Galaxies and jets as flows of space

Interactive models

Four models run the equations live in your browser. Drag to rotate, scroll or pinch to zoom, and change the parameters yourself.

• Spiral galaxy with polar jets
• The Hubble sequence from cone occupation
• The M 87 jet in a single PG geometry
• The riverbed of space

Two models grow one galaxy through cosmic time:

• CL galaxy through cosmic time: the derived baseline
• CL galaxy growth with core resets: an exploratory extension

Three data explorers test the CL fits against the SPARC galaxies:

• CL fits and the baryonic Tully–Fisher relation
• CL fits against photometry
• CL fits and the Lelli central-gradient relation, with rotation curves predicted from the light

Open the models

Videos

Animations of how space flows through a galaxy, step-by-step fits to SPARC rotation curves, and polar jets in 3D. The Python scripts that render them are open.

• The galaxy model: inflow, nested spirals, the Hubble sequence
• Fitting real rotation curves: one Lagrangian, two, and an offset
• Jets: Painlevé–Gullstrand cones and the jet of M 87

Watch the videos

SPARC rotation-curve fits

All 175 SPARC galaxies fitted with the constant-Lagrangian law, now derived from the rapidity rotor, and its extensions, sorted into seven rotation-curve shapes. Which shape a galaxy has follows the size of its bulge. The SPARC explorers test the single-Lagrangian fits against the BTFR, photometry and the Lelli relation.

See the fits

Why the constant-Lagrangian law

The rotation law fitted to SPARC follows from a single gravitational rapidity field. A radial (Newtonian plus Hubble) and an azimuthal rapidity feed one local Lorentz rotor G. Three constructions from that rotor arrive at the same condition, isorapidity: the composed Lorentz factor Γ = cosh χ_r cosh χ_φ is conserved along the flow.

• Kinematic: the time channel of the convective four-acceleration is c DΓ/Dτ. Isorapidity is zero power along the flow, and the river is turned but never sped up.
• Maurer–Cartan: the time channel of the rotor’s own connection, transported with the flow, is (c/2) DΓ/Dτ, exactly. The identity A_g = Ω_τU_g + U_gΩ_τ^† makes the two one statement.
• Metric: in the Painlevé–Gullstrand metric of the same flow, the condition reads, to slow-flow order, as flow along levels of the redshift factor √(−g_tt).

With the Newtonian virial condition at the bulge radius R as boundary data, the conserved value gives the rotation curve v_φ² = (3/2)v_r(R)² − v_r(r)², with v_r = √(2GM/r) − Hr: no halo and no interpolation function.

Lifted to the Dirac level. All three routes are repeated with 4×4 matrices in the Weyl representation of the Clifford algebra Cl(1,3), and the condition is unchanged: the β₀ channel of the transport product U_gΩ_τ is −(c/2) DΓ/Dτ, exact in the rapidities. The lift adds one structural result: the vector part of U_gΩ_τ is −½A_g, the kinematic acceleration itself, so the kinematic and Maurer–Cartan routes are one object rather than two calculations that agree. At this level the virial closure gives the same rotation law.

The same routes at the Dirac level:

The BQ algebra behind the models

The galaxy and jet models are built on biquaternion (BQ) algebra: four-vectors as 2×2 complex matrices, where one product of two four-vectors gives scalar, space and spin channels at once. A separate site builds it up in eleven interactive pages and four themes, then lifts every step to the Dirac level in thirteen more, with Python scripts that check every result exactly.

• Algebra only: the BQ-Pauli multiplication table
• Vectors and their products: the bilinear A^TB, the triple product D(A^TB), field quadratics
• A global rapidity: Lorentz transformations of vectors, bilinears and the triple-product equations
• A local rapidity: the gravity rotor and its metric, and the three routes to the constant-Lagrangian law
• The Dirac level: the same matrices placed in 2×2 blocks give the 16-element Clifford algebra Cl(1,3); the Weyl and Dirac representations, the boosts, and the gravity rotor Q_g with the CL law derived again