Understanding Einstein’s special theory of relativity

In imagining a Sci-Fi scenario, I found a simple way of describing, if not explaining, Einstein’s special theory of relativity.

Imagine if a flight to the moon was no different to flying half way round the world in a contemporary airliner. In my scenario, the ‘shuttle’ would use an anti-gravity drive that allows high accelerations without killing its occupants with inertial forces. In other words, it would accelerate at hyper-speeds without anyone feeling it. I even imagined this when I was in high school, believe it or not.

The craft would still not be able to break the speed of light but it would travel fast enough that relativistic effects would be observable, both by the occupants and anyone remaining on the Earth or at its destination, the Moon.

So what are those relativistic effects? There is a very simple equation for velocity, and this is the only equation I will use to supplement my description.

v = s/t

Where v is the velocity, s is the distance travelled and t is the time or duration it takes. You can’t get much simpler than that. Note that s and t have an inverse relationship: if s gets larger, v increases, but if t gets larger, v decreases.

But it also means that for v to remain constant, if s gets smaller then so must t.

For the occupants of the shuttle, getting to the moon in such a short time means that, for them, the distance has shrunk. It normally takes about 3 days to get to the Moon (using current technology), so let’s say we manage it in 10 hrs instead. I haven’t done the calculations, because it depends on what speeds are attained and I’m trying to provide a qualitative, even intuitive, explanation rather than a technical one. The point is that if the occupants measured the distance using some sort of range finder, they’d find it was measurably less than if they did it using a range finder on Earth or on the Moon. It also means that whatever clocks they were carrying (including their own heartbeats) they would show that the duration was less, completely consistent with the equation above.

For the people on the Moon awaiting their arrival, or those on Earth left behind, the duration would be consistent with the distance they would measure independently of the craft, which means the distance would be whatever it was all of the time (allowing for small variances created by any elliptic eccentricity in its orbit). That means they would expect the occupants’ clocks to be the same as theirs. So when they see the discrepancy in the clocks it can only mean that time elapsed slower for the shuttle occupants compared to the moon’s inhabitants.

Now, many of you reading this will see a conundrum if not a flaw in my description. Einstein’s special theory of relativity infers that for the occupants of the shuttle, the clocks of the Moon and Earth occupants should also have slowed down, but when they disembark, they notice that they haven’t. That’s because there is an asymmetry inherent in this scenario. The shuttle occupants had to accelerate and decelerate to make the journey, whereas the so-called stationary observers didn’t. This is the same for the famous twin paradox.

Note that from the shuttle occupants’ perspective, the distance is shorter than the moon and Earth inhabitants’ measurements; therefore so is the time. But from the perspective of the moon and Earth inhabitants, the distance is unchanged but the time duration has shortened for the shuttle occupants compared to their own timekeeping. And that is special relativity theory in a nutshell.


Footnote: If you watch videos explaining the twin paradox, they emphasise that it’s not the acceleration that makes the difference (because it’s not part of the Lorentz transformation). But the acceleration and deceleration is what creates the asymmetry that one ‘moved’ respect to another that was ‘stationary’. In the scenario above, the entire solar system doesn’t accelerate and decelerate with respect to the shuttle, which would be absurd. This is my exposition on the twin paradox.

Addendum 1: Here is an attempted explanation of Einstein’s general theory of relativity, which is slightly more esoteric.

Addendum 2: I’ve done a rough calculation and the differences would be negligible, but if I changed the destination to Mars, the difference in distances would be in the order of 70,000 kilometres, but the time difference would be only in the order of 10 seconds. You could, of course, make the journey closer to lightspeed so the effects are more obvious.

Addendum 3: I’ve read the chapter on the twin paradox in Jim Al-Khalili’s book, Paradox: The Nine Greatest Enigmas in Physics. He points out that during the Apollo missions to the moon, the astronauts actually aged more (by nanoseconds) because the time increase by leaving Earth’s gravity was greater than any special relativistic effects experienced over the week-long return trip. Al-Khalili also explains that the twin who makes the journey, endures less time because the distance is shorter for them (as I expounded above). But, contrary to the YouTube lectures (that I viewed) he claims that it’s the acceleration and deceleration creating general relativistic effects that creates the asymmetry.

Are natural laws reality?

In a not-so-recent post, I mentioned a letter I wrote to Philosophy Now challenging a throwaway remark by Raymond Tallis in his regular column called Tallis in Wonderland. As I said in that post, I have a lot of respect for Tallis but we disagree on what physics means. Like a lot of 20th Century philosophers, he challenges the very idea of mathematically determined natural laws. George Lakoff (a linguist) is another who comes to mind, though I’m reluctant to put Tallis and Lakoff in the same philosophical box. I expect Tallis has more respect for science and philosophy in general, than Lakoff has. But both of them, I believe, would see our ‘discovery’ of natural laws as ‘projections’ and their mathematical representation as metaphorical.

There is an aspect of this that would seem to support their point of view, and that’s the fact that our discoveries are never complete. We can always find circumstances where the laws don’t apply or new laws are required. The most obvious examples are Einstein’s general theory of relativity replacing Newton’s universal theory of gravity, and quantum mechanics replacing Newtonian mechanics.

I’ve discussed these before, but I’ll repeat myself because it’s important to understand why and how these differences arise. One of the conditions that Einstein set himself when he created his new theory of gravity was that it reduced to Newton’s theory when relativistic effects were negligible. This feat is quite astounding when one considers that the mathematics, involved in both theories, appear on the surface, to have little in common.

In respect to quantum mechanics, I contend that it is distinct from classical physics and the mathematics reflects that. I should point out that no one else agrees with this view (to my knowledge) except Freeman Dyson.

Newtonian mechanics has other limitations as well. In regard to predicting the orbits of the planets, it quickly becomes apparent that as one increases the number of bodies the predictions become more impossible over longer periods of time, and this has nothing to do with relativity. As Jeremy Lent pointed out in The Patterning Instinct, Newtonian classical physics doesn’t really work for the real world, long term, and has been largely replaced by chaos theory. Both classical and quantum physics are largely ‘linear’, whereas nature appears to be persistently non-linear. This means that the Universe is unpredictable and I’ve discussed this in some detail elsewhere.

Nature obeys different rules at different levels. The curious thing is that we always believe that we’ve just about discovered everything there is to know, then we discover a whole new layer of reality. The Universe is worlds within worlds. Our comprehension of those worlds is largely dependent on our knowledge of mathematics.

Some people (like Gregory Chaitin and Stephen Wolfram) even think that there is something akin to computer code underpinning the entire Universe, but I don’t. Computers can’t deal with chaotic non-linear phenomena because one needs to calculate to infinity to get the initial conditions that determine the phenomenon’s ultimate fate. That’s why even the location of the solar system’s planets are not mathematically guaranteed.

Below is a draft of the letter I wrote to Philosophy Now in response to Raymond Tallis’s scepticism about natural laws. It’s not the one I sent.


Quantities actually exist in the real world, in nature, and they come in specific ratios and relationships to each other; hence the 'natural laws'. They are not fictions, we did not make them up, they are not products of our imaginations.

Having said that, the wave function in quantum mechanics is a product of Schrodinger's imagination, and some people argue that it is a fiction. Nevertheless, it forms the basis of QED (quantum electrodynamics) which is the most successful empirically verified scientific theory to date, so they may actually be real; it's debatable. Einstein's field equations, based on tensors, are also arguably a product of his imagination, but, according to Einstein's own admission, the mathematics determined his conclusion that space-time is curved, not the other way around. Also his famous equation, E= mc², is mathematically derived from his special theory of relativity and was later confirmed by experimental evidence. So sometimes, in physics, the map is discovered before the terrain.

The last line is a direct reference to Tallis’s own throwaway line that mathematical physicists tend to ‘confuse the map for the terrain’.

What makes humans unique

Now everyone pretty well agrees that there is not one single thing that makes humans unique in the animal kingdom, but most people would agree that our cognitive abilities leave the most intelligent and social of species in our wake. I say ‘most’ because there are some, possibly many, who argue that humans are not as special as we like to think and there is really nothing we can do that other species can’t do. They would point out that other species, if not all advanced species, have language, and many produce art to attract a mate and build structures (like ants and beavers) and some even use tools (like apes and crows).

However, I find it hard to imagine that other species can think and conceptualise in a language the way we do or even communicate complex thoughts and intentions using oral utterances alone. To give other examples, I know of no other species that tells stories, keeps track of days by inventing a calendar based on heavenly constellations (like the Mayans) or even thinks about thinking. And as far as I know, we are the only species who literally invents a complex language that we teach our children (it’s not inherited) so that we can extend memories across generations. Even cultures without written scripts can do this using songs and dances and art. As someone said (John Hands in Cosmo Sapiens) we are the only species ‘who know that we know’. Or, as I said above, we are the only species that ‘thinks about thinking’.

Someone once pointed out to me that the only thing that separates us from all other species is the accumulation of knowledge, resulting in what we call civilization. He contended that over hundreds, even thousands of years, this had resulted in a huge gap between us and every other sentient creature on the planet. I pointed out to him that this only happened because we had invented the written word, based on languages, that allowed us to transfer memories across generations. Other species can teach their young certain skills, that may not be genetically inherited, but none can accumulate knowledge over hundreds of generations like we can. His very point demonstrated the difference he was trying to deny.

In a not-so-recent post, I delineated my philosophical ruminations into 23 succinct paragraphs, covering everything from science and mathematics to language, morality and religion.  My 16th point said:



Humans have the unique ability to nest concepts within concepts ad-infinitum, which mirror the physical world.

In another post from 2012, in answer to a Question of the Month in Philosophy  Now: How does language work?; I made the same point. (This is the only submission to Philosophy Now, out of 8 thus far, that didn’t get published.)

I attributed the above ‘philosophical point’ to Douglas Hofstadter, because he says something similar in his Pulitzer Prize winning book, Godel Escher Bach, but in reality, I had reached this conclusion before reading it.

It’s my contention that it is this ability that separates us from other species and that has allowed all the intellectual endeavours we associate with humanity, including stories, music, art, architecture, mathematics, science and engineering.

I will illustrate with an example that we are all familiar with, yet many of us struggle to pursue at an advanced level. I’m talking about mathematics, and I choose it because I believe it also explains why many of us fail to achieve the degree of proficiency we might prefer.

With mathematics we learn modules which we then use as a subroutine in a larger calculation. To give a very esoteric example, Einstein’s general theory of relativity requires at least 4 modules: calculus, vectors, matrices and the Lorentz transformation. These all combine in a metric tensor that becomes the basis of his field equations. The thing is, if you don’t know how to deal with any one of these, you obviously can’t derive his field equations. But the point is that the human brain can turn all these ‘modules’ into black boxes and then the black boxes can be manipulated at another level.

It’s not hard to see that we do this with everything, including writing an essay like I’m doing now. I raise a number of ideas and then try to combine them into a coherent thesis. The ‘atoms’ are individual words but no one tries to comprehend it at that level. Instead they think in terms of the ideas that I’ve expressed in words.

We do the same with a story, which becomes like a surrogate life for the time that we are under its spell. I’ve pointed out in other posts that we only learn something new when we integrate it into what we already know. And, with a story, we are continually integrating new information into existing information. Without this unique cognitive skill, stories wouldn’t work.

But more relevant to the current topic, the medium for a story is not words but the reader’s imagination. In a movie, we short-circuit the process, which is why they are so popular.

Because a story works at the level of imagination, it’s like a dream in that it evokes images and emotions that can feel real. One could imagine that a dog or a cat could experience emotions if we gave them a virtual reality experience, but a human story has the same level of complexity that we find in everyday life and which we express in a language. The simple fact that we can use language alone to conjure up a world with characters, along with a plot that can be followed, gives some indication of how powerful language is for the human species.

In a post I wrote on storytelling back in 2012, I referenced a book by Kiwi academic, Brian Boyd, who points out that pretend play, which we all do as children (though I suspect it’s now more likely done using a videogame console) gives us cognitive skills and is the precursor to both telling and experiencing stories. The success of streaming services indicates how stories are an essential part of the human experience.

While it’s self-evident that both mathematics and storytelling are two human endeavours that no other species can do (even at a rudimentary level) it’s hard to see how they are related.

People who are involved in computer programming or writing code, are aware of the value, even necessity, of subroutines. Our own brain does this when we learn to do something without having to think about it, like walking. But we can do the same thing with more complex tasks like driving a car or playing a musical instrument. The key point here is that they are all ‘motor tasks’, and we call the result ‘muscle memory’, as distinct from cognitive tasks. However, I expect it relates to cognitive tasks as well. For example, every time you say something it’s like the sentence has been pre-formed in your brain. We use particular phrases, all the time, which are analogous to ‘subroutines.’

I should point out that this doesn’t mean that computers ‘think’, which is a whole other topic. I’m just relating how the brain delegates tasks so it can ‘think’ about more important things. If we had to concentrate every time we took a step, we would lose the train of thought of whatever it was we were engaged in at the time; a conversation being the most obvious example.

The mathematics example I gave is not dissimilar to the idea of a ‘subroutine’. In fact, one can employ mathematical ‘modules’ into software, so it’s more than an analogy. So with mathematics we’ve effectively achieved cognitively what the brain achieves with motor skills at the subconscious level. And look where it has got us: Einstein’s general theory of relativity, which is the basis of all current theories of the Universe.

We can also think of a story in terms of modules. They are the individual scenes, which join together to form an episode, which form together to create an overarching narrative that we can follow even when it’s interrupted.

What mathematics and storytelling have in common is that they are both examples where the whole appears to be greater than the sum of its parts. Yet we know that in both cases, the whole is made up of the parts, because we ‘process’ the parts to get the whole. My point is that only humans are capable of this.

In both cases, we mentally build a structure that seems to have no limits. The same cognitive skill that allows us to follow a story in serial form also allows us to develop scientific theories. The brain breaks things down into components and then joins them back together to form a complex cognitive structure. Of course, we do this with physical objects as well, like when we manufacture a car or construct a building, or even a spacecraft. It’s called engineering.