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CL Vacuum Flow: Bernoulli, Rayleigh and the Boundary of a Rotating Bulge

A rotating bulge of radius R drags space with it. At its border the flow of space must be continuous; from there outward the vacuum continues on its own terms. Three continuations of the same boundary value are compared: the constant-Lagrangian (CL) flow, GR’s vacuum frame dragging (Lense–Thirring), and a Kepler-like rotation, all carried by the same radial river vr = √(2GM/r). Only one of them keeps the boundary energy along its streamlines and stays laminar.

The postulate

The argument rests on one postulate: the vacuum behaves as an ideal, barotropic, non-dissipative flow. The river and analogue-gravity picture is taken as physics rather than as an analogy.

For dissipation, matter is needed, as the source of turbulence. Inside R the bulge is matter: its flow may be rigid, may exchange and lose energy. Outside R the flow is pure vacuum: the energy it receives at the border has no way to be dissipated, so it must be carried along the streamlines.

The boundary

Units GM = R = 1, so velocities are in √(GM/R) and the radial river is vr = √(2/x), x = r/R. The CL papers take vφ(R) = √(GM/R) (Kepler at the border), so the Bernoulli value, the constant Lagrangian of the CL papers, is L = vr² + vφ² = 3GM/R, and the plateau speed is vL = √L. Inside R: rigid rotation of matter.

Three outward laws from the same boundary value

The flow of space, seen from above

Tracers follow the space flow (inflow vr, rotation vφ); their colour is the Bernoulli value L along the streamline relative to the border: green = the border value kept, red = changed by 50% or more. Inside R the matter of the bulge rotates rigidly and is turbulent.

Motion model

27 s: the postulate, the three continuations from the same border value, and the conclusion. File: CL_Vacuum_Flow.mp4 · on YouTube ↗ · all videos.

The tests, radius by radius

Rotation of space vφ(r)
Bernoulli L = vr² + vφ² along the streamline
Rayleigh specific angular momentum ℓ = r vφ
GR source needed (r⁴Ω′)′/(2r³), zero in GR vacuum

Verdict

Each criterion belongs to a different law. Bernoulli and Rayleigh are the laws of an ideal flow, and only CL passes both. The GR vacuum condition is GR’s momentum constraint for an azimuthal shift, and only Lense–Thirring passes it. Which column decides is the postulate.

The argument

  • No dissipation. Outside R there is no matter, so no viscosity and no turbulence source. A steady, ideal flow conserves the Bernoulli value along each streamline, and the border fixes that value: L = vr² + vφ²(R) = 2GM/R + vφ²(R).
  • The river makes it unique. For purely circular flow (vr = 0) Bernoulli would hold for any vφ(r) and fix nothing. The radial river turns the streamlines into spirals that cross radii, so the border value is carried outward, and vφ = √(L − 2GM/r) is the only profile: the CL flow (exactly, cosh χr cosh χφ = Γ0).
  • Laminar. Rayleigh’s criterion: a rotating ideal flow is stable when r vφ increases outward. CL does, toward vφ → √L; Lense–Thirring, with r vφ ∝ 1/r, does not, and read as a flow it would become turbulent.
  • Matter is different. Inside R the matter can dissipate and rotate rigidly, and its Bernoulli value varies with radius. The premise applies to the vacuum outside only.

Where it departs from GR

  • GR’s outward law. In river gauge the slowly rotating vacuum obeys (r⁴Ω′)′ = 0, so Ω = Ω∞ + 2GJ/(c²r³): frame dragging falls off as r−3. The gravitomagnetic field outside the sources is a dipole.
  • CL in GR. The CL branch makes (r⁴Ω′)′ nonzero: at first order in the rotation GR would need a mass current outside R. CL is therefore not a hidden GR vacuum; it replaces GR’s momentum constraint by Bernoulli. (At second order every differential rotation, Lense–Thirring included, also changes the energy density, −r²Ω′²/4 in this flat-slice form, which GR balances by second-order metric corrections; that term does not distinguish the branches.)
  • Why GR does not see it. In GR the shift is not a fluid and its energy is not localized; Lense–Thirring is stable as geometry. The difference is the postulate, not an error on either side.
  • Caveats. Slow flow; the radial river is taken unchanged by the rotation it carries (no back-reaction); the GR continuation matches Ω at R but not Ω′, which needs a surface current there.

Related: the GR literature that seeks flat rotation curves from frame dragging within GR’s own equations, Cooperstock & Tieu (2005) and Balasin & Grumiller (2006), van Stockum-type rotating dust. The CL condition: de Haas, the CL papers and CL in the metric. The motion model: CL_Vacuum_Flow.mp4 (script bq-video-scripts/scripts/cl_vacuum_flow_video.py); exact checks in CL-vacuum-flow/code/cl_vacuum_flow.py. Part of Galaxies and jets as flows of space.