As I understand it the space-time interval $\sigma$ is defined as $\sigma^2=\eta_{\alpha \beta}x^\alpha x^\beta$. Why is it that some books define the metric with the signs (-,+,+,+) and some with (+,-,-,-)? In the book Spacetime and Geometry the Minkowski metric $\eta_{\alpha \beta}$ is defined as
$\eta_{\alpha \beta}=\begin{pmatrix} -1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1 \end{pmatrix}$
while Wikipedia uses reversed signs (+,-,-,-). When do you choose which definition of the metric? Also the book Spacetime and Geometry uses $\Delta \sigma^2 = -c^2t^2+(x^2+y^2+z^2) $ to calculate the space-time interval and defines the proper-time interval as $\Delta \tau^2=-\Delta \sigma^2$ while Wikipedia and the book Introduction to the Theory of Relativity define $\Delta \tau^2=-\frac {\Delta \sigma^2}{c^2}$? If I want to calculate the time a moving object is measuring, which definition of the proper time and which metric signature do I use? I am very confused about this, could somebody explain this to me?