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Speed of Light, as a speed limit?
There's a popular misconception that the speed of light is some sort of speed limit, so when you travel in space you can get almost up to the speed of light and then you just can't go any faster, and then you have to wait four years before you get to Alpha Centauri which is 4 light years away. This is not true. You can get to Alpha Centauri a lot quicker than that, provided you don't mind much what other people think. The idea that the speed of light is a speed limit is a misconception a bit like thinking that people who live in Australia are upside-down.
Before I try to explain Relativity and what really happens when you travel very fast, let's look at that idea about Australian people being upside-down. Well, are they upside-down? If you're in the UK, then in a literal sense, yes, Australians are upside-down. However, from an Australian perspective, they are not upside-down at all, and kangaroos leap UP in the air and then come DOWN again, etc. So, the moral of the story is: It depends on from whose perspective.
The situation with the strangeness about the speed of light is another thing where it depends on from whose perspective we are considering. If you are standing on Earth watching the passing spaceships as they fly across the sky, you notice that none of them seem to be travelling faster than light, and they always take years to get from one star system to another, according to how many light-years it is. It would be easy to imagine that the people aboard those spaceships get very bored waiting to get to their destinations. However, if you get on a flight to a distant star system, the journey time is nothing like as long as was expected. You can get from Earth to Alpha Centauri in a few hours! How can that be?!
The fact is that when you travel very fast in space you also travel in time. Going from Earth to Alpha Centauri, your watch says it's only taken a few hours, but when you arrive it's four years in the future. This fact makes it very convincing for people on the ground to think you've taken four years to get there.
After a couple of weeks' sightseeing tour of Alpha Centauri you get on the flight back to Earth. This also only takes a few hours from your own perspective, but when you arrive on Earth it's now eight years and two weeks since you set off. People you knew are now eight years older than when you saw them two weeks ago, whereas you're only two weeks older. They've experienced eight years of life on Earth, whereas you've experienced two weeks of Alpha Centauri and a few hours' flight time.
Honestly I am not making this up! It's not sci-fi, and this really is the way space and time work, according to Albert Einstein!
If you're wondering how much energy it takes to go at these speeds, yes, it's a lot. But it's not infinitely big, and the amounts of energy are reasonably similar to the Newtonian energy amounts. In effect, it's possible to travel around in a Steampunk style in space as if Newtonian physics applies, provided you don't mind much about travelling in time as well.
Here's another example: Suppose you like to drive very fast, but the cops don't want you to go faster than the speed limit of the speed of light. You've got a very fast car and you put your foot down and watch the speedometer go higher and higher. Strange as it may seem, the scenery is going past at a rate that would mean you must be travelling faster than light, as you're covering that much ground in that short a time. Mile posts are going past at a million every second*. From your own perspective, the way the scenery is going past, would suggest you are travelling much faster than light. There's a police speed-trap ahead, so should you worry? No, because from the cops' perspective, you are only travelling at 0.95 times the speed of light or some such figure. They can clock you with a roadside radar gun and by scientific measurement you're in the clear.
*Note that although the mile-posts are going past at a million every second, the scenery looks all squashed-up, and they aren't a mile apart. Oddly, the objects in the scenery go past you at 0.95c but it's all compressed. The whole scene is contracted in the direction of travel, so the mile posts are bunched up and are no longer one mile apart. Each one of them passes you at 0.95c, but they are so bunched up that a huge number of them pass you each second. Thanks to Don of Relativistic Rocket for this adjustment.
The Andromeda Galaxy is two million light years away. So when you look at it through a telescope, you're seeing the history of the Andromeda Galaxy, as it was two million years ago. The light from it takes two million years to get to Earth, so, that's what you'd expect. You've got light from the Andromeda Galaxy coming into your telescope, and you know where it's come from and when it set off, so what would happen if you could ask the light how long it took to cover the distance? "Well, light, how long did it take you to get here?". Light responds "It took no time at all. I travel infinitely fast". It's not lying, you know. From light's own perspective, it really does travel infinitely fast, and covers any distance in an infinitesimal instant. However, from a ground-based perspective, light has to be seen to travel at the speed of light. That's the way it works.
One of the reasons why the laws of physics about Relativity and the speed of light seem so odd and tricky to explain, is that the speed of light is so fast that it's outside everyday experience. So, it's usual in these thought-experiments to have a hypothetical universe where the speed of light is 10MPH, or some similar speed which would fit with experience. In the world of 10MPH speed of light, you could ride a bicycle and see for yourself the way relativity distorts time and space. What you observe is very different from normal, and quite different from the misconception world of a speed limit speed of light. As you start peddling the bicycle, it's harder than expected, but you don't seem speed-limited at all. When you get up to 7MPH (which in that hypothetical universe is 0.7 times the speed of light), you think you are travelling at 10MPH, the speed of light. Lamp posts in the street are going past at 10MPH, or at least they would be if the lamp posts stayed the same distance apart. What happens is the lamp posts seem to be squashed-up closer together and are going past at 7MPH*.
*They're still going past you at 7 miles per hour. I know you are talking about a gamma factor of 1/sqrt(2) here, but that's immaterial. The posts always go past you at the same speed as you go past them. Again, thanks to Don of Relativistic Rocket for this.
Peddling harder to put on a bit more speed, you are covering even more terrain, and when you look at people in the street they are moving as if they are on video in fast-forward mode! You wave at them and shout "Hey, look! I'm travelling faster than light!", but what they see is someone going past on a bicycle doing about nine and a half MPH waving very slowly and shouting in a tape-recorder-slowed-down voice "H e y , l o o k ! I ' m t r a v e l l i n g f a s t e r t h a n l i g h t !". The other thing is, people look at each other and say in their chipmunk-sped-up voices that the reason you're having to pedal so hard to go at almost 10MPH is because you have become very heavy! Yet, from your own perspective, you don't feel particularly heavy at all! The reason for the difference in perception of mass is so the maths add up ok. You have to have the momentum, so therefore the different perceived speeds have different perceived masses associated with them. Anyway, having travelled at what you feel is 20MPH or 30MPH on a bicycle, you might assume that you can use this ability to get to meetings on time and not be late, even if the meeting places are so-many miles away. However, if you try this, you'll still be late, because when you travel fast, you also travel forward into the future. So, you can speed around for fun and to see the sights, but you can't use the ability to avoid being late for meetings!
Don't blame me for space and time being bent like a banana! It's just the way it is!