## Special Relativity Overview - Quick Menu

- Intro
- The "ether" hypothesis
- The ugly duckling Lorentz solution
- The Resemblance to Pythagoras and trigonometry
- From wombat 1 to wombat 3
- How does it work both ways?
- So do non inertial frames have slower time?
- Does that mean wombats can travel faster than light?
- Any More Questions?

## Intro

The universe does not always do things in the most easy way possible, and that might be just as well. In the nineteenth century a clever mathematician name Maxwell deduced that light and radio waves were essentially related to electricity and magnets. In fact, he showed that the speed of light could be deduced from what was already known about magnets and electricity. His equations led scientists on one of the greatest wild-goose chases in history - the quest to define the 'ether' of space.

Sure, Maxwell's equations did not say "though must now seek out and find the ether", in fact it said nothing about this mysterious invisible substance, but his equations did say, the speed of light is 3 X 10^{8} ms^{-1} without any useful footnotes to explain how in a universe where there is no way to define an absolute velocity, light could just arrogantly move at a particular speed without any consideration to the movements of the observer.

To appreciate Special Relativity, it is important to appreciate this contradiction. Let us therefore take a nostalgic journey back to the 1980s and consider Marty McFly driving a DMC-12 equipped with a flux capacitor and a small nuclear power source. Apparently. to go back in time, you can set the dials and when the car reaches 80mph (128km/hr) the time machine kicks in and the car disappears into the past. To the critical eye, this ability to travel in time is not as problematic (well it is problematic of course) as the fact that it happens when the car reaches an arbitrary speed of 128km/hr or 35.56 ms^{-1}.
So what is special about 35.56 ms^{-1}? and what is this velocity in reference to? Is it the surface of the Earth which is moving at different speeds depending on how far you are from the equator?

If the central plot driving device of the Back to the Future movies disturbs you and makes you want to jump on to internet forums to rant, then that is exactly how late nineteen century scientists felt about Maxwell's equations - like the "Back to the Future" movies, you had to admit they were fantastic, but under the scrutiny of logic, they were infuriating!

## The "ether" hypothesis

Obviously Maxwell's equations needed a sci-fi patch, and that patch was the "ether". A mysterious ghostly invisible substance that allows us to mark out space with road-signs and speed cameras. If such a substance existed it should provide the medium for light to propagate through. Just as oceans waves would move at speeds relative to the ocean currents, then so should light move at a fixed speed relative to the ether. If the Earth moves in different directions over the course of its orbit, then surely our relative velocity to this ether must change, and so to should the speed of light? yet, all the experiments stuck to script - the speed of light was the same in all directions and in all seasons.

## The ugly duckling Lorentz solution

If you cannot pin down this ether, then you just have to massage the mathematics. Hendrik Lorentz could not let the idea of an ether go, but nevertheless he wasn't going to let it tie him down. If the speed of light is the same regardless of how you are moving, then simple linear arithmetic is not going to work. Consider these three wombats flying through space:

(click to replay)

The wombat on the left is not moving according to us. The wombat in the middle is moving from the first wombat at two thirds of the speed of light, ie 2X10^{8} ms^{-1}. The third wombat on the right is moving away from the second wombat also at two thirds the speed of light 2X10^{8} ms^{-1}.

So how fast is the third wombat on the right moving from the stationary wombat on the very left?

OK, let's not get too far ahead of ourselves here. Let us start with the obvious answer which you know is wrong first:

Yes, just add the velocities together as they are all in the same dimension and the third wombat is traveling faster than light. Simple, case closed!

OK - it is horribly wrong, even wombats cannot travel faster than light. Sure, Maxwell's equations don't say anything about marsupials, but if a wombat can travel faster than light, then she can overtake a light-beam, and that light-beam would not be traveling at the speed of light according to 2 out of three wombats! Quite clearly this speed of light has implication for marsupials as well!

If we cannot just add things up, then we will need some other sort of relationship for wombats with rockets.

The Lorentz transformation discovered by Hendrik Lorentz and later championed by Albert Einstein defines the relationships between the three wombats. As far as the first wombat is concerned, wombat number 3 is going faster than wombat number two, but she is certainly not going as fast as light. It does not matter how many more wombats with rockets we imagine, each going faster than the next, they will not surpass the speed of light, just appear to be getting closer to it. Mathematicians call this type of relationship an asymptote - a barrier that can only be approached.

And yet an asymptote is also an inadequate description, the perception of the third wombat getting closer to the speed of light is merely that of the first wombat, the third wombat herself considers herself stationary and sees the first wombat receding away to the left at close to the speed of light. The speed of light is always a horizon that one can never reach or even get closer to, unless you are light, a gravity wave or other mass-less phenomena, for which it is the only speed you can have.

We measure time in seconds and distances in metres, but the speed of light tells us how to compare time and distance. One light-second is 3X10^{8} metres.
If the first wombat has a watch, she sees the second wombat travel ⅔ of a light second in one second.

The Lorentz transformation formula, tells us how to calculate the time experienced by different wombats:

Δt - time (seconds) elapsed on wombat 1 clock

Δx - distance (light seconds) traveled by wombat 2

Δs - time (seconds) elapsed on wombat 2 clock

So applying the Lorentz formula from wombat 1 to wombat 2 where wombat 2 traveled ^{2}/_{3} of a light second in a second we get:

\mathrm{\Delta s}=\mathrm{\sqrt{\frac{5}{9}}}\simeq \mathrm{0.7454}

So this can be crudely described as wombat 2's clock records 0.7454 seconds for every second of wombat 1's clock. Yet, I will also point out that wombat 2 sees wombat one moving away to the left at ^{2}/_{3}C and so from her perspective wombat 1's clock is running at 0.7454 seconds for every one of wombat 2's seconds.

## The Resemblance to Pythagoras and trigonometry

You see, no one's clock is actually going slower in this case, just time is pointing in different directions. The clue lies in that formula above - what does it remind you of?

OK - I will give you a clue, what would you say if the the formula looked like this?

So what about that angle ν above? If velocity is distance divided by time, then:

\mathrm{tan\; \nu}=\frac{\mathrm{\Delta x}}{\mathrm{\Delta s}}

\mathrm{cos\; \nu}=\frac{\mathrm{\Delta s}}{\mathrm{\Delta t}}

\mathrm{sin\; \nu}=\frac{\mathrm{\Delta x}}{\mathrm{\Delta t}}

In the above relationships, the velocity of our moving wombat looks the sin of an angle in a right-angle triangle. The factor that compares the two times is the cosine (the inverse of the Lorentz factor). But, what about this angle ν?

Well, we can see that as the moving wombat gets faster, this angle approaches 90° or \frac{\mathrm{\pi}}{2} radians, and this shows how the time of the stationary observer is completely pointing in a different direction to the time that the moving wombat experiences Δs.

So is this triangle useful? well the angle ν might convince you that different velocities through time and space are a bit like rotating our time coordinate system to a different direction, but it is a bit of a mathematical dead-end. Proper time (Δs) is the invariant factor, the value that is constantly assigned to the moving wombat while the other values of distance Δx and reference time Δt vary according to different observers in different frames of reference.

If the Δs was the hypotenuse of the above triangle, it would be a much simpler universe, but it would mean that the time dimension is indistinguishable from the other three space dimensions. However, trigonometry has a dark secret - an evil twin that it hides away from the un-initiated - the hyperbolic functions! The hyperbolic functions sinh, cosh and tanh do a superior job of describing the 'rotations' we see in special relativity, but like the dark occult section of the library, I would not send young joeys there! (oh dear - what have I done?)

Δs - is also called proper time. Like a fixed length stick that is rotated in different ways, it remains constant while the other coordinates change. We can use this fixed length stick to calibrate our perspective no matter what our relative speed is. From wombat 1's perspective she sees 1 second pass and wombat 2 move ⅔ of a light second. Another wombat moving at a completely different speed might see the same process occurring in two seconds, in which case, we can deduce that if Δs (proper time for wombat 2) is \sqrt{\frac{9}{5}} (using the same value from earlier) then:

So using the same proper time of \mathrm{\Delta s}=\mathrm{\sqrt{\frac{9}{5}}} But now using Δt=2, we can deduce the the relative speed from this hypothetical observer:

{2}^{2}-\mathrm{{\mathrm{\Delta x}}^{2}}=\mathrm{\frac{5}{9}}

\mathrm{{\mathrm{\Delta x}}^{2}}=4-\mathrm{\frac{5}{9}}

\mathrm{\Delta x}\simeq 1.856 \mathrm{\frac{\mathrm{\Delta x}}{\mathrm{t}}}\simeq \frac{1.856}{2}\simeq \mathrm{0.928\; C}

# From wombat 1 to wombat 3 (and back again)

Because I said wombat one measured one second, the time passed on wombat 2's clock was 0.7454 seconds. The Lorentz factor is simply the rate of one clock divided by the other. In this case the Lorentz factor is 1 / 0.7454 = 1.3416.

In general you calculate the Lorentz factor that tells us how two wombats at different speeds see each other's time dimensions tilting away from each other.

Using light seconds for distance

Δt^{2} - Δx^{2} = Δs^{2}

Δs^{2} / Δt^{2} = 1 - Δx^{2}

^{Δs} / _{Δt} = √ 1 - Δx^{2}

γ = ^{Δt} / _{Δs} = 1 / ( √ 1 - Δx^{2} ) (The Lorentz Factor)

\mathrm{{\mathrm{\Delta t}}^{2}}-{\mathrm{\Delta x}}^{2}=\mathrm{{\mathrm{\Delta s}}^{2}}

\mathrm{\frac{\mathrm{{\mathrm{\Delta s}}^{2}}}{\mathrm{{\mathrm{\Delta t}}^{2}}}}=\mathrm{1-\mathrm{{\mathrm{\Delta t}}^{2}}}

\mathrm{\frac{\mathrm{\Delta s}}{\mathrm{\Delta t}}}=\mathrm{\sqrt{1-\mathrm{{\mathrm{\Delta x}}^{2}}}}

\mathrm{\gamma}=\mathrm{\frac{\mathrm{\Delta t}}{\mathrm{\Delta s}}}=\mathrm{\frac{1}{\mathrm{\sqrt{1-\mathrm{{\mathrm{\Delta x}}^{2}}}}}} (The Lorentz Factor)

But because we are using a distance of light seconds for one second here, Δx is really a speed in light seconds per second. We can express the same formula using speeds of metres per second and the speed of light as the conversion factor:

The Lorentz factor from wombat one to wombat 2 is 1.3416, and so too is the lorentz factor between wombat 2 to wombat 3. So how do we figure out the lorentz factor between wombat 1 and wombat 3?

Adding velocities in special relativity is not straight forward, but this is at least the simplest case
where all three wombats are moving in the same direction (ie collinear).

The collinear addition law is as follows:

Where:

ω_{3} is the velocity of Wombat 3 in the frame of wombat 1

V_{2} is the velocity of Wombat 2 in the frame of wombat 1

V_{3} is the velocity of Wombat 3 in the inertial frame of wombat 2

As V_{2} = V_{3} = ^{2}/_{3} therefore

The Lorentz factor for a velocity of ^{12}/_{13} is γ = ^{13}/_{5}

(A very neat result because 12 and 13 makes a right angle triangle with 5! ie 12^{2} + 5^{2} = 13^{2})

So in summary, we have three wombats that see each others time dilation and speeds respectively as:

Wombat 1 | Wombat 2 | Wombat 3 | |

Wombat 1 | γ = 1 V = 0C |
γ = 1.3416 V = 0.667C |
γ = 2.6 V = 0.923C |

Wombat 2 | γ = 1.3416 V = -0.667C |
γ = 1 V = 0C |
γ = 1.3416 V = 0.667C |

Wombat 3 | 2.6 -0.923C |
1.3416 -0.667C |
1 0C |

So our three wombats should actually move like this (according to wombat 1):

(click to replay)

## How does it work both ways?

OK, so wombat 1 thinks wombat 3's clock is ticking slower, but wombat 3 thinks wombat 1's clock is ticking slower - that does not make any sense!

Well spotted young joey!

As I said before, time is ticking in a different direction for each three wombats, but no one's time is going any slower. You would therefore conclude that wombat 1 is aging quicker than wombat 3 but also wombat 3 is aging quicker than wombat 1. So here is the Possum paradox, or "Twin paradox" as you humans call it. Sure, wombat 3 is apparently aging slower from wombat 1's perspective. And yet, while wombat 3 keeps heading away from wombat 1 then how is that going to play out? You see, if there is a contradiction here it is irrelevant because wombat 3 will never ever again meet wombat 1 to compare their clocks while they are on these straight line trajectories. They can of course send signals to each other that will take time to reach each other, and these will just reinforce to each other their own perspectives.

But, what if wombat 3 turns around and comes back?

OK - but now you are now dealing with paths that are not straight lines and the situation is no longer symmetric from one wombat to another. Wombat 1 sticks to her position or frame as we call it, and wombat 3 follows a path has her moving away and then returning. She experiences accelerations in the process and her frame of reference is said to be non inertial. On wombat 3 returning to wombat 1's frame of reference her time direction becomes reconciled again to that of wombat 1's. Wombat 3 will have covered long distances according to wombat 1's perspective but experienced less time. The Lorentz transforms calculated from the inertial frame (wombat 1's) will predict the time difference and wombat 3 will have aged less.

## So do non inertial frames have slower time?

OK - I don't want to say yes, because what does slower time even mean? But you have been told that the shortest distance between two points is a straight line, but if I take the long way I can take a short-cut to the future. In fact, if I travel to the nearest star and back at close to the speed of light, my Lorentz factor becomes very large indeed. I could visit the colony of Hairy nosed wombats on Proxima Centauri B and come back so that for me it only took a few days, but for those of you on Earth it will seem like 9.8 years have passed.

## Does that mean wombats can travel faster than light?

No, but light speed is a deceptive speed limit. There are no Emu-police out there patrolling space to tell you to slow down, you can go as fast as your engines allow you, and that means if you have enough carrots to feed the engine you could theoretically get to the Hairy-nose colony in as little time as your rocket drive allows. So yes, based on your initial estimate you did on Earth, you could be traveling faster than light towards Proxima Centauri and breaking the laws of physics, but actually the distance between my cosy little hole and my hairy nose friends on Proxima Centauri B is less, by the same factor as the Lorentz factor. It is only to you suckers stuck on Earth that speed seems to be limited to the speed of light!

And here is the other catch, the amount of energy required to get a 25kg wombat to a speed that would make Proxima Centauri B only a few days away for the traveler is huge.

The relativistic kinetic energy is given by:

Ek = (γ - 1) × mc^{2}

If my clock tells me I took 1 day, and Earth clocks suggest 4.3 years, then the Lorentz factor (γ) is 4.3×365 / 1, therefore my kinetic energy must have been about:

(4.3×365 - 1) × 25 × (3 × 10^{8})^{2} joules

Quite clearly a rocket engine that burns carrots is not going to get you there, and that is assuming you are using anti-carrot/carrot drives that convert most of the mass of carrot into useable propulsion energy and only a little bit of carrot juice on the side to feed the crew! (don't use the anti-carrots for juice or you will annihilate yourself as one hairy-nose wombat space explorer once did!)

So let's face it - even with anti-matter propulsion (or anti-carrot drives) it is difficult to reach speeds approaching the speed of light. Even a modest Lorentz factor of 2 means converting your body weight in carrots/anti-carrots into pure propulsion energy. The quicker you want to get there, the more carrots you need to burn, which is even more weight to bring on. So you might be wondering, how do I know there is a colony of hairy-nosed wombats on Proxima Centauri B?

well, that my human friends is where I will leave the lesson.

## Any More Questions?

**Did Einstein or Lorentz first discover special relativity?**

OK - so you mean who was the first human to discover this? well that is easy, it was the Dutch scientist Hendrik Lorentz ... or maybe it was Albert Einstein? Well Hendrik Lorentz still believed in an ether, but he thought the ether itself was doing something crazy and stretching time and space so that light appeared to be traveling at the same speed regardless of the observers frame of reference. Hendrick Lorentz was no fan of these equations, and he no doubt was thinking about the speed of light literally as nothing more than the speed that light appears to move at. Hence why I describe this as Lorentz's ugly duckling, his equations seem to him like a fudge more than a theory.

Albert Einstein however made a bold claim, that the speed of light is a universal constant that is invariant (unchanged) in all frames of reference (yes, Lorentz could have said that is what his equations imply, but he thought it was an illusion of the ether). The speed of light came to mean much more than the speed of light, and furthermore, he discovered that these equations define a relationship between mass and energy (E=MC^{2}) and even shed some more light on how magnetism and electricity are related - as if Maxwell had not quite had the final world.

Lorentz's ugly duckling had transformed into a beautiful elegant swan!

**If an event A occurs before event B in my perspective (reference frame), does it mean it occurs in that order in all reference frames**

Now you are getting it! Yes, there are situations where two observers will apply a different order to the same events, but this cannot be used to tell an observer the future, because these events can only be retrospectively determined by information that arrives at the speed of light - so way too late to give me hot tips on the which Goanna will win the Lizard Cup. In fact this situation is so interesting it is going to need four wombats with rockets!

(click to replay)

Two wombat on the left are stationary and are 1 light second apart, while two wombats on the right are also 1 light second apart but are moving towards the stationary wombats on the left. With a Lorentz factor of about 1.75, the wombats on the right appear to be 1.75 light seconds apart from the perspective of the stationary wombats on the left.

For the wombats on the left, the following events occur in this order:

- Wombat 3 passes wombat 2
- Wombat 3 passes wombat 1
- Wombat 4 passes wombat 2
- Wombat 4 passes wombat 1

But now we watch the same events from the perspective of the wombats on the right

(click to replay)

From this perspective the events occur as follows:

- Wombat 2 passes wombat 3
- Wombat 2 passes wombat 4
- Wombat 1 passes wombat 3
- Wombat 1 passes wombat 4

The middle two events have swapped around in order, because respective distances between wombats has changed with the change of frame! Wombat 3 passes wombat 1 and wombat 4 passes wombat 2 in different order depending on the frame. The separation distance between the two swapped events means the phenomenon can only be appreciated retrospectively.

**Is it possible to go faster than light?**

Everything about Lorentz's equations and Einsteins interpretations say no. To accelerate requires propulsion energy, and even an infinite amount of energy only allows us to keep approaching the speed of light with respect to your original frame. The equations also suggest that if the speed was greater than the speed of light, the proper time becomes the square root of a negative number. This would seem to suggest proper time to the traveler would behave space like, and space-like dimensions don't tick, they let you go back and forwards! This might be why there exists a sci-fi trope that traveling faster than light makes you go backwards in time. However it is a fanciful interpretation of an impossible situation.

Yet, despite this the universe has black-holes where one can fall through an event horizon which has similarities to going faster than light. It is also generally believed that the universe itself is expanding such that distances greater than the Hubble distance (14.4 billion light years) are stretching faster than light speed. If time is running backwards 14.4 billion light years away, it is certainly not noticeable to any sentient life-forms out there, any more than we might imagine that polar bears must be walking upside down because they live near the North pole! Yes, even if the time dimension does point in opposite directions in some part of the universe, it still behaves like time locally and these parts of the universe are now cut off from us.