Study Guide

# Special Relativity Themes

### Time Dilation of Mu-Mesons

Earth is constantly bombarded by cosmic rays. A delicious variety of cosmic events produces these particles (mostly protons). FACT: The cosmos is full of things that go BOOM. Examples are supernova explosions, which release tons of cosmic-rays.

Once they reach Earth, cosmic rays interact with the atmosphere and produce secondary cosmic-rays, dominated by particles called mu-mesons (µ-mesons), or simply muons. Yes, more Greek. Scientists ran out of English letters long ago.

As µ-mesons enter the Earth's atmosphere, they decay into other particles.

Depending on the charge, a µ-meson will decay into an electron or anti-electron, along with a neutrino and an anti-neutrino. Note: these two neutrinos are actually of different kinds, but that's not our topic of interest at the moment.

The goal today isn't to review particle physics, though, but to point out that these mu mesons travel at speeds very close to the speed of light as they zoom into Earth's atmosphere. What happens at relativistic speeds?

If "time dilation" is one of your answers, give yourself a pat on the back.

Mesons decay into their products at a certain rate, just like radioactive isotopes have half-lives. However, we can count the number of mesons at the top of a mountain, based on the height L of a mountain and on their decay rate of mesons per second, and we can predict the number of mesons at the bottom of the mountain after they travel for where v is nearly c.

See mesons shouldn't have time to travel all the way down to earth because they decay so quickly, but they do. If you suspect wizardry is involved, good guess, but guess again.

We haven't taken time dilation of the muon in its own frame of reference into account. Since their decay rate is based on time, a lot more mesons will survive the trip than non-relativistic calculations predict.

In 1941, B. Rossi and D. B. Hall conducted a muon experiment10. With the use of a sophisticated particle detector, Rossi and Hall detected a very large number of μ-mesons at the bottom of Mount Washington in New Hampshire. Instead of the predicted rate of 25 mesons per hour, the rate was 400 mesons per hour.

This is because according to Earth, the trip lasted about t ~ 6.5 μsec, assuming the velocity of the mesons was near the speed of light. Taking into account time dilation, the trip actually lasted only about 0.7 μsec according to the muons.

Special relativity strikes again.

• ### General Relativity

After spending all this time discussing special relativity, it's only fair to give general relativity some time.

Remember when we talked about time as the fourth dimension? We defined an event as having space-time coordinates of (x, y, z, t). Space-time, then, is simply the unification of space and time. We said at the time that special relativity is special because it ignores gravity.

Without gravity, space-time is flat, an infinite sheet, though in more than two dimensions. However, we have massive planets and stars and galaxies that revolve around each other in defined orbits. If we picture space-time as a type of malleable fabric affected by mass, we'll see that it's not flat around planets and the rest. In other words, gravity curves space-time, just like standing on a trampoline bends the surface. We call this phenomenon space-time curvature.

In the above illustration, space-time is drawn as a malleable grid. Without the effect of gravity, Earth would simply rest on a flat surface, like a rock half buried under the ground on a field. The grid would simply be, well, a grid. Its squares would all be the same size. The curvature of space-time changes the length of each square, depending on the influence of gravity. The closer we get to a massive object, the longer the length of each grid square, or the more stretched a grid becomes. This is a simplified explanation of how physicists measure the curvature of space-time.

In flat space, the shortest distance between two points is a straight line. What about in curved space? Ah, now we get to the meat of the matter.

Since space and time are combined, as mass bends space, time gets stretched too. That's right, time dilation. Space-time curvature causes time to flow more slowly. Mathematically, this is because light has to travel a lesser distance between two points in curved space.

The effects of gravity may also be explained with general relativity. Newton viewed gravity as an attractive force between two objects, such as the Earth and the Sun. Einstein explained the same phenomenon using space-time curvature. The Earth rotates around the Sun following the curvature they each form in space-time.

Both explanations are accurate, but Newton's equations don't hold once we start to deal with super massive objects like black holes, just like his equations don't hold near the speed of light.