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Topic 7 of 10

Four-Vectors & Tensors

A Lorentz-covariant formulation combines space and time into four-vectors that transform simply under Lorentz boosts. Physical laws written in four-vector form are automatically valid in all inertial frames. The invariant inner product of two four-vectors — formed with the Minkowski metric — is the relativistic generalization of the dot product.

Key Concepts

Minkowski Metric

The metric tensor

\eta_{\mu\nu}=\text{diag}(+1,-1,-1,-1)

(or

-,+,+,+

in the other convention). The invariant interval:

ds^2=\eta_{\mu\nu}dx^\mu dx^\nu=c^2dt^2-dx^2-dy^2-dz^2

Four-Velocity

u^\mu=dx^\mu/d\tau=(\gamma c,\gamma\vec v)

. Its invariant square:

u_\mu u^\mu=c^2

Four-Momentum

p^\mu=(E/c,\vec p)=(\gamma mc,\gamma m\vec v)

. Invariant:

p_\mu p^\mu=(mc)^2

. This gives

E^2-(pc)^2=(mc^2)^2

Four-Gradient

\partial_\mu=(\partial/c\partial t,\nabla)

(covariant). The d'Alembertian

\Box^2=\partial_\mu\partial^\mu=\partial^2/c^2\partial t^2-\nabla^2

is Lorentz invariant and appears in the wave equation.

Key Equations

Four-momentum invariant

p_\mu p^\mu = \frac{E^2}{c^2} - |\vec{p}|^2 = (mc)^2

Lorentz invariant; gives energy-momentum relation E² = (pc)² + (mc²)².

Four-velocity

u^\mu = \gamma(c,\,\vec{v})

Tangent to worldline; magnitude always c.

Minkowski inner product

a_\mu b^\mu = a^0b^0 - \vec{a}\cdot\vec{b}

Lorentz-invariant combination of two four-vectors (+−−− convention).

Worked Example

Invariant Mass from Decay Products

Problem

A particle decays into two photons with energies $E_1=300$ MeV and $E_2=200$ MeV emitted at $90°$ to each other. Find the rest mass of the parent (in MeV/c²).

Solution

Four-momenta of photons: $p_1^\mu=(300,300,0,0)$ MeV/c and $p_2^\mu=(200,0,200,0)$ MeV/c.

Total four-momentum: $P^\mu=(500,300,200,0)$ MeV/c.

(Mc)^2 = P_\mu P^\mu = 500^2 - 300^2 - 200^2 = 250000 - 90000 - 40000 = 120000\text{ MeV}^2/c^2

Mc^2 = \sqrt{120000}\approx346\text{ MeV}

Answer Rest mass

\approx346

MeV/c².

Practice

Exercises

7 problems

1 of 7

Drag the slider to explore the four-velocity hyperbola. The blue vector sweeps along the amber curve $u_\mu u^\mu = c^2$ regardless of β. What is the magnitude |u^μ| of the four-velocity (in m/s)?

4-velocity hyperbola

Speed β (γβ, γ) = (0.00, 1.00)

As β changes, the four-velocity vector (blue) sweeps along the amber hyperbola — but its Minkowski magnitude is always c.

|u^μ| in m/s

|u| = m/s

Enter e.g. 3e8 or 300000000

2 of 7

Watch the proton four-momentum vector sweep the mass-shell hyperbola as γ changes. A proton has $m_pc^2 = 938$ MeV. Find the invariant $p_\mu p^\mu = (m_pc)^2$ (in MeV²/c²).

mass shell

Lorentz factor γ E=938 MeV, pc=0 MeV

Drag γ — the point climbs the amber hyperbola. E and p change, but E²−(pc)² stays constant at (mc²)².

p_μp^μ

= MeV²/c²

Proton: mc² = 938 MeV. Enter e.g. 879844

3 of 7

Two photons, each with energy $E=511$ MeV, travel in opposite directions. Find the invariant mass $M$ of the system (in MeV/c²). Total $P^\mu=(2E/c,0,0,0)$ .

Unlock Exercise 3