A couple of days ago, I answered a question from a donor to the Uncertain Principles challenge page in this year’s Social Media Challenge (we’ve raised $1,807 thus far– thank you all). If you’d like a question of your own answered on the blog, all you need to do is send me the confirmation email for a donation of at least $20, and your question.
The donor from the other day, Lauren Uroff, had a second question as well, also on behalf of her teenager:
He’s an avid Discworld fan (where the speed of light seems to be variable) and wants to know what would happen in our real world if the speed of light was infinite.
There are a bunch of things that would change, some obvious, some less obvious. The most obvious thing that would change is that we wouldn’t have any need for relativity. The equations describing special relativity reduce to the equations of classical mechanics when the speed of the object being described is much less than the speed of light– that is, Newton’s Laws are a very good approximation of reality for a slow-moving object. For a slow-moving object with a mass of 1kg, a 1N force produces a 1m/s2 acceleration. Accelerate that object to 50% the speed of light, though, and the acceleration from a 1N force is significantly less than 1 m/s2.
If the speed of light was actually infinite, there wouldn’t be any limit to that approximation– any speed you like would be small compared to the speed of light, and Newton’s Laws would always work.
There are a number of less obvious consequences to an infinite speed of light, though. My favorite of these is that you would be able to read a paper outside at midnight with no problem.
What does relativity have to do with whether it’s dark at night? It’s a thing called Olbers’ Paradox, after an astronomer who asked in 1826 “Why is the sky dark at night?”
This may seem like a silly question– “Because the Sun is on the other side of the Earth”– but it’s not. The argument goes like this: If we live in an infinite universe (which we do, or near enough as makes no difference), and matter in the universe is distributed more or less uniformly (which is is, on a large enough scale), then no matter what direction you look in, you should be looking straight at a star. It may be a distant star, but there should be a star somewhere along your line of sight.
If that’s the case, then the night sky ought to be a brilliant white, thanks to the light of all those stars. So, Olbers asked, why don’t we see all that light?
The answer is, basically, “because the speed of light is finite.” It takes time for the light from distant stars to reach us, and the Universe isn’t old enough for all of the light from all of the stars to get here. The universe is roughly 14 billion years old, which means we can only see out to a distance of 14 billion light-years. While that’s mind-bogglingly big, it’s not infinite. You’re not guaranteed to see a star within 14 billion light-years in the line of sight.
In a universe with an infinite speed of light, Olbers would be exactly right. No matter how big the universe, the light from even the most distant stars would reach us instantaneously, and you really would see a star no matter where you looked. So, the Sun would be a brighter spot against a bright background, and night would only be a little bit darker than day.
So, a universe with an infinite speed of light wouldn’t be a very pleasant place to live. Happily, that’s not the universe in which we live…
I hope that answers your son’s question, Lauren.