If one has a Killing vector that turned out to be a timelike Killing vector field because of negative norm. Can we set this Killing vector field equal to $\partial/\partial t$?

  • 3
    $\begingroup$ Yes, you can choose the coordinates (locally) in that way. $\endgroup$ – MBN Jan 31 '15 at 10:15

@Phoenix87 is spot on, but I'll elaborate a bit.

Definition 1 A spacetime $(M,g)$ is stationary if there exists a timelike Killing field $K$, i.e. a vector field $K$ such that $\langle K,K\rangle<0$ and $\mathcal{L}_Kg=0$.

We shall show that Definition 1 implies the existence of local coordinates for which $g_{\mu\nu}$ is independent of time.

Choose a spacelike hypersurface $\Sigma$ of $M$ and consider the integral curves of $K$ passing through $\Sigma$. In $\Sigma$ we choose arbitrary coordinates and introduce local coordinates of $M$ as follows: If $p=\phi_t(p_0)$, where $p_0\in\Sigma$ and $\phi_t$ is the flow of $K$, then the Lagrange coordinates of $p$ are $(t,\vec{x}(p_0))$. In terms of these coordinates, we have $$K=\frac{\partial}{\partial t}$$ and $\mathcal{L}_Kg=0$ implies $$\partial_tg_{\mu\nu}=0$$ We call such coordinates adapted to the Killing field.


The rough idea: take the local flow of the vector field and use it to get a new "time" coordinate. In general this will work locally, so you have to patch your manifold with small enough open subsets where you can then define the new set of coordinates where now the Killing vector field corresponds to $\partial_t$.


Comments to the question (v3):

  1. Given a manifold $M$, if a smooth vector field $X\in \Gamma(TM)$ does not vanish in a point $p\in M$, then one may choose a local coordinate neighborhood $U\subseteq M$ of $p$, with local coordinates $(x^1, \ldots, x^n)$, so that $X=\frac{\partial}{\partial x^1}$. This procedure is sometimes called stratification or straightening out of a vector field. It is a special case of Frobenius theorem.

  2. A timelike vector field $X$ does not vanish by definition, so it can be locally stratified $X=\frac{\partial}{\partial x^1}$, cf. 1. Since $X$ is timelike, one would call $x^1$ a time-coordinate.

  3. The Killing vector field property is irrelevant for local stratification.

  • $\begingroup$ Is stratification the same as adaptation? $\endgroup$ – Ryan Unger Jan 31 '15 at 14:41
  • $\begingroup$ Possibly. Do you have a reference? I guess the notion of adaptation depends on context and author. $\endgroup$ – Qmechanic Jan 31 '15 at 14:55

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