Light and Special Relativity

In short: Special relativity is Einstein's theory that the laws of physics, and the speed of light, are the same for every observer moving at a steady velocity. As a result, moving clocks run slow, moving objects get shorter along their motion, and mass and energy are linked by $E = m c^{2}$ .

Frames of reference

A frame of reference is a point of view from which you measure positions and times.

An inertial frame is one that is not accelerating: it is at rest or moving at a constant velocity. In an inertial frame, a ball dropped falls straight down and nothing pushes you back into your seat. You cannot tell whether you are moving by any experiment inside it. Think of a smooth plane cruising at a constant speed.

Einstein's two postulates

In 1905, Einstein built special relativity on two statements:

The laws of physics are the same in all inertial frames.
The speed of light in a vacuum, $c$ , is the same for all observers, whatever their motion or the motion of the light source.

The second one is strange. If a spaceship moving at $0.8 c$ shines a light forwards, an observer on Earth measures the light at $c$ , not $1.8 c$ . For the speed of light to stay the same for everyone, time and length must be different for different observers.

Simultaneity is relative

Two events that happen at the same time for one observer may happen at different times for another observer moving relative to them.

A lamp flashes in the exact middle of a moving train carriage. For a passenger, the light reaches the front and back walls at the same time. For someone on the platform, the back wall is rushing towards the light and the front wall is running away from it, so the light reaches the back first.

Time dilation

A clock that is moving relative to you runs slow, as you measure it.

$t = \frac{t _{0}}{1 - \frac{v ^{2}}{c ^{2}}}$

$t_{0}$ = proper time: the time measured by a clock at rest relative to the events (the moving person's own clock)
$t$ = the time measured by the observer who sees the clock moving
$t$ is always longer than $t_{0}$

The factor $γ = \frac{1}{1 - v ^{2} / c ^{2}}$ is called the Lorentz factor. It is 1 at low speeds and grows without limit as $v$ gets close to $c$ .

Length contraction

A moving object is shorter along its direction of motion, as you measure it.

$l = l_{0} 1 - \frac{v ^{2}}{c ^{2}}$

$l_{0}$ = proper length: measured by an observer at rest relative to the object
$l$ = the length measured by an observer who sees it moving
Only the length along the motion shrinks. Height and width do not change.

Evidence for special relativity

Muons: many more reach the ground than classical physics predicts. From the ground, their clocks run slow (time dilation). From the muon's view, the atmosphere is shorter (length contraction). Both views agree on how many reach the ground.
Hafele-Keating experiment (1971): very accurate atomic clocks were flown around the world on aeroplanes and compared with clocks on the ground. They disagreed by tiny amounts that matched the predictions of relativity.
Particle accelerators: particles live longer at high speed, and they need far more energy to reach speeds near $c$ than classical physics predicts.
GPS satellites: their clocks have to be corrected for relativity, or GPS positions would drift by kilometres each day.

Mass and energy

Einstein showed that mass is a form of energy:

$E = m c^{2}$

A tiny amount of mass holds an enormous amount of energy. One gram of mass is equivalent to $0.001 \times (3.00 \times 1 0^{8})^{2} = 9.0 \times 1 0^{13}$ J, about the energy of a 20-kilotonne bomb. This is the energy source of the Sun and of nuclear power.

Special Relativity

Inertial frames

Not accelerating
Same laws of physics

Two postulates

Same physics in all inertial frames
c same for all observers

Simultaneity

Depends on observer

Time dilation

t = t₀ / √(1 − v²/c²)
Moving clocks run slow

Length contraction

l = l₀ √(1 − v²/c²)
Only along the motion

Evidence

Muons
Hafele-Keating
Accelerators, GPS

E = mc²

Year 12Relativistic momentum and the speed limit

Momentum at high speed is:

$p_{v} = \frac{m _{0} v}{1 - \frac{v ^{2}}{c ^{2}}}$

where $m_{0}$ is the rest mass. As $v$ approaches $c$ , the momentum grows without limit. A constant force keeps adding momentum, but the speed creeps closer and closer to $c$ and never reaches it. That is why no object with mass can reach the speed of light: it would need infinite energy.

Kinetic energy at high speed is $K = (γ - 1) m_{0} c^{2}$ , not $\frac{1}{2} m v^{2}$ . The total energy is $E = γ m_{0} c^{2}$ , which is the rest energy $m_{0} c^{2}$ plus the kinetic energy.

Both sides are right. In the muon example, the ground observer says "your clock runs slow". The muon says "your atmosphere is short". Each measures from their own inertial frame, and both predict the same result. Neither frame is special.

Finding the speed from $γ$ . Rearranging $γ = \frac{1}{1 - v ^{2} / c ^{2}}$ gives $v = c 1 - \frac{1}{γ ^{2}}$ . If a lifetime grows from 2.2 µs to 6.6 µs, $γ = 3$ and $v = c 1 - \frac{1}{9} = 0.943 c$ . For a length to shrink from $l_{0}$ to $l$ , $1 - v^{2} / c^{2} = \frac{l}{l _{0}}$ .

A trip, from both sides. A ship flies at $0.60 c$ ( $γ = 1.25$ ) to a star 9.0 light-years away by Earth's measurement. Earth says the trip takes $\frac{9.0}{0.60} = 15$ years. The crew's clocks (proper time) show $\frac{15}{1.25} = 12$ years, and in the crew's frame the distance is contracted to $\frac{9.0}{1.25} = 7.2$ light-years. Both agree that $7.2 \div 0.60 = 12$ years. If you know both clock readings, $γ = \frac{Earth time}{crew time}$ gives $v$ , then distance $= v \times$ Earth time.

Simultaneity in practice. A flash at the middle of a moving capsule reaches both ends together for the crew. An outside observer sees the back end move towards the light, so the back hatch opens first.

Energy and mass. $m = \frac{E}{c ^{2}}$ : releasing $9.0 \times 1 0^{15}$ J converts 0.10 kg of mass. Speed from kinetic energy: $γ = 1 + \frac{K}{m _{0} c ^{2}}$ . An electron (0.511 MeV rest energy) given 1.0 MeV has $γ = 2.96$ , so $v = 0.94 c$ .

Objects at an angle. Only the part of the length along the motion shrinks. A 2.0 m rod at 30° to the motion has parts $1.73$ m along and $1.0$ m across; at $0.80 c$ the part along becomes 1.04 m, so the rod looks 1.44 m long at about 44° to the motion.

Light keeps the lead. A radio message sent at the same moment as a ship leaves always travels at $c$ in every frame, so it arrives first. Each observer sees the other's clocks running slow; ship B sees ship C (going faster in the same direction) with a slower clock than its own, by an amount set by their relative speed.

Quick summary

An inertial frame is not accelerating. The laws of physics are the same in all of them.
The speed of light is the same for every observer.
Simultaneity depends on the observer.
Moving clocks run slow: $t = \frac{t _{0}}{1 - v ^{2} / c ^{2}}$ .
Moving objects shrink along their motion: $l = l_{0} 1 - v^{2} / c^{2}$ .
Evidence: muons, Hafele-Keating clocks, particle accelerators and GPS.
$E = m c^{2}$ : mass and energy are equivalent. Nothing with mass can reach $c$ .

Quiz

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An inertial frame of reference is one that is:
1. A Accelerating
2. B At rest or moving at a constant velocity
3. C Moving in a circle
4. D Always at rest
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B: At rest or moving at a constant velocity
Inertial frames do not accelerate.
A spaceship moving at 0.5c fires a laser forwards. An observer on Earth measures the laser light's speed as:
1. A 0.5c
2. B c
3. C 1.5c
4. D 2c
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B: c
Einstein's second postulate: everyone measures the speed of light as c.
Compared with a clock at rest, a clock moving past you at high speed appears to run:
1. A Fast
2. B Slow
3. C At the same rate
4. D Backwards
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B: Slow
Time dilation: moving clocks run slow as seen by a stationary observer.

Frames of reference

Einstein's two postulates

Simultaneity is relative

Time dilation

Length contraction

Evidence for special relativity

Mass and energy

Quick summary

Quiz

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