A scientific theory hasn’t really arrived until the cynical and unscrupulous find a way to use it to extract money from the credulous and gullible. This has posed a significant obstacle for general relativity, dealing as it does with gravity, which requires really gigantic masses to produce measurable effects. That makes it a little difficult to sell wacky general relativity-based schemes to people.
Until now, anyway– recent advances in atomic clocks have made it possible to see relativistic effects on a human scale. There was a really nice talk on this experiment in the fundamental symmetry session at DAMOP, and talking about it afterwards revealed a great get-rich-quick scheme using general relativity. It follows the time-honored strategy of tying dodgy science to health, and promising people that they can extend their lives through crank physics.
In this case, you can think of it as a sort of Bat Diet (no hyphen– this isn’t a superhero comic): you can exploit general relativity to extend your life by sleeping underground and upside down.
“How does that make any sense?,” you ask. Well, general relativity tells us that clocks that are lower down in a gravitational field “tick” more slowly than clocks at higher altitudes. So, when you’re standing upright, your brain is aging more rapidly than your feet, and when you’re in an office on an upper floor, you’re aging more rapidly than a person on the street below. In order to live longer, then, you want to stay at the lowest altitude possible, and if you care about the age of your brain, you want to keep it lower down. Thus, bats have the right idea: if you sleep upside down in an underground cave, you’re helping your brain age more slowly.
“That’s completely ridiculous,” you say. “Why would a clock care about the altitude?” The easiest way to understand it is by thinking about the Equivalence Principle, which tells us that the effects of gravity are indistinguishable from the effects of acceleration. The equivalence principle was the key insight that led Einstein to develop general relativity, extending his theory of special relativity from a simple description of motion to a complete theory of gravity.
The key insight for Einstein was the realization that an observer falling freely in a gravitational field would feel no force of gravity– everything else in their vicinity would fall at exactly the same rate, and thus appear to be floating in the air at rest. A dog sealed in an elevator, then, would be completely incapable of distinguishing between the case where the elevator was falling freely (say, because some dastardly cat cut the cable) and a case where the elevator was floating by itself off in outer space (say, because the dog was in a Roald Dahl novel). All the laws of physics must appear exactly the same to the dog in the free-falling elevator as in an elevator floating in space.
So, let’s think about a dog in a falling elevator shining a light from the top of the elevator to the bottom. She sees the light beam follow a straight-line path from top to bottom, and the beam hitting the floor is exactly the same color as the beam that left the ceiling. That’s what you would expect in an elevator at rest, and is just common sense.
This presents a problem for the cat outside the elevator, though, cackling maniacally while watching the dog fall. From the cat’s perspective, the floor of the elevator is accelerating downward, and when the light reaches the floor, the floor is moving away from the light at a higher speed than when it was emitted. This means that the light hitting the floor should be Doppler shifted to the red. But the dog doesn’t see the frequency change, and the cat has to agree that the dog sees the same frequency at both ends of the elevator. So, how do we reconcile these?
The solution to the problem is a gravitational frequency shift: According to the cat, the light must increase in frequency by an amount that exactly counters the Doppler shift due to the acceleration of the elevator. Light moving down in a gravitational field gets more blue as it falls, while light sent upward in a gravitational field gets more red. this “gravitational redshift” has been experimentally confirmed in a number of different experiments.
This still leaves a problem, though, namely that the shift must also occur when the source and the detector are not moving relative to one another. If a cat at the top of the elevator shaft shines a light down, a second cat at the bottom must see a higher frequency for the light coming down, and the top cat must see a lower frequency for the light coming up from the bottom. And yet, the cats aren’t moving relative to one another, and each sees his own source emitting the same frequency at all times. So how do we explain the shift in that situation?
In order to make everything work out right, we are forced to conclude that the rate at which time passes is different at different points in a gravitational field. The high-altitude cat looking down at his crony on the ground sees the lower cat’s clocks ticking more slowly. This explains not just why the light coming up is at a lower frequency (because the lower cat’s clock ticks too slowly, and he’s sending lower-than-expected frequencies up), but also why that cat sees the light coming down at a higher frequency (each “second” on his clock is a little more than one second according to the cat at the top, and thus include a few more oscillations of the light from the top cat’s source). And it can be turned around to explain the bottom cat’s results in terms of a too-fast clock for the top cat.
This gravitational time dilation is one of the strangest consequences of relativity, but it’s been confirmed in numerous experiments. The most sensitive test is the atomic-clock experiment mentioned above, which includes this graph showing a clear difference between the rates of two identical clocks when one of them was hoisted up by one foot:
This is also the prediction of general relativity with the most direct technological consequence: the atomic clocks orbiting the Earth as part of the Global Positioning System must be corrected for the effects of relativity, with the largest correction coming from the fact that the clocks are at high altitude. Without that correction, the GPS clocks would drift out of synch with clocks on the ground by almost 40 μs per day, which amounts to a position error of around 11 km/day. The current accuracy of GPS– my cell phone can get my position on the surface of the Earth to within several meters– is possible only because the engineers who designed the system built in a correction for the gravitational time dilation.
So, again, general relativity tells us that clocks that are lower down in a gravitational field tick more slowly than clocks at higher elevation. Which means that a person who spends their entire life at or a little below ground level will live longer than a person who spends a lot of time in tall buildings and airplanes. And also that your head ages slightly faster than your feet. Thus, the Bat Diet (which isn’t really a diet, but it’s a catchy name): if you spend your time underground and upside down, your brain will experience time moving more slowly than surface-based people standing upright.
How big an effect is this? Well, the difference near the surface of the Earth goes like the difference in height multiplied by the gravitational acceleration g=9.8m/s2, divided by the speed of light squared. Which means that for every meter of altitude you manage to lower yourself, on average, you’ll slow the rate of time by a bit less than 300 nanoseconds over an 80-year lifespan.
OK, fine, that’s not much of a difference. But then, the whole point of this sort of scam is to prey on people who are bad at math, so I bet you could still fool some people with more money than sense into constructing expensive subterranean dwellings, and shelling out big consulting fees for advice on the most relativistically appropriate way to live.
Of course, I’ve just given this idea away for free on the Internet. Hmmm… Maybe I need to rethink my business model…