Why is a spacetime interval defined as $(\Delta s)^2$ = $\eta_{\mu\nu}\Delta x^{\mu} \Delta x^{\nu}$? I'm currently studying SR and GR and I noticed that all the books introduce spacetime interval without any motivation as $$(\Delta s)^2 = -c^2 (\Delta t)^2 + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2$$ and than motivate that expression defining the Minkowski metric $\eta_{\mu\nu}$.
Is there a physical reason why the metric of spacetime is the Minkowski metric?
 A: Simple explanation: Its the only type of metric that conserves the speed of light.
Mathematical explanation: Let us assume an inertial observer, $O$, measures the speed of light as $dx/dt = c$. Let us assume another inertial observer, $O'$, measuring the speed of light in his coordinates, $dx'/dt' = c$. 
Naturally at this point one can ask, What is the connection between $(t,x)$ and $(t',x')$ ?
The crucial point is to keep $c$ a constant while doing this transformation. Calculations show that Galilean transformation does not satisfy this condition since $c$ varies. 
Let us take $dx/dt = c$ and write in the form of $dx^2 - c^2dt^2=0$. 
(PS: We need to square because light can move in either poisitive or in negative direction. Such that the equations should both satisfy for $dx/dt = c$ and $dx/dt = -c$. For this reason we should square both sides)
Similarly by using $dx'/dt' = c$,  we can write $dx'^2 - c^2dt'^2=0$
This implies that $$dx^2 - c^2dt^2=dx'^2 - c^2dt'^2$$
This transformation is the one that conserves the speed of light and that's what we're looking for. 
In general we write this as $$ds^2 = -c^2dt^2 + dx^2$$ 
And later on, you can also show that 
$ds^2 = -c^2dt^2 + dx^2 = -c^2dt'^2 + dx'^2$
