Minkowski everything -- invariants

Some philosophers often say silly things like "truth is relative" or worse, "relativity implies that truth is relative".

Even before relativity, there would be people who gave obviously insincere explanations of this axiomatically incorrect statement -- e.g. "the number 6 viewed from the opposite direction looks like the number 9, therefore truth is relative" or "some people like doughnuts, some people don't, therefore truth is relative". The answer to these kinds of arguments is "someone who sees the number as 6 agrees the other guy sees it as 9, and vice versa", "someone who likes donuts agrees the other person doesn't". The statement donuts are good is not meaningful, except in terms of the donut-liker's neurobiology -- it's equivalent to saying "when you put a donut in his mouth, dopamine is released in his brain". All observers agree that this is the case with him, it's just that dopamine isn't released in the donut-disliker's brain. These statements of absolute truth are absolute.

Perhaps this gives too much credit to these nonsensical arguments, but the response is similar with relativity. If your parents were bored of raising two children so decided to send your twin brother to Trappist-1 at close to the speed of light, then you would be 80 years old when he returns as a newborn baby. But you do see him as a newborn baby, not an old man, and if you could understand his unintelligible babbling, you would hear that he sees you as an old man on the verge of death, not a kid his age he can play with.

So biological age is an invariant. Even though you see him as having lived 80 years, you also think that his clock moved a lot slower, which is why he's still an infant.

But there's nothing special about human biology or biological clocks. Even if the newborn took a clock with him, the time recorded on that clock is an invariant -- all observers agree on what it is.

Let's try to extract this biological time -- we will call this the "proper time" from the co-ordinate measurements of any arbitrary observer.

We have:

$$\Delta t = \frac{{\Delta t'}}{{\sqrt {1 - {v^2}} }}$$
We write ${\Delta t'}$ as ${\Delta \tau }$, the general proper time according to the moving observer himself.

$$\begin{array}{l}\Delta \tau  = \Delta t\sqrt {1 - {v^2}} \\\Delta \tau  = \sqrt {\Delta {t^2} - {v^2}\Delta {t^2}} \\\Delta \tau  = \sqrt {\Delta {t^2} - \Delta {x^2}} \\\Delta {\tau ^2} = \Delta {t^2} - \Delta {x^2}\end{array}$$
One may check that this result is always invariant by Lorentz-transforming $t$ and $x$ and showing $t'^2-x'^2=t^2-x^2$. In a general orthonormal co-ordinate system of spatial co-ordinates (i.e. we don't necessarily take $x$ to be the direction of motion), we may write:

$$\Delta {\tau ^2} = \Delta {t^2} - \Delta {x^2} - \Delta {y^2} - \Delta {z^2}$$
Note the resemblance to the Euclidean norm/Pythagorean theorem! If only the minus signs were pluses, this would be the Euclidean norm. This norm is called the Minkowski norm, and the proper time $\Delta\tau$ (or sometimes $\Delta s=c\Delta\tau$, which is the same thing when we set $c=1$) is called the spacetime interval.

This equation summarises the non-dynamical results of special relativity, and can be treated as an alternative axiomatic foundation for the theory (the "Minkowskian formulation", as opposed to the Einsteinian one we've been discussing so far) -- it's the Pythagorean theorem on spacetime. Unlike in Galilean relativity, where time and space are individually invariant, in special and general relativity, spacetime is invariant -- time and space simply transform between each other leaving the norm of $(\Delta t,\Delta x,\Delta y,\Delta z)$ invariant. This is indeed a rotation ("skew") of this vector, but in Minkowski spacetime, rotations are across hyperboloids, called invariant hyperboloids (or in 2D, hyperbolae), not spheres (or circles). Changing the observer changes the spacetime vector (called four-position), but doesn't take it off this invariant hyperbola.

Indeed, this means that Minkowski spacetime doesn't have the geometry of Euclidean geometry -- instead, it has a geometry called "hyperbolic geometry", which cannot be embedded in Euclidean space (i.e. we have no way to visualise it).

Here's another possible motivation for studying invariants:
Lorentz boosts are essentially rotations in the t-x plane (hyperbolic rotations, actually, or skews, but stick with the analogy for now), so it's often useful to get an intuitive feel for them in special relativity by comparing boosts to rotations on some other plane, like the x-y plane. So let's do that.

Consider if you were measuring the y-length of a stick on the x-y plane -- clearly, this depends on your frame of reference. A co-ordinate system in which the stick lies on the y-axis clearly gives you the maximum value of this y-length, a co-ordinate system in which it lies on the x-axis clearly gives you a value of 0.

So the specific co-ordinate dimensions $(x, y)$ of the stick depend on your reference frame. But we can also be interested in the real lengths of sticks, because this is invariant in all reference frames. This can be calculated easily using the Pythagorean theorem:

(Note that the invariance is not the only thing that is important, but also that it allows you to define a polar co-ordinate system where $x=\psi\cos\theta$, $y=\psi\sin\theta$.)

If you accept that it can be useful to know the dimensions of objects on their own axes, it's clear that the same principle applies on the t-x plane. Here, the "rotations" are skews, the trigonometry is hyperbolic trigonometry, the Pythagoras theorem is $\tau=\sqrt{t^2-x^2}$ and instead of the proper time being the highest point of a circle it is the lowest point of a hyperbola.

But the same principles still apply -- if you see someone blast a toddler off into outer space at a high speed then return, you might measure the toddler as having taken a hundred years to return, but you and the toddler both agree (assuming he isn't dead yet from starvation) that he's only aged a year. This biological time, or proper time, is an invariant.
(From my answer on Physics Stackexchange to Why is invariance important?)

A related fact is an intuitive explanation for the speed of light being the maximum achievable speed -- all observers have a fixed speed ($ds/d\tau$) through spacetime, which is the speed of light -- this is essentially a tautology. A stationary object has no speed through space, so $dx^2+dy^2+dz^2=0$ so it moves at $c$ through time ("co-ordinate time" $t$ -- as opposed to proper time), i.e. $d(ct)/d\tau=c$. On the other hand, when an object moves at the speed of light, its clock has stopped -- we see $d(ct)/d\tau=0$. The velocity cannot exceed the speed of light, because the object simply doesn't have that much speed -- it doesn't have any more speed to take from its time-speed. Another way of saying this is that an invariant hyperboloid never crosses the light cone.

It's important to keep in mind that in our argument above, time, position and velocity are always with respect to some other observer (again, this is also implied by the Minkowskian formulation, as $dx$, $dt$ etc. are in the frame of some observer). So the point is really that "no observer can see an object going faster than light, because to keep the speed through spacetime fixed, the Lorentz transformation would have to map the time to an imaginary number ($\Delta t^2 < 0$).

We will see later that there are other quantities that transform between each other like time and space. Then we will see that the four-position is just another vector among a class of vectors called four-vectors.

(Note of caution: often, $\Delta s^2$ instead of $\Delta s$ is called the spacetime interval. When you hear the phrase "negative spacetime interval", this is typically what is being referred to.)

(Note: Because both $\Delta s^2$ and $-\Delta s^2$ are invariants, sometimes $- {c^2}d{t^2} + d{x^2} + d{y^2} + d{z^2}$ is called the spacetime interval instead. This choice is called the "metric signature" and is denoted by $(+---)$ and $(-+++)$ respectively. The first is also called the particle physics convention, the quantum field theory convention, the West coast convention, the time-like convention and the mostly-minus convention. The second is also called the cosmology convention, the general relativity convention, the East Coast convention, the space-like convention and the mostly-plus convention. However, $\Delta\tau^2$ is always defined via the time-like convention, as it is the proper time.)

You might be tempted to say that Minkowski spacetime is simply 4-dimensional Euclidean spacetime with one of the dimensions being $ict$ instead of $ct$. However, this doesn't actually make Minkowski spacetime Euclidean -- for instance, Minkowski spacetime allows distinct points in spacetime to have a zero spacetime interval between them, something not possible with a Euclidean distance function. After all, the norm of a complex number $t + ix$ is still $\sqrt{t^2+x^2}$, not $\sqrt{t^2-x^2}$.

You might be tempted to rewrite the equation as $d{t^2} = d{\tau ^2} + d{x^2} + d{y^2} + d{z^2}$. But since $d{t^2}$ is not an invariant, this obscures the true geometry of Minkowksi spacetime, which is hyperbolic, not Euclidean. Similarly, equations like $m^2 = E^2-p^2$ (where $m$, $E$ and $p$ are the proper mass, relativistic mass and momentum respectively -- we will later derive this) should not be written as $E^2=m^2+p^2$.

You might recall some equations in physics that seem to exhibit the same kind of symmetry between space and time as the spacetime interval -- $-c^2t^2$ and $x^2$ showing a symmetry. An example is the wave equation for light, $\frac{1}{c^2}\frac{\partial^2u}{\partial t^2}-\frac{\partial^2u}{\partial x^2}=0$. This is actually the reason why Maxwell's equations are already Lorentz invariant, and indeed, we will see that this symmetry will be our criterion for Lorentz invariance.

(Technical note: Formally speaking, Minkowski spacetime doesn't actually have hyperbolic geometry itself. What it does have are sub-manifolds with a hyperbolic geometry.)

We may divide spacetime intervals into three categories: space-like (outside the light cone), light-like (on the light cone) and time-like (inside the light cone), corresponding to the cases $\Delta s^2<0$, $\Delta s^2=0$ and $\Delta s^2>0$ respectively (in the cosmology convention, it is exactly disrespectively). The fact that you cannot influence space-like separated events, i.e. cannot travel faster than light is the same as saying "you cannot transverse an imaginary proper time".

Saying the speed of light is fixed for all observers is equivalent to saying that the statement $\Delta s^2=0$ is invariant, since $\Delta s= \sqrt{c^2\Delta t^2-\Delta x^2}$ and $x=ct$. We now know that $\Delta s^2=n$ is invariant for all $n$, not just 0.

The image above shows invariant some hyperbolae plotted -- $\Delta s^2=-3$, $\Delta s^2=-2$, $\Delta s^2=-1$, $\Delta s^2=0$, $\Delta s^2=1$, $\Delta s^2=2$, $\Delta s^2=3$. Note how the hyperbolae never cross the light cone -- implying the existence of an absolute future, an absolute past, an absolute left and an absolute right.

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