Beauty in Physics - Discovering Special Relativity

Beauty and simplicity as guiding principles

A giant of 20th century physics Paul Dirac said that ‘‘It seems that if one is working from the point of view of getting beauty in one’s equations, and if one has really a sound insight, one is on a sure line of progress’’. It is a curious quote. At schools when learning physics the word “beauty” does not tend to come up, which is unfortunate. Fundamentally physics is an experimental science, and it describes the world as it is. World is a place with huge amount of phenomenon - sun traveling across sky, stars showing up in nights, skies having blue color, water flowing etc etc. It is a huge diversity and one might think that physics could very much be a science like butterfly collecting - describing and cataloging all the phenomenon. But astonishingly - it is not so.

Physics has turned out to be a unifying science. There is no pre-determined reason it should be so. With hindsight we talk about “laws of nature”. Phenomenon like moon phases, seasons, star positions and tides all turn out to be related and connected. The unification of a wide range of phenomenon into a few principles was achieved by Newton, with whom pretty much all modern teaching of ‘‘physics’’ begin with. As if all the disparate phenomenon, like dancing lights in a room, has turned out to be reflections from a single crystal in middle of the room. Simplicity from complexity, beauty from disorder. Due to some reason, nature can be described simply, and the language used to described the nature elegantly is mathematics. If nature yields itself to elegant descriptions with mathematics, then one might be inclined to use that as guiding principle in guessing the ‘’laws of nature’’. So without further ado, lets do so.

Point of view

As starting point for physics, we need to assign some numbers to describe phenomenon. Most simple thing to describe is a point which moves. The point could represent a swing swinging, or spaceship flying in space, or anything else. The point has four numbers associated with it - three numbers describing its position, and fourth number is time. E.g. for example, a tennis ball is in front of player - 1m above, 1m to right, 1m in front, 3600 seconds after the start of match. Notice, that position is specified with respect to one of the players. For the other player ball can be 1m above, 1m to left, 22m in front. Both sets of numbers describe the same point - but from different players point of view. So it is in physics in general - position numbers of x, y, z, t are specified with respect to some “point of view”, like the tennis player in example. It is important to note, that none of the tennis players are more special than any other - while numbers from each players point of view were different, they described the same point.

As we are doing physics, the laws of physics should not depend on “point of view” - the equation of physics should be the same for everyone. This leads to question - are there other quantities which are not dependent on point of view? Such quantities would be very convenient when formulating our physics equations.

Distance and rotation

Lets call two points of view x (“Bob’s” point of view) and x’ (“Alice’s” point of view). One way we could go from one “point of view” to a different one would be by a translation. So they are related numerically with x’ = x + number. Another way they could be related is by rotation. I.e. if Alice is facing south, while Bob north - Bob could align will Alice by 180 degree turn. Now, what is the distance for a points with coordinates x and y? We know from geometry lessons (“Euclidean” geometry more specifically) that the distance can be calculated with Pythagoras theorem: $$ x^2 + y^2= distance^2 $$ Let say that Alice and Bob is in same location. However, they might be not facing the same direction, so numbers x,y and x’,y’ would be different. Key point is that when Alice calculates the distance with x and y, and when Bob calculates distance with x’ and y’ numbers, they both get the same number for distance. When points of view are related by rotation, the distance measured is the same from all points of view. $$ x^2 + y^2= x’^2 + y’^2 = distance^2 $$ We could prove this with some algebra, but it should be relatively intuitive - e.g if the stick end is at x,y, then if we rotate point of view so that its endpoint is at x’,y’ - the distance the stick is still the same.

To wrap it up - what if the starting point of the stick is not located at the same point as Alice and Bob is standing at? It really changes nothing much but introduces just a bit more notation. If stick’s endpoint value is \(x_{end}\), and stick’s start point is \(x_{start}\) then we can calculate a new number $$ \Delta x = x_{end} - x_{start} $$ And all our above discussion still applies $$ \Delta x^2 + \Delta y^2 = \Delta distance ^2 $$ To make notation a bit more simple, instead of writing “distance” we will write \(\Delta s\) $$ \Delta x^2 + \Delta y^2 = \Delta x’^2 + \Delta y’^2 = \Delta s ^2 $$ Note, that while the \(\Delta x\), \(\Delta y\) and \(\Delta x'\), \(\Delta y'\) are different numbers, the calculated distance of the stick from both points of view is the same.

More on Pythagoras theorem

Shortly on vocabulary on what we mean when saying how many “dimensions” a space has. If we describe a point only with one number (e.g. x), we will say that point lives in “1 dimensional space”. If point is described with 2 numbers (e.g. x and y), we way it lives in “2 dimensional space”, if point is described with 3 numbers, we say “3 dimensional” etc. Now we have talked about Pythagorean theorem, but we only have talked about 2 dimensional case, ie. our stick lives in on some surface, like a table. Two numbers are sufficient for such point. But what is distance to a point in 3D, e.g. if the point is somewhere above the table too. Let’s quickly derive 3D Pythagoras theorem from 2D Pythagoras. Having a look at Fig 2. from the 2D Pythagoras we get $$ \Delta s_{xy}^2 = \Delta x^2 + \Delta y^2 $$ But having a look at Fig 2 we see that sides \(\Delta s_{xy}\) and \(\Delta z\) are two sides of another 2D triangle and we can again apply 2D Pythagoras $$ \Delta s^2 = \Delta s_{xy}^2 + \Delta z^2 $$ Plugging equations together we see that we get a straighfroward extension form 2D to 3D - just an extra \(\Delta z^2\) term shows up in the sum in contrast to 2D Pythagoras theorem $$ \Delta s^2 = \Delta x^2 + \Delta y^2 + \Delta z^2 $$

The leap of faith

Lets summarise Pythagoras for all the dimensions going up to three - as we live in 3D space:

• 1 dimension: \(\Delta s^2 = \Delta x^2\) (which is self evident)
• 2 dimensions: \(\Delta s^2 = \Delta x^2 + \Delta y^2\) (which we could show with a bit of geometry)
• 3 dimensions: \(\Delta s^2 = \Delta x^2 + \Delta y^2 + \Delta z^2\) (which we derived)

There is a clear pattern - as we go to higher dimension, we introduce extra term in the sum for the dimension. Now we are ready to take the leap of faith. We have not yet talked about time, which is another number describing a point - as in our example with tennis ball. We have 3 points describing where ball is in space, and 1 number describing time when. Now, going back to Dirac’s quote, what could be the “point of view of getting beauty in one’s equations”? What is the most simple thing we could try, and not treat “time” number as something more special than “positon” numbers? Well, lets roll with it, go crazy and live a full life, take risks and die young, and do the most obvious thing possible - lets continue extending Pythagoras to 4D by following the pattern above!

$$ \Delta s^2 = \Delta x^2 + \Delta y^2 + \Delta z^2 + a^2\Delta t^2 $$

The x is measured in “meters”, but t in “seconds”, so \(a^2\) is just a constant to allow us to add something measured in “meters” to something measured in “seconds” - the units of “a” is “meters per second”. There is nothing special about it - just getting our units to match. This is just our 4D Pythagoras where one of the numbers is time!

Now, what we have done is clearly crazy. The only motivation was not to treat time number as anything special - and handle it in the same way as we handle position numbers. In daily experience that is not how time seems to be - time feels to be very different from position. For one, all 3 numbers in position we measure with ruler, while time we measure with a clock. We can move physically in space, but we don’t really “move” in time in the same sense - clock just keeps ticking along. But we have started a crazy journy, so lets see if we can interpret what we have done, what are the consequences, and at the end of the day - do the consequences match experimental observations. It won’t be a spoiler to say - that consequences match the obervations!

Nature has indeed turned out to be simple, elegant, but also - bonkers.

A fix

Lets take a point of view, and then shift to other points of view x and t $$ \Delta s^2 = \Delta x^2 + a^2\Delta t^2 = constant $$ Lets consider continuously changing our point of view, with \(\Delta x\uparrow\). As \(\Delta s^2\) must stay constant as we are only shifting point of view, it implies \(\Delta t\downarrow\). Lets say we start with \(\Delta t > 0\). As we continue gradual shift to a different perspective, we arrive at perspective when \(\Delta t = 0\). Now we can continue continuously shift perspective further, and \(\Delta t\) becomes negative \(\Delta t < 0\). This is wacky! When we say \(\Delta t > 0\), it means \(\Delta t = t_{after} - t_{before} > 0\), ie. \(t_{after} > t_{before}\). But we are allowing gradually shift to a perspective from which \(\Delta t = t_{after} - t_{before} < 0\), ie. \(t_{after} < t_{before}\).

This might be too brave. It should not be possible for two points of view diagree about ordering of events in time! If something happened “before” - it must happen “before” from all points of view. Lets see if we can fix this. The problem was that \(\Delta t\) had to go down to keep the sum constant. How could we modify our Pythagoras that we can keep the sum constant, even when \(\Delta x\uparrow\)? We can see that one of the squared terms need a minus sign in front $$ \Delta s^2 = \Delta x^2 - a^2\Delta t^2 = constant $$

To write out our 4D Pythagoras in its full final form $$ \Delta s^2 = \Delta x^2 + \Delta y^2 + \Delta z^2 - a^2\Delta t^2 $$

Rotate in time

We know what a normal rotation is in space physically. But what does it mean to "rotate" when one of the directions is time? Lets consider a perspective of an ant, who is standing on a tennis ball and has a "wristwatch". As much as ant is concenred, the location of the teniss ball is not changing, so in this perspective \(\Delta x' = 0\), while \(\Delta t' > 0\) as measured by its wristwatch. Lets shift to a different perspective as we have done above. In this perspective \(\Delta x\) will not be zero anymore. Lets call this perspective Alice's. $$ \Delta s^2 = \Delta x^2 - a^2\Delta t^2 = - a^2\Delta t’^2 $$ In this perspective the ant has moved distance \(\Delta x\) in time \(\Delta t\). Ie. the ball on which ant is standing on is moving away from Alice with velocity $$ \frac{\Delta x}{\Delta t} = v $$

We now have an intepresaton of what “rotation” means when one of the direction of rotation is time. To “rotate” perspective with one coordinate being time, is to shift to point of view which has a velocity v.

Some consequences

Lets continue with Alice throwing the ball to the right, and she measures continously the position of the ball, and the time shown by her watch. Lets also imagine an ant on tennis ball standing on it. For the ant, the position of the ball in not changing, as ant is on the tennis ball, however the ant’s “wristwatch” is still ticking. So going to our 4D Pythagoras theorem $$ \Delta s^2 = \Delta x^2 - a^2\Delta t^2 = -a^2\Delta t’^2 $$ As for the ant \(\Delta t' > 0\), then $$ \Delta x^2 - a^2\Delta t^2 = -a^2\Delta t’^2 \le 0 $$ $$ \Delta x^2 - a^2\Delta t^2 \le 0 $$ $$ \Delta x^2 \le a^2\Delta t^2 $$ $$ \frac{\Delta x}{\Delta t} \le a $$ Term on the left is just velocity (ie. distance travelled per time). We have discovered physical meaning of “a” - it is the maximum speed limit of how fast an object can travel at! Now, this is counter intuitive. E.g. if Alice was travelling on train, and threw a tennis ball, then one would intuitively think that the person on the platform would measure the speed of the ball to be “speed of Alice” + “speed of ball”. But turns out that this is not right - if “speed of Alice” was 90% of “a” and ball’s speed was 90% of “a”, then the total speed of ball according to law above would be \(90\% \times a + 90\% \times a = 180\% \times a\) . But this is wrong. Nothing can travel faster than “a”, thus the intuitive “add velocities rule” is wrong. Having a maximum travel speed matches experimental observations.

As second example, lets imagine the ball is moving aways from Alice, and ant is still sitting on the ball. The speed of the ball is $$ \frac{\Delta x}{\Delta t} = v $$ $$ \Delta x = v\Delta t $$ Then plugging in our equation relating ant’s and Alices perspectives $$ \Delta s^2 = \Delta x^2 - a^2\Delta t^2 = -a^2\Delta t’^2 $$ $$ v\Delta t^2 - a^2\Delta t^2 = -a^2\Delta t’^2 $$ $$ \Delta t^2(a^2 - v^2) = a^2\Delta t’^2 $$

Now, notice that when \(v \ge 0\), then \(\Delta t \ge \Delta t'\). When ant has velocity v, then for ant time is passing slower than for Alice! Literally. For Alice it would seem as if ant is moving in slow motion. This matches experimental observations.

Note also how regular world reemerges. Lets consider that objects velocity is much, much slower than “a” (“\(\ll\)” means “much smaller than”). $$ v \ll a $$ $$ \Delta t^2(a^2 - v^2) = a^2\Delta t’^2 $$ $$ a^2\Delta t^2 \approx a^2\Delta t’^2 $$ $$ \Delta t \approx \Delta t’ $$

When velocity is low, then for Alice and ant the time is passing in same manner. This matches the normal intuition - time passes in same manner for all.

Hint of General Relativity

When we talk about 2 dimensions, we can imagine it as some sheet. One direction on sheet is e.g. “x” and perpendicular direction to it is “y”. We can pick any two points on sheet, and measure distance \(\Delta s\) between them. See Fig 3. If an ant is on this sheet, it can use normal 2D Pythagoras theorem to relate distances on this sheet. Now, we can imagine in thee dimensions instead of sheet, there is some sort of volume, and again we can measure \(\Delta s\) between any two points in this space. Lets relax our head a bit, and keep moving forwards - in similar manner we extend to four dimensions, where the new dimension is time. And as before we can measure \(\Delta s\) between any two points. And physicist in this 4D space can use our a bit funky 4D Pythagoras to relate distances in this space.

Now, the idea of General Relativity is that things which, e.g. have mass, stretch this 4D space. Imagine the Fig 3. sheet to be made out of rubber. Then if we applie pressure to some point, the sheet gets stretched. Note, it will affect the distances between nearby points in comparison with unstreched sheet. Thus so in general relativity - where e.g. objects with mass stretch the 4D spacetime “sheet”. Then the motion of physicist in this 4D space will be affected by this distortion. This distortion is what we call gravity.

Beauty in one’s equations

Pythagoras theorem for space has been known for more then two thousand years, while special relativity has been around only a hundred. Ie. it took almost two thousand years to arrive at this insight that space and time are not separate things, but one - “spacetime”. And it’s most amazing that they join together in almost most natural way imaginable $$ \Delta s^2 = \Delta x^2 + \Delta y^2 + \Delta z^2 - a^2\Delta t^2 $$

Above has been quite a whirlwind. We started with very little - making in a way a simple leap of extending Pythagoras from 2D to 3D, and then to 4D, where fourth dimension is time. This lead us to rich and remarkable results, which have experimentally turned out to be true. Fundamentally physics is experimental science through wich we discover character of nature. And surprisingly due to whatever reason nature turns out to be beautiful.