# Physical intuition for the Minkowski space?

As the title suggests, I am looking for physical intuition to better understand the Minkowski metric.

My original motivation is trying to understand the necessity for distinguishing between co-variant and contra-varient indices.

For example, the quantity: $$A^{\mu}A^{\mu}$$ Could still be computed in Minkowski space.

I am suspecting the necessity of $$g_{\mu\nu}$$ is to separate the space into two regions, one where $$A^2>0$$ and one where $$A^2<0$$. These corresponding to timelike and spacelike regions, which are further dependant on the specification of a constant $$c$$.

For example, I would interpret a metric

$$m_{ij}=\begin{bmatrix} 2 & 0 \\ 0 & 1\end{bmatrix}$$

in the Euclidian $$[x,y]$$ basis, to be a space where one dimension grows twice as fast as the other. Alternatively,

$$m_{ij}=\begin{bmatrix} y & 0 \\ 0 & 1\end{bmatrix}$$

again in the Euclidian $$[x,y]$$ basis, would be a space where the $$x$$ dimension changes proportional to the location in the $$y$$ dimension.

How could the Minkowski space metric,

$$m_{ij}=\begin{bmatrix} -1 & 0 \\ 0 & 1\end{bmatrix}$$

now in the basis $$[t,x]$$, be interpreted?

As someone pointed out in the comments, this could be viewed as a mapping $$t \to -t$$, but this doesn't seem to be the full picture.

• Your example with $\begin{bmatrix} 2 & 0 \\ 0 & 1\end{bmatrix}$ is physically undistinguishable from the ordinary Euclidean plane space with matrix $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Its just that whoever is in charge of placing the distance markings on the $x$ axis changed their mind and decided to place them twice as densely. Commented Jul 5 at 13:17
• Contravariance and covariance can be physically motivated even purely within the Newtonian mechanics framework. It had simply been hidden from students. Commented Jul 5 at 13:20
• Note that the components of the metric depend not only on the geometry of the space but also on the coordinate system. Your example is simply Minkowski spacetime with a different coordinate system. You have not changed anything at all. In order to specify a tensor, you need to supply both components and basis. Simply writing down components does not adequately specify a geometry. Commented Jul 5 at 13:43
• @naturallyInconsistent how are contra and co variance naturally motivated in a Newtonian framework ? Commented Jul 5 at 14:30
• @user10709800 They are mathematical concepts and do not arise out of physics. Commented Jul 5 at 15:24

I don’t think that people can give you intuition. Intuition is something that you develop through experience. Since we basically live non-relativistic lives, such experience is gained through working problems, drawing spacetime diagrams, and so forth.

That said, here is how I think of things.

First, the motivation: experimentally we see that the speed of light is frame invariant. This means that if $$-c^2 dt^2+ dx^2+dy^2+dz^2= 0$$ then $$-c^2 dt’^2+ dx’^2+dy’^2+dz’^2=0$$ We notice that this looks a lot like the Pythagorean theorem except that time is also a part of it. So we make an intuitive leap that maybe space and time are part of a united spacetime and the quantity is a sort of spacetime “distance” $$-c^2 dt^2+ dx^2+dy^2+dz^2=ds^2$$

With that leap, we notice that the above quantity has a lot of desirable properties:

1. it has positive (spacelike) $$ds^2$$ values which are measured by rulers and negative (timelike) $$ds^2$$ values measured by clocks.
2. the timelike values are separated into future and past loci that are frame invariant. This gives an invariant causal structure to time. There is no similar separation in spacelike loci.
3. it has 0 (null or lightlike) $$ds^2$$ values which all frames agree on.
4. Newton’s first law is easily represented by straight lines in this spacetime. The transform between inertial frames is an affine transform.

For me, these desirable properties are enough to give some intuition. By understanding why we like this sort of structure, I understand how it behaves a little.

The distinction between covariant and contravariant quantities is not strong in flat (Minkowski) spacetime. You do need it in curved spacetime.

But the general meaning is not particularly illuminating: at each point your coordinates can be considered to establish a set of basis vectors or a set of basis planes perpendicular to those vectors. The contravariant components are projections onto the vectors and the covariant components are projections onto this planes. They don’t really form different physical quantities, but different ways of representing the same quantity. I.e. referencing the basis vectors or the basis planes.

Anyway, intuition is built by experience. So just get busy solving problems and actually using them.