The speed of light can help win you the lottery!

A layman’s understanding (without equations or jargon!) of the fascinating implications of a simple universal law of physics:

The speed of light is constant for everyone.

A universal law of physics

Sometimes reality is stranger than fiction. For me, one of the greatest examples of this is the amazing implications of the simple fact that the speed of light is constant for all observers. This is in contrast to our day-to-day experiences. If I am standing still and throw a ping-pong ball at you, you will feel it hit with a certain force. If I am running towards you and throw the same ball you will feel it hit you harder because is moving faster with the added speed of my running.

This isn’t how light works, though. If I am standing still and shine a light at you, you will measure the light as moving at 300,000 kilometres per second (in a vacuum). If I am moving towards you when I shine the light – even at very high speeds – you will still measure the light as travelling at 300,000 kps. This is the basic difference between the classical mechanics of Newton and Einstein’s special relativity which superseded it.

The implications of a universal speed of light

That’s all very interesting – but why is it so very fascinating? Well really it is the logical implications of this fact, confirmed by experiment, which are mind-blowing. If the speed of light is always constant for different observers then this implies that both distance and time are not necessarily constant for all observers since speed is defined as distance over time. Yes, they can really be different. The stopwatch and ruler of one person can measure time and distance differently from another person with identical tools – not because of some error in observation but because time and distance really are different for them both if one is moving at a different speed with respect to the other.

Visualising time dilation and length contraction

A thought experiment can show why this is the case. Imagine a pool table with a ball bouncing widthways in a straight line back and forth between the two cushions. Now imagine that same table on a train with a clear roof moving past below you at some constant speed as you stand still on a bridge. You would see the pool table with the ball bouncing back and forward on the table and the ball would appear to you to be tracing a saw-tooth pattern, ΛΛΛΛΛ. To someone on the train, though, it would be moving backwards and forwards in a straight line as before and the distance travelled by the ball is the width of the table by the number of times it crosses. For you stationary outside the distance you see travelled by the ball is greater because the ball has been tracing out a saw-tooth pattern and the diagonal is always longer than the length of a side.

Now imagine that that pool ball was in fact a photon of light bouncing between two perfect mirrors. We know that both the person on the train and the observer outside will judge that the photon moves at 300,000 kps. So in order for this to be true time must be running slower for the person on the train from your point of view on the bridge if the distance covered is greater. The opposite is true too. If the pool table were on the bridge then someone on the train would see a ball on that table moving in a saw tooth motion as the bridge passed overhead and think time for you was running slow. Similar reasoning can show that the length of the train has also contracted in the direction of its movement from your point of view, and the length of the track contracted from the point of view of someone on the train.[1]

Implications of time dilation and length contraction

The implications of this are fantastic. The closer you travel to the speed of light, which you cannot overtake, from your point of view the slower time runs for the stationary points around you and the more distance shrinks for you in the direction of your motion. This means you can easily travel many light-years within your life-time. If you are travelling at 80% of the speed of light then a star 100 light years away from Earth becomes only 60 light years away for you because of length contraction[2]. You can therefore reach it in 75 years of your lifetime (so you still haven’t travelled faster than light from your point of view. Time seems to be ticking as normal but the distance you need to travel is much less). The people on Earth would see you as being able to achieve this because although the distance hasn’t changed from 100 light years, time for you has slowed down to such an extent that your life-span can last the journey. They will see you pouring a coffee that takes almost twice as long to fill the cup.

Because two people moving relative to each other have a different measure of time and distance there can be no such meaningful concept as “now”. That doesn’t mean that their points of view can’t be reconciled since both are perfectly correct[3]. It does mean that any intergalactic community would have trouble deciding when to hold the bake-sale.

There are also no universal concepts of “before” and “after”. Imagine a train travelling along a track at near light-speed approaching a warehouse sitting over the track with doors at the front and back where the train can be housed over night if not in use. The length of the warehouse is just long enough for the train to fit in when at rest. A warehouseman at the front of the warehouse sees the train coming and knows that the train isn’t stopping and needs to continue along the track. As the nose of the train meets the front door he opens it so the train can enter. When the end of the train passes the front door so the train is completely inside the warehouse (it can fit comfortably since it is approaching light-speed its length has contracted from the warehouseman’s point of view) he closes the front door. A second warehouseman sees the front of the train approach the back door and opens it so the train can exit. When the back of the train passes the back door he shuts it again. All well and good.

However, for the train driver it is the warehouse that has shrunk and so the train, from his point of view the same length as it ever was, will no longer fit in the warehouse with both doors shut. Will the train crash through the back door? No. From the train driver’s point of view the order of events has also changed. As he approaches the front door he sees it open as before so he can enter. However, the back door opens next before the front door closes again and so the front of the train can leave the warehouse. As the back of the train passes the front door it closes behind it, and when the back of the train leaves the warehouse through the back door that closes also. The order of events has changed saving a crash. If the train driver and warehouseman were to meet up later down the pub they would disagree on the order of what happened when.[4]

A universal speed of light can make you rich!

So the train driver could win the lottery! Imagine that lottery tickets go on sale when the warehouseman opens the front door. The sale ends when the front door is closed. The lottery is drawn when the back door opens, and the period to collect any winnings ends when the back door closes again. This is perfectly normal from the warehouseman’s point of view. But from the train driver’s point of view the order is “ticket sale start” “lottery drawn” “ticket sale end” “winnings collection time end”. So he has a window of opportunity to buy a ticket and claim his winnings after the draw has taken place! This is perfectly true but he will not necessarily become a millionaire.

The apparent contradiction can be explained since for relativity to have a meaningful effect on the order of events, the train has to be moving at a significant percentage of the speed of light and so for a decent time for the sale of tickets the length of the warehouse and train would have to be mind-bogglingly huge. On reaching the back door both the train driver and second warehouseman are not local to the front door and are using the facts they know to be true – the speed of the train, the length of the warehouse, and the time taken for the train to cross it – to calculate when the front door opened and closed (the window of opportunity to buy a ticket) but neither is actually at that door. So both are perfectly correct within their own frame of reference to calculate the order of events according to the distance and time they know to have elapsed and the driver gets the winnings only if the lottery ombudsman is also on the train and so also in that frame of reference.

Time dilation and length contraction in real life

The engineering necessary to move anything the size of a person to near the speed of light is still a long way from us. However, this effect has been observed both in experiment and in nature. Identical clocks have been synchronised before one has been flown at great speeds around the Earth before returning home. This clock has then been confirmed as out-of-sync with the stationary clock by the expected amount.[5]

There also exist particles called muons. These have a very short life before they decay. They have a very small mass and so don’t travel at the speed of light but just below it. Their life is so short that they haven’t the time to travel through the Earth’s atmosphere to hit the ground at below light-speeds before they would decay. Yet they have been observed hitting the ground. How? Because for them the distance to the Earth has shrunk and from our point of view time for the muons is running so slowly that their extended life-span allows them the time to hit the ground.[6]

Is this the future of space-travel?

If special relativity allows us to travel great distances in our life time – distances as great as we wish, as long as we can approach the speed of light sufficiently closely – is this then the future of space exploration? Possibly. But the problem is that although time slows down on an interstellar space craft from the point of view of someone on Earth and time slows down on Earth from the spaceman’s point of view, if an interstellar space craft turns around and returns to Earth then the acceleration of the turning back will break the symmetry of movement and crystalise the time difference[7]. They will find that all their friends and family are now long dead.

Links to detailed calculations and explanations for those who would like them:
[4] I couldn’t find a space-time diagram on line for specifically the example I use above. The attched is for a very similar situation of the “pole in the barn” paradox. The spacetime diagrams show how the order of events can change at relativistic speeds. astro.berkeley.edu/~jcohn/phy7c_03/pb.html