Question on Lorentzian geometry

Question

I apologize in advance if this is a too basic question.

Let $(M,g)$ be a Lorentzian manifold with signature convention $(-,+,\dots,+)$. Now, lets suppose $X\in\Gamma(TM)$ defines a global time-orientation, i.e. a global timelike vector field. In other words, $g_{p}(X_{p},X_{p})<0$ for all $p\in M$. Now, it is well-known that this allows me to split the space of timelike tangent vectors $v\in TM$ into two classes:

• future-directed (or future-pointing) if $g(v,X)<0$
• past-directed (or past-pointing) if $g(v,X)>0$

So far so good. Now, my confusion arises because in many references, people use the concept of "future-directed" and "past-directed" also for covectors $\xi\in T^{\ast}M$, without explaining the concept. I was trying to search the internet for quite some while but it was never explicitely explained. My problem is essentially that there are two possibilities:

1. You call $\xi$ future (resp. past) directed if $\xi^{\sharp}$ is future (resp. past) directed;
2. You call $\xi$ future (resp. past) directed if $\xi^{\sharp}$ is past (resp. future) directed;

Both definitions, which are exactly the opposite, seem meaningful to me. While Definition 1. seems a bit more "symmetric", the second one also kind of makes sense, since lowering/raising indices somehow involves a sign flip. Furthermore, in the second definition, $\xi(v)>0$ for $\xi,v$ future-directed and $\xi(v)<0$ for $\xi,v$ past-directed.

Example: To make it a bit more clear: Consider Minkowski spacetime $\eta=-dt\otimes dt+\sum_{i}dx^{i}\otimes dx^{i}$

Then, a global time-orientation is provided by the canonical choice $X:=\partial_{t}$. In particular, $X^{\flat}=-dt$ due to the sign convention. Now, Definition 1. would hence imply $dt$ to be future-directed, while Definition 2. implies that $dt$ is past-directed. So, which one is usually used?

Answer 1

The signature convention $(−,+,\cdots,+)$ is more commonly used in General Relativity and Lorentzian geometry because of the desire among their practicioners to make a closer parallel to Riemannian geometry. This is not the case in the remainder of Mathematical Physics (particularly in PDE's and field theory) precisely because of issues such as this. So my guess is, you usually do not find it in papers in the latter areas because there usually one adopts the $(+,−,\cdots,−)$ signature convention and then it becomes a non-issue. The latter choice also simplifies matters regarding spinors in Lorentzian manifolds, but that is another story...

That being said, if you look e.g. at Chapter 3 of the book of J. K. Beem, P. E. Ehrlich and K. L. Easley, Global Lorentzian Geometry (second edition, CRC Press, 1996), there one can infer the following rationale for choosing the OP's second convention (as suggested by the OP itself and Igor in the comments above) if the signature convention for our Lorentzian manifold $(M,g)$ happens to be $(-,+,\cdots,+)$ - that is, $\xi\in T^*\!M$ is a future directed timelike covector if $g^{-1}(\xi,\xi)<0$ (here $g^{-1}$ is the Lorentzian metric on $T^*\!M$ induced by $g$) and $\xi^\sharp=g^\sharp(\xi)=g^{-1}(\xi,\cdot)$ is past directed. The rationale is the following: consider a (real-valued) smooth function $t$ on $M$ such that $dt$ is timelike. If $t$ is a global time function on $(M,g)$, that is, a (continuous real-valued) function $t$ on $M$ that strictly increases along any future directed causal curve on $(M,g)$ (see pp. 64 of ibid.), this means that $T(t)=dt(T)=g(dt^\sharp,T)>0$ for any future directed causal vector field $T$, which amounts to saying that $dt^\sharp$ is past directed.

In other words, if we are to understand a global time function on $(M,g)$ as a global time coordinate that grows from the past into the future according to our choice of time orientation on $(M,g)$, it is much more natural to set the timelike covector field $dt$ as future directed. This also seems to be the reasoning adopted in R. M. Wald's General Relativity (University of Chicago Press, 1984), see e.g. Theorem 8.2.2, pp. 198-199 therein and the discussion that precedes it. Since these two books are quite often used as references in their field, it seems reasonable to me to reckon the above choice for the time orientation on $T^*\!M$ as the most common (albeit not the only) one with the signature convention $(-,+,\cdots,+)$.

If one applies this reasoning to the signature convention $(+,-,\cdots,-)$, one concludes instead that if a (smooth) global time function $t$ on $(M,g)$ has a timelike differential $dt$ as in the above paragraph, then $dt^\sharp$ becomes a future directed timelike vector field on $(M,g)$. This, of course, is consistent with the (resp. absence of) "sign flipping" of timelike (co)vectors when "raising and lowering indices" (that is, respectively applying the musical isomorphisms $g^\sharp$ and $g^\flat=(g^\sharp)^{-1}$) with the signature convention $(-,+,\cdots,+)$ (resp. $(+,-,\cdots,-)$). Using time functions (which, more generally, always exist locally on $(M,g)$) to set the time orientation on $T^*\!M$ is then also seen to have the advantage of being independent of the signature convention on $(M,g)$.

Edit (December 1st 2024): another perspective on why setting a causal covector $\xi\in T^*\!M$ to be future directed when $\xi^\sharp$ is past directed might be the most natural convention when the signature convention is $(-,+,\cdots,+)$ was also suggested by the OP and it has to do directly with the bundle of cones of future directed timelike vectors. Recall that, given a real vector space $V$ and a (convex) cone $\varnothing\neq C\subset V$ - that is, a (convex) subset such that $\alpha C\subset C$ for all $\alpha>0$ -, the dual cone $C^*\subset V^*$ of $C$ is given by $C^*=\{\xi\in V^*\ |\ \xi(v)\geq 0\text{ for all }v\in C\}$. The above convention ensures that corresponding bundle of dual cones consists of future directed causal covectors, regardless of the signature convention. This is useful when considering cone bundles generalizing Lorentzian metrics, given e.g. by the principal symbol of a general symmetric hyperbolic partial differential operator which is not necessarily associated with a Lorentzian metric.