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The Janus point: the maths

A swarm of 60 stars under Newtonian gravity (softened), started at the Janus point — its moment of least size — and integrated both forwards and backwards in time. Total energy is set exactly to zero, as in Barbour, Koslowski & Mercati's model. Computed live.

Equations

E = Σ ½ m v² − Σ G m_i m_j / √(r_ij² + ε²) = 0

Kinetic plus (softened) gravitational energy: zero for the whole swarm, so it can expand for ever in both directions.

C = (size) × (tightness)

A measure of clumpiness (the 'complexity'): lowest at the Janus point, growing both ways.

Constants

N	60	stars
ε	0.04	softening length

What's simplified

• Two dimensions; equal masses; softened gravity so close passes don't blow up the integration.
• The energy is zero by construction, so there's no worked example that would test more than the definition; the code's own check (energy stays within about 10⁻⁶ of zero through the run) is noted in the source.

Where it breaks

As a model of our universe the Janus-point idea is contested: it's a proposal for where the arrow of time comes from, not established physics.

Sources: J. Barbour, T. Koslowski & F. Mercati, PRL 113, 181101 (2014); J. Barbour, The Janus Point (2020).