Special Relativity

Nov 20, 2020

⚠️ Post in development but still contains useful information.

This is a semi-poetic short book from 2018 written by a real physicst and is very recommended. The main thing we learn from it is that there is no absolute order of events. You cannot always meaningfully say if something happened before or after something else.

The Order of Time
Time is a mystery that does not cease to puzzle us. Philosophers, artists and poets have long explored its meaning while scientists have ...

I stop and do nothing. Nothing happens. I am thinking about nothing. I listen to the passing of time.

This is time, familiar and intimate. We are taken by it. The rush of seconds, hours, years that hurls us toward life then drags us toward nothingness … We inhabit time as fish live in water. Our being is being in time. Its solemn music nurtures us, opens the world to us, troubles us, frightens and lulls us. The universe unfolds into the future, dragged by time, and exists according to the order of time.

In Hindu mythology, the river of the cosmos is portrayed with the sacred image of Shiva dancing: his dance supports the coursing of the universe; it is itself the flowing of time. What could be more universal and obvious than this flowing?

And yet things are somewhat more complicated than this. Reality is often very different from what it seems. The Earth appears to be flat but is in fact spherical. The sun seems to revolve in the sky when it is really we who are spinning. Neither is the structure of time what it seems to be: it is different from this uniform, universal flowing. I discovered this, to my utter astonishment, in the physics books I read as a university student: time works quite differently from the way it seems to.
First page of the book The Order of Time by the physicist Carlo Roveli

Why cannot we always say that something happened before, after or at the same time as something else?

Imagine two events that occur "close together" (we usually mean one shortly after the other):

Two events: event a and event b.

We are used to thinking that we can always determine with absolute certainty (with enough masurement) which one happened first. This is untrue.

If events are also rather far apart in space, then we can be even less certain which one happened first. The reason is that even the Nature does not now. There is no notion of absolute ordering of events.

What is time?

Time does not exist as a separate quantity outside of our regular 3D space. Time is a part of 4D space-time fabric.
We are not yet concerned with the curvature of space time. This would be a part of General Relativity (1915) which expands on Special Relativity (1905). We can gladly forget about any curvatures and be only concerned with nice, usual, always perpendicular dimensions in whatever coordinate system "we invent from scratch" in the following paragraphs.

Time as a special 4th dimension just sounds abstract but is one of the simplest things to understand at the conceptual level with just a little bit of step-by-step effort! Distributed systems make this topic very relevant and worth (re)learning. Remember that any abstract ideas you heard about any proven theory turn out not to be so abstract but in fact very exact when looked at a closer mathematical theory. So far this mathematical theory has been proven to hold to any imaginable precision we are currently capable of measuring as a human race. So, special relativity can be considered as a valid, buletproof theory that survived more than 115 years of attacks on it by groups of  some of the smartest people in our galaxy.

Partially ordered sets of events

If there is no absolute ordering of events, what ordering there is then, there must be some order, right? Yes! Definitely, the answer is called partial ordering of events, after reading the book and / or the Time, Clocks and Ordering of Events paper you will have a clearer notion of what is meant by this, we will not go into details on this topic here.

The basics of Space-Time construct

We can give the basic intuition about space-time though so that partial ordering of events is clearer once you read the basic material about it and take some time for thinking. You will soon have an 'aha' moment with enough persistence and light mental effort.

Why do we think about time as something separate from usual space we find ourselves in? It is because it seems to tick the same for everyone and anywhere we go. This turned out to be untrue, especially when you start going very high speeds, nearing the speed of light which is

\[ v_{light}=c=300.000 km/s \]

We don't usually travel that fast and so we don't know how the world looks from this perspective. But how is this connected with relative time?

Scientist have discovered by careful measurements and also some initial thought experiments that whatever speed anyone (particle, person, etc.) travels with, they will always measure the same speed of light. This is not how speeds of regular objects behave. If a train goes by with 40 km/h and you run besides the tracks with 5 km/h, you will measure the speed of train to be 35 km/h. If the train was a light beam, you would still measure it at exactly 40 km/h even if you bicycled by the tracks with 35 km/h! As you tried to approach 40 km/h you would get heavier and heavier until you couldn't pedal anymore so that you could never be the same speed as the "light-speed" train. This mass increase is not very relevant for our observations though, let's go to basic algebra.

If you take this measurement / discovery:

The speed of light is always the same (c)onstant for all ovservers, no matter how fast they move.

as an axiom / fact / basic truth then what follows is that the only quantity from equation \( v = distance / time \) that can expand or contract to compensate for stubborness of the \( c \) constant is (\ time\).

Every object has its own local clock instead of one absolute time ticking "somewhere outside of the system". The basic notion is not time but order of events (as far as we can distinguish it).

What we can do already is to build the intuition of time as a part of 4D space-time construct instead of something existing outside of our usual 3D space.

This is actually rather easy if you drop the notion that space has to always be 3D.

You know about 2D space which is a sheet of paper for example.

Drop one dimenstion lower: what is 1D space? It is this:

1-dimensional space

Fine! There are absolutely no dilemmas here. This kind of space is still infinite (in both directions) but only as long as it stays on this one line.

Since we are still writing on 2D medium (sheet of paper of computer screen), we have one extra free dimenstion available to us. Have you noticed?

There is no space where these circles are positioned:

Circles are outside of our defined space

So we placed the circles outside of our space. We said we have one extra dimension though. What if we placed the other dimension like this (perpendicular to our space dimension):

What should we call this new dimension?

Let's call it time! This would be a great idea.

What can we call our circles now that they are finally part of something? And let's call this something: a space-time construct.

Let's call circles events, this would be yet another great idea. It just seems so fitting.

Events always happen in space and also in time. We are already familiar with that.

This space-time diagram now represents a snapshot of various events.

How do we represent communication or rather travelling of events (information) through space-time?

By arrows of course. Let's make them wiggly (or at least dashed ;), this is a great idea from the 1978 paper Time, Clocks and the Ordering of Events in a Distributed System discussed in:

Distributed Systems
My plan with this post is to give the initial pointers for everyone’s individualfurther research if they are so inclined. I’m clarifying that I’m learning allof this in a more formal way myself and am inviting other previouslyinexperienced (younger?) programmers and thinkers in this area to consi…
Traveling of events. Or rather causal relation.

Here is the secret: events don't travel anywhere. They just happen. The effects do travel, they are called messages. Causal relation somehow connects two events. In physics we think about messages that could have be sent while in the basic theory of Distributed Systems we are only concerned and try to order in time the messages that actually were sent. We are not concerned about ordering the internal events inside different processes that have no effect (and don't produce any messages) that have to travel to the other processes. As far as each process is concerned there is no relation order of events for such events. The notion happened before does not apply. The messages that we do receive though mostly have to be in guaranteed order or we can end up with inconsistend state. Read more about the concept of state in this Distributed Systems course installment:

State Machines
This and the next post [/blog/distributed-systems] are the longest posts of our Distributed Systems series. The first one about Computation and Mathematics[/blog/computation-and-mathematics/] was the shortest. We plan to stabilizesomewhere in between as we progress. This post also touches almost …

Here is an example of a space-time diagram from the mentioned Turing award winner Leslie Lamport's paper:

space-time diagram for three spatially separated processes P, Q and R communicating by sending messages. Horizontal direction is space, verical direction is time.

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