$ intervals are positive.

In physics, space-like intervals means that two events $A$ and $B$ cannot be linked with light-speed signal. In this reasoning, I think what you have found is right, that a future point and a past point (inside the light cone) always gives a positive inner product.

Physically a particle in a space-time diagram moves from the past to the future, so it should be time-like, no? That's not the right way to think, the particle moves one step at a time. That is, it moves in infinitesimal steps, so it never comes from a $x^0<0$ event to a $x^0>0$ one. In other words, the particle path (world line) is continuous, so nothing physically wrong is happening with your answer.

This clears your doubts?

Answer 2

With some efforts, I have answered my own question. The point is that in the Lorentzian geometry the following statement is true (by using the reversed Cauchy-Schwarz inequality):

Time-like vectors $X$ and $Y$ are in the same time-cone (future-directed or past directed) iff $g(X,Y)<0$. In other words, the reversed Cauchy-Schwarz inequality reads $$ g(X,Y)\leq - |g(X,X)||g(Y,Y)| \,;\quad \text{for future-directed timelike vectors $X$ and $Y$} $$ and $$ g(X,Y)\geq |g(X,X)||g(Y,Y)| \,;\quad \text{for a past-directed timelike vector $X$ and a future-directed time-like vector $Y$}\,. $$