In one of Jeff Shallit's recent posts on the Panda's Thumb, he mentioned that Tom Bethel, aside from being a creationist, was also a relativity denier. In general, relativity denial is a veritable mine of bad math. So I went looking - and found Bethel's anti-relativity site. As I expected, we've got extremely silly bad math. In fact, it's the worst kind of bad math - it's a lack of math masquerading as being math. It's also, sadly, full of pathetic errors.
For instance, there's this:
The argument that gravity must travel faster than light goes like this. If its speed limit is that of light, there must be an appreciable delay in its action. By the time the Sun's "pull" reaches us, the Earth will have "moved on" for another 8.3 minutes (the time of light travel). But by then the Sun's pull on the Earth will not be in the same straight line as the Earth's pull on the Sun. The effect of these misaligned forces "would be to double the Earth's distance from the Sun in 1200 years." Obviously, this is not happening. The stability of planetary orbits tells us that gravity must propagate much faster than light. Accepting this reasoning, Isaac Newton assumed that the force of gravity must be instantaneous.
Astronomical data support this conclusion. We know, for example, that the Earth accelerates toward a point 20 arc-seconds in front of the visible Sun -- that is, toward the true, instantaneous direction of the Sun. Its light comes to us from one direction, its "pull" from a slightly different direction. This implies different propagation speeds for light and gravity.
The problem here is that this isn't math - it's presenting things as if they were mathematical results, but without actually showing the math. In effect, it's trying to pretend that there's math backing them up, when what they're really doing is bullshitting.
The math doesn't support them at all. To see why, you just need to remember for a moment what gravity is, in terms of relativity. Gravity is a warp in the shape of space. Space is literally bent around objects with mass. Gravitational fields change the shape of space. Objects are affected by the shape of space that they pass through.
That's not what the argument above says. To anthropomorphize a bit, it says that the sun is constantly looking around itself in space, and sending little packets of gravity to the things that it sees. So at a given time t0, it looks at the earth, which is in position p0. At that time, the gravitational force that should be exerted on the earth is pointed toward the center of mass of the sun. (See the diagram to the side.) At t0, the sun creates a packet of gravity, with a force vector pointing at the center of the sun. The sun sends this packet of gravity to the earth - when it arrives, it will accelerate the earth in the direction that it's pointing. But it takes 8 minutes to get there. So when it gets to earth at time t1=t0+8 minutes, the earth has moved to a new position - position p1. In this new position, the packet of gravity isn't pointing at the center of the sun anymore - because the earth has moved, the packet of gravitational force is pointing a little bit ahead of the center of the sun. That's what they're arguing - that there should be a directional error in the gravitational force because of the time delay.
That's not how gravity works. The sun doesn't look at where the earth is, pick a packet of gravity for that position, and send it to wherever the earth will be 8 minutes later. According to relativity, the earth is accelerated by a gravitational force created by the shape of space that it's passing through. There's no magically aimed packets of gravity. The sun doesn't emit specific bits of gravity for the earth - it creates a field permeating the space around it; the earth is affected by the field that it passes through. The earth is accelerated at any moment by the gravity field that it's passing through at that moment. It doesn't matter at what rate that field propagated through space - what matters is what field the earth is passing through. The rate of propagation of gravity only matters when something changes - then the change takes time to propagate. But that's not what they're talking about.
But there's another error - an even more obvious one - in the nonsense argument above.
According to Bethel, The sun sends a packet of gravity containing direction to the earth at time t0. It arrives at time t1. But the earth isn't where it was when the sun sent the packet of gravity. If the sun sent it to the position where the earth was at t0, then the packet of gravity will miss the earth. So the sun needs to send a packet of gravity matching the earths position at t0 - but it needs to aim it at t1. In other words, the sun needs to correct for the kind of direction error that they're arguing should exist, in order to make that direction error exist.
That's the kind of error that you make when you're not actually doing the math.
One other error that I can't resist pointing out, which comes later in the article, when he's talking about GPS.
Both theories, Einsteinian and local field, would yield the same results. So far. Now let's turn back to the Global Positioning System. At high altitude, where the GPS clocks orbit the Earth, it is known that the clocks run roughly 46,000 nanoseconds (one-billionth of a second) a day faster than at ground level, because the gravitational field is thinner 20,000 kilometers above the Earth. The orbiting clocks also pass through that field at a rate of three kilometers per second -- their orbital speed. For that reason, they tick 7,000 nanoseconds a day slower than stationary clocks.
To offset these two effects, the GPS engineers reset the clock rates, slowing them down before launch by 39,000 nanoseconds a day. They then proceed to tick in orbit at the same rate as ground clocks, and the system "works." Ground observers can indeed pin-point their position to a high degree of precision. In (Einstein) theory, however, it was expected that because the orbiting clocks all move rapidly and with varying speeds relative to any ground observer (who may be anywhere on the Earth's surface), and since in Einstein's theory the relevant speed is always speed relative to the observer, it was expected that continuously varying relativistic corrections would have to be made to clock rates. This in turn would have introduced an unworkable complexity into the GPS. But these corrections were not made. Yet "the system manages to work, even though they use no relativistic corrections after launch," Van Flandern said. "They have basically blown off Einstein."
The GPS satellites do have to make a correction in their clock speeds to adjust for the differing rate of time in a different gravitational field. That, in itself, is an astonishing thing: we can predict, with incredible accuracy, just how much you need to modify the clock rate of a satellite for the difference in the relative speed of time between the surface of the earth and the orbit of the satellite. Bethel waves past that - an incredible feat, a spectacular example of how precise relativity is when you do the math correctly - as if it's a trivial and unimportant thing. And he moves on to his supposed point - that GPS doesn't need to do any other relativistic corrections depending on the location of a GPS receiver on earth. Except that he's wrong - and he's wrong because he doesn't do the math.
You see, modern GPS receivers do do relativistic corrections.
But back when the article was written, GPS was using something called SA, which was a deliberate degredation of the signal. This was done so that non-military users couldn't take advantage of the full precision of the GPS. The relativistic errors caused by receivers being in different locations on the earth was dwarfed by the amount of imprecision introduced by SA. So in the early receivers, there was no point in doing the relativistic correction - not because there was no effect, but because the signal quality was degraded too much for the correction to be doable.
Once again, if Bethel sat down and did the math (which he describes as no more complicated than high school algebra), actually plugged the numbers into the equations to see how much error could be introduced by the predicted effects of relativity, and compared it to the known precision of SA GPS, he would have known how stupid this argument was. But he didn't.
As usual - the worst math is no math.