Minkowski Space

Minkowski space is the flat four-dimensional spacetime manifold of special relativity. It replaces the Galilean separation of absolute time and Euclidean space with a single geometric structure where temporal and spatial separations enter on equal footing — but with opposite sign.

The metric

In coordinates $(t, x, y, z)$ with $c = 1$, the Minkowski metric is

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

The line element (invariant interval) between two events is

$$ ds^2 = \eta_{\mu\nu}\thinspace dx^\mu dx^\nu = -dt^2 + dx^2 + dy^2 + dz^2 $$

Einstein summation convention applies. The signature $(-+++)$ is common in particle physics; $(+—)$ appears in some GR texts — physics is unchanged up to an overall sign convention.

Event classification

For a displacement $\Delta x^\mu$ from one event to another:

The light cone at an event $P$ divides spacetime into future, past, and elsewhere:

flowchart TB
    P((Event P))
    P --> F[Future: ds² < 0, Δt > 0]
    P --> PA[Past: ds² < 0, Δt < 0]
    P --> E[Elsewhere: ds² > 0]
    P --> L[Light cone: ds² = 0]

Lorentz transformations

Transformations preserving $\eta_{\mu\nu}$ form the Lorentz group $O(1,3)$. A boost with velocity $v$ along $x$:

$$ \Lambda^\mu_{\ \nu} = \begin{pmatrix} \gamma & -\beta\gamma & 0 & 0 \\ -\beta\gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}, \quad \beta = \frac{v}{c},\ \gamma = \frac{1}{\sqrt{1-\beta^2}} $$

Proper Lorentz transformations ($\det\Lambda = +1$) include continuous boosts and rotations; discrete parity and time reversal are in $O(1,3)$ but not the connected component.

Four-vectors

Quantities transforming as $V’^\mu = \Lambda^\mu_{\ \nu} V^\nu$ include:

Four-position: $x^\mu = (t, \mathbf{x})$

Four-velocity: $u^\mu = \frac{dx^\mu}{d\tau}$ where $\tau$ is proper time. Normalization: $u^\mu u_\mu = -1$.

Four-momentum: $p^\mu = m u^\mu = (E, \mathbf{p})$ with

$$ p^\mu p_\mu = -m^2 \quad \Leftrightarrow \quad E^2 = |\mathbf{p}|^2 + m^2 $$

Four-gradient: $\partial_\mu = \frac{\partial}{\partial x^\mu}$

Raising and lowering indices uses $\eta_{\mu\nu}$: $v_\mu = \eta_{\mu\nu} v^\nu$.

Relativistic mechanics in geometric form

Newton’s second law becomes the geodesic equation in flat spacetime (trivial Christoffel symbols):

$$ \frac{d u^\mu}{d\tau} = \frac{q}{m} F^{\mu\nu} u_\nu $$

where $F^{\mu\nu}$ is the electromagnetic field tensor. Energy-momentum conservation for a closed system:

$$ \partial_\mu T^{\mu\nu} = 0 $$

The stress-energy tensor $T^{\mu\nu}$ encodes energy density, momentum density, and stress in a single symmetric (for perfect fluids and EM) rank-2 tensor.

Maxwell equations

In natural units ($\varepsilon_0 = \mu_0 = c = 1$), Maxwell’s equations unify as

$$ \partial_\mu F^{\mu\nu} = J^\nu, \quad \partial_{[\lambda} F_{\mu\nu]} = 0 $$

where $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ is the field strength and $J^\mu = (\rho, \mathbf{j})$ is the four-current. The second equation is the Bianchi identity (homogeneous Maxwell equations).

Why Minkowski space matters

1. Covariance — physical laws take the same form in all inertial frames without ad hoc length contraction or time dilation factors.
2. Causality — the metric defines light cones and forbids superluminal influence.
3. Bridge to GR — Minkowski space is the tangent space at any point of a curved Lorentzian manifold; special relativity is local flatness.

The transition from Minkowski space to curved spacetime is developed in Curved Spacetime.