Hey, so you know when I said that in order for the universe to accommodate the speed of light being invariant something else has to pick up the slack, and that something is time?

I lied.

Or at least I was economical with the truth because that post on time dilation was getting on for 3000 words. The invariance of light speed in fact has knock-on effects for most of the dimensions of space-time as well as things like mass and energy, and in this post I’m going to focus on what relativistic speeds do to an object’s size – otherwise known as Lorentz contraction.

We’re mostly going to stay away from the horrible maths this time because I get the feeling most people aren’t able – or willing – to follow that all the way through, so I’ll do the whole ripping-off-a-plaster thing and get what we have to do out of the way quickly. After looking at the light clock we derived the following expression for time dilation

The 1/√(1-v²/c²) term is actually something called the Lorentz factor γ, which crops up a hell of a lot in equations dealing with special relativity. For example, when dealing with the length of an object in a stationary moving frame x versus a moving (dilated/relativistic) reference frame x_d, we say that

which honestly makes the whole thing much easier to deal with if it’s a simple matter of multiplying the stationary, or “rest” mass/length/time by this common Lorentz factor to get the dilated/contracted version.

Anyway, what this implies is that just as time slows down compared to a stationary observer as an object reaches relativistic speeds, the object will also contract in size along the axis of travel (i.e. a train would contract down the long axis of its length rather than in width or height). This phenomena is known as length contraction, or Lorentz contraction, and I’m bringing it up because it poses an interesting paradox that has consequences for our concept of simultaneity (i.e. the concept that it can be 5 o’ clock everywhere at the same time).

Wikipedia explains this paradox as the Ladder paradox; it was taught to me with a medieval knight riding a charger carrying a lance in place of the ladder, but honestly any long horizontal object will do for this thought experiment. The idea is, you have a man carrying a ladder running very very quickly towards a garage. The garage has a door at either end, and the length of the garage is just slightly shorter than the length of the ladder when it is at rest. The front door of the garage is open and the rear door of the garage is shut. The man runs into the garage. When the ladder is fully inside the garage, the front door immediately slams shut and the rear door opens, allowing the man to exit the garage without running into the door and becoming a nasty relativistic smear across its surface.

Now, this thought experiment is already a bit of a brain twister; the length of the garage is smaller than the rest length of the ladder, so if the front and rear doors open/close simultaneously and they are not both open at once there is no possible way the ladder can fit inside the garage – unless the ladder is moving at a relativistic velocity sufficient to contract its length so that it can. Everyone with me so far?

The problem here – and the paradox – lies in the frames of reference I was jabbering about last week. From the point of view of the garage, the ladder shrinks in length and can easily fit inside it with no problems. However, just because we happen to think of the garage as the stationary object doesn’t mean that this is the one true reference frame. The ladder is also valid frame of reference, and from the point of view of the ladder it’s the garage which is moving towards it at a relativistic velocity, and hence the garage which shrinks and contracts in length.

The paradox is this, then: from the point of view of the garage, the ladder shrinks in length and everything is okay, but from the point of view of the ladder the already-too-small garage contracts in length even further while the ladder remains far too long to fit inside it. How on earth is the ladder supposed to make it through the garage in this reference frame?

There is an answer to this (or else special relativity would have come crashing down by now) and that answer is to throw out the concept of absolute simultaneity. In the garage’s reference frame the front door closes and the back door opens at exactly the same time – simultaneously. However, this is not the case for the ladder’s reference frame. The ladder sees things differently. From the ladder’s point of view, the back door opens first, allowing the front end of the ladder to pass through without any problems. Then, when the back end of the ladder has passed the front door, the front door slams shut. The back door opening and the front door closing happen at different times; the simultaneous events in one frame of reference happen at different times in another. This demonstrates the relativity of simultaneity, or as Wikipedia puts it:

According to the special theory of relativity, it is impossible to say in an absolute sense whether two distinct events occur at the same time if those events are separated in space, such as a car crash in London and another in New York. The question of whether the events are simultaneous is relative: in some reference frames the two accidents may happen at the same time, in other frames (in a different state of motion relative to the events) the crash in London may occur first, and in still other frames the New York crash may occur first.

In order to demonstrate the whys behind this we’re going to use a thought experiment similar to last week’s light clock, only this time our moving object is a train whizzing past a stationary platform.

There is a lamp in the centre of the train carriage. The lamp is turned on and light emerges which illuminates the train carriage. This seems fairly uncontroversial, but the way this happens will be perceived slightly differently by an observer standing next to the lamp and an observer standing on the station platform.

The observer standing next to the lamp is moving at the same velocity as the lamp and is stationary with respect to it. He sees the light moving as we would expect; the light covers the equal distances to either end of the train carriage at the same speed, and the light reaches the ends of the train carriage simultaneously.

However, to the observer standing on the station platform the lamp is moving. Once the lamp is switched on the rear of the traincar will be moving towards the emitted light, while the front of the traincar will be moving away from it. Since the speed of light c is invariant in all reference frames (as we saw last week) the light will travel towards the rear of the carriage at the same speed as it does for the moving observer, except because the rear of the carriage is moving towards the light the light will have a shorter distance to travel. Thus to the observer on the station platform the light will appear to hit the rear of the train carriage first, and the front of the train carriage second. QED.

This, I hope, dovetails nicely with the time dilation post to show that time isn’t the immutable constant we’re all used to in everyday life. Depending on where you are and how fast you’re going it can be bent and stretched and tied in knots. You pays your money and you takes your frame of reference.

The post Lorentz Contraction And The Ladder Paradox. appeared first on The Scientific Gamer.

Noah Rhee asks

Dear Professor,

A question professor, what is time dilation and how does it work? Furthermore, if time indeed is relative, how would this play into futuristic space colonization?

Thanks,
Noah Rhee

Oh, this is a doozy. Time dilation is something I’ve tried to avoid talking about because it’s one of those things that involves counterintuitive concepts that are particularly difficult to explain without using reams and reams of awful, awful maths. However this is by far the most polite question I’ve been asked so far, and so I shall make a special effort just for you.

There are two types of time dilation: time dilation which arises from differences in relative velocity (i.e. going really really fast compared to a “stationary” observer), and gravitational time dilation caused by particularly heavy celestial bodies such as planets and stars. The gravitational type is pretty much impossible for me to explain since it arises as a consequence of general relativity, and I don’t understand that anywhere near enough to feel comfortable describing it. Happily the type you appear to be asking me about is time dilation due to a difference in relative velocity, which I do understand fairly well and which furthermore has several simple prepackaged metaphors and thought experiments to aid in its explanation.

This particular type of time dilation is a result of the invariance of the speed of light, and the usual analogy used to get this across is the hilariously quaint boy with a peashooter riding a bike. The boy is riding his bike at five metres per second. To a stationary observer moving at zero metres per second, the boy appears to be riding his bike at five metres per second. Sounds straightforward, right? The boy then fires a pea out of his peashooter which is also travelling at 5 ms^-1. To the stationary observer this pea was already going at 5 ms^-1 before it was fired since the boy was carrying it along with him. After being fired it has a combined velocity of 5 ms^-1 + 5 ms^-1 = 10 ms^-1 – with respect to the stationary observer’s frame of reference. With respect to the boy’s frame of reference, the pea is moving away from him at the 5 ms^-1 at which he fired it, and so as far as he’s concerned the pea has a velocity of 5 ms^-1.

This analogy explains how frames of reference work and how different observers can see the same object moving at different relative velocities. As far as light is concerned, though, it travels at exactly the same velocity relative to all observers and in all frames of reference. Light travels at 300,000 km s^-1, or c. We can explain the invariance of the speed of light using our above analogy, except this time we replace the boy’s peashooter with a flashlight and accelerate him up to a relativistic velocity of, I don’t know, 0.1c (10% of the speed of light). To the stationary observer the boy is travelling at 0.1c. The boy turns on his flashlight. He sees the light travelling away from him at c, the speed of light, just as he saw the pea travelling away from him at 5 ms-1. But the stationary observer doesn’t see the light travelling at the combined velocity of the boy and the light; that is, he doesn’t see it travelling at a velocity of c + 0.1c = 1.1c. He sees the light travelling at c, just like the boy. This is what we mean when we say the speed of light is invariant; everyone sees it travelling at c no matter where they are or how fast they are moving.

In order for the universe to accommodate this invariance something else somewhere has to make some concessions so that the fabric of the universe doesn’t fall askew like a fat man’s bath towel, and that something is time. Time dilation is the universe’s way for compensating for the invariance of the speed of light; light can be the same speed for everyone just so long as they perceive the passage of time at different rates depending on where they are and how fast they are going. If you want a why as to the invariance of the speed of light I’m afraid I can’t help you. It’s one of those deep cosmological things that has been proven experimentally (most notably by the Michelson Morley interferometer experiment) but for which we don’t quite know the explanation. However, now that we know that it is invariant we can figure out the whys and hows of time dilation using another simple thought experiment: the light clock. I’m cribbing this particular example from Wikipedia but it’s one which is used to teach first-year undergraduate students the world over, so it’s not like they’ve got copyright on the method.

Imagine a beam of light bouncing between a pair of mirrors. Each time the light bounces from one mirror to the other and back again, the clock “ticks” once. The time period between ticks will be the time it takes for the light to do this, i.e. twice the distance L separating the two mirrors (since the light has to cover this distance twice) divided by the speed of light, C.

That is the length of the tick when both mirrors are stationary with respect to the observer. When two things are stationary with respect to one another they have the same reference frame, and there is no time dilation apparent between the two. In order to observe the effects of time dilation we’re going to have to set one mirror in motion.

In this setup the bottom mirror is moving from left to right with a velocity v. The top mirror remains stationary. The path of the light is now angled in comparison to the light from the first clock, meaning the light has to travel a longer distance D. The time it takes for this clock to “tick” once will therefore be

If we know the velocity of the bottom mirror v we can use simple trigonometry to work out the value of t_d in terms of L, v and c – i.e., in terms comparable to the tick of the stationary clock.

The perpendicular distance separating the mirrors is L. The bottom mirror is moving at velocity v. The bottom mirror is moving from left to right at the same rate as the beam of light moves from left to right (it has to in order to be there to receive the light at the end of the “tick”), so when the beam of light strikes the top mirror the bottom mirror will be directly underneath it. Exactly half the tick time period t_d will have elapsed once this happens, so the total distance the bottom mirror will have moved from left to right at this point will be its velocity multiplied by the time it has spent travelling at this velocity, or

These three vectors form a right-angled triangle, with D as the hypotenuse. Pythagoras’s theorem states that the square of the length of the hypotenuse will be the sum of the square of the lengths of the other two sides of the triangle, or

Taking the square root gives

And we can then substitute this value for D back into our original equation for t_d above (t_d = 2D/c).

We then do some extremely convoluted rearranging and cancelling out to get an equation of the form

But wait, didn’t we say earlier that 2L/c was equal to the tick period of the first stationary clock t?

And that is how you get your time dilation of a moving body as it appears to a “stationary” observer. Not seeing it? Well, hopefully I can help you out there.

First, why does this happen? It’s to do with the invariance of c and the different reference frames mentioned above. If we lived in a world where time was invariant (i.e.  t_d = t without any of that other crap) then the speed of light would not be absolute, and we’d have a similar situation to the one with the boy and his peashooter. To a stationary observer the light in the stationary clock would travel at c, while the light in the moving clock would travel at c plus some component of the left-to-right velocity v. However, it does not do this. The light in the moving clock moves at c as well. In order for the light in both the stationary and moving clocks to move at c even though you have this additional velocity vector v added in to the moving clock, you have to slow down time in the moving reference frame. It’s not quite a correct way of thinking about it, but you could say that light moving with the “faster” velocity c + v at this slower rate of time appears, to an outside observer, to be moving at the normal speed of light c. The time dilation cancels out any added velocity vectors from the perspective of a stationary reference frame and ensures light always moves at c. It’s a physical necessity even though it seems ridiculously counterintuitive to our sluggish human perception of time.

What the above equation is saying is the tick period of a moving clock t_d will be equivalent to the tick period of the stationary clock t divided by the term in the square root. C is a really, really big number (3 × 10⁸ ms^-1) so if the velocity of the moving clock v is the sort of thing we’re likely to see in our everyday lives (velocities on the order of 10-1000 ms^-1) then v²/c² is going to be really goddamn small — it’s practically nothing, and when you subtract nothing from something you’re still left with the original something. This means 1 – v²/c² is going to be effectively 1, the square root of 1 is 1, and so the equation shakes out to t_d = t. In other words, for relatively small velocities (small in comparison to c) time dilation will be almost non-existent. This is why we don’t notice time dilation as we walk around on the surface of the Earth, and even satellites orbiting at several kilometres per second need extremely sensitive clocks to measure the effect of time dilation (although it is relevant). It is only when you get up to velocities approaching a significant fraction of the speed of the light that the v²/c² term in that equation is anywhere near big enough to put a significant dent in the 1 when subtracted from it.

Once you’re there, though, things start getting interesting. The 1 – v²/c²  term in the square root becomes appreciably smaller than 1. When you divide t by a number smaller than 1, it has exactly the same effect as multiplying it by a number larger than 1. In other words, for every tick t_d the dilated clock makes, a stationary clock will make 1.1 or 1.25 or 2 ticks. And if a stationary clock is ticking twice for every tick of a dilated clock, then if you’re in the same reference frame as the dilated clock time will effectively be moving forward at half the rate for you as it would for somebody in a stationary reference frame. The stationary person will age at twice the rate you do and live their life at twice the pace. That’s time dilation. I was going to make my own graph of this using FABULOUS EXCEL TECHNOLOGY but it turns out Excel sucks, and so I resorted to stealing this one off the usual source.

The X-axis shows the speed of an object expressed as a fraction of the speed of light c (i.e. 0.1 is equivalent to 0.1c). The Y-axis shows t_d/t (or how many “ticks” a stationary clock will make in the time it takes for a dilated clock moving at the relevant speed to make one tick) as the speed increases. The graph shows that even somebody moving at 0.5c – half the speed of light — will experience very little time dilation. It’s only when you get to 0.6c that it becomes noticeable, and 0.9c that it becomes significant. However, once you’re past 0.9c the amount of time dilation you experience begins to increase very very quickly, trending to infinity as you approach the speed of light.

So that’s time dilation. It is a thing that happens, and that we know happens. We have measured it many times. We have to take time dilation into account when plotting the long-term trajectories of interplanetary probes, even though the amount of time dilation the probes experience is tiny. I think the GPS satellites even have to compensate for time dilation when doing their routine station-keeping maneuvers. But what does it mean for space colonisation?

At the moment, very little. We do not need to move at relativistic velocities to get around the solar system – indeed, the drastic acceleration required to get to relativistic velocities in the timescale required would crush any human crew on board a relativistic spacecraft to paste. Time dilation, if it is a factor at all, will only become a factor if and when we get around to sending out colonisation parties to nearby stars – and this is a journey that will take so long that some form of suspended animation will be necessary no matter how we do it. It would take decades (or longer) just to accelerate to a velocity where time dilation becomes relevant (and this is leaving aside the fact that you have to slow down again before you can stop at your destination), so while we’re likely to be travelling interstellar distances at relativistic velocities, time dilation is not a get out of jail free card for people on board to age so slowly they can weather the trip in a human lifetime. It will be a factor, but it will be a factor in the same way it is for the GPS satellites: a technical hurdle that must be taken into account rather than a neat form of time travel. After all, if your trip is taking longer than a human lifetime it doesn’t much matter for the person on board whether it takes 300 years or 30,000; they’re still going to be dead at the end of it without some kind of space-magic stasis technology.

Wow, this really reminded me why I don’t do sums on this blog. Anyway, I hope this at least made time dilation a little clearer for those of you who have stuck with me through to the end. Thanks for reading!

The post When This Baby Hits 88 Miles Per Hour. appeared first on The Scientific Gamer.