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Minkowski Spacetime can be treated as either having the signature $(+,-,-,-)$ or more commonly $(+,+,+,-)$ as in Minkowski Spacetime the spacetime interval between two events is the quantity that is the same for all observers, and with $ct=w$, and $\Delta{s^2}$ giving the spacetime interval between two events $\Delta{s^2}=\Delta{x^2}+\Delta{y^2}+\Delta{z^2}-\Delta{w^2}$. There is also a Euclidean Signature with the signature $(+,+,+,+)$, in which the spacetime interval between any two events is just given by the euclidean distance formula $\Delta{s^2}=\Delta{x^2}+\Delta{y^2}+\Delta{z^2}+\Delta{w^2}$.

In a sense Galilean Spacetime is in between Minkowski and Euclidean Spacetime as in Minkowski Spacetime the spacetime path of a body accelerating at a constant rate is a hyperbole, in Euclidean Spacetime the spacetime path of an object with constant acceleration is a semi circle, and in Galilean Spacetime the path of an object accelerating at a constant rate is a parabola.

In Galilean Spacetime, if two events aren't simultaneous, then the time between the two events events is the same in all reference frames, but the spatial distance between two events is not the same in all reference frames, and the time between two events does not depend on their separation in the x, y, or z dimensions.

So can we label Galilean Spacetime as having the signature $(0,0,0,+)$?

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    $\begingroup$ This is maybe equivalent to considering Galilean space by thinking of the spacetime co-ordinates as $(x/c,y/c,z/c,t)$ with a metric $g=diag(-1,-1,-1,1)$ and $c \rightarrow \infty$. $\endgroup$
    – jacob1729
    Aug 25 '20 at 15:03
  • $\begingroup$ galiean space+time is not a vector space $\endgroup$ Aug 25 '20 at 21:07
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The notion of signature makes sense for any quadratic form, for elements of a vector space. In the case of the metric, $g$, we define $g_{ab}=g(e_a,e_b)$ where $\{e_a\}_a$ forms a basis of the vector space.

In the case of a smooth manifold, the relevant vector space to consider is the tangent space at a point. Then the signature is the number of positive, negative and zero eigenvalues of $g_{ab}$, counting multiplicities.

If you write $(+,-,-,-)$ for example, this means one positive and three negative eigenvalues. In your case, $(0,0,0,+)$ means an eigenvalue of zero with multiplicity three and one positive eigenvalue.

Note that $g$ is defined for elements of the tangent space at a point $p$ on the manifold, so technically your signature can differ point to point. However, this is only a situation to be concerned by if you are dealing with some kind of discontinuity or degeneracy.

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