In a vacuum, the speed of light is a universal constant: 299,792,458 m/s exactly. For light traveling through a substance like water or glass the speed is lower. The light hasn’t actually slowed down. Instead, the interaction with each successive atom in the material takes a little bit of time and so the average speed within the material is slower than c. We call the ratio of slowing (c/v) the refractive index n. This is the stuff intro physics classes spend a chapter or two on, and optics classes spend months on. Essentially none of those classes are concerned with relativity, because it’s always assumed the various materials are standing still. But what if they’re not? Consider the following problem. Please forgive the far fetched-ness of it.
A stream of water is flowing at a speed c/2 relative to the laboratory. A beam of light enters the stream along its direction of flow. From the perspective of someone in the lab, how fast is the light in the stream? Assume the refractive index of water is 1.33.
Hmm. Try thinking about it for a bit if you don’t already know how to do it, and see if you can come up with some ideas. Solution below.
If the world were not relativistic we could just add the speed of light and the speed of the water and get an answer. So what’s the speed of light in the water? Well, from the definition of the refractive index above,
Using that we see that the velocity of light in water is v = (3/4)c, and adding that to c/2 for the speed of the water itself, we would get that the speed of the submerged light in the lab frame would be (7/4)c. This clearly isn’t right. But Einstein showed us that in fact the the sum of velocities is not just the first velocity plus the second velocity. If the motions are along the same line as they are here, the correct formula happens to be
Where v is the summed velocity, and v1 & v2 are the individual velocities. We don’t have space to prove this formula (that’s a project for another day), but notice that’s it’s exactly the classical formula but with a strange term in the denominator. Since v1 = (3/4)c and v2 = (1/2)c, we can just plug and and see what we get.
Which is comfortably below the speed of light, as expected.
Relativity is a little weird. It’s very counterintuitive that velocities don’t add like they seemingly do in daily life. But this is just because all the velocities we deal with in everyday life are so much smaller than c. The extra term in the denominator is so small that it might as well be 0 in those situations.
2 responses so far ↓
1 meichenl // Jun 4, 2008 at 12:06 am
This isn’t just frivolity; the speed of light in moving water was an important empirical observation that hinted at relativity.
Fresnel actually had an equation for it he derived through an argument about the mechanics of aether.
c’ = c/n + v(1-1/n^2)
In 1851 Fizeau measured it to confirm it was correct. Why could he confirm an incorrect formula? Turns out it’s right to first order in v/c, (in the relativistic law, take the first term in the binomial expansion of (1+v1*v2/c^2)^-1) and when v/c is that small first order is all you need.
This is from chapter 6 of “Subtle is the Lord” by Abraham Pais. In fact, that’s the only chapter of the book I had read, but here it is coming in handy!
2 Tom // Jun 4, 2008 at 10:45 am
Instead of water, consider a solid bar of the same index. You send in the pulse of light (assume a really good AR coating so there’s no reflection). What happens to the speed of the bar?
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