# Sign of the Minkowski metric and proper time

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?

• Haha well I read two introductory textbooks and none had a section on this. What about calculating the time a moving object is experiencing? Do I use the definition $\Delta \tau^2 = -\Delta \sigma^2$ or $\Delta \tau^2 =-\frac {\Delta \sigma^2}{c^2}?$ – Jannik Pitt Dec 4 '16 at 21:38
• @AccidentalFourierTransform Oh okay, I see. So when working in units where $c$ is not equal to $1$ the space-like interval $\Delta \sigma^2$ and the time-like interval $\Delta \tau^2$ are always connected by $\Delta \sigma^2 = -c^2\Delta \tau^2$ no matter which metric signature I'm using? – Jannik Pitt Dec 4 '16 at 21:43
• @JannikPitt as AccidentalFourierTransform said, this is another bit of convention. If you are working in so called 'natural units,' then these are both the same expression as $c=1$. Are you sure you are looking at introductory books? I believe Griffith's book on particle physics has a gentle and clear discussion about this, and also in the later chapters in his E&M book when he gets to the relativistic formulation. – Bobak Hashemi Dec 4 '16 at 21:45