March ?, 2020. Stuff
Contents
1. The parable of the surveyors
I like to imagine that, a thousand years before Pythagoras, Mesopotamian surveyors trying to measure out parcels of land discovered a curious fact. For two measuring sticks at right angles, and measurements $\Delta x$ and $\Delta y$ as shown below, they found that a particular combination $\Delta x^2 + \Delta y^2$ never changed, however they oriented their sticks. In other words,
Wherever you place your rulers, $\Delta x^2 + \Delta y^2$ is the same.
This quantity is naturally interpreted as length squared:
\[d^2 = \Delta x^2 + \Delta y^2,\]so the surveyor’s have discovered that length is preserved when we move our system of measuring sticks. This is an empirical rather than a mathematical fact! This may be surprising, given that Pythagoras’ theorem, $a^2 + b^2 = c^2$, is perhaps the one mathematical fact that everyone knows! And indeed, you can prove it mathematically.
But this proof only works on a flat, two-dimensional plane. So really, we should view the empirical observation that $\Delta x^2 + \Delta y^2$ is invariant as evidence that, to good approximation, the surface of the earth is a flat plane.
1.1. Rotating rulers
Our two surveyors now measure the same length of fence, but with differently-oriented measuring sticks. For simplicity, let’s assume that the stick systems are identical, and all meet at a point, as well with one end of the fence.
The first surveyor takes a pair of measurements $(x, y)$, while the second surveyor takes $(x’, y’)$. Given that
\[\Delta x^2 + \Delta y^2 = (\Delta x')^2 + (\Delta y')^2,\]how are these two sets of measurements related? It will depend on the relative orientation, so we need a way to gauge this. Here is a simple and familiar option. We rotate one measuring-stick system anti-clockwise into the other, and trace out the end of one stick with a piece of string. If the length of the string is $d_1$, and the length of the stick is $d_2$, then we can define the “relative orientation parameter”
\[\theta = \frac{d_1}{d_2}.\]Of course, this is just an elaborate way to describe the angle, measured in radians. In fact, we can choose a system of units where $d_2 = 1$, and the stick has unit length. As we rotate the stick through a full revolution, we can label the endpoints as a function of $\theta$:
\[x(\theta) = \cos(\theta), \quad y(\theta) = \sin(\theta).\]Since the stick’s length is $1$, these two functions obey, for any value of $\theta$,
\[1 = x^2 + y^2 = \cos^2(\theta)+\sin^2(\theta).\]With these tools in hand, we can finally determine how the measurements $(x, y)$ and $(x’, y’)$ are related. Say the relative orientation is $\theta$ if the $x$-axis and $y$-axis are rotated clockwise by $\theta$ to get the $x’$-axis and $y$-axis. A point a distance $d$ along the $x$-ruler, i.e. with coordinates $(x, y) = (d, 0)$, subtends an anti-clockwise angle $\theta$ in the new coordinates. It follows from our definitions that
\[x' = d \cos \theta = x \cos \theta, \quad y' = d \sin \theta = x \sin \theta.\]Similarly, we can see that a point with coordinates $(x, y) = (0, d)$ becomes
\[x' = -d \sin\theta = -y \sin\theta, \quad y' = d \cos\theta = y \cos\theta.\]For a general point $(x, y)$, we just add these two together (demonstrated in the picture) yielding
\[x' = x \cos(\theta)+y \sin(\theta), \quad y' = -x \sin(\theta) + y \cos(\theta).\]We can check that the length is unchanged:
\[\begin{align*} (x')^2 + (y')^2 & = \left[x \cos(\theta) +y \sin(\theta)\right]^2 + \left[-x \sin(\theta) + y \cos(\theta)\right]^2 \\ & = (x^2 + y^2) \left[\cos^2(\theta) + \sin^2(\theta)\right] + 2xy \left[\cos(\theta)\sin(\theta)-\sin(\theta)\cos(\theta)\right]\\ & = (x^2+y^2) \cdot 1 + 2xy \cdot 0 \\ & = x^2 + y^2. \end{align*}\]Let’s summarise the story so far. Surveyors observe that $\Delta x^2 + \Delta y^2$ is unchanged, however they choose to orient their measuring sticks. This suggests they model the land as a flat plane, since in this setting, they can prove the invariance from Pythagoras’ theorem. Finally, by thinking more carefully about relative orientation, they can precisely relate measurements in their different measuring-stick systems. This may seem like an odd way to think about plane geometry, but it will pay off when we come to study spacetime!
1.2. Trip-planning
We know that Mesopotamian surveyors really did discover Pythagoras’ theorem. At this point, however, we will depart from the historical record and speculate about what other discoveries the surveyors might have made.
A single side trip
Imagine that our two surveyors are commissioned to plan roads between two Mesopotamian cities, Assur ($A$) and Babylon ($B$). The first surveyor thinks a straight line between $A$ and $B$ is shortest, but the second argues that a side trip to Charax Sidae ($C$) can provide a shortcut. Without even knowing where these cities are in relation to one another, the first surveyor can prove the straight line is shortest using their newfound knowledge of circles.
For convenience, they orient measuring sticks so that Assur $A$ is at the origin and Babylon $B$ on the vertical stick. The first surveyor then draws circles from $A$ and $B$ which pass through $C$, using long ropes with ends fixed at $A$ and $B$ respectively. This lets us compare the length $AB$ (the length of the straight line from Assur to Babylon) to the lengths $AC, BC$ traversed when you travel via Charax. The circles overlap with some length $L$, in which case
\[AC + BC = AB + L.\]It’s clear geometrically that $L$ is always non-negative, so that $AB \leq AC + BC$, and $L = 0$ just in case Charax Sidae lies directly between Assur and Babylon.
Multi-stop trips
The second surveyor has an even crazier suggestion. Although a single side trip cannot decrease the length, they propose that a sequence of $n$ intermediate stops, $C_1, C_2, \ldots, C_n$, can conspire to somehow become shorter than the direct route. The first surveyor disagrees once more, and argues against the multi-trip proposal as follows. If we visit cities in order, the total length is
\[AC_1 + C_1C_2 + \cdots C_{n-1}C_n + C_nB.\]We can make the trip from $A$ to $C_2$ shorter by eliminating a stop at $C_1$, since we have already established that the direct route between two locations is shorter than the route with a single side trip. Thus,
\[AC_1 + C_1C_2 \leq AC_2.\]But once we’ve eliminated $C_1$, we can repeat this reasoning for $A$ and $C_2$, then $A$ and $C_3$, and so on, until we have no cities left except for Assur and Babylon. An extreme version of this is a curved path, which consists of infinitely many intermediate stops. I won’t prove it formally, but hopefully it’s clear that such a path is always longer than the straight line as well. Putting it all together, we learn a basic fact about route planning:
Any departures from a straight line increase length.
1.3. Over the mountains
The second surveyor says to the first: “Your reasoning is all well and good for a completely flat plane, but there are mountains between $A$ and $B$. It’s not a plane at all! The side trip to $C$ avoids them, and in general, we may have to make many departures from a straight line in order to minimise the actual time taken to travel from $A$ to $B$.” The second surveyor’s proposals weren’t so crazy after all! The two surveyors now have a much harder problem before them: what do the shortest paths between $A$ and $B$ look like when there are mountains? And how long are these paths?
The surveyors go out into the mountains with some pieces of glass that allow them to magnify the ground. They discover an important fact: when you zoom in with a magnifying glass, even the mountains look flat. On a small, flat patch of mountain, the first surveyor’s reasoning works, and straight lines gives the shortest distance between two points. A local property is one that holds when you zoom in enough, and a curved surface is one which isn’t flat. So the surveyors have learned the following important fact:
A curved surface is locally flat.
This actually allows them to determine which paths are shortest, and even how long they are! To see why, consider the shortest route between $A$ and $B$ on a curved surface. Pick any intermediate locations visited on this route, say $C_1, C_2, \ldots, C_n$, in order of visiting. Not only does the shortest route tell us how to get from $A$ to $B$ as quickly as possible, but also tells us the shortest route between intermediate locations! If it didn’t, and we took an unnecessarily long path from $D_1$ to $D_2$ for instance, we could always replace this segment of the path with the shortest route from $D_2$ to $D_3$ and reduce the overall length.
Let’s pick many, many points $D_i$, and place them close enough together that every pair of successive points $D_i$ and $D_{i+1}$ is contained in a small, flat patch. The shortest path from $A$ to $B$ also gives the shortest path between each pair $D_i$ and $D_{i+1}$, and as the surveyors realised, the shortest path in flat patch is just a straight line. This leads to the key result about lengths on curved surfaces:
Shortest paths on a curved surface are locally straight.
In mathematics, these locally straight paths are called geodesics. The distance between $A$ and $B$ is just the length of the shortest path, which as you might expect, is just the sum of the lengths of small line segments making up the path:
\[d(A, B) = AD_1 + D_2D_3 + \cdots D_{n-1}D_n + D_nB.\]In general, this will only be an approximation to the true length, but we can make the approximation as good as we like by picking more and more points. (In the limit of an infinite number of points, the sum becomes an integral, and the length is exact, but we won’t worry about these technicalities.)
2. The surveyors of spacetime
In Newtonian physics, time and space are independent. You can rotate your measuring stick in different directions, but you can never rotate a measuring stick into a clock! In Newton’s own and beautiful words:
Absolute, true and mathematical time, of itself, and from its own nature, flows equably without relation to anything external.
This was the official view until Einstein came along, $200$ years later. By combining empirical observation and mathematical reasoning, just like the Mesopotamian surveyors above, which can recover some of Einstein’s radical insights into the nature of spacetime.
2.1. Einstein’s outrageous proposal
The laws of electromagnetism predict that, in vacuum, light travels at the speed
\[c = 3.0 \times 10^8 \text{ m/s}.\]Most physicists of the 19th century were convinced that light had to travel through a medium, which they called aether, in the same way that sound travels through air. If you move relative to the aether, you will measure a different speed of light.
But the laws of electromagnetism make no mention of the aether, or relative motion. Albert Einstein viewed these laws as so beautiful, and self-evidently complete, that if they made no mention of the aether then none was required. Since the earth should move relative to the aether (it executes circular motion, and the aether presumably does not), Michelson and Morley were able to devise a clever experiment to detect differences in the speed of light in different directions. Famously, they found no difference at all. This also suggests the aether does not exist.
But if there is no aether, and no change in the speed of light as we move at different speeds, we are left with an outrageous proposition:
The speed of light is the same in all reference frames.
This smells very similar to the statement that length does not depend on where we place our measuring sticks. We can mathematise accordingly! Instead of two measuring sticks at right angles, we’ll measure length in a single direction with a ruler, and duration with clocks attached to the ruler.
To calculate the speed of light, we shoot a laser beam along the ruler, with two points separated by $\Delta x$, and time how long it takes using the clocks. Einstein’s outrageous proposal is that, provided the ruler is moving at constant velocity (an inertial reference frame), we always find that
\[c = \frac{\Delta x}{\Delta t}.\]To make things simpler, let’s choose a system of units in which $c = 1$. This just corresponds to choosing a scale on the graph below, where a light ray travels a single horizontal unit of distance in a single vertical unit of time:
Now, Einstein’s proposal can be written
\[1 = \frac{\Delta x}{\Delta t} \quad \Longrightarrow \quad \Delta x^2 - \Delta t^2 = 0\]for a light ray in any inertial reference frame. This should remind you of the discovery of the surveyors, namely that $\Delta x^2 + \Delta y^2$ does not depend on where we put the measuring sticks. It’s clear that Einstein’s outrageous proposal follows from a more general invariance result:
Whatever the speed of your ruler-clock system, $\Delta x^2 - \Delta t^2$ is the same.
2.2. Spacetime rotations
Just as two surveyors measuring the same length of fence may want to relate their measurements, two spacetime surveyors measuring the time and distance between two events may want to relate their measurements. For the Mesopotamian surveyors, “relative orientation” meant a regular old rotation, which we labelled with points on the unit circle:
\[x^2 + y^2 = 1.\]Now, “relative orientation” will be labelled by points on the unit hyperbola (in the north-south quadrants):
\[t^2 - x^2 = 1.\]Recall that angle (in radians) was the length of the arc on the circle. Here, we instead use the distance $\eta$ measured along the graph from the time axis, and rather than sine and cosine, we define the hyperbolic sine and cosine:
\[t(\eta) = \cosh(\eta), \quad x(\eta) = \sinh(\eta).\]By definition, for any $\eta$ these functions satisfy
\[1 = t^2 - x^2 = \cosh^2(\eta) - \sinh^2(\eta).\]We can parameterise a hyperbola $t^2 - x^2 = \tau^2$ by scaling up the hyperbolic functions:
\[t(\eta) = \tau \cosh(\eta), \quad x(\eta) = \tau \sinh(\eta).\]With circles, we imagined rotating the $x$ coordinate into the $x’$ coordinate by an anti-clockwise rotation through $\theta$, which we could perform by simply shooting the $x’$ axis through an appropriate point on the unit circle. Now, we will “spacetime rotate” in the same way. We pick a point $\eta$ on the unit hyperbola, and send the new $t’$-axis through that point. This maps a point on the $t’$-axis $(x’, t’) = (0, \tau)$ to a point with
\[x = \tau \sinh(\eta) = t' \sinh(\eta) , \quad t = \tau \cosh(\eta) = t' \cosh(\eta).\]To complete the transformation, we need to know where the $x’$-axis lies. Define the east-west hyperbola $x^2 - t^2 = 1$. It is plausible that $x’$ should lie at $\eta$ above or below the $x$-axis, with a point $(x’, t’) = (d, 0)$ having coordinates
\[x = d \cosh(\eta) = x' \cosh(\eta), \quad t = \pm d \sinh(\eta) = \pm x' \sinh(\eta), \quad.\]We can’t really tell whether $+$ or $-$ is correct just yet, so let’s plow on. We can combine these two special transformations to obtain the general case:
\[x = x' \cosh(\eta) + t'\sinh (\eta) , \quad t = t' \cosh(\eta) \pm x' \sinh (\eta).\]Now, we simply choose $\pm$ to ensure $x^2 - t^2$ is preserved:
\[\begin{align*} x^2 - t^2 & = \left[x' \cosh(\eta) + t' \sinh(\eta)\right]^2 - \left[\pm x' \sinh(\eta) + t' \cosh(\eta)\right]^2 \\ & = (x^2 - t^2) \left[\cosh^2(\eta) - \sinh^2(\eta)\right] + 2xt \left[\cosh(\eta)\sinh(\eta)\mp \sinh(\eta)\cosh(\eta)\right]\\ & = x^2 - t^2 + 2xt \cosh(\eta)\sinh(\eta) (1 \mp 1). \end{align*}\]We need to pick the upper sign $-$ in $\mp$ to make sure we respect Einstein’s outrageous proposal, and this corresponds to the upper sign $+$ in $\pm$. Hence,
\[x = x' \cosh(\eta) + t'\sinh (\eta) , \quad t = t' \cosh(\eta) + x' \sinh (\eta).\]
On the spacetime plane, this looks as follows:
2.3. Turning rulers into clocks
Although we have all the mathematics we need to relate measurements on the spacetime plane, we haven’t explained what a spacetime rotation really is!
2.4. The Twin Paradox
Xtra
Higher dimensions
First, let’s imagine that our destinations are located in three-dimensional, rather than two-dimensional, space. The extension actually works for higher dimensions, but we’ll stick to three for ease of visualisation.
As before, I want to compare a direct route from $A$ to $B$ to the route via the candy store $C$. In two dimensions, we drew circles, but in three dimensions, the set of points at a fixed distance from $A$ or $B$ is a sphere. So it seems like we’ll have to draw some complicated picture with spheres. Thankfully, this is unecessary! We can simply work in the plane containing $A$, $B$ and $C$. Three points are always contained in a plane, so this is a reasonable thing to do. We then draw circles, and the argument above goes through, word for word.
Since our result for multiple candy stores (including infinitely many!) simply iterated the result for a single candy store, we can do the same iteration here. The quickest route between $A$ and $B$ in three (or indeed any number of) dimensions is the direct route, and any additional stops make it longer.
Other
When you rotate a stick, it doesn’t magically change length, right? In fact, you can think of this as a definition of length: it is the thing that doesn’t change under rotations. This just seems to be a basic physical fact about the universe. Here is a setup for measuring sticks. Take two identical rulers and put them at right angles. Call measurement along the first ruler called $\Delta x$, and the measurement along the second ruler $\Delta y$. We picture this below:
You find that, no matter how you orient the stick, these two measurements obey
The constant $\ell$ is the length of the ruler, and this result is of course just Pythagoras’ theorem. But it is telling us an important fact:
The quantity $\Delta x^2 + \Delta y^2$ is preserved by rotations.
When we place the end of the stick at the point the rulers meet (the “origin”), this becomes $x^2 + y^2$, and the equation $x^2 + y^2 = \ell^2$ describes a circle of radius $\ell$. Of course, this has to be the case, since geometrically speaking, this equation describes what happens when you rotate a stick of length $\ell$ with one end fixed.
There is a convenient way of referring to rotations using circles. Imagine a stick of unit length, $\ell = 1$, revolved around the point our rulers meet, so it obeys
\[x^2 + y^2 = 1.\]For some rotation, use a tape measure to find the length of the arc from the $x$ ruler to the end of the stick, and call this length $\theta$. This is the angle measured in radians, rather than degrees. We can view $x(\theta)$ and $y(\theta)$ as functions of $\theta$, and give them special names:
\[x(\theta) = \cos(\theta), \quad y(\theta) = \sin(\theta).\]It follows immediately, just from the way we’ve defined things, that
\[\cos^2(\theta) + \sin^2(\theta) = 1.\]For a circle of radius $\ell$, we can still refer to angles using $\theta$ — we just choose a ruler for which the circle has unit radius, and measure $\theta$ as before. Returning to the original ruler, where the circle has radius $\ell$, we will find that
\[x(\theta) = \ell \cos (\theta), \quad y(\theta) = \ell \sin (\theta),\]and we can check, from our earlier observation about sine and cosine, that $x^2 + y^2 = \ell^2$.
The point of having a nice way to discuss rotations is not so much because we want to rotate sticks. Rather, we want to rotate rulers. Imagine that I have some measurements $(x, y)$ on my pair of rulers. Now, you have a second pair of rulers which meet at the same point, but are rotated through some angle $\theta$. Then your measurements, call them $(x’, y’)$, will be related to mine as follows:
\[x' = x \cos(\theta) + y \sin(\theta), \quad y' = -x \sin(\theta) + y \cos(\theta).\]Length is indeed unchanged when we rotate rulers:
\[\begin{align*} (x')^2 + (y')^2 & = \left[x \cos(\theta) + y \sin(\theta)\right]^2 + \left[-x \sin(\theta) + y \cos(\theta)\right]^2 \\ & = x^2 \left[\cos^2(\theta) + \sin^2(\theta)\right] + y^2 \left[\sin^2(\theta) + \cos^2(\theta)\right] + 2xy \left[\cos(\theta)\sin(\theta)-\sin(\theta)\theta(\theta)\right]\\ & = x^2 \cdot 1 + y^2 \cdot 1 + 2xy \cdot 0 \\ & = x^2 + y^2. \end{align*}\]Let’s summarise the story so far. For physical reasons, rotating something shouldn’t change its length. Pythagoras’ theorem tells us that
\[\Delta x^2 + \Delta y^2\]doesn’t change under rotations. Thus, we can define the Pythagorean length by
\[d_\text{P}(\Delta x, \Delta y) = \sqrt{\Delta x^2 + \Delta y^2}.\]We can draw circles and define trigonometric functions, and use them not only to rotate sticks, but to relate measurements between ruler systems.
Exercise 1 (polar coordinates). We can label a point in a measuring stick system by the two measurements $(x, y)$. But for performing rotations, there is a more convenient system where we label with the distance from the origin and the angle from the $x$-axis, $(d, \theta)$. These are called polar coordinates.
(a) Using the definition of $\sin(\theta)$ and $\cos(\theta)$, show that $(x, y)$ can be expressed in terms of $(d, \theta)$ by
\[x = d \cos\theta, \quad y = d\sin \theta.\](b) Invert the answer in (a) to show that
\[d = \sqrt{x^2 + y^2}, \quad \tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{y}{x}.\]Rotating $(x, y)$ into $(x’, y’)$ by an anti-clockwise rotation through $\alpha$ is called a passive transformation, since we just transform coordinates. But equivalently, we can perform an active transformation where we rotate points, and leave the coordinates alone.
(c) Argue that the active transformation takes
\[(d, \theta) \to (d, \theta - \alpha).\](d) Show