Sunday, January 22, 2012

Rapid slow space travel

Credit: Craig Crawford on APoD
I have posted in the past about mundane space travel such as might be used with the solar system (or another star system if we're talking aliens or whatnot). However, with speeds that slow, it would take an extremely long time to reach another star, even the closest. To have any hope of reaching another star, we need to be able to travel much faster.

Right now, we aren't technologically equipped to do so and that's not what this post is about. What I'm going to talk about is what happens when we (or rabbits or clocks or whatever) travel at high speeds. Because strange and interesting things do happen. Welcome to the weird and wonderful world of Einstein's special relativity.


We live in a world with three spatial dimensions and one time dimension. All this really means is that we can define a co-ordinate system (for example x-, y- and z-axes) which can define any point in space by listing three numbers (the x, y, z co-ordinates) and which can define any point in time with a single number (although it doesn't look like a single number, that's how we can think of "11:00 am on 21 January 2012"). Any point in spacetime (that is to say, our universe, past and present) can be defined by combining those two co-ordinate systems to give four numbers, unique to each point.

Now, say you're in a long corridor. There are several ways you might try to measure how long it is. You might walk along it and count steps or use a measuring tape. You might jog or walk at a know speed and time how long it takes to get to the other end. If you were particularly eager and the corridor sufficiently long, you could bounce light (or radio waves) off the far end and time how long it takes to complete a round trip.

Of these three options, I'd hazard that most people would use a length-based measurement as per the first option.

Now suppose your room is actually a space ship traveling close to the speed of light with you inside it. (For now we're ignoring how it got up to that speed.) You can still use the same three methods to measure it. Remember, when you're moving at a constant speed, you don't feel the movement. Aside from bumpiness due to uneven roads/train tracks/turbulence, the only sort of movement you can detect without looking out a window are the periods of acceleration and deceleration. So, if you're traveling at a constant velocity in a spaceship with no windows, you would have no way of checking how fast you're going, but other than that, nothing weird would seem to be happening.

On the other hand, if you were outside the spaceship watching it go past, how could you measure how long it was? Being on the outside rules out walking along it with a tape measure (unless it's stationary, but then it's not going past, is it?), but the other two methods more or less work. If you know how fast it's going, you can time how long it takes to go past. If you know how long it is, you can time how long it takes to go past and work out how fast it's going.

Intuitively, we might expect that spaceship length doesn't change and that the speed of the spaceship is the only thing that determines the time taken for it to go past. This isn't strictly true.

The one immutable quantity when we're talking about moving objects in a vacuum (that is, spaceships in space) is not how long they are or, strictly speaking, how fast they're going. It is, in fact, the speed of light. The old mantra of special relativity is:

The speed of light is constant in all inertial reference frames.

A definition before I go on: inertial reference frame is a set of co-ordinates which isn't accelerating. If you are in an inertial reference frame, you can define your spacetime position with respect to those co-ordinates.

Also, an important point is that it's not possible for any object with mass to move at the speed of light (or faster). The only reason light gets away with it is because photons, particles of light, are massless.

Goin' fast

Say your fancy long spaceship is constantly going at half the speed of light. Because the speed of light is constant in all inertial reference frames, light from a torch you shine inside the spaceship will still travel at the same speed of light as it would anywhere else. Furthermore, just because you're traveling at half the speed of light doesn't mean the light from your torch will appear to travel at one and a half times the speed of light to someone outside your spaceship who can look inside.

Sounds paradoxical, doesn't it?

To make up for the apparent paradox, two things happen. Remember that speed or velocity is basically the amount of distance covered in a stretch of time. To keep the speed of light constant, both distance and time change, depending on how fast you're observing from.

A fast moving object appears to be shorter than it would were both object and observer in the same reference frame (that is, traveling at the same speed in the same direction). This applies to the outside distance for a fast-moving spaceship -- the distance traveled/left to go appears shorter than if the spaceship was stationary with respect to it. This is called length contraction.

Quick side note: this means that all distance is relative and there is no such thing as being truly stationary, just stationary with respect to some other reference frame.

The faster your spaceship goes, the more slowly time passes for you. Well, actually, to you it would seem that between starting your journey and ending it, time passed more quickly planetside than it did for you. (It's all relative, see?) This is called time dilation.

The amount by which time slows down or distance shrinks is dictated by the relative speed of your spaceship. There's a mathematical quantity called the Lorentz factor, represented by the Greek letter gamma, which tells us how much.

Gamma, the squiggle on the left, is the Lorentz factor, v is the speed the spaceship or whatever is traveling, and c is the speed of light, equal to 3 x 108 metres per second.

So if you're traveling at half the speed of light, gamma would be equal to about 1.15, so time would pass 1.15 times more slowly. An hour on the spaceship would take about 69 minutes to pass on Earth. The length of the spaceship, to someone on Earth, would be 1.15 times shorter. One metre would appear to be about 87 cm long.

Some other values of gamma for speeds which are significant fractions of the speed of light are:
  • Speed: 0.75c, gamma =1.51
  • Speed: 0.867c, gamma = 2
  • Speed: 0.9c, gamma = 2.3
  • Speed: 0.95c, gamma = 3.2
  • Speed: 0.99c, gamma =7.1
  • Speed: 0.9999c, gamma =70.7
To work out the time dilation, multiply by gamma, to work out the length contraction, divide by gamma.

This isn't quite all there is to know. For example, objects traveling at relativistic velocities (at an appreciable fraction of the speed of light) also increase in mass by the same factor of gamma and accelerating up to high speeds is also a bit strange. More on that next week.

And in case you're wondering how fast you have to go for these relativistic effects to kick in, or even how we know they're real... well, they exist no matter how fast you're going, it's just that at the sort of speeds we experience on a day to day basis, the time differences are entirely negligible. We have been able to test relativity, however, in a couple of ways. Flying super-precise atomic clocks around on aeroplanes has shown that time passes more slowly for them relative to us. The same has been shown for GPS and possibly other satellites. On a much larger scale, measurements of binary pulsars have also confirmed Einstein's theory of special relativity.

Stay tuned for more next week.

No comments:

Post a Comment

Have a question or comment? Feel free to leave a response, even on old posts.


Related Posts Plugin for WordPress, Blogger...