15.1 Special Relativity, Mass–Energy Equivalence, Pair Production & Annihilation

Special Relativity, Mass–Energy Equivalence, Pair Production & Annihilation

At ordinary speeds Newtonian mechanics works extremely well. Near the speed of light it fails, and Einstein's special theory of relativity (1905) replaces it. The same theory gives $E = mc^2$, which explains how photons can turn into particles and particles into photons. Section 15.2 then develops the photon picture through the photoelectric and Compton effects.

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1. Frames of Reference

• A frame of reference is a coordinate system from which positions and times are measured.
• An inertial frame moves with constant velocity; Newton's first law holds in it.
• A non-inertial frame accelerates (or rotates); in it, objects appear to experience fictitious forces. The Earth is very nearly inertial for most laboratory purposes.

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2. The Postulates of Special Relativity

1. Principle of relativity: the laws of physics are the same in all inertial frames. No experiment inside a closed laboratory moving uniformly can reveal its velocity.
2. Constancy of the speed of light: light in vacuum travels at $c = 3.0 \times 10^8\text{ m/s}$ for every inertial observer, regardless of the motion of the source or the observer.

These postulates force space and time measurements to depend on the observer's motion. All the results below contain the Lorentz factor

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

which is almost exactly 1 at everyday speeds.

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3. Consequences

Time Dilation

A clock moving with speed $v$ relative to an observer runs slow. If $t_0$ is the proper time measured by a clock at rest relative to the event,

$t = \frac{t_0}{\sqrt{1 - v^2/c^2}} = \gamma t_0$

Evidence: muons created high in the atmosphere have a mean life of about $2.2\,\mu\text{s}$ in their own frame, too short to reach the ground at nearly $c$ in Newtonian terms. Because their clocks run slow in the Earth's frame, many survive to sea level. Precise atomic clocks flown in aircraft and the timing corrections built into satellite navigation systems confirm the effect.

Twin paradox: a twin who travels to a distant star at high speed and returns is younger than the twin who stayed on Earth. The situation is not symmetrical, because the travelling twin changes frames (accelerates) to turn around.

Length Contraction

An object of proper length $L_0$ (measured at rest) moving with speed $v$ is shorter along its direction of motion:

$L = L_0\sqrt{1 - v^2/c^2} = \frac{L_0}{\gamma}$

Dimensions perpendicular to the motion are unchanged.

Mass Variation

The mass of a moving body increases with speed:

$m = \frac{m_0}{\sqrt{1 - v^2/c^2}} = \gamma m_0$

where $m_0$ is the rest mass. As $v \to c$, $m \to \infty$, so no material body can be accelerated to the speed of light. This increase is observed in particle accelerators, where electrons must be pushed ever harder for tiny gains in speed.

Worked Example

A spacecraft passes the Earth at $v = 0.8c$.

$\sqrt{1 - (0.8)^2} = \sqrt{0.36} = 0.6, \qquad \gamma = \frac{1}{0.6} \approx 1.67$

• A 1 hour journey measured by the crew's clock lasts $\frac{1}{0.6} \approx 1.67$ hours by Earth clocks.
• A craft 100 m long at rest measures $100 \times 0.6 = 60\text{ m}$ for an Earth observer.
• A 1000 kg (rest mass) craft has a relativistic mass of about 1667 kg.

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4. Mass–Energy Equivalence

$E = mc^2$

Mass is a form of energy. A body at rest has rest energy $E_0 = m_0c^2$; its kinetic energy is the extra energy due to motion:

$K = mc^2 - m_0c^2 = (\gamma - 1)m_0c^2$

which reduces to $\frac{1}{2}m_0v^2$ at low speeds.

The $931\text{ MeV}$ per atomic mass unit is the conversion used for mass defect and binding energy in Section 15.2. Converting $1\text{ g}$ of mass entirely into energy would release $E = 10^{-3} \times (3 \times 10^8)^2 = 9 \times 10^{13}\text{ J}$.

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5. Interaction of Photons with Matter

A photon can interact with matter in three main ways, depending on its energy:

The energy ranges overlap and depend on the absorbing material; heavy elements favour the photoelectric effect and pair production.

Pair Production

A high-energy photon passing close to a heavy nucleus can disappear and create an electron and a positron (the electron's antiparticle: same mass, positive charge):

$\gamma \rightarrow e^- + e^+$

• Threshold energy: $hf_{\min} = 2m_0c^2 = 2 \times 0.511 = 1.02\text{ MeV}$. Any extra energy appears as kinetic energy of the pair: $hf = 2m_0c^2 + K_{e^-} + K_{e^+}$.
• The corresponding maximum wavelength is $\lambda_{\max} = \frac{hc}{2m_0c^2} \approx 1.2 \times 10^{-12}\text{ m}$, in the gamma-ray region.
• A nucleus is required: a lone photon cannot produce a pair in empty space, because energy and momentum could not both be conserved. The nucleus absorbs the recoil momentum but takes very little energy.
• Charge is conserved: the photon has zero charge and the pair has $-e + e = 0$.

Example: a $3.02\text{ MeV}$ photon produces a pair with $3.02 - 1.02 = 2.00\text{ MeV}$ of kinetic energy shared between the electron and positron.

Annihilation of Matter

When a positron meets an electron, the two annihilate, converting their mass into electromagnetic energy:

$e^- + e^+ \rightarrow 2\gamma$

• If both are nearly at rest, total momentum is zero, so two photons are emitted in opposite directions, each of energy $0.511\text{ MeV}$ (wavelength about $2.4 \times 10^{-12}\text{ m}$).
• A single photon cannot be produced, because it would carry momentum where there was none.
• Positron emission tomography (PET) scanners detect these back-to-back photon pairs to locate a positron-emitting tracer in the body.