E.P.J. de Haas · Kandinsky College, Nijmegen
From one matrix algebra to the rotation of galaxies
A physical four-vector written on a basis of 2×2 complex matrices: the biquaternion (BQ) algebra. Multiplying such vectors is ordinary matrix multiplication, and every product sorts itself into channels that carry Maxwell’s equations, the Lorentz force and relativistic fluid dynamics. Lifted to 4×4 matrices it becomes the Dirac level. Made local, its rapidity gives a gravitational rotor, and from that rotor the constant-Lagrangian rotation law of galaxies follows. Three sites, each with interactive models that compute everything live from the explicit matrices.
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Level 1 · 2×2 complex matrices
The BQ algebra
The BQ-Pauli algebra: four-vectors on the space-time basis T̂, Î, Ĵ, K̂, and their products in the norm, rotation and boost channels. Eleven interactive models in four themes, each adding one ingredient:
- the bare algebra: the 8 × 8 multiplication table
- vectors and their products: bilinear, triple product, field quadratics
- a global rapidity: Lorentz transformations and covariance
- a local rapidity: the gravity rotor and three derivations of the constant-Lagrangian law
Open the BQ algebra11models4themes8basis matrices -
Level 2 · 4×4 complex matrices, Weyl representation
The Dirac level
The same algebra doubled: β0 = [[0, T̂], [T̂, 0]] and βi = [[0, K̂i], [−K̂i, 0]] generate sixteen elements sorted by grade, from the scalar to the pseudoscalar χ. Every Dirac model shows the Pauli result inside as a block:
- the 16 × 16 table with the Pauli content of every product
- bilinear, triple product and field quadratics, with Maxwell, the Lorentz force and the stress–energy map
- Lorentz transformations: the Pauli U doubled into Λ = diag(U−1, U), up to the Thomas–Wigner rotation
Open the Dirac level5models3themes16basis elements -
Application · gravity as a flow of space
Galaxies and jets
The constant-Lagrangian metric-inflow galaxy model and the three-rapidity Painlevé–Gullstrand jet geometry. The rotation law follows from one gravitational rapidity rotor, along three routes that meet in the same condition.
- interactive galaxy, jet and growth models, and SPARC data explorers
- videos of the flow and of the fits
- constant-Lagrangian fits to all 175 SPARC rotation curves, in seven shapes
Open galaxies and jets175SPARC galaxies3routes to the law7fit reports
How the three fit together
One line of argument runs through all three sites, and each step has its own models.
- The algebraFour-vectors on 2×2 matrices; products sort into channels. BQ algebra, themes 1–2.
- Global rapidityA boost is a rotor: AL = U−1AU−1, and the products are covariant. BQ algebra, theme 3.
- The Dirac liftThe same content as blocks of 4×4 matrices, sorted by grade, with U doubled into Λ. Dirac level.
- Local rapidityA rapidity field gives a gravity rotor, isorapidity and the constant-Lagrangian galaxy. BQ algebra, theme 4, and galaxies.