### The Idiot's Guide to Relativity

I've been spending a bit of time lately drumming up traffic by engaging in discussions on Twitter on a range of scientific topics. As is often the case, I've come up against a fair bit of silliness on reasonably well-established scientific fronts. I want to address one of them here: Relativity.

Some of what will be covered in this post has already been covered in previous posts but, in order to ensure completeness here, I'm going to break with my usual convention and include it, so apologies for any repetition.

Let's start, as always, with a little useful background. In this case, a little perspective on what it is that physicists do. Yes, I realise that seems a bit silly but, hopefully, the motivation will become clear.

Much of physics is concerned with finding quantities and the relationships between them. I've posted previously on why this is a good way of going about things, so I won't belabour the point here. The really interesting things in physics happen when we find quantities that are constant. So this is where we should start.

What is a constant?

This might be a source of some confusion, as a constant has slightly different definitions dependent on context. In mathematics, a constant is simply a fixed numerical value (although generally taken to be of some significant interest), without reference to any physical measurement. An obvious example is π (pi), also previously known as Archimedes’ Constant, which gives the ratio between the circumference and the diameter of a circle.

In physics, a constant is any quantity that is taken to be universal in nature and unchanging over time. Since it will have relevance later in the post, we might as well stick with the example of c, the speed of light in a vacuum, which is given as 299,792.458 km/s.

There are many such examples, and we should also note that ALL constants are unit-system independent, so it doesn’t matter whether we’re talking about feet or metres, the ratios will still be the same (and indeed all constants are actually descriptions of ratios). In the case of the speed of light (and actually light is a bit of a red herring here, because there’s nothing really special about light, it’s simply that this is the thing we can point to that travels at this speed), the ratio is that between space and time, i.e. for any photon in a vacuum, it will travel a certain distance (variable) in a certain time (variable) such that the ratio between these two variables will always be the same (constant), regardless of who’s doing the observing or what their state of motion (inertial frame).

Constants are really important for physics, because they give the fixed relationships between things we can measure. In fact, we can even reasonably think of physics as being the search for constants. When we use a constant in an equation, things get really interesting. In the case of the most famous equation in physics, E (energy; a variable) is directly related to m (mass; a variable) with the ratio given by the square of c (light-speed; a constant). Because there’s a constant in there as the conversion factor, it’s crystal clear that, although the things being related are variable, the relationship between them is constant.

Another useful constant, expressed by something we don’t necessarily think of as being a constant, is given by Pythagoras’ Theorem, which expresses the ratio between the lengths of the adjacent sides in a right-angled triangle and the length of the hypotenuse. As with all equations, this one describes a fixed relationship between quantities (it’s why they’re called equations; whatever’s on one side of the = sign must be equal to what’s on the other side). This particular example has implications for our current discussion, because the relationship given by this theorem is almost exactly the same as another relationship in relativity. In Pythagoras, the relevant equation is $a^2+b^2=c^2$. Geometrically, of course, it looks like this:

In natural language, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

There's an interesting consequence to be noted here, and it will bring us nicely to our main topic. It's to do with how that hypotenuse relates to the other long axis.

To make it intuitive, let's picture some racing cars on a track:

Here, you can see that the lines of travel, along with the start line, make up the sides of a right-angled triangle. To relate to the Pythagoras diagram above, car 1 is travelling along the b side, which we've rotated to occupy the x axis, and car 2 is travelling the hypotenuse, while they're separated along side a, now occupying the y axis.

Imagine that they both hit the start line at exactly the same time, and exactly the same speed. For simplicity, we'll say that the cars are travelling at 10 m/s and that the track is 100 metres long. If they start 10 metres apart, we can do a quick beermat calculation, which is how all the best, most exciting physics is done. $100^2 + 10^2=10,100$  So now we need the $\sqrt {10,100}$, which gives us ~100.05 metres.

From this, we can easily see that car 1 is travelling 100 metres at 10 m/s on the x axis, so it will take 10 seconds. Car 2, however, because some of its travel is taken up with motion along the y axis, will take slightly longer. It's travelling a distance of ~100.05 metres, so it will take ~10.05 seconds to traverse the distance. It's easy to see that travel through the second dimension takes away slightly from travel through the first dimension, so it takes longer to travel the same distance.

So that's how it works in space, but what about time? That's where we're going to head next, but we first need to do something with all three of our spatial dimensions: We're going to amalgamate them. This shouldn't pose any problems for our intuition and it's going to be important later. From here on in, all travel through space will be represented by a single axis, the x axis. We're also going to swap our y axis label for t, denoting time.

What does relativity tell us?

The most important thing to take away about how relativity works is to understand the relationship between space and time, and why it’s a mistake to think of them as different entities. The principle of relativity is actually a good deal older than SR and GR, and comes to us from Galileo, who asserted that any experiment carried out under a range of translations should yield the same results (in the technical jargon, we say that it has translational symmetry). A translation is any change in the spatial position or orientation of the experiment. Thus, if I conduct an experiment at the top of the Tower of Pisa, it should yield the same result as an identical experiment conducted in the centre circle at Wembley stadium (for the really interested, Emmy Noether showed that this symmetry is just an expression of the law of conservation of momentum). Further if I conduct an experiment facing West, it should yield the same result as an identical experiment conducted facing East, or indeed any other direction (as you might expect, Noether’s Theorem tells us that this is an expression of the law of conservation of angular momentum).

The real distinction between Galileo’s treatment of relativity (upon which Newtonian mechanics was based) and Einstein’s is that you may not separate space and time, but you must treat them as a single entity. This means replacing Galilean transformations with Lorentzian transformations. In Einstein’s scheme, everything in the universe travels through spacetime at a single, fixed rate (which we will denote s, following a fairly widely-used convention). Regardless of how one is moving through space, one’s rate of travel through spacetime does not change. What this means in practice is that, if one is travelling through space, the equation for the distance in spacetime is $s^2=(ct)^2-x^2$, where c is our exchange rate (the speed of light), t is the distance through time and x is the distance through space.

You’ll note that there is one slight difference between this formulation and the Pythagorean one, namely the minus sign. This protects causality and ensures that no effect can precede its cause. It also tells us something about the nature of spacetime. To show how this works, let's look at what happens if we leave it alone.

Going back to our racetrack, we can plot key events on a spacetime graph. Since we've now amalgamated our spatial dimensions, we only have to worry about car 1, travelling 100 metres  at 10 metres per second on the x axis. Using the simpler Pythagorean formulation, we can give the equation for the distance in spacetime as
$s^2=(ct)^2+x^2$. We'll plot the start line as the origin, where the axes meet, and we can also plot on the diagram all the places that are the same distance from the origin (not to scale).

You can immediately see what the issue is here, only noting that everything 'North' of the origin on the y axis is in the future, and everything 'South' is in the past. This means that anywhere on that circle lower than the origin means that you arrive at the finish line before you've crossed the start line!

However, if we use the minus sign version:
$s^2=(ct)^2-x^2$ we get a slightly different picture of spacetime, namely a non-Euclidean, hyperbolic spacetime.

The hyperbola, like the ring in the basic Pythagorean version, represents all the points the same distance in spacetime from the origin, except that now, all points lie in the future, so the law of causality is adhered to, and no effect can precede its cause.

As an aside, this is the only thing like a law of causality in science. I often come across citations to the effect that everything has to have a cause, but this is rooted in Aristotelian notions of causality, which have no place in either physics or the 21st century.

Anyway, there's one final thing we need to with our diagram to our diagram to make it complete, and it's to do with how long it takes to get anywhere. Because we have a maximum speed at which we can travel, we need to plot past and future light cones. These sit at 45 degrees to the x and t axes, thus:

Anything that is travelling along those lines is travelling at the maximum velocity through space, thus doesn’t experience time at all, because all its motion through spacetime is exhausted in the space dimensions. So, when we plot our finish line, we can see that all possible points the same distance in spacetime lie within our future light cone.

So the equation tells us that the square of our travel through spacetime s is the square of ct (time multiplied by the speed of light), minus the square of our travel through space x

So, we travel through spacetime at a single, fixed speed. The nearer you get to the diagonal lines representing the limits of the future light cone by going faster through space, the slower time runs. The maximum speed you can travel is the speed of light. If you think of time as just like another spatial dimension, you can see that, given a fixed speed through spacetime, any travel through space will reduce your travel through time. This speed limit applies to ALL dimensions, which means that if you're travelling through one dimension at light speed, you must be standing still in all others. So, when you travel through space, you are reducing the amount of travel through time. This is why time slows when you are in motion, simply because, although you're always moving through spacetime at the same rate, that rate is divided between three dimensions of space and one of time. When you're not moving, all your motion is in the time dimension. It is also why photons don't age, because they move at light speed, meaning that their travel through time is nil.

Another interesting consequence of this is that, as well as time slowing down when you're in motion, distances in space are contracted in the direction of travel, which is what allows an observer in motion to measure the same distances in space and time. These effects combined are the reason that the same speed of light will be measured by any observer, regardless of the motion of source or observer.

So what happens when two observers, in differential motion, observe the same event? Let's look at a clock to find out. In this case, this will be the simplest clock we can find.

This is a light clock. It's a simple device, consisting only of a photon bouncing between two mirrors a fixed distance apart. Picture this clock, being watched by a passenger, let's call her Tami, on a train. She sees the photon happily bouncing up and down, ticking away the seconds. Nothing very interesting or exciting is happening here. Something spectacular happens, though, when she passes a station.

On the platform, Joe is watching the train go past, and he spots the clock. This is what he sees. As we can see, from Joe's perspective, the clock is moving, but of course the ticker is a photon, meaning that it's moving at the same speed the entire time. It's moving at c in Tami's frame, and it's moving at c in Joe's frame, but it's clearly travelling further in Joe's frame, because it's travelling in two dimensions at once where, in Tami's frame, it's only moving in one dimension the entire time. So which one of them is correct?  This being Tami and Joe, you might be forgiven for thinking that they've both got it horribly wrong but, in a twist of weirdness as you'd expect from relativity, they're actually both correct! From Joe's perspective, time will appear to be running slow for Tami, as per our equation above.

That's special relativity in a nutshell, but there was another problem namely that, in the Newtonian framework, because space and time were independent of each other and absolute, gravity was transmitted instantaneously. Special relativity presents a major problem because, in Einstein's scheme, no information can be transmitted at greater than lightspeed. Enter General Relativity:

Newton's theory of gravity was based on some assumptions. Among these were the assumptions that space and time were absolute and immutable. In other words, if it's 9am on Monday 4th October 1672 in Cambridge, it's 9am Monday 4th October 1672 on an as-yet-undiscovered planet in the Andromeda galaxy and in the heart of a star in the Kalium galaxy (strictly, although this picture has been fairly concretely undermined, we still have a universal standard of time namely UTC, which is pretty much Greenwich Meantime universalised). Furthermore, because simultaneity existed between all bodies in the universe, and because the range of gravity was infinite, it meant that gravity propagated instantaneously. This has some interesting implications, not least that, if the sun were to pop out of existence right this second, Earth, along with all the other bodies that are gravitationally bound to the sun, would go careening instantly off into space, likely with a few collisions along the way (although not as many as you might think; space is really big; I mean, you might think it's...)

Einstein's paper changed all that. The Special Theory of Relativity comprehensively demolished the assumption that space and time were immutable and absolute. Einstein saw the term for the speed of light in Maxwell's equations for electromagnetism, a term that had been introduced purely for mathematical consistency, as far as we can tell, with no term for how the source or the observer might be moving, and ran with the conclusion that light must travel at the same speed for all observers, and tried to work out what that might mean.

The result was that space and time must move around, and stretch and squeeze, in order to accommodate this constancy of light. It wasn't about gravity, but now we had a new picture of space and time in which neither space nor time existed independently but were different facets of the same entity, spacetime, and Newton's theory was simply not compatible with it.

This troubled Einstein, and for the next 10 years he worked on producing a theory of gravity that was compatible with this new picture of space and time. He said that the moment when it all made sense was when he thought about an elevator falling in its shaft, and the implication for an observer inside the elevator. He worked out that being immersed in a gravitational field and acceleration were basically the same thing. He extrapolated this to the General Theory of Relativity, which was published in 1915.

The effects of GR are far easier to measure for us for the most part, because we have the instruments to measure them in our pockets, in our mobile phones and satnav systems. SR effects require particle accelerators and short lived particles to show (when accelerated close to light-speed, they live longer before decay), or kinky experiments involving very fast planes and atomic clocks.

Relativistic time dilation appears in two forms; gravitational time dilation and velocity-related time dilation. To take the example of GPS satellites, they orbit at an altitude of approximately 20,000 kilometres, which means that the time dilation they experience is somewhat lower than the time dilation experienced on the Earth's surface. In other words, they experience time faster (about 45 microseconds per day, give or take) because they are less immersed in the Earth's gravitational field, which is a source of time dilation due to the warping of spacetime. This is general relativity at work.

The second is that, due to orbital velocity (14,000 kilometres per hour, in the case of GPS satellites), they experience time at a slightly reduced rate (about 7 microseconds per day). This is special relativity at work. The time dilation due to orbital velocity is thus considerably less than the time dilation due to immersion in the planet's gravitational field.

Thus, GPS satellites have relativistic error correction of -38 microseconds per day. This doesn't sound like much but, without this correction, the GPS system would drift out by approximately 10 kilometres per day.

For further information on any of the above, I recommend the brilliant Why Does E=mc2 (and why should we care) by Brian Cox and Jeff Forshaw. Some really good stuff also in the two excellent books The Elegant Universe and Fabric of the Cosmos, by Brian Greene, The ABC of Relativity by Bertrand Russell, and many other excellent books for the layman.

For the more mathematically inclined, my favoured source is the stupendous video series on relativity by Youtuber Ozmoroid on his Viascience channel, who presents many everyday examples of these principles in action, as well as giving a far more detailed account of the mathematics. I should note that they are still extremely approachable even if you don't have the maths.

You can also find more information (as well as some of this repeated) in the other topics on this blog dealing with the Big Bang and Quantum Mechanics.

Nits, crits and corrections welcome, as always.E=mc2