Every now and again, I like to share fun questions that aren’t right for the column, but which you guys might find interesting. For example, check out this, this, or this. As I’ve done in the past, I’m going to add a couple of equations here and add the “technical” tag. You can skip the few equations if you like.
Not every question is phrased in just the right way, but oftentimes, there are some interesting ideas in there, anyway. For example, I recently got an email from a reader named Winsor:
Is it possible that gravity and time are really just opposite ends of the same spectrum? I want to compare it to uncertainty principle, where you can never know both exact location and speed.
It’s difficult to answer this question directly. Gravity and time aren’t really comparable things. I’d say it’s like comparing apples and oranges but it’s really along the lines of comparing apples and love. Both are nouns, but beyond that, it’s very difficult to compare the two.
However, gravity and time do have a lot to do with one another, and Winsor is on the right track in that there’s an uncertainty principle involving time and energy. Just as you can’t simultaneously measure the momentum and position of a particle, the universe allows an uncertainty in energy for short periods of time. Particles (and antiparticles, they always come in pairs) can pop into and out of existence and the greater their mass (or energy) the shorter the uncertainty in time — that is, the shorter the period of time that they exist for:
where is the famous “reduced Planck constant.”
Or, since , it means that we can create particles of mass, m for short periods of time:
These particle/antiparticle pairs could, at least in principle, contribute to gravity. In principle, gravity should respond to all of the energy in the universe, although because of how this “vacuum energy” works, it actually should be repulsive, very much like the cosmological constant that seems to be accelerating the universe.
That’s the good news.
The problem, though, is that if you work out the numbers in a naive way, the vacuum energy is something like times larger than the cosmological constant that we actually observe, so there is clearly something wrong with our theory. As I frequently tell my students, this is one of the biggest unsolved problems in physics.
There is another way that time and energy are related to one another (but not, I’m afraid, on any sort of spectrum): gravity changes the flow of time. One of the coolest predictions (which has since been observed) of general relativity is that time moves slower near massive bodies than far away. Jeff and I go through the full reasoning in Chapter 5 of the “User’s Guide” (though we shamelessly steal the example from Richard Feynman.
The basic idea is that if you fire a beam of light from a position near a massive body to somewhere further away, the light will lose energy. The frequency and wavelength of light are related to the energy. One way of expressing it is:
If light loses energy, then it also decreases in frequency.
But wait! Light is a wave, and every peak sent from (say) the surface of the earth has to be received up at the top of a tall cliff. If the rate seen at the top really slower than the rate seen at the bottom, another way of interpreting this is that the clocks at the bottom of the cliff really are running slower than at the top. Or to put it mathematically:
where is the gravitational potential. Near the surface of the earth, this is simply:
You can even test this near the surface of the earth, though the effect from the bottom of a tall building to the top is incredibly tiny. In 1960, Robert Pound and Glen Rebka did this experiment by firing a laser up the height of Jefferson Lab at Harvard. Chug through the numbers and you’ll find that time runs faster at the top than at the bottom by one part in . An incredibly tiny effect, but a real one!
Near black holes the effect gets even more dramatic, to the point where time at the event horizon runs infinitely slowly. In other words, if you want a really crappy time machine, all you have to do is fly down outside the event horizon of a black hole, hang out for a short while, and fly away. Pow! You’re in the future!