What is the exact motivation for the Minkowski metric? In introductory texts about Lorentz Geometry, one always learns about the Minkowski space, i.e. $R^4$ with the Minkowski metric
$$ m(x, y) := -x_0 y_0 + x_1y_1 + x_2y_2+ x_3 y_3  $$
Using this metric, one can define lightcones $\{x \in R^4 \mid m(x, x) = 0 \}$ and timecones $\{x \in R^4 \mid m(x, x) < 0\}$. The existence of these cones is then physically interpreted as the fact that no particle can move faster than light. 
But - thinking the other way around - why exactly do I have to use the Minkowski metric to formalize this fact? These cones can be defined also without reference to any metric. I could use $R^4$ with Euclidean metric as my mathematical model of reality. The fact that no particle moves faster than a certain bound is then modelled by the fact that the tangent vectors of curves are within these cones. 
Is there an instructive way to see why one has to take the Minkowski metric on $R^4$? 
Idea: Maybe the Euclidean metric conflicts with the fact that the speed of light is the same in any reference frame? But how to make this formally precise? How to define speed then?
 A: Let's use natural units where the speed of light is $c=1$, and let's denote the time coordinate with $x_0$, and spatial coordinates  with $x_1$, $x_2$ and $x_3$.
If light is emitted from the origin at time zero (i.e. emitted at spacetime point $(0,0,0,0)$), distance it transverses is equal to amount of time passed (because its velocity is $c=1$). Distance is $\sqrt{x_1^2+x_2^2+x_3^2}$ and time is $x_0$, so they are related by:
\begin{equation}
x_0^2=x_1^2+x_2^2+x_3^2
\end{equation}
or, in a more suggestive form:
\begin{equation}
-x_0^2+x_1^2+x_2^2+x_3^2=0
\end{equation}
Speed of light is equal in all the inertial frames, and changing the frame is one of the transformations of spacetime coordinates, therefore, this equation must be invariant under transformations of coordinates, just like in Euclidean space $x^2+y^2+z^2$ is invarinat.
It is trivial to check that in the case of spacetime, only Minkowski metric allows for such an invariance and then, more generally, any combination of form
\begin{equation}
-x_0^2+x_1^2+x_2^2+x_3^2
\end{equation}
is invariant under coordinate transformations, not only zero ones, and even combinations like
\begin{equation}
-x_0y_0+x_1y_1+x_2y_2+x_3y_3
\end{equation}
are invariant.
Those are called Lorentz scalars.
A: I'll do a silly computation that might be helpful. Take a particle in space that goes straight from a point $p$ to a point $q$ in time $\Delta t$, with velocity direction $v= (\Delta x, \Delta y, \Delta z)$. We know that the speed is less than the speed of light $c$, so $$\sqrt{\left(\frac{\Delta x}{\Delta t}\right)^2 + \left(\frac{\Delta y}{\Delta t}\right)^2 + \left(\frac{\Delta z}{\Delta t}\right)^2} < c,$$which can be rewrited as $$(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 - (c\Delta t)^2 < 0.$$There you go: put geometric units for which $c=1$ and look at the bilinear form $$\langle (x_1,y_1,z_1,t_1),(x_2,y_2,z_2,t_2)\rangle_L = x_1x_2+y_1y_2+z_1z_2 - t_1t_2.$$Movements which can be realized at speed less than $c$ give rise to timelike vectors, lightlike vectors represent motions of photons (at spped $c$), and spacelike vectors lack physical significance.
A: Consider a 2d world. There are basically three useful cases to consider: Lorentzian (-,+) signature, Galilean (0, +) signature, and Euclidean (+, +) signature. [Yes, perhaps there is a (0,0) signature, but this is especially boring.]
Each of these spaces has a "rotation-like" operation, generated by exponentials of various quantities that are not real numbers but are geometrically significant to the space.
In Euclidean space, that quantity is something like $i$ in that it squares to $-1$ and thus, the rotation-like operation is rotation and the exponential is trigonometric in nature.
In Galilean space, that quantity is instead some $\epsilon$ such that $\epsilon^2 = 0$, and the exponential power series degenerates to $\exp(\epsilon) = 1 + \epsilon$.  This generates the simple velocity-addition law of Galilean relativity.
In Lorentzian space(time), the fundamental quantity is some $j$ such that $j^2 =1 $.  This is how the exponential generates hyperbolic functions; the rotation-like operation is Lorentz boosting.
What is unique to Lorentzian space is the presence of non-zero vectors that are unaffected by rotation-like operations in planes they occupy.  Every vector in Euclidean space is rotated by some angle as long as it lies in the rotation plane.  The same goes for Galilean space.  But Lorentzian space has the lightlike or null vectors, which go along light cones, which are totally unaffected by boosts.
This is just part of what makes Lorentzian space(time) an attractive model for the physical world; it reduces the constancy of the speed of light to a purely geometric phenomenon.  We need not ask ourselves why some object with a given speed should be perceived as having the same speed regardless of our orientation (reference frame).  Choosing Lorentzian space(time) as our model makes the existence of such directions of travel patently obvious.  The question then becomes why light should travel along those directions, which opens up new avenues of exploration for physics.
