Imaginary and split-complex time: SMM/DTT and Presentism and the growing-block universe

01The question behind the comparison

The expression imaginary time can refer to different mathematical constructions. Ordinary complex numbers use i² = −1. Split-complex numbers use j² = +1 and have a quadratic form with indefinite sign. A substitution t = −iτ in an analytic expression is another operation again: it can turn oscillatory evolution into exponential damping under suitable assumptions. These distinctions matter for SMM/DTT because its terminology connects inner and outer time with complex geometry. The interpretation becomes clearer when the algebra, bilinear form and physical observables are stated separately.

An ordinary complex coordinate z = a + ib has squared modulus a² + b². A split-complex coordinate z = a + jb has product with its conjugate a² − b². The second expression can vanish for a nonzero element and can be negative; its square root is therefore not a norm on the whole algebra in the usual positive-definite sense. Hyperbolic rotations preserve this indefinite form. This is a precise mathematical affinity with Lorentzian geometry, but it does not establish that a pair of temporal roles constitutes two physical timelike dimensions.

The nested interpretation helps here. Inner generation and outer completion can be represented using different algebraic components without becoming independently traversable axes. A model still needs a rule connecting its components with clocks and records. If ordinary quantum phase also appears, a compatible complex structure with square −1 must be distinguished from the split unit with square +1. Treating the two symbols as interchangeable would change the multiplication law and invalidate the inference. DTT can gain precision by using each structure only for the function it actually performs.

02The single-source and nested-time proposal

SMM begins with one nonspatial generative source. DTT distinguishes its inner production from the outer order in which completed results become accessible. The six inner levels are nested functional dependencies within that production; the seventh denotes outer completion. They are not six independent clocks, six spatial directions, or six simultaneously occupied temporal axes. This distinction makes the framework more specific than a general claim that everything changes: it asks how a world of persistent relations can be generated before its spatial representation is taken for granted. ^[2]

03The comparator’s explanatory structure

Presentism restricts concrete temporal existence to the present; the growing-block view includes the past and present while excluding the future. The distinction is decisive for a record-based ontology. A present record of yesterday and yesterday’s continuing existence are different claims. Broad’s treatment of becoming supplies the historical growing-block comparator; the presentist alternative sharpens what an archive does and does not establish.

Suppose a stone bears an inscription describing a vanished ceremony. A presentist can treat the inscription as a current object with a history. A growing-block theorist can additionally include the ceremony itself in reality. Both can agree about what the inscription says. The disagreement therefore cannot be settled by pointing to a stored record alone. SMM/DTT gains precision by stating whether renewal preserves past events, preserves only present traces, or uses another form of persistence. Relativity adds the separate question of how a privileged present, if proposed, relates to spatially separated events. ^[1]

04The precise agreement and difference

Neither presentism nor a growing block selects an ordinary complex or split-complex algebra. A presentist can use real-valued clocks, and a block-growth model can be formulated with a discrete order alone. DTT’s algebraic time therefore contributes a separate hypothesis about representation or dynamics. The fact that a diagram has an inner and an outer direction does not determine what exists in its other regions.

A useful distinction is between the domain of a model and its ontology of actuality. A complexified equation may be evaluated at values that are not physical events, just as a generating function can contain coefficients for unrealized possibilities. A growing-block interpretation could use these values to represent future potential, but it would need an explicit rule identifying completed states. The algebra does not supply that rule by itself. DTT becomes stronger when completion is defined independently and the chosen algebra is shown to preserve or constrain it. This avoids identifying imaginary coordinates with a metaphysical future simply because both are not directly observed. ^[1]

05Worked example: what follows and what does not

For z = a + jb, multiply by exp(jη) = cosh η + j sinh η. The new components are a′ = a cosh η + b sinh η and b′ = a sinh η + b cosh η. Direct expansion gives a′² − b′² = a² − b². In contrast, multiplying a + ib by exp(iθ) preserves a² + b². Thus a quarter-turn in the ordinary complex plane and a finite hyperbolic boost have different invariants. No real rapidity η produces the circular operation i. The calculation locates the exact algebraic similarity to relativity and prevents a picturesque rotation from being mistaken for a derivation of physical evolution.

06What this comparison contributes

The positive research question is which operational distinction selects a particular algebra. If completion singles out a compatible ordinary complex structure while a separate indefinite bilinear form organizes clock comparisons, the two roles can coexist coherently. If the distinction is merely a notation chosen for convenience, it cannot determine the observed signature or the probability rule. Clarifying this issue makes comparisons with philosophical theories of time more illuminating, because those theories concern existence and ordering rather than an unexplained multiplication table.

THE SAME QUESTION, DIFFERENT STARTING POINTS

The comparison at a glance

Continue on SMONAD

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