Understanding Einstein’s Special Relativity: A Beginner’s Guide to the Mathematical Physics

Introduction

This article provides a simple introduction to Einstein’s Special Relativity, using Prof Hermann Bondi’s k-calculus, which he introduced to us at King’s College, London Mathematics department in the late 1960s.

Special Relativity explains why time passes at different rates for observers moving at different speeds relative to each other, and why the length of objects is measured differently by those observers. Those speeds must be a substantial fraction of the speed of light for the effects to be noticeable, but they are always there, however low the relative speed, as we shall see.

This article might(!) lead to a later article on General Relativity, a far more difficult theory, evidenced by the fact that it took Einstein 10 years from 1905, when he introduced Special Relativity, until 1915/16 when General Relativity was introduced, employing curved space-time, using Riemann’s mathematics of curved spaces, rather than “flat” spacetime (the standard Cartesian geometry we all used at school) where Special Relativity finds its setting.

Even Einstein had relied on colleagues to explain Riemannian geometry to him when he realised he needed something more than Cartesian geometry to make progress with General Relativity.

Consequently, there is no gravity in Special Relativity. John Archibald Wheeler, a renowned theoretical physicist, famously (and possibly apocryphally) summarised Einstein’s General Theory of Relativity in the single, elegant sentence: “Spacetime tells matter how to move; matter tells spacetime how to curve.”

How do we observe things?

In the following discourse, we shall describe the use of light rays to communicate and observe what is happening. Rather than “smoke and mirrors”, our observers will use torches and mirrors.

The torches will allow observers to send signals, and/or mirrors will allow recipient observers to send a signal back as soon as it is received.

These light rays all travel at the “invariant” speed of light, which decades of experiments have confirmed is always observed to travel at the same speed, no matter the position or movement of an observer. It’s actually quite a counter-intuitive concept, and is absolutely at the heart of the theory of relativity; it is counter-intuitive, because we are all familiar with the concept and observation of relative motion of physical objects in our daily lives. This is why time dilation and length contraction are called “paradoxes” by some, because of their counterintuitive nature.

Relative motion

If I walk at 5kph (kilometres per hour) along a train carriage in the direction the 60kph train, then someone standing at the side of the track will measure my speed relative to them at 65kph; if I turn around and walk the other way, opposite to the train’s direction of motion, that person will see me moving past them at 55kph.

If that person is travelling in a car at 55kph in the same direction as the train, that person (or their passenger!) sees me as not moving relative to them as I walk back along the train. That is relative motion as we normally experience it. But the following diagram shows how Relativity changes the perceptions of different observers.

But Einstein’s pivotal assumption, corroborated by innumerable experiments ever since the famous Michelson-Morley experiment of 1887, was that we always see light passing us at the same speed, irrespective of our own speed, whether towards or away from the light source. In that sense, this apparently simple assumption is not only counterintuitive but, as we will see, is fundamental to all that follows.

Ironically, that 1887 Michelson-Morley experiment was a failed experiment, in that the scientists expected to measure a shift in light interference patterns as the Earth moved through space, because their prevailing assumption had been that light travelled though a medium called the “aether”, and thought that the Michelson-Morley experiment would demonstrate that our own Earth’s motion through the aether would show different interference patterns as the light moved in one direction or the other, relative to the aether, i.e. comparing light rays along the Earth’s direction of motion, or against it.

Instead, they found no significant difference, demonstrating that the speed of light is constant, regardless of the Earth’s motion. This “null result” disproved the aether theory (as did many other subsequent experiments) and paved the way for Einstein’s theory of special relativity,

Light cones

For most of the time in this article, I will present diagrams in the (t,x) plane, as most points can be made clearly that way. But you will often read about or hear the term “light-cone”, so I will explain what it means.

Light travels at just under 300,000 kms/sec (kps), but in agreement with tradition, we will call this speed “c” and renormalise it to 1. Thus, someone travelling at 30,000 kps would be travelling at 0.1 kps in these renormalised units.

With this choice of c=1, if we were to plot light’s “motion” on a simple x–t chart (labelling the vertical axis as “t” for our purposes), then light travels on lines at 45° to these x and t axes.

Forgive the gaudy colours in the first image – I used WordPress’s new Jetpack image creation tool to render this light cone. I probably don’t like it any more than you do! But it does convey the double cone I was trying to describe in words. Ignore everything else in the image!

Here is a more sober version of a light cone image.

Here is a one-dimensional version from my original notes (from years ago) for this article.

In the diagrams above, light always travels at 45° to the gridlines – at 1 x-unit per unit time t interval, a speed of c=1.

The meaning of “simultaneous” events

I will take some time to explain this clearly, including how we measure time and distance using light beams in Special Relativity, as it is fundamental to understanding how simultaneity differs for observers in relative motion to each other, a crucial outcome of the theory of Special Relativity.

We will find that different observers see different events as “simultaneous”, and can also disagree whether the same two events are simultaneous, depending on their relative motion.

But first, let us define simultaneity. In what follows, I will talk about two observers – Abel (usually me!) with coordinates (t,x), and, later on, Bar, whose time t̅ and spatial x¯ coordinates will be barred, as in (t̅,x¯).

What is the vertical t-axis here? It is my “world line”. If I don’t move, in my coordinates (t,x), my life moves up the t-axis with no “spacelike” x-axis movement. The points P (a TV screen, perhaps) and Q (a wall mirror across the room, say) have world lines too, shown dotted in the diagram.

P₀ and Q₀ are snapshots of their state at my time t=0 (P₀ is maybe a 9am breakfast TV programme snapshot). Many seconds (4 hours) later, at t=14,400 seconds, P₁₄₄₀₀ might be showing the 1 o’clock news.

P₀ and Q₀ are called events, in the language of relativity, and they are simultaneous for me, at t=0. Why is this so?

I can use my torch (all observers have a torch and a mirror in this experimental world) to illuminate firstly Q₀ (which is further away so I shine my torch earlier) and then a little later towards P₀ (maybe the TV screen!), and then receive my light beams back (later on my world-line) from mirrors at Q₀ and P₀, (or by light beams sent by the observers with whom I have arranged for them to signal back at those points P₀ and Q₀ as soon as they receive my light signal), as shown in the diagram above by the squiggly lines.

I can calculate the coordinates of both P₀ and Q₀. I need to know 3 things for both: their distance away, how long it takes from my emission of the light beam to my reception of the return beam, and the speed of light, using the simple formula d=ct, as I’ll show in a moment.

In this diagram, P₁ and Q₁ are two later events – maybe the TV and the mirror 1 second later, and we can see that they are also simultaneous for me. These could have been at the time of the 1 o’clock news at t=14,400 rather than t=1.

Non-simultaneous events

In the following diagram, P₂ and Q₀ are NOT simultaneous for me.

The method of calculation of the time for the round trip of the light beam (and therefore the distance away of P₂ and Q₀, at the time they are illuminated), and the calculation of the time of reception of the light beams, is shown below.

Suppose, as in the diagram, that I sent light out at time t=0 and received it back at time t=T. Thus, the time for the round trip is T, measured by my clock, and so I calculate that the time for the one-way trip out (or back) is T/2.

Since light has speed c=1 in the units we are using, and distance (d), speed (v) and time (t) are related by v=d/t, or d=vt, my distance D away from the point P is given by

D = 1xT/2 = T/2,

The point P₁ in the diagram below is T/2 units up the t-axis, with coordinates (T/2, T/2).

See above that the events at P₁ and Q₁ are simultaneous (for me) because the calculation for each gives the same t-coordinate, although Q₁ is further away than P₁; P₁ is at (T/2, T/2) and Q₁ is at (T/2, T), twice as far away (remember the t-coordinate is always the first coordinate mentioned).

My line of simultaneity – “los”

We can see from the foregoing diagrams that any events connected by a line parallel to the x-axis are simultaneous for me: the x-axis and all lines parallel to it are my lines of simultaneity “los“.

P, Q and R on the x-axis all lie on a los for me, but any line parallel to the x-axis, such as t=3, is also a los for me, connecting events P’, Q’ and R’, which are simultaneous for me.

Each los can be constructed using light beams (and reflected light beams) to establish that any points on the los are simultaneous for me, just as are P, Q and R, or P’, Q’ and R’ above.

Someone else’s line of simultaneity (los)

Now we turn to a crucial construction: the lines of simultaneity (los) for another observer moving relative to me.

At this point, we should note that Special Relativity focuses on “inertial” observers, who move at constant relative speed to each other. Acceleration (changing velocity) and, in particular, gravity (that causes acceleration), are the province of General Relativity, which deals with “curved space” that corresponds with gravity, as per the John Wheeler quotation mentioned in the Introduction.

Einstein’s Special Relativity deals with “flat spacetime” – flat (Cartesian, not curved) space and “spacetime”, as illustrated by our (t,x) coordinate diagrams for the time dimension and one space dimension.

In general, and for real space travel, we have to deal with time and three spatial dimensions (t, x, y, z). Still, for the purposes of this article, to explain simply the fundamentals of Special Relativity, we can deal more easily (and sufficiently) with one space dimension. Imagine we are on a straight railway track – a 1-dimensional world (plus time).

Another Inertial Observer’s world line

Imagine another inertial observer, Bar, moving at constant speed v out of (say) Central Station in Glasgow at the Origin O of our 2D spacetime, on the rail track, having waved goodbye to me (Abel) as she left from O on the noon train.

Her world-line is t̅ as shown in the diagram above. As time progresses for her, she moves along the railway track (which I see as my x-axis). Her world line t̅ might make an angle ⍺ with my own world line in this diagram.

I remain sitting in Costa Coffee on the station concourse, so my own world line t remains defined as the t-axis at x=0. The x-axis itself, as discussed before, is one of my lines of simultaneity, the first one.

What is Bar’s line of simultaneity?

This construction is absolutely fundamental to Special Relativity, because it relies on the constancy of the speed of light for all observers.

For Bar, what events are simultaneous with O, Central Station at noon? For example, does she see the Queen Street (Glasgow’s other main terminus) noon train (on my clock) departing at noon as well?

Surprisingly, the answer is no, even though we both agree on the noon departure of her train from Central Station, where we were both located at the moment of her departure.

Bar’s line of simultaneity for the Central noon train

To find Bar’s x-axis, we do exactly what we did for Abel’s (my) x-axis.

On Bar’s world line, t̅, she emits a light beam at B₁ and receives it back at B₂ at equal times before and after her world line passes through Central Station. (It’s a terminus, so she must have walked there! She wouldn’t have taken Glasgow’s subway (known locally as the ‘clockwork orange’) as light beams wouldn’t reach her, being underground, from the main concourse, where Abel is located.)

These points on her world line are shown as B₁ and B₂, but any two points would do, as long as they are at equal times before and after noon (in Bar’s time) at Central Station.

The two barred points B¯₁ and B¯₂ are points we can construct, defined by postulating mirrors at those points that can reflect back the light sent by Bar from B₁ to be received on her world line at B₂, at the same time interval after noon as she sent them before noon.

By this construction, the points B¯₁ and B¯2 lie on Bar’s line of simultaneity through Central Station (the Origin O), at noon, and that straight line is therefore her x-axis, x¯, as labelled in the diagram above.

You can see that x¯ is not (as I see it from my world) perpendicular to her own world line t̅, and this is key to the so-called “paradoxes” of Special Relativity, Time Dilation and Length Contraction, as I will show later. The logic of those discussions, incidentally, will show that these aren’t paradoxes at all; they are part of the unavoidable logic of Special Relativity.

Don’t miss that Queen St train!

Look at Queen St station at Q from Bar’s perspective in the diagram above.

Her line of simultaneity (LoS) through Q, parallel to x¯, cuts her timeline t̅ before she reaches Central Station (i.e. on the negative t̅-axis). No wonder she carried on to Central Station; she would have missed the noon Queen St train (in my timeframe) because in her timeframe t̅, it left before noon.

Bar and I agree on the time of the noon train at Central Station, because we were both there at the same time, but we disagree on the time of the train from Queen St, because we were not there at the same time.

Constructing Lines of Simultaneity LoS

You can also see from the previous diagram that any LoS for Bar (starting with her x¯-axis) can easily be drawn; if Bar’s world line t̅ is at an angle ⍺ to mine (t), then her axis of simultaneity x¯ must be at an angle ⍺ to mine (my x-axis), in that same quadrant of the diagram. Every LoS for Bar is a line parallel to her x¯.

Remember that the angle ⍺ must be less than 45º (π/4 radians) because nothing in Special Relativity theory can travel faster than light, which travels on those light cones we discussed, whose surfaces are at 45° to the x-axis (and to the t-axis).

In our daily human world, speeds are far less than a substantial fraction of the speed of light. We might call them “Newtonian” speeds. This means that our measurements of t and t̅ are very nearly coincident (and ⍺ is extremely small), and everyone tends to agree on train times at any station, wherever we are – fortunately – and therefore we don’t usually miss our connections.

Let’s now see how ⍺ relates to Bar’s relative velocity to me, v, using Hermann Bondi’s k-calculus.

Hermann Bondi’s k-calculus

Given the assumed constant relative speed v of Bar to Abel involved in this analysis, it is reasonable to observe that the time on Bar’s world line OB is in a constant ratio to the time t measured by the originator of the light beam, Abel. Let’s call that ratio k.

From the diagram, we can see that Abel measures the time taken for the light beam’s one-way trip to Bar as half of the difference between the time of emission and the time of reception of the return signal:

(k²t – t)/2 = t* (k² – 1)/2 = d, say.

And, since light speed c = 1, in our units, this is also the distance to Bar when she receives the light signal, which is why I have called it d.

The time that Bar receives the light signal, T, measured by Abel, is

T = t + (k² – 1)t/2 = (k² + 1)t/2.

Bar’s speed, therefore, measured by Abel, is

v = d/T = {(k² + 1)t/2}/{t* (k² – 1)/2} = (k² – 1)/(k² + 1).

We can now express k in terms of v, the relative speed between Bar and Abel. Multiplying out the equation above, we find

k²v + v = k² – 1, or k²(1-v) = v+1,

And rearranging, we find

k = √{(1+v)/(1-v)}.

This is the ratio of time measured by Bar to the time measured by Abel. From this equation, we see that if v = 0 (Bar remains at the same place as Abel, then k = ±1 (which are both the square root of 1).

From the definition of k in the diagram, we can see that if k>1, v is positive, and for k <1, v is negative. from the calculation of k in terms of v, we see that if v –> –v, then k –> 1/k.

Composition rule of k factors

Let’s call the k-factor between Abel and Bar k_AB, and suppose there is an additional observer, Charlie, travelling at relative speed to Bar, with k-factor k_BC between Charlie and Bar, as shown in the diagram below.

It is clear that if t and t̅ are related by t̅ = k_AB t, and Charlie’s time t̅¯ = k_BC t̅, then the k-factor between Abel and Charlie must be given by

k_AC = k_AB x k_BC .

A numerical example

Let us put some numbers on the relationships between v, ⍺, k, t and t̅.

In this diagram, from the right-angled triangle where ⍺ appears, we can relate ⍺ and v, where v is equal to the distance Bar travels in unit time:

v = tan(⍺) = (k² -1)/(k²+1).

Note that if ⍺ = 0, k = 1, since ⍺ = 0 means that Bar remains at Central Station with Abel (maybe sipping coffee with me in Costa Coffee!).

But, if, say, ⍺ = 30º, then tan(⍺) = 1/√3 = v. With this value of v,

k = √{(1+v)/(1-v)} = √(2.732/0.732) = 1.9, approximately.

So, for this value of k, if a year (t) passes for Abel, and, as before, t̅ = kt years pass for Bar, about 1.9 years.

This is an example of the “space traveller” twin paradox; that a space traveller, if they can travel at a significant fraction of the speed of light, c, which we call a relativistic speed, such as v = 1/√3, or about 0.577 of the speed of light, a space traveller Bar ages nearly 2 years, while her twin back on Earth, Abel, only ages 1 year.

Another view of how the line of simultaneity relates to v and ⍺

Consider t̅ = 7 years this time.

We see from the diagram above that if Bar travels at a speed v = tan(𝝷), (such that our previous ⍺ = 𝝷), then Abel sees the angle between Bar’s x¯ and t̅ axes as 90º minus 2𝝷, or (π/4 – 2𝝷).

Remember that the maximum speed of anything, or anyone, is less than the speed of light, which travels on the light cone at 45º (π/4) to the (t,x) axes in whoever’s time frame we are considering. If someone travels at just under the speed of light, their x and t axes as Abel sees them are virtually coincident, with 𝝷 just under 45º, or π/4 radians.

This also means that the light cone, the invariant in any oberver’s space-time, must bisect the angle between any oberver’s t and x axes, so that the speed of light measures the same in anyone’s coordinates. The faster the observer travels, the narrower the angle between your t and x axes.

Space traveller “paradox” continued

This time, in the diagram, we set t̅ = 7 years, and ⍺ = 30º as before. This value ⍺ again sets tan(⍺) = 1/√3 = v = 0.577 of the speed of light, c (which has the value c = 1 in our units).

This time, we have arranged with Bar to travel away from Earth for 7 years on her clock, and then to turn back and make the return journey to Earth at the same speed, also taking 7 years.

The angle 2ϴ in the diagram is the angle, as Abel sees it, between Bar’s x¯ and t̅ axes, which is not 90º (π/2 radians) as it is for Abel’s own space-time axes, as explained above.

Bar’s lines of simultaneity on her outward journey are the x¯-axis and lines parallel to it.

The other “double-barred” axes shown in the diagram are the (x¯¯, t̅¯) axes corresponding to Bar’s return trip, starting after being on her outward space journey for 7 years.

Construction of (t, x) frame simultaneity

The x-axis is the locus of points of the light beam received from Abel and reflected back to Abel.

Now we construct the line of simultaneity for Bar, x¯ axis, given t̅, Bar’s world-line of her uniformly moving inertial motion.

As before, we use light rays emitted by Bar from her world line towards Abel, at equal intervals before and after meeting him at the origin O, in both directions from her light cone at those points. As we see, Bar’s line of simultaneity, x¯, is distinct from Abel’s; all events on Bar’s LoS are simultaneous for Bar with her meeting with Abel at O.

Disagreement on Simultaneity between Abel and Bar

How does Bar perceive events that are simultaneous for Abel?

Events P and Q are simultaneous for Abel, but Q happens before P for Bar.

P and Q are what we call “spacelike separated”, meaning that no light beam (or observer) can travel between the two events. An observer who travels through P (or Q) cannot observe what happens at Q (or P), respectively.

**Q͞ and P͞ are the times at which Bar sees the events Q and P happen; these are, as we see, different from Abel’s perception of the events and their timing; she sees them as events that happen at times Q͞ before (for Q) and P͞ after (for P) Abel sees them at Q and P respectively.

How does Bar perceive events that are simultaneous for Abel?

Conversely, events M͞ and N͞ are simultaneous for Bar but for Abel, M͞ happens before N͞.
Events M and N are also spacelike separated for Bar, and also for Abel and all other observers. An observer travelling through Mbar (Nbar) cannot see what happens at Nbar (Mbar) respectively.

Another way of putting this is that M͞ and N͞ are not in each other’s light cones, which also applies to the pair M and N, and also for the pair P and Q, and for the pair P͞ and Q͞ that we discussed before.

Even worse, Abel and Bar can disagree on which of the two events might happen first.

For Abel, event G happens before H, but Bar sees H happen before G, at H͞ and G͞ respectively on her timeline, points that we identify by using light beams from G (as Abel sees it) to G͞ on Bar’s timeline, and from H (as Abel sees it) to H͞ on Bar’s timeline.

NB For Abel, G & H are spacelike separated. We will see that for two events with timelike separation, all observers must agree on the order of such events. For spacelike separation, they can disagree, as we have seen.

Another view – absolute past & future

All observers agree that event E occurs after O, because they can see the torch pressed at O and the beam arrive at E; similarly for any event E’ inside the light cone. Breakfast at O and lunch at E’ is observable by all observers, even if the time between them is different for different observers.

Relative past and future

We now look at events outside the light cone at O, and consider the event H. We can find an observer for whom O and H are coincident by plotting light rays at right angles (backwards and forwards) from H. An observer whose timeline is the line P–>Q sees H as coincident with O; light emissions and receptions are equally either side of O; H is on PQ‘s line of simultaneity.

Absolute & Relative past & future

Timelike and Spacelike separation

The two events Q and R are timelike separated; an observer could travel between them, i.e. an observer can make the journey between Q and R at a speed less than the speed of light.

The two events L and M are spacelike separated, and an observer cannot travel between them; they are in each other’s “elsewhere”.

Because of this dichotomy between timelike and spacelike separation, we can also understand that some observers can see events that others cannot. Some experiments, therefore, are not decidable by all observers; and, as we have seen, observers can even disagree on the sequence of events they can both see.

In the diagram below, we can see that observers at L and M have some common future and common past that they can both see, the overlap between the areas within their respective light cones. For events outside those shaded areas, L and M have no common view of such events.

Timelike separation

Events at R and S have timelike separation for Abel, with R happening before S.

For Bar, her view of the first event R at R͞ also happens after her view of S at S͞.

In general, there is no speed that Bar can attain that would change her perception of the order of the events R and S. The more slowly Bar travels relative to Abel, the more nearly her perception agrees with that of Abel. See the diagram below where Bar is travelling faster.

Events at R and S are timelike-separated for Abel, and this time, Bar is travelling faster relative to Abel than before. But we still have R͞ happening before S͞, in Bar’s view, so that Abel and Bar still agree on the sequence of events R and S. The faster that Bar moves relative to Abel, the greater the times she measures between the events R and S, approaching infinity as Bar’s relative speed to Abel’s speed approaches (but cannot reach, of course) the speed of light.

The Lorentz Transformation (via k-calculus)

The Lorenz transformation, named after the Dutch physicist Hendrik Lorenz, is the mathematical formula that allows us to transform time and distance between two inertial frames of reference, and is the mathematical description of all that we have been discussing so far.

Let us calculate the coordinates of event P from the point of view of both Abel and Bar in their respective coordinate systems, (t, x) and ( t̅, x¯). Their clocks are synchronised when they were together at the Origin O. By our previous k-calculus,

t̅ –x¯ = k(t–x) (1)

t+x = k( t̅+x¯) (2)

or t̅ +x¯ = (1/k)(t+x) (2′)

Then adding (1) and (2′), we see that

t̅ = (1/2){kt –kx +(1/k)t + (1/k)x}, or

t̅ = (1/2)(k+(1/k)t – (1/2)(k-1/k)x, and (3)

x¯ = (1/2){(1/k)t +(1/k)x –kt +kx} or

x¯ = (1/2)(k + 1/k)x – (1/2)(k – 1/k)t (4)

Note, as a sanity check, that if k = 1 in the above equations (3) and (4), , ie no relative motion between Abel and Bar, then t̅=t and x¯=x, as they should.

Time Dilation

The term “time dilation” describes the effect that time travel progresses more slowly for a space traveller, Bar, compared with her twin, Abel, here on Earth, one of the two so-called paradoxes in Special Relativity, the other being length contraction, which I shall cover in the next section.

If Bar ticks off a time interval T on her world line to the event P, Abel will measure the time to this point differently, to P’: by our (Bondi’s) k-calculus, P’ occurs at the time given by

t(P’) = (1/k)T + (1/2){kT – (1/k)T} = (T/2)(k – 1/k)

But as we saw before,