Oops, my mistake, as Clinger pointed out, I wrote 0.2 instead of 0.02 - a fifth instead of a fiftieth. The square root of a
fiftieth is appoximately a seventh. And there was a typo where I put a ² outside the brackets. Aw, it was late, and I was using the size option, it's fiddly. Shall we try again? The real reason why there’s so much crackpot physics is conviction. And arrogance, and pride, and hubris. Some people will use anything they can to dismiss something that proves them wrong, be it experiment and evidence and explanation, and even Einstein. They come out with accusations like
they never use any mathematics, but when you do, they dismiss that too. Then they make some new accusation, or change the subject, or pull some other slippery stunt. Let's try again with that invariant Lorentz interval:
[qimg]http://www.forkosh.com/mimetex.cgi?ds^2 = -dt^2 + dx^2 + dy^2 + dz^[/qimg]
You can work it through using a pair of
parallel-mirror light-clocks. Event
1 is when we set them running, keeping clock
1 as the local clock whilst sending clock
2 travelling on an out-and-back trip. Event
2 is when they meet back up. We see the light moving like this
ǁ in the local clock
1, and each reflection adds 1 to t
1. But because it’s just sitting there, x
1 y
1 and z
1 are zero. We see light moving like this /\ in the travelling clock
2. Treat one side of the angled path as a right-angled triangle, and the hypotenuse is the lightpath where c=1 in natural units, the base is the speed v as a fraction of c, and the height gives the Lorentz factor γ = 1/√(1-v²/c²). If the travelling clock
2 is doing .99c the Lorentz factor is 1/√(1-0.99²/1²) = 1/√(1-0.98) = 1/√0.02 = 1/0.142 = 7. So there's a sevenfold time dilation. That means t
2 is a seventh of t
1. And because it’s only on a straight-line out-and-back trip, x
2 is non-zero whilst y
2 and z
2 are zero.
When the two clocks meet back up we can be confident that delta s is the same for both clocks because the travelling clock covered a total distance x
2 = vt
1 = 0.99t
1. Whatever the value of t
1, delta s
1= t
1 and delta s
2 = √(-0.142t
1²) + √(0.99t
1²) = 0.02t
1 + 0.98t
1 = t
1. Simple. What’s even simpler is to look at what you’re dealing with, and realise the two total light-path lengths between event
1 and event
2 are the same in both clocks. That’s what underlies the invariant interval. Macroscopic motion comes at the cost of a reduced local rate of motion. That’s why there’s a minus on the t.
Note that there's no literal time flowing in those parallel-mirror light-clocks, just light moving back and forth between the mirrors. All a clock does is “clock up" some kind of regular cyclic local motion. That’s really useful for when you’re looking at things like black holes, when you’re really up against the crackpots. They will tell you about the
waterfall analogy, where space is supposedly falling into a black hole. That's abject nonsense. It's like saying
the sky is falling in, and it absolutely contradicts general relativity. But if you point that out, the crackpots will call you a crackpot. Or a heretic, or something else. And as ever they will dismiss experiment and evidence and explanation, and even Einstein.