Chapter 53B: Minkowski Space and Lorentz Groups¶

Prerequisites: Inner Product Spaces (Ch08) · Symplectic Matrices (Ch53A) · Basics of Special Relativity

Chapter Outline: From Euclidean to Minkowski Space → Indefinite Inner Product and the Metric Matrix $\eta$ → Matrix Definition of Lorentz Transformations → The Lorentz Group $O(1, 3)$ and its Branches → Proper Lorentz Transformations, Rotations, and Boosts → Classification of Vectors: Time-like, Space-like, and Null (Light-like) → Causal Structure and the Light Cone → Polar Decomposition of Lorentz Matrices → Applications: Spacetime Coordinate Transforms, Doppler Effect, and Momentum Tensors in Particle Physics

Extension: Minkowski space is the algebraic vehicle for the "unification of spacetime" in physics; it weaves time and space into a four-dimensional manifold with a negative-signature metric, proving that the invariance of physical laws is essentially the manifestation of the Lorentz group at the operator level. It is the only mathematical foundation for understanding relativity.

In everyday experience, distance is always positive. However, in special relativity, the interval between time and space requires a specific type of combination. Minkowski Space introduces a metric with a negative sign, allowing "distances" to be zero or even negative. The matrices that preserve this interval form the Lorentz Group. This chapter introduces the linear algebraic language describing the four-dimensional skeleton of the universe.

53B.1 Minkowski Metric and Indefinite Inner Product¶

Definition 53B.1 (Minkowski Metric $\eta$)

In four-dimensional spacetime, the metric matrix is typically taken as: $$\eta = \operatorname{diag}(-1, 1, 1, 1)$$ The Minkowski Inner Product of two four-vectors $u, v$ is defined as: $$\langle u, v \rangle_\eta = u^T \eta v = -u_0 v_0 + u_1 v_1 + u_2 v_2 + u_3 v_3$$

Classification of Vectors

Time-like: $\langle v, v \rangle_\eta < 0$.
Space-like: $\langle v, v \rangle_\eta > 0$.
Null / Light-like: $\langle v, v \rangle_\eta = 0$ (the trajectory of a light signal).

53B.2 Matrix Representation of Lorentz Transformations¶

Definition 53B.2 (Lorentz Matrix)

A square matrix $\Lambda \in M_4$ is a Lorentz Transformation if it preserves the Minkowski inner product: $$\Lambda^T \eta \Lambda = \eta$$ The group of all such matrices is denoted by $O(1, 3)$.

53B.3 Boosts and Rotations¶

Lorentz Boost

A boost along the $x$-axis with velocity $v$ is represented by the matrix: $$\Lambda = \begin{pmatrix} \gamma & -\beta\gamma & 0 & 0 \\ -\beta\gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$ where $\beta = v/c$ and $\gamma = 1/\sqrt{1-\beta^2}$. This corresponds to a "rotation" (hyperbolic rotation) between time and space coordinates.

Exercises¶

1. [Basics] Calculate the norm squared of vector $v = (1, 1, 0, 0)^T$ under the metric $\eta = \operatorname{diag}(-1, 1, 1, 1)$.

Solution

Steps: 1. $\langle v, v \rangle_\eta = v^T \eta v = -v_0^2 + v_1^2 + v_2^2 + v_3^2$. 2. Substitute components: $-1^2 + 1^2 + 0^2 + 0^2 = -1 + 1 = 0$. Conclusion: The norm squared is 0. This is a Null vector (light-like), representing a path traveling at the speed of light.

2. [Lorentz Group] Prove that the determinant of a Lorentz matrix must be $\pm 1$.

Solution

Proof: 1. Given $\Lambda^T \eta \Lambda = \eta$. 2. Take the determinant of both sides: $\det(\Lambda^T) \det(\eta) \det(\Lambda) = \det(\eta)$. 3. Since $\det(\eta) = -1 \neq 0$, we divide to get $(\det \Lambda)^2 = 1$. Conclusion: $\det \Lambda = \pm 1$. Matrices with $\det = 1$ are called proper Lorentz transformations.

3. [Calculation] Verify that the $2 \times 2$ boost matrix $\begin{pmatrix} \cosh \phi & -\sinh \phi \\ -\sinh \phi & \cosh \phi \end{pmatrix}$ satisfies the Lorentz definition.

Solution

Steps: 1. Let $\eta = \operatorname{diag}(-1, 1)$. 2. Multiply: $\Lambda^T \eta \Lambda = \begin{pmatrix} \cosh \phi & -\sinh \phi \\ -\sinh \phi & \cosh \phi \end{pmatrix} \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} \cosh \phi & -\sinh \phi \\ -\sinh \phi & \cosh \phi \end{pmatrix}$. 3. The top-left entry is $-\cosh^2 \phi + \sinh^2 \phi = -1$. 4. The bottom-right entry is $-\sinh^2 \phi + \cosh^2 \phi = 1$. 5. Cross terms: $\cosh\sinh - \sinh\cosh = 0$. Conclusion: The result is exactly $\eta$. The parameter $\phi$ is known as rapidity.

4. [Causality] Why is the angle between time-like vectors undefined, but they still have "future" and "past" designations?

Solution

Algebraic Reason: For a time-like vector $v$ ($v_0^2 > v_1^2+v_2^2+v_3^2$), the sign of the $v_0$ component is invariant under continuous Lorentz transformations. This splits spacetime into two disconnected regions: $v_0 > 0$ (the future light cone) and $v_0 < 0$ (the past light cone). This forms the mathematical basis for causality in physics.

5. [Invariants] Does the rest mass of an object (the norm of the momentum four-vector) change under a Lorentz transformation?

Solution

Conclusion: No. Reasoning: The squared norm of the momentum four-vector $P^T \eta P = -E^2 + p^2$ corresponds to $-m^2 c^2$. Since Lorentz transformations preserve the Minkowski inner product, the rest mass is a Lorentz Invariant agreed upon by all observers.

6. [Property] Prove that the inverse of a Lorentz transformation is $\Lambda^{-1} = \eta \Lambda^T \eta$.

Solution

Proof: 1. From $\Lambda^T \eta \Lambda = \eta$, right-multiply by $\Lambda^{-1}$. 2. $\Lambda^T \eta = \eta \Lambda^{-1}$. 3. Since $\eta^{-1} = \eta$, left-multiply by $\eta$. 4. $\eta \Lambda^T \eta = \Lambda^{-1}$.

7. [Classification] If a vector has only spatial components $(0, 1, 0, 0)$ in a given frame, what type of vector is it?

Solution

Determination: $\langle v, v \rangle = -0^2 + 1^2 + 0^2 + 0^2 = 1 > 0$. Conclusion: It is a Space-like vector. Such vectors represent points that cannot have a causal connection in any reference frame.

8. [Application] What is a "Hyperbolic Rotation" in spacetime?

Solution

Standard rotations preserve $x^2+y^2$ (circles). Lorentz boosts preserve $-t^2+x^2$ (hyperbolas). Thus, a boost is mathematically equivalent to a hyperbolic rotation in the spacetime plane, with the rotation "angle" being the rapidity $\phi$.

9. [Spectral] What are the characteristics of the eigenvalues of a Lorentz matrix?

Solution

Conclusion: 1. If $\lambda$ is an eigenvalue, $1/\lambda$ must also be an eigenvalue (similar to symplectic matrices). 2. For a boost, the eigenvalues are $e^\phi$ and $e^{-\phi}$. These correspond to the Doppler shift factors.

10. [Application] Briefly state the role of the Lorentz group in particle collider data analysis.

Solution

In collision experiments, detectors are in the lab frame, while physics is simplest in the center-of-mass frame. Using Lorentz matrices to transform measured four-momenta between frames is the only algebraic path to calculating invariant masses and confirming the discovery of new particles (like the Higgs boson).

Chapter Summary¶

Minkowski space and Lorentz groups reconstruct our understanding of reality:

Absoluteness of Interval: By introducing an indefinite inner product, the spacetime interval replaces Euclidean distance as the only physical entity independent of the observer's state.
Unity of Symmetry: The Lorentz group unifies spatial rotations and velocity boosts into a single algebraic framework, proving that time is just a special direction with a negative metric in the spacetime manifold.
Boundaries of Causality: The null vectors and light-cone structure establish the ultimate limit for information propagation, proving that the topological causality of the universe is a direct consequence of the Minkowski metric signature.