Allow me to give a brief introduction to the topic, which has to do with physics; my question will still be a mathematical one. I think my question is aimed at people with a background both in physics and in mathematics.
I am familiar with the Theory of Relativity, in which spacetime is described as a four-dimensional manifold with a notion of distance (invariant under the Poincare group). This distance is described by an object that is called the metric or the metric tensor. A simple example of such a metric is the Minkowski metric. This metric (tensor) is (at least as evaluated in some coordinate frame and up to certain conventions) described by the 4x4 matrix $\eta = diag(-1,1,1,1)$. As a result, the spacetime distance between two points $x,y$, described by their components $x^\mu, y^\mu$, $(\mu=0,1,2,3)$, is given by \begin{equation} d(x,y)=\eta_{\mu\nu} (x^\mu - y^\mu)(x^\nu - y^\nu)~, \end{equation} where summation over repeated indices is implied. Now comes my question. When we take $x$ and $y$ to be given, for instance, by $x = (2,0,0,0)$ and $y=(1,0,0,0)$, then we arrive at a negative distance: \begin{equation} d(x,y) = -1~. \end{equation} Now, regarding the Theory of Relativity, this is quite possible, but recently I came across the formal definition of a metric. And one of the conditions stated that a metric maps each pair of points in the space to a non-negative real number. But in our example of the Minkowski metric, this is clearly not the case. So the Minkowski metric does not seem to satisfy the definition of a metric.
So, is the metric in the Theory of Relativity just a different kind of metric, or does it have to do with a distinction between metrics and metric tensors? Or am I missing something here?