Is higher faster? Does time pass more quickly when gravity is reduced?

May 11, 2007

Dear Cecil:

I read recently that "time . . . passes more quickly when gravity is reduced." Assuming gravity is reduced at higher altitudes, that means time goes by faster in Santa Fe than in Poughkeepsie. What's up with that?

You heard right, Chris. But unless you're way more anal than anybody I want living in my reference frame, you won't have to reset your watch. Due to the "warpage of time," clocks run slower in Poughkeepsie than Santa Fe by about a millisecond. Per century. (Get friendly with a black hole and it's another story - gravitational time dilation approaches infinity as you near the event horizon. However, notwithstanding sporadic distortions of space-time due to Taos, Los Alamos, Roswell, etc, we'll assume that's not a problem for you in New Mexico.) Time dilation affects not just ordinary clocks but any measure of time, including how long it takes to say "one Mississippi, two Mississippi." So you'll never notice anything odd about SF time, only about that of people living under different gravity conditions, e.g., Poughkeepsie, Hoboken, or other low burgs.

The effect was first hypothesized by Albert Einstein in 1907 as a consequence of his "happiest thought," the equivalence principle, which says gravity is locally indistinguishable from acceleration. Think of an elevator - as it accelerates you upward, you're squashed to the floor, which feels like an increase in gravity. That's no trick of the senses, says Einstein. Experiments performed over short range and brief time can't differentiate between acceleration and gravity. Originally just a cool idea, the EP and its consequences, including gravitational time dilation, have since been thoroughly confirmed.

It's probably not obvious to you why the EP results in time dilation, and I'll admit steam was rising off the diodes by the time I got the whole thing processed. But let's give it a shot:

1. Imagine you're in a spaceship far from any source of gravity. The ship is moving in a straight line at constant speed, so you float in the center of the cabin.  Now imagine your idiot brother at the controls unexpectedly turns on the rocket booster, accelerating the ship.  The rapidly approaching back wall is now indistinguishable from a floor you're falling toward under gravity.

2. Now consider a light source on this "floor" that emits a photon (light particle) of a certain frequency.  Because the speed of light is finite, it takes time (albeit very little) to make the trip to the "ceiling."  By that time, the light receiver, along with the rest of the ship, has slightly increased its speed due to the ship's acceleration.  The ceiling receiver (at the moment of reception) is always moving a bit faster than the floor emitter was (at the time of emission), even though the distance between them never changes.  This invokes the Doppler effect, more familiar to us in sonic form: because of the aforesaid speed difference, the receiver will record the photon's frequency on arrival as slightly lower than it was on departure from the emitter.

3. Frequency, whether of clock ticks, pendulum swings, or photon pulsations, is a basic measure of time - a second officially is the time it takes for certain photons emitted by cesium-133 atoms to vibrate 9,192,631,770 times. If you and I measure the frequency of a given photon differently, we'll measure the flow of time differently too. So if I'm on the ceiling when the photon arrives, I time its vibrations and say, woo, that pup is slow. Meanwhile, an observer on the floor will say, nah, your stopwatch is fast.

4. Likewise, since the EP tells us acceleration = gravity, and gravity decreases with elevation above sea level (the "floor"), you in your mountain fastness will say sea-level time runs slow, while I in my shoreline cabana will say mountain time runs fast.

Anyway, that's the theory. Does it really work that way? You bet. Einstein used general relativity (which is bound up with gravitational time dilation) to explain a known oddity in Mercury's orbit. More recent experiments involved atomic clocks on jet flights. Here both gravity- and speed-dependent special relativity effects must be taken into account. After a westward around-the-world jet flight, flying clocks gained 273 nanoseconds, of which about two-thirds was gravitational.

Mere nanoseconds, you say - who gives a flying clock? You do, if you use the global positioning system. Because of their altitude, the clocks on GPS satellites run about 30 nanoseconds fast per minute due to gravitational effects. Since the system works by timing light signals and the distances involved are great, an uncorrected time error would mean a distance error growing at about 9.5 meters per minute. You may think it's amazing you can hike in the Sangre de Cristos with a $300 GPS receiver that tells you exactly where you are. What's more amazing is that the geniuses who designed it needed a rough knowledge of general relativity to get it to work.

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