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The Wikipedia article on metric signatures says that the signature of a metric can be written $(v,p,r)$, where $v$ is the number of positive eigenvalues, $p$ is the number of negative eigenvalues, and $r$ is the number of zero eigenvalues.

Explanations of the metric signature usually give short shrift to 'r', as if it is an unimportant mathematical artifact....

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    $\begingroup$ I have edited your question to include the relevant context. In the future, please include this information yourself. As a general rule of thumb, you should define all of the symbols you use, either explicitly via text or implicitly by their appearance in a universally-recognized equation (e.g. $F=ma$). $\endgroup$
    – J. Murray
    Commented May 27 at 12:34

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In general relativity, the metric is assumed to be Lorentzian, which means that it has signature $(1,n-1,0)$ or $(n-1,1,0)$ depending on convention. Because the metric does not have any zero eigenvalues, we generally don't refer to $r$ at all, and simply refer to the metric signature as $(+---)$ (mostly minus) or $(-+++)$ (mostly plus).

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Just to add something, in some contexts, e.g., asymptotic symmetries, the relevant quantity is the pullback metric $q_{ab}$ of the conformally rescaled metric to $\mathcal{I}^+$, which has signature $(0,+,+)$.

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