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I took a relativity class as an undergraduate but lost contact with the theory many years ago. Recently I took some old notes to revisit some concepts. I am a layman in the subject, so I apologize in advance for the basic question but I have never seen any satisfactoric explanantion on the matter.

The metric tensor $g$ tells you how spacetime curves and, consequently, how one measures distances. In $\mathbb{R}^{3}$ the norm $\|x\|$ has the natural interpretation of the lenght of the vector $x$ or the distance between the point $x$ and the origin. In relativity, we talk about four dimensional vectors ${\bf{x}}=(ct,x,y,z)$. Whats does $g({\bf{x}}, {\bf{x}})=(ct)^2-x^2-y^2-z^2$ means? What does it measure? In other worlds, why is the metric tensor defined as: \begin{equation} g=\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0& -1 & 0 & 0 \\ 0& 0 & -1& 0 \\ 0& 0 & 0 & -1 \end{pmatrix} \end{equation} in flat spacetime?

Prahar
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JustWannaKnow
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1 Answers1

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It measures the "wristwatch time" $\tau$ between two events. This is the time read on a wristwatch that passes through both events at constant velocity. This is because $\tau$ is invariant and $dx=dy=dz=0$ in that frame so $d\tau=cdt$. (You made a little error in your metric. The space intervals should all have the same sign.) But there is a lot more to say about the metric. It's subtle. Check out Exploring Black Holes by Taylor and Wheel. The entire book is about the metric. More advanced books tend to gloss over the physical meaning of the metric. For example, that an object moving in a gravitational field takes the path of maximal aging. If there is no gravity then this path is a straight line. The metric also allows you to compare space and time intervals in different reference frames such as stationary and rotating or far away from a massive object and close to that object.

jelly ears
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