Special Relativity

The short version

Einstein's two postulates:

• The laws of physics are the same in every inertial frame.
• The speed of light in a vacuum, $c$, is the same for every inertial observer.

Together they force time and length to depend on the observer. With the Lorentz factor

$\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

a moving clock runs slow, $t=\gamma t_0$, and a moving object is shorter along its motion, $l=l_0/\gamma$. The proper time $t_0$ is measured by the clock present at both events; the proper length $l_0$ is measured in the object's rest frame.

Relativistic momentum is $p=\gamma m_{0}v$. It grows without limit as $v\to c$, so nothing with mass can be accelerated to the speed of light.

Mass and energy are equivalent: $E=m{c}^2$. Any process that releases energy $E$ loses mass $E/c^2$, whether it is fusion in the Sun, annihilation or burning petrol.

Frames of reference

An inertial frame is one that is not accelerating: a laboratory at rest, or a train carriage gliding at constant velocity. Inside one, Newton's first law holds and no experiment can tell you whether you are moving. Drop a ball in a smooth train and it falls straight down, exactly as it would on the platform.

Before Einstein, physicists combined velocities by simple addition. A ball thrown forwards at $10$ m s${}^{-1}$ from a train moving at $30$ m s${}^{-1}$ travels at $40$ m s${}^{-1}$ relative to the ground. Time was the same for everyone.

The problem with light

Maxwell's equations predict that light travels at $c=3.00\times 10^8$ m s${}^{-1}$, but they do not say relative to what. Most physicists assumed light was a wave in a medium, the aether, and that $c$ was its speed relative to the aether. Earth moves around the Sun at about $30$ km s${}^{-1}$, so light should travel at slightly different speeds relative to Earth in different directions.

Einstein's postulates

In 1905 Einstein built special relativity on two statements:

1. The laws of physics are the same in all inertial frames. There is no special frame at rest, so no experiment can reveal absolute motion.
2. The speed of light in a vacuum is the same for every inertial observer, whatever the motion of the source or the observer.

The second one is what makes relativity strange. A spacecraft passing at $0.60c$ shines a laser forwards, and both the crew and an observer on Earth measure the light at $c$, not $1.6c$. For that to be true, the two observers must disagree about distances and times.

Evaluating the evidence

Time dilation

The key to every calculation is identifying the proper time. It is measured by the one clock that is present at both events, so the events happen at the same place in its frame. Proper time is always the shortest measurement of the interval; everyone else measures $\gamma$ times longer.

The effect is symmetric. The astronaut sees the Earth's clocks running slow by the same factor, because to the astronaut it is the Earth that moves.

Below about $0.1c$, $\gamma$ differs from $1$ by less than one per cent, which is why nobody noticed any of this before 1905.

Length contraction

If moving clocks run slow, moving lengths must shrink, or observers would disagree about whether a trip was completed. A moving object is measured shorter along its direction of motion:

The proper length $l_0$ is measured in the frame where the object is at rest. Only the length along the motion changes; the width and height are unchanged.

Muons: both effects at once

Cosmic rays striking the upper atmosphere create muons, heavy relatives of the electron, which rain down at about $0.995c$. At rest, a muon has a mean lifetime of $2.2\mu$s. Without relativity, it could travel only about $660$ m in one lifetime, yet large numbers reach sea level.

More evidence

Atomic clocks. In 1971 Hafele and Keating flew caesium clocks around the world, east and then west, and compared them with clocks at the US Naval Observatory. Viewed from a frame that does not rotate with Earth, the ground clock moves east at Earth's rotation speed. An eastbound plane moves faster than that, so its clock ran slower and lost time; a westbound plane moves slower than the ground, so its clock gained time. The differences, tens to hundreds of nanoseconds, matched the predictions once the effect of altitude, from general relativity, was included.

Particle accelerators. Unstable particles in accelerators live exactly as long as time dilation predicts. At CERN in the 1970s, muons circulating in a storage ring at $\gamma \approx 29$ lived about $29$ times longer than muons at rest, confirming the formula to about one part in a thousand.

Cosmology. A Type Ia supernova brightens and fades over a characteristic time in its own frame. Distant supernovae, receding from us rapidly with the expansion of the universe, are seen to brighten and fade more slowly, stretched by the factor predicted from their redshift.

Relativistic momentum

Newton's momentum $p=m_{0}v$ is not conserved in collisions at high speed when measured from different frames. The version that is conserved everywhere is

At low speed $\gamma \approx 1$ and it reduces to $m_{0}v$. Near $c$, $\gamma$ grows without limit, so the momentum does too.

This sets the speed limit. A force changes momentum at a finite rate, $F=\Delta p/\Delta t$, and since the momentum needed to reach $c$ is infinite, no finite force acting for a finite time can get there. Pushing harder adds momentum, but past about $0.9c$ each extra unit of momentum adds very little speed. Particle accelerators see this directly: the magnets that steer protons in a circle must provide a force set by $\gamma m_{0}v$, and it rises far faster than the speed does.

Mass and energy

Einstein showed that a body's rest mass is a store of energy:

Whenever a system releases energy $E$, whether as light, heat or kinetic energy of its products, its mass falls by $E/c^2$. Because $c^2=9.0\times 10^{16}$ m${}^2$ s${}^{-2}$ is enormous, a tiny mass is worth a huge energy.

The annihilation photons are the basis of PET scanning: a tracer releases positrons in the body, and detectors on opposite sides record the pairs of $0.511$ MeV photons.

Relativistic mass

Many older books, and some notes, say a moving object's mass increases to $m=\gamma m_0$. That gives the right momentum, but physicists now keep $m_0$ as the one mass of an object and put the $\gamma$ into the momentum and energy instead. The syllabus follows this, with its formula written using $m_0$. In an exam, "the momentum grows without limit" is always safe; "the mass becomes infinite" invites a marker to ask which mass.

Simultaneity

The deeper reason for time dilation is that two events that happen at the same time in one frame need not in another, if they are separated along the direction of motion. Each observer's clocks are synchronised differently, and that is what lets both see the other's clocks running slow without contradiction.

Total energy

The full relativistic energy of a moving particle is $E=\gamma m_0{c}^2$. At rest it is $m_0{c}^2$; the kinetic energy is the extra, $(\gamma -1)m_0{c}^2$, which reduces to $\frac{1}{2}{m}_0{v}^2$ at low speed. This also shows why reaching $c$ needs infinite energy, not just infinite momentum. Nuclear physics uses $E=m{c}^2$ constantly: the next module finds the energy of fission and fusion from the mass lost.