Gravitational Lensing

Light Deflection as an Independent Test of Gravitational Realization

Gravitational lensing measures how gravitation redirects light. A galaxy rotation curve instead measures the motion of matter through a gravitational environment. These distinct observables test different aspects of the same gravitational description. Agreement with stellar velocities alone does not establish agreement with lensed images. ^[1]

This distinction is particularly important for Measurement Quantization (MQ). MQ begins with one axiom, “measure is discrete.” From that starting point, its construction geometrically anchors an expanding frame of information. The Internal Frame describes discrete count configurations; the System Frame describes their realized relational geometry and physical observables. The Frames mapping connects these descriptions. These are roles within MQ’s construction, rather than additional measured starting axioms. ^[2]

The fundamental-measure and constant reconstruction uses four measured inputs—the fine-structure constant, speed of light, electron mass, and Bohr radius—to select and normalize the physical realization. The speed of light has an exact numerical value in SI by definition; the four named quantities are not four statistically independent empirical constraints. An epoch selector, such as the cosmic microwave background temperature, has a separate role. A particular lens calculation also needs its source distribution and observational geometry. Thus the four-input statement does not mean that a galaxy’s light profile, distances, or gravitational metric follow without further information. ^[2]

The lensing question is whether the same gravitational realization used for galactic dynamics yields a metric and photon trajectories consistent with observation. MQ’s proposed construction does not add an independently adjustable lensing response. Establishing that common description requires the relevant covariant solution, source coupling, boundary conditions, and weak-field limit; it does not follow from discreteness or the kinematic reconstruction of a metric alone. ^[3]

From Gravitational Acceleration to Light Deflection

In the conventional weak-field exterior point-mass limit, let $\hat{\alpha }$ denote the physical bending angle at the lens, $b$ the impact parameter of the unperturbed ray, $M$ the gravitating mass, and $c$ the speed of light. $\mathrm{G}$ denotes the macroscopic gravitational constant, with the upper-count-limit typography used here. The leading weak-field result is

The expression appears in Weak Gravitational Lensing by Bartelmann and Schneider, Section 3.1.1, equation (3.1). It is the general-relativistic exterior point-mass benchmark, not an exact strong-field formula. It should not be applied to a ray inside an extended stellar distribution by inserting a projected aperture mass without deriving the appropriate deflection. A modified gravitational response must establish the metric that light traverses. ^[1]

MQ approaches this requirement through its discrete treatment of gravitation. Let $g_{\mathrm{bar}}(r)$ denote the Newtonian baryonic radial acceleration at radius $r$, and $g_{\mathrm{real}}(r)$ the realized radial acceleration. Let $g_{\mathrm{real}}$ denote the acceleration scale fixed upstream by MQ’s cosmological construction, without fitting it to the lens. The galactic realization law is

The scale and response function are carried over from the galactic construction rather than reconstructed from the Einstein ring. In Jody A. Geiger’s A First Principles Derivation of the Galactic Radial Acceleration Relation from Measurement Quantization, Section 4.2 and Appendix D develop that law, while Section 5.3 applies it to the lensing consistency calculation. The adopted Newtonian source and the realization assumptions remain part of this calculation. ^[3]

For a static, spherically symmetric weak field, assume the negligible-slip specialization in which the temporal and spatial metric potentials coincide in the standard convention. The null-geodesic reduction then gives the following physical deflection at impact parameter $b$. This is a conditional weak-field result, not a derivation of negligible slip for an arbitrary MQ source.

This integral accumulates the transverse gravitational response along the unperturbed light path. To obtain this radial form from the line-of-sight integral, use spherical symmetry and substitute the radial distance for the line-of-sight coordinate on each half of the path. The coordinate change supplies the square-root denominator. The equality is within the stated leading weak-field description. ^[3]

Why the Outer-Galaxy Limit Matters

Outside a finite spherical baryonic source, let $M_{\mathrm{bar}}$ be its total baryonic mass. Its Newtonian radial field is $g_{\mathrm{bar}}=\mathrm{G}{M}_{\mathrm{bar}}/r^2$. At low acceleration, the realization law has an inverse-radius outer limit, which yields asymptotically constant circular speed. Within the same isolated spherical, negligible-slip benchmark, the exact limiting relation for the leading weak-field deflection is

Let $v_{\infty }$ denote the asymptotic realized circular speed. The exact outer-limit fourth-power relation is ${v_{\infty }}^4=\mathrm{G}{M}_{\mathrm{bar}}{g}_{\mathrm{real}}$. The asymptotic physical deflection is therefore two pi times this speed squared divided by light speed squared. This limiting relation does not place a central Einstein aperture in the outer-galaxy regime. The infinite-radius operation belongs to the idealized spherical weak-field model; its slowly decaying field requires an environmental or cosmological completion for a global spacetime. ^[3]

The Relativistic Requirement

The spherical result is not the complete gravitational-lensing problem. A real galaxy is an extended three-dimensional source. Its projected stellar distribution may be elliptical, nearby structures may contribute external gravitational fields, and the observable arcs sample a two-dimensional lensing geometry. More fundamentally, photons respond to the metric rather than to a scalar acceleration considered in isolation.

In the stationary scalar weak-field description, temporal and spatial gravitational potentials enter the metric. Their difference is described as gravitational slip in a fixed convention. Slow matter primarily probes the temporal potential, while bending probes their combined effect. Rotation-curve agreement therefore leaves an additional metric relation to establish. ^[3]

The MQ requirement is correspondingly stringent. The System Frame metric must arise consistently with the same frames construction used for gravitational realization. In the proposed MQ variational principle, admissible metric and source configurations are constrained by a common action. The photon coupling and null-geodesic limit must be established for that solution. The required relation between weak-field potentials must be derived, rather than selected solely to recover a desired bending angle. ^[3]

Extended Lenses, Convergence, and Shear

Once a weak-field metric has been solved, the thin-lens projection connects it to observable maps. Let $\mathit{\theta }$ denote the two-component angular position, and let $D_L$, $D_S$, and $D_{LS}$ denote observer-to-lens, observer-to-source, and lens-to-source angular-diameter distances. Let $z$ be physical distance along the unperturbed line of sight, ${\Phi }_{\mathrm{sys}}$ the common potential in the negligible-slip specialization, and $\psi$ the dimensionless projected lensing potential. With the line-of-sight integral taken across the lens,

The reduced angular deflection field is $\mathit{\alpha }={\nabla }_{\mathit{\theta }}\psi$. It includes the lens-to-source over observer-to-source distance factor through the projected potential and differs from the physical bending angle defined earlier. Define convergence $\kappa$ as one half of the sum of the two diagonal angular second derivatives of $\psi$. It describes isotropic focusing. The shear components ${\gamma }_{\mathrm{1}}$ and ${\gamma }_{\mathrm{2}}$ are respectively one half of the difference of those diagonal derivatives and the mixed angular second derivative. Shear describes anisotropic distortion; magnification depends on both convergence and shear. These are projection relations within the specified metric limit, not new MQ response laws. Section 5.3, equations (175)–(179), of A First Principles Derivation of the Galactic Radial Acceleration Relation from Measurement Quantization gives the source expressions. ^[3]

ESO 325-G004 as a Relativistic Benchmark

ESO 325-G004 is a nearby elliptical lens with an Einstein aperture in a stellar-dominated central region. In A giant elliptical galaxy with a lightweight initial mass function, Smith and Lucey report a CMB-frame lens redshift of 0.0347 and an Einstein radius of 2.85 arcsec, corresponding to 1.96 kpc in their adopted cosmology. The aperture is about one quarter of the effective radius. ^[4]

The same paper’s Table 3 gives an F814W aperture luminosity of (4.07 ± 0.08) × 10¹⁰ solar luminosities and a conventional projected lensing mass of (1.50 ± 0.06) × 10¹¹ solar masses. Its Table 2 default stellar mass-to-light ratio is 3.01 ± 0.27 in solar units in the same band, including index systematics. The abstract quotes ±0.25. The normalization assumes a Milky Way-like Kroupa stellar initial mass function, the distribution of stellar birth masses, and remains dependent on stellar-population modeling. ^[4]

The stellar source can be normalized from photometry and population modeling without using the lensing-inferred total mass. This is independence from the lensing target, not freedom from astrophysical assumptions or shared observational uncertainties. A lensing-derived mass, inferred dark-matter fraction, or normalization adjusted to fit the arcs must not be fed back into a prediction presented as fixed before comparison. ^[3]

The benchmark leaves the covariant extended-lens test open. It omits the resolved two-dimensional stellar-light distribution and does not independently derive the extended solution’s gravitational slip. The central comparison therefore sets a normalization problem for the completed calculation rather than establishing that the full theory passes or fails the lensing test. ^[3]

The remaining difference cannot properly be filled with a lens-specific acceleration scale or an arbitrarily selected dark halo while still calling the result the parameter-frozen MQ prediction. It must emerge, if the theory is correct, from the covariant extended-source solution, its metric potentials, the actual three-dimensional source geometry, and independently constrained environmental contributions.

Lensing as a Cross-Observable Test of MQ

Gravitational lensing occupies a special position in Measurement Quantization. The gravitational constant, galactic gravitational realization, frame structure, and relativistic metric meet in a single observable trajectory.

For spherical systems in the weak-field negligible-slip limit, MQ provides a definite bridge. The galactic realization law determines the radial acceleration, the radial acceleration determines the integrated photon deflection, and the finite-mass outer limit links that deflection directly to the asymptotic orbital velocity. No new lensing scale is introduced.

For general extended lenses, the full covariant System Frame equations must determine both weak-field metric potentials from an independently specified baryonic source. Those potentials must then be projected through the lensing geometry to predict the deflection, convergence, shear, critical curves, and Einstein-ring morphology before comparison with observation.

Gravitational lensing is therefore more than another application of the MQ gravitational relation. It tests whether the discrete measurement structure proposed by MQ survives the transition from the motion of matter to the geometry traversed by light.