Consequences of and evidence for special relativity

By the end of this topic you should be able to:

•solve problems with relativistic energy, E = γm0c2;•describe muon decay experiments as evidence for time dilation and length contraction;•Appreciate the significance of the Michelson-Morley experiment,

Rest energy

The theory of relativity predicts that, to a particle of mass m0 that is at rest with respect to some inertial observer, there corresponds an amount of energy E0 that the observer measures to be E0=m0c2. This energy is called the rest energy of the particle.

Relativistic energy

If the particle moves with speed v relative to some inertial observer, the energy corresponding to the mass of the particle that this observer will measure is given by

E = γm0c2

This is energy that the particle has because it has a mass and because it moves. If the particle has other forms of energy, then the total energy of the particle will equal γm0c2 plus the other forms of energy.

Example questions

In Newtonian mechanics, a constant force produces a constant acceleration, and so the speed increases uniformly without limit, exceeding the speed of light. In relativistic mechanics, the speed increases uniformly as long as the speed is substantially less than the speed of light. However, as the speed increases, so does the energy. Because it takes an infinite amount of energy for the particle to reach the speed of light, we conclude that the particle never reaches the speed of light.

Muon decayLifetime when the muon is at rest: 2.2 x 10-6s

Lifetime from the point of view of the lab frame:

In this time the muon travels a distance:

Without the time dilation effect, the muon would have travelled a distance of only:

Muon decay

The distance of 3 km measured by the observers on Earth is a proper length for them but not for an observer in the muon’s frame of reference.

The Earth’s surface is coming up to this observer with a speed of 0.99c and so the time when they will meet is

The Michelson-Morley experiment

The constancy of the speed of light

The first conclusive experiment that demonstrated the constancy of the speed of light with great accuracy was performed at CERN in 1964.

In this experiment, neutral pions moving at 0.99975c decayed into a pair of photons moving in different directions.