Recovering the Tully-Fisher Relation (TFR) from First Principles

(image AI generated)

The TFR is an empirical relationship that links galaxy rotational velocity with its luminosity. The underlying link here is that higher-mass objects will generally have higher rotation velocities due to the higher gravitational potentials associated with these more massive systems; more mass means more stars, and therefore higher luminosity. An underlying, zero-free-parameter, first-principles derivation for why this must be the case has never been offered in standard literature, despite the TFR being verified in numerous studies across the decades since it was first proposed by astronomers R. Brent Tully and J. Richard Fisher in 1977.

Within the ETF, however, we can successfully recover this relationship from first principles and zero free parameters through the acceleration floor, in turn derived from first principles yielding . We demonstrate that the TFR is not an empirical fit of the data but a fundamental requirement if the Hamiltonian with respect to the galaxy rotation velocity is to be preserved across the galaxy disc.

Rather than relying on a kinematic transition function—which would introduce circularity—this derivation originates directly from the system’s energy boundary conditions. In “A Non-Dark Matter, Zero Free Parameter Solution to the Galaxy Radial Acceleration Relation”, the Lagrangian was used to identify the energy boundary condition for a stable system. At the metric coupling radius (), the local Newtonian baryonic acceleration () and the background metric acceleration () reach a state of parity I.e., they are in balance.

For high-mass systems such as NGC 2841, the plateau velocity () represents the asymptotic state where the system has fully transitioned from local metric isolation in the core into the background inertial locking zone. At this boundary, the principle of stationary action dictates that the observed centripetal acceleration () is geometrically constrained by the coupling of these two fields.

Because the system is locked at the threshold where local baryonic gravity meets the universal background stiffness, the kinematic equilibrium is established by the geometric mean of the two interacting acceleration fields

Substituting the explicit terms for and , the fundamental balance of the metric coupling isolates as

Squaring both sides to evaluate the operational action across the spatial manifold removes the radial dependence entirely, like this

yielding the classic TFR

This derivation confirms that the Baryonic Tully-Fisher Relation (BTFR) is a direct consequence of the energy boundary condition identified in the ETF Lagrangian, rather than an empirical curve fit. It demonstrates that for a given mass , there is only one specific velocity at which the inertial lock can maintain a stable, synchronised metric coupling with the universal temporal background.

The recovery of the TFR within this framework shows that the relation is a direct physical consequence of the metric stiffness at the boundary. The analysis of gas-rich galaxies by Stacey McGaugh provides empirical support for this derivation by confirming that the relationship remains remarkably tight and linear even in systems where the baryonic mass is dominated by gas rather than stellar populations.

This invariance across several orders of magnitude, from HSB giants to gas-dominated dwarfs, further validates the use of – also derived from first principles – as a fundamental universal acceleration constant. Unlike CDM, which requires the complex fine-tuning of dark matter halo parameters to mirror the baryonic distribution (the halo-baryon conspiracy), the ETF shows that the BTFR is the inevitable result of a 2D orbital manifold reaching energetic equilibrium with the background metric floor.