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.
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Excuse me, but there would be no magnetic field since it's a reativistic effect. so i guess no, you couldn't really read outside during night, even assuming that humans could exist without magnetic fields (e.g. would the hydrogen bond in water still work without the electrons spin?).
I pulled Purcell off my shelf to look up the CGS form of Maxwell's equations. All of the terms on the right hand side of the curl equations are proportional to 1/c, so they would vanish as c -> infinity. This implies that both E and B must be expressible as gradients of scalar fields. Contra Chris, the magnetic field is not forced to vanish identically. However, there would be no magnetic component to propagating free space waves. You can deduce this from noting that in SI units E/B = c for a propagating free space electromagnetic wave, and if c is infinite and E is finite then B must be 0.
I have a problem with the idea that simply having a star in each direction would result in us seeing a white sky. Even if the speed of light is infinite, wouldn't the intensity of that light still vary inversely with the square of the distance? That is, farther away stars would have very little of their light actually reach us, so that we would barely see most of them.
Secondly, if I understand correctly we do NOT live in an infinite universe. While it is very, very large, and there are many, many stars, there are still finitely many of them. So we would not see a star in every direction. Am I wrong?
How can it be true that the sky would be bright at night with infinite speed of light? We can defenitely not see all the stars within a 14 billion light year radius at night today - we need telescopes and stuff to do that because the light is so faint.
What am I missing? Why wouldn't we have the same situation with c=inf?
Ok, I think Susan was a bit quicker and put it a bit better than me.
Susan and stormen_per - 14 billion light years is the size of the "visible universe", i.e. the size of the universe according to us here on Earth in 2009, given that the speed of light is c and that the universe is roughly 14 billion years old. But if the speed of light was infinite, the concept of the "visible universe" would change dramatically -- we could see everything in the universe at once no matter how far away.
In other words, light falls off as inverse r squared (good), but that's easily made up for by the fact that we've now added radiation from ALL the stars in the universe (bad). Every mathematical point in the sky is the surface of a star. Basically we all fry.
And no, dust clouds and whatnot blocking those stars won't save us. They'll just heat up and re-radiate at the same temperature. We still fry.
Infinity isn't a number. It has to be approached as a limit, and how that limit is taken can totally change the resulting physics. If you just set 1/c equal to zero, there's no magnetism, no light, a probably no us. If you handle the limit differently, you can get something different.
Susan asks: wouldn't the intensity of that light still vary inversely with the square of the distance?
This is offset by the fact that the average number of stars in a spherical shell would increase with the square of the distance. So the amount of light reaching us from a given distance would, to lowest order, be independent of distance, resulting in an infinite incoming photon flux. Or as Evan put it, we fry.
Redshift of light from more distant stars would not help, either. Doppler shift depends on the ratio of source velocity to wave velocity, and if the latter is infinite the Doppler shift will be zero.
There's a common misconception about Olber's Paradox, which Eric got right above, but everyone else seems to be missing.
The sky would not be as bright as a star everywhere. It would be infinitely bright everywhere. If you look in any direction and see a star, then you see another star behind it, and another behind that, and so on for infinitely-many stars.
A common counter-argument is that the stars in front shield the light of the stars behind, but this is wrong.
Think of this from the point of view of a star. It sees starlight from everywhere as well. So if the star were a simple blackbody, it would heat up to star-heat. But it produces energy by fusion. So it has to get hotter than mere thermal equilibrium. But this is true of all stars. Conclusion: all stars are hotter than all other stars. The stars are thus infinitely hot, and infinitely bright.
Wow, that's a complex answer to the question. We'll save this discussion for 'in the car, going someplace far away' because presenting these ideas and Olber's paradox is going to be a lot of talk.
Thank you, Dr. Orzel, and thank you all for this information.
Physics is just cool.
Eric Lund,
what waves are you talking about? there is no propagation, there will be action at a distance, quasistatic electric fields. and there might be magnetic ones, but they will be decoupled. and i don't see any order of limits problem here either. c->oo is just that, there is no other limit involved causing any ambiguity.
so a good first consequence would be that with an infinite speed of light there would be no light. now how's that for a start :-)
The universe doesn't HAVE to be infinitely large. Isn't is possible that the universe if somewhere between 14 billion light years and infinitely large? If the speed or light were increased and we saw that the universe is not substantially larger than what we currently see, the night wouldn't be all that much see now. Of course, we would also get light from any nearby stars which happen to be very new as well.
Someone is bound to quibble at the notion one could send a constant with units to infinity, so I may as well get the ball rolling. The statements regarding Maxwell's equations taking such-and-such a limit give this or that result are more than likely nonsense. The form of Maxwell's equations depend on the unit system one uses. If I use "natural units" in which I set c = 1, then how can I send c to infinity? In a rough sense we might be able to talk about what would happen if this or that fundamental constant changed, but strictly speaking, to make definite claims, we should talk about taking limits of dimensionless quantities only, no?
(This may be a touchy subject - there's a preprint by M. J. Duff (hep-th/0208093) in which he discusses such issues and how experiments which are trying to measure variations in constants like c or G are flawed, and it is only meaningful to measure the time variation of dimensionless constants like alpha.)
how can you even use natural units if c were infinite? The whole idea of setting c=1 revolves around special relativity existing. if c is infinite, special relativity doesn't exist, ergo no natural units, and therefore c cannot be set to 1. If c cannot be set to 1, is there another unit set where c's limit cannot be taken to infinity, and therefore we cannot tell how maxwell's equations will behave?
The statement "setting c = 1" is absolutely not dependent on special relativity existing. It is only dependent on the fact that c is a constant with units, and I'm choosing to measure speeds in units such that c = 1. It isn't special to c; it applies to any theory with a "fundamental" constant with units. If I have a theory with c in it I am free to choose units in which c = 1, 2, 5, or any other finite number I want, because really all I'm doing is changing the units I'm measuring distance and time with. You might well be able to imagine a universe where the "speed" of light is infinite (or perhaps more precisely, light travels instantaneously from one point to another), but you can't attain such a theory by simply sending c to infinity in the formulas we all know and love, for exactly the reasons stated - if I choose to work in units in which c = 1, then sending c to infinity is impossible! Really what I'm doing is changing the units in which I'm measure space or time (making my unit of length smaller or my unit of time larger).
Wouldn't your unit of length have to be zero?
I am free to choose units in which c = 1, 2, 5, or any other finite number I want, because really all I'm doing is changing the units I'm measuring distance and time with.
I thought the reason that the sky is dark at night is because the universe has a finite age and finite size. If we had an infinitely old universe and an infinitely large universe the sky would be light at night even if light were no faster than c.
Does an infinite speed of light imply an infinitely old and infinitely large universe?
My reference on this is Harrison's Darkness at Night, an excellent layman's treatment of the dark sky problem.