Overview#

• Relativistic momentum \(\vec{p} = \gamma m\vec{v}\) replaces \(\vec{p} = m\vec{v}\) so that momentum is conserved in all inertial frames. As \(v \to c\), \(p \to \infty\).

• Rest energy \(E_0 = mc^2\): mass is a form of energy. Total energy \(E = \gamma mc^2 = E_0 + K\). Kinetic energy \(K = (\gamma - 1)mc^2\).

• The relation \(E^2 = (pc)^2 + (mc^2)^2\) is invariant—the same in all frames. It connects \(E\), \(p\), and \(m\) without reference to \(v\).

• In nuclear reactions, mass can be converted to other forms of energy. The \(Q\) value measures the energy released or absorbed.

Relativistic momentum#

In classical mechanics, \(\vec{p} = m\vec{v}\) and total momentum is conserved in collisions. In relativity, if we keep that definition, conservation of momentum fails when we transform between frames. The correct definition that preserves the conservation law is

where \(\gamma = 1/\sqrt{1 - v^2/c^2}\). For \(v \ll c\), \(\gamma \approx 1\) and we recover the classical formula. As \(v \to c\), \(p \to \infty\)—another reason no massive particle can reach \(c\).

Rest energy and mass-energy equivalence#

Einstein showed that mass is a form of energy. The rest energy (or mass energy) of an object of mass \(m\) is

This is the energy the object has simply because it has mass, even when at rest. In chemical reactions the mass change is negligible; in nuclear reactions it can be significant and measurable.

Units: In particle physics, masses are often in MeV/\(c^2\) or GeV/\(c^2\), and energies in MeV or GeV. The conversion \(c^2 \approx 931.5\) MeV/u (atomic mass unit) is useful.

Total energy and kinetic energy#

The total energy of a particle is the sum of its rest energy and kinetic energy:

It can also be written as

Solving for kinetic energy:

For \(v \ll c\), \(\gamma \approx 1 + v^2/(2c^2)\), so \(K \approx \frac{1}{2}mv^2\) (classical limit). For \(v\) near \(c\), \(K\) grows without bound; an infinite amount of work would be needed to accelerate a massive particle to \(c\).

Energy-momentum relation#

Eliminating \(v\) between \(p = \gamma mv\) and \(E = \gamma mc^2\) yields the invariant relation:

This holds in all inertial frames. For a photon (\(m = 0\)): \(E = pc\).

Reactions and the \(Q\) value#

For a system undergoing a chemical or nuclear reaction, the \(Q\) value is the negative of the change in total rest energy:

where \(M_i\) and \(M_f\) are the total masses before and after the reaction.

• \(Q > 0\): mass decreases; energy is released (e.g., fusion, fission).

• \(Q < 0\): mass increases; energy must be absorbed.

Example: Hydrogen fusion in the Sun: two protons fuse to form deuterium and release energy. The total mass of the products is less than the total mass of the reactants; \(Q > 0\).

Summary#

• Momentum \(\vec{p} = \gamma m\vec{v}\); rest energy \(E_0 = mc^2\); total energy \(E = \gamma mc^2\); kinetic energy \(K = (\gamma - 1)mc^2\).

• Invariant: \(E^2 = (pc)^2 + (mc^2)^2\). For photons, \(E = pc\).

• \(Q\) value \(= (M_i - M_f)c^2\); \(Q > 0\) means energy released.