The original (c,0) label specifies a proposed temporal structure. A pressureless dark component can be constructed from persistent collective modes, with its stress, abundance and clustering calculated explicitly.
The gravitational component that light does not reveal
Galaxy motions, gravitational lensing and the growth of cosmic structure indicate more gravitating material than the luminous matter alone supplies in the standard gravitational description. The microwave background also distinguishes a component that clusters differently from the baryonic plasma. The physical question is what carries this additional density and why its electromagnetic response is weak.
A dark-matter model must explain its distribution as well as its total amount. Clowe and collaborators’ study of the Bullet Cluster is a useful example: the main lensing signal is spatially separated from the dominant hot-gas component in a merging system. This is a demanding test for any proposed substrate because a perfectly uniform background would not produce those localized lensing peaks.
The original proposal: the (c,0) structure
The original dark-matter article introduces four schematic temporal-component structures: (0,0), (c,0), (0,c) and (c,c). It describes ordinary matter as (c,v), with continuing inner formation and an outward component. The limiting (c,0) state is identified with the ground state of matter: the original description sets both overall and microscopic outward motion to zero and associates the result with absolute-zero vacuum or aether. This vacuum is distinguished from the complete absence represented by (0,0).
The same article associates dark energy with the complementary (0,c) structure and describes (c,c) as a fully symmetric limiting state. These assignments belong to the author’s temporal ontology in Duality of Time; the component labels are not independently measured velocities. Their distinctive proposal is that apparently empty space retains organized inner activity capable of contributing gravitationally. A physical realization must specify how this retained activity acquires a matter-like stress and a nonuniform density. The collective-mode construction below develops that possibility as an illustrative extension of the original ground-state idea.
From a persistent substrate to a clustered component
Separate the underlying source state from the persistent patterns it supports. A spatially uniform vacuum contribution has a different gravitational behavior from a population of localized or coherently occupied modes. The source can remain in a stable regime while its retained collective degrees of freedom carry spatially varying energy. The observable dark component would then be those occupations and their response to gravity.
In this candidate interpretation, the original zero outward component describes the absence of ordinary thermal motion in the ideal source background. Collective structures can still change their spatial distribution and develop bulk motion in the completed world. Their dynamics must be supplied explicitly; the temporal label alone determines neither pressure nor a halo profile.
Let χ be a slowly varying outward description of one retained source mode. The six inner stages determine the dependencies that prepare and maintain this mode, while outward completion determines when its energy and position become available. They remain parts of one generative construction rather than independent physical time axes.
A worked example with nearly pressureless energy
Use units with ℏ = c = 1 and consider an effective canonical scalar mode with potential V(χ) = m2χ2/2. In an expanding homogeneous region its equation and energy density are
χ̈ + 3Hχ̇ + m2χ = 0, ρχ = χ̇2/2 + m2χ2/2.
When m is much larger than the expansion rate H, many coherent oscillations occur during one expansion time. Averaging over them gives nearly equal kinetic and potential energies. The pressure pχ = χ̇2/2 − m2χ2/2 therefore averages to approximately zero. Conservation then yields
⟨pχ⟩ ≈ 0, ρ̇χ + 3Hρχ ≈ 0, ρχ ∝ a−3.
This is the dilution law for nonrelativistic matter. It demonstrates how coherent internal activity can yield a cold gravitational component without a thermal gas of visible particles. The identification of χ with a source mode and the selection of m are additional model ingredients. Hu, Barkana and Gruzinov provide a primary scalar-field example in which wave dynamics also affects halo structure.
Why the spatial pattern matters
For a spherical halo in the weak-field regime, a circular orbit obeys vc2(r) = GM(<r)/r. An approximately flat rotation curve therefore requires M(<r) to grow approximately with r over that interval, giving a density proportional to r−2. A source realization must generate an appropriate density profile through its dynamics and initial conditions.
For a wave-like collective mode, gradients resist compression at sufficiently short wavelengths. The characteristic scale depends on its mass and background density. This connects the microscopic source choice with halo cores, small-scale structure and the abundance of low-mass objects. A mode that is too smooth cannot reproduce the observed clustering; a sufficiently heavy or suitably interacting mode can approach the cold-matter limit over a wider range of scales.
Electromagnetic invisibility also needs a mechanism. A source selection rule could suppress the relevant coupling to charged matter. That rule should be consistent with the gravitational response and any permitted interactions with ordinary particles.
A coherent route from the source to observations
The initial calculation should derive the collective-mode action, its mass, its gravitational coupling and its production or occupation history. These determine the relic abundance and pressure. A simulation can then evolve the same component through structure formation and compare its lensing distribution with its predicted dynamics.
Use complementary tests: the early-universe matter density, the large-scale power spectrum, halo structure, and systems where gas and collisionless components separate. Planck’s cosmological-parameter analysis supplies a primary reference for the early-universe comparison, while the Bullet Cluster tests localized gravitational behavior.
The source-ground-state proposal becomes substantial when one specified mode accounts for both persistence and clustering. It then connects the original idea of active vacuum with a definite stress tensor, a conserved abundance and a measurable distribution of gravitational mass.
Books, papers and further reading
- Mohamed Haj Yousef — Duality of Time
Chapter III: original (c,0) dark-matter proposal; Chapter V, Complex-Time Hyperbolic Geometry; Chapter VII, Alchemy and Eternal Symmetry. Native edition and contents record.
- Wayne Hu, Rennan Barkana and Andrei Gruzinov — Cold and Fuzzy Dark Matter
Primary scalar-field dark-matter model; wave support, halo cores and suppression of small-scale power. Used as an illustrative physical realization, not an identification with DTT.
- Douglas Clowe and collaborators — A direct empirical proof of the existence of dark matter
Primary weak-lensing study of the Bullet Cluster and the separation of lensing mass from hot gas.
- Planck Collaboration — Planck 2018 results. VI. Cosmological parameters
Primary CMB inference of cosmological matter content within specified models.