4-Vectors & Invariants

The Lorentz transformations mix space and time coordinates, just as rotations mix the $x$ and $y$ coordinates. In ordinary 3D physics, we handle rotations elegantly using vectors: objects with three components that transform predictably under rotations. The natural generalization to spacetime is the 4-vector: an object with four components that transforms predictably under Lorentz transformations. 4-vectors are the language of relativistic physics, and once you master them, the laws of nature practically write themselves.

Index Notation and the Einstein Convention

Before diving into specific 4-vectors, we need the notation that makes relativistic calculations efficient. A 4-vector has components labeled by a Greek index $\mu$ running over $0, 1, 2, 3$:

$$A^\mu = (A^0, A^1, A^2, A^3)$$

The index $0$ is the "time" component, and $1, 2, 3$ are the spatial components. We use Latin indices $i, j, k$ when we mean only spatial components ($1, 2, 3$).

Einstein introduced a powerful notational shorthand: whenever an index appears once upstairs and once downstairs in the same term, it is summed over. This is the Einstein summation convention:

$$A^\mu B_\mu \equiv \sum_{\mu=0}^{3} A^\mu B_\mu = A^0 B_0 + A^1 B_1 + A^2 B_2 + A^3 B_3$$

This convention eliminates countless summation signs and makes equations far more readable. It is said that Einstein considered this his greatest contribution to mathematics.

The Metric Tensor

The spacetime interval $ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2$ defines the geometry of Minkowski spacetime. We can write it compactly using the metric tensor $\eta_{\mu\nu}$:

$$ds^2 = \eta_{\mu\nu} \, dx^\mu \, dx^\nu$$

Contravariant and Covariant Vectors

There are two types of 4-vectors, distinguished by where the index sits:

• Contravariant (upper index): $A^\mu = (A^0, A^1, A^2, A^3)$
• Covariant (lower index): $A_\mu = (A_0, A_1, A_2, A_3)$

The metric tensor converts between them, it lowers indices:

$$A_\mu = \eta_{\mu\nu} A^\nu$$

Explicitly: $A_0 = -A^0$, $A_1 = A^1$, $A_2 = A^2$, $A_3 = A^3$. The time component flips sign; the spatial components don't. The inverse metric $\eta^{\mu\nu}$ (which has the same numerical values as $\eta_{\mu\nu}$) raises indices:

$$A^\mu = \eta^{\mu\nu} A_\nu$$

The contraction of a contravariant and covariant vector gives a Lorentz scalar (invariant):

$$A^\mu B_\mu = -A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3$$

The 4-Position

The most basic 4-vector is the position 4-vector (or event vector):

$$x^\mu = (ct, x, y, z)$$

Under a Lorentz boost along the $x$-axis, this transforms as:

$$x'^\mu = \Lambda^\mu{}_\nu \, x^\nu$$

where $\Lambda^\mu{}_\nu$ is the Lorentz transformation matrix. The invariant "length" of the displacement 4-vector gives the spacetime interval:

$$\eta_{\mu\nu} \, \Delta x^\mu \, \Delta x^\nu = -c^2\Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2 = \Delta s^2$$

The 4-Velocity

In Newtonian mechanics, velocity is $d\mathbf{x}/dt$. But $dt$ is not Lorentz-invariant, so this ratio doesn't form a 4-vector. The fix is simple: differentiate with respect to proper time $\tau$ instead, since $\tau$ is a Lorentz scalar.

The fact that $U^\mu U_\mu = -c^2$ is a constant is deeply meaningful: it says that every massive particle moves through spacetime at the same "speed." When you sit still in space, all your motion is through time (at rate $c$). When you move through space, some of your motion is diverted from time, which is why moving clocks run slow.

The 4-Momentum

Just as 3-momentum is mass times 3-velocity, the 4-momentum is rest mass times 4-velocity:

The 4-momentum packs energy and momentum into a single object. Its invariant magnitude gives the most important relation in relativistic physics:

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

Rearranging:

$$E^2 = (pc)^2 + (mc^2)^2$$

This is the energy-momentum relation, which we will explore in detail in Lesson 4.4. For a particle at rest ($\mathbf{p} = 0$), it gives $E = mc^2$.

The 4-Force

Newton's second law $\mathbf{F} = d\mathbf{p}/dt$ generalizes to the Minkowski force:

$$f^\mu = \frac{dp^\mu}{d\tau} = \gamma\left(\frac{1}{c}\frac{dE}{dt}, \mathbf{F}\right)$$

where $\mathbf{F}$ is the ordinary 3-force. The time component relates the power delivered by the force to the rate of change of energy. The constraint $f^\mu U_\mu = 0$ (which follows from $p^\mu p_\mu = -m^2c^2$) means the 4-force is always perpendicular to the 4-velocity in the Minkowski sense.

How 4-Vectors Transform

Any 4-vector $A^\mu$ transforms under a Lorentz boost along the $x$-axis as:

$$A'^0 = \gamma(A^0 - \beta A^1)$$ $$A'^1 = \gamma(A^1 - \beta A^0)$$ $$A'^2 = A^2, \quad A'^3 = A^3$$

This is the same transformation law as for the coordinates, that is what makes it a 4-vector. Any quantity that transforms this way under Lorentz transformations is a 4-vector.

Building Lorentz-Invariant Laws

The power of the 4-vector formalism is that any equation written as an equality between 4-vectors (or 4-tensors) is automatically Lorentz-invariant. If it holds in one frame, it holds in all frames.

For example, conservation of 4-momentum:

$$\sum_{\text{initial}} p^\mu_i = \sum_{\text{final}} p^\mu_f$$

This single equation contains both conservation of energy (the $\mu = 0$ component) and conservation of 3-momentum (the $\mu = 1, 2, 3$ components). And because it's a 4-vector equation, it's valid in every inertial frame.

Similarly, the equation of motion $f^\mu = dp^\mu/d\tau$ is a single 4-vector equation that replaces both Newton's second law and the work-energy theorem.

Other Important 4-Vectors

The 4-Wavevector

For a plane wave with angular frequency $\omega$ and wave vector $\mathbf{k}$:

$$k^\mu = \left(\frac{\omega}{c}, \mathbf{k}\right)$$

The invariant $k^\mu k_\mu = -\omega^2/c^2 + |\mathbf{k}|^2$ equals zero for light (the dispersion relation $\omega = c|\mathbf{k}|$) and $-m^2c^2/\hbar^2$ for massive particles (de Broglie waves). The Doppler effect and aberration of light emerge naturally from the Lorentz transformation of $k^\mu$.

The 4-Gradient

The spacetime derivative forms a covariant 4-vector:

$$\partial_\mu = \frac{\partial}{\partial x^\mu} = \left(\frac{1}{c}\frac{\partial}{\partial t}, \nabla\right)$$

The d'Alembertian operator (the Lorentz-invariant wave operator) is:

$$\Box = \partial^\mu \partial_\mu = -\frac{1}{c^2}\frac{\partial^2}{\partial t^2} + \nabla^2$$

This is the relativistic generalization of the Laplacian and appears in the wave equations of electromagnetism and quantum field theory.

Tensors: Beyond Vectors

4-vectors are the simplest tensors (rank 1). Some physical quantities require higher-rank tensors. The most important example is the electromagnetic field tensor $F^{\mu\nu}$, a rank-2 antisymmetric tensor that combines the electric and magnetic fields into a single relativistic object. We encountered this in the electromagnetism phase.

The energy-momentum tensor $T^{\mu\nu}$ is another rank-2 tensor that describes the density and flow of energy and momentum. It will be the source term in Einstein's field equations of general relativity.

Key Insights

• 4-vectors are objects with four components that transform under Lorentz transformations like the spacetime coordinates
• The Minkowski metric $\eta_{\mu\nu} = \text{diag}(-1, +1, +1, +1)$ defines the geometry of flat spacetime and converts between upper and lower indices
• The Einstein summation convention (repeated indices are summed) makes relativistic calculations compact and elegant
• Key 4-vectors: position $x^\mu$, velocity $U^\mu = \gamma(c, \mathbf{v})$, momentum $p^\mu = (E/c, \mathbf{p})$
• The squared magnitude of any 4-vector ($A^\mu A_\mu$) is a Lorentz invariant
• Physical laws written as 4-vector equations are automatically valid in all inertial frames
• This formalism is the essential language for electromagnetism, quantum field theory, and general relativity

Looking Ahead

Now that we have the mathematical machinery of 4-vectors, we can formulate relativistic mechanics properly. In the next lesson, we will use the 4-momentum to derive the energy-momentum relation, analyze relativistic collisions, and understand how particle physics experiments work in the center-of-momentum frame.