The problem isn’t with how many dimensions we’re seeing the universe in – basically, in thinking that Gallilean relativity (I’ll explain this below) applies to all cases, we’re mistaking a limiting case for a general rule. Don’t worry if that’s confusing, I’ll ellaborate. I made a brief reference to this when I mentioned “chasing a beam of light at half the speed of light”, but I can go into a bit more detail.
Basically, according to Gallilean relativity (named for Gallileo Gallilei), if you are moving in the x-direction relative to me at a speed v, and we both measure the x-component of a velocity, the value that you measure will be v less than the value that I measure.
If that’s not clear, let me give an example:
Suppose I’m sitting next to the road, and I see a car go by at 45 mph. (I used kilometers in my previous examples, but since I’m an American I’m used to seeing road speeds in miles per hour, so that’s what I’ll use here. Anyway, the units don’t really matter for the purposes of this example.) Then I see another car go by at 30 mph. Suppose you are in the second car, and are trying to judge how fast the first car is moving. To you, it will only appear to be moving at 45 mph, because it is only going 45 mph faster than you.
You might say, “Hold on, I can just look down at my speedometer and see that I’m traveling at 30 mph, so I know the first car is really going 45 mph!” And yes, it’s true that you can easily compute that the first car is traveling at 45 mph relative to the road (or relative to me, the bystander standing next to the road). But it is meaningless to talk about how fast it’s really going. It’s going at 45 mph in my frame of reference and at 15 mph in your frame of reference. In other words, it’s going at 45 mph relative to me and 15 mph relative to you. Neither frame of reference is inherrently “more correct” than the other. You might think the “person who’s standing still” has the more correct measurement, but there really is no “person who’s standing still.” I’m moving relative to some things, and you’re moving relative to some things. For instance, we’re both moving relative to the Sun, since the Earth is orbiting it.
At any rate, the formula for converting the speeds I measure to the speeds you measure (assuming, for simplicity, that everything is moving along the same line, say the x-axis) is:
v[sub]2[/sub] = v[sub]1[/sub] - v.
Here v[sub]1[/sub] is the speed of the first car in my frame of reference, v[sub]2[/sub] is the speed of the first car in your frame of reference, and v is your speed in my frame of reference. (Obviously, your speed in your own frame of reference is zero, since you aren’t moving relative to yourself.) In the above example, v[sub]1[/sub] is 45 mph, v is 30 mph, and thus v[sub]2[/sub] is 15 mph.
So that’s the formula we use in Gallilean relativity. For a long time, it was basically considered “common sense” that this formula always works. (Again, it’s a little more complicated if we aren’t all moving along the same line, but I’m trying to keep things simple.)
But let’s say we replace the first car with a beam of light. And we’ll say that the beam of light is moving relative to me at 300,000 km/s, a value typically represented by the letter c. Let’s try to figure out how fast the beam of light is moving relative to you. If Gallilean relativity is to be believed,
v[sub]2[/sub] = c - v.
So v[sub]2[/sub] should be a bit less than c. In fact, in my previous post, where I talked about someone chasing a beam of light while traveling at half the speed of light, v would be c / 2, so v[sub]2[/sub] = c / 2.
However, this is wrong. According to Einstein, v[sub]2[/sub] should also be equal to c. The speed of light is the same in your reference frame as mine, even though this goes against Gallilean relativity! If my explanation of Gallilean relativity made complete sense to you, then the fact that it is wrong here should seem very odd indeed. This is what I meant when I said that Einstein’s theory of relativity defies common sense.
(Aside: I always find it amusing when people summarize Einstein’s theory of relativity by saying “Everything is relative.” It’s true that certain things that we thought were absolute, like distances and time intervals, actually depend on your frame of reference. But this is all based on the fact that the speed of light isn’t relative! It’s the same in all inertial reference frames. Basically, “inertial” means “unaccelerated” – to deal with accelerated reference frames, you need Einstein’s General Theory of Relativity.)
Anyway, according to Einstein’s relativity, the correct equation for calculating these speeds is:
v[sub]2[/sub] = (v[sub]1[/sub] - v) / (1 - v v[sub]1[/sub]/c[sup]2[/sup])
Note that if I plug in c for v[sub]1[/sub], then the equation becomes:
v[sub]2[/sub] = (c - v) / (1 - v/c)
Then I can multiply the right side by c / c (which is of course 1). I get:
v[sub]2[/sub] = c (c - v) / (c - v)
But (c - v) / (c - v) = 1, so I’m left with:
v[sub]2[/sub] = c
Thus, the value you measure for the speed of light is also c, just as I claimed. Note that I didn’t need to specify the value of v. In other words, you will still measure light to be traveling at speed c, regardless of how fast you are moving relative to me.
So, if this is the correct equation, how come everyone thought that Galilean relativity was right for so long? Well, look at what happens when the velocities involved are much smaller than c. In that case, v v[sub]1[/sub] / c[sup]2[/sup] is very small, in fact it’s basically zero. So if we set this to zero, our equation goes from:
v[sub]2[/sub] = (v[sub]1[/sub] - v) / (1 - v v[sub]1[/sub]/c[sup]2[/sup])
to
v[sub]2[/sub] = (v[sub]1[/sub] - v) / 1
Dividing by 1 doesn’t do anything, so we can just write this as:
v[sub]2[/sub] = v[sub]1[/sub] - v
which is the equation we had for Gallilean relativity. So Gallilean relativity works just fine for small velocitities. This is what I meant by saying that Gallilean relativity is a limiting case of Einstein’s relativity. The reason our common sense is wrong is because we’re only used to dealing with small velocities, so our intuition tells us Gallilean relativity should work all the time, which is wrong.