Light Deflection · From Optical Path to Rays

01Why light turns toward a mass source

Light from a distant star changes direction slightly when it passes near the Sun. The first question concerns propagation through a nonuniform spatial background: if a central source changes the spatial length state at different positions, how can that change be expressed as a light path, and how can the path determine the direction seen at the receiver?

ASD begins with the spatial scale relations associated with gravitational redshift. A matter source establishes static longitudinal stretching. The wavelength associated with a standard process grows with its scale, while frequency and period change accordingly. The local propagation speed of light remains $c$, but both the true spatial length traversed by the light and the conversion of the local propagation process to a common reference vary with position. Their combined effect defines the reference optical path.

The optical-path gain is greater on the side nearer the central source. Propagation phase therefore accumulates differently across the wavefront, continuously turning the propagation direction toward the source. The construction below first establishes this optical path and then obtains rays from phase stationarity. Its results depend on the stated spatial length law and standard-process response; constancy of $c$ alone does not determine them.

02True length and reference optical path

Consider an isolated, static, spherically symmetric, uncharged source of mass $M$, with background gravitational coupling coefficient $G_0$. Define $\ell_g=G_0M/c^2$. The radial coordinate $r$ is defined by the sphere area $A=4\pi r^2$ and is called the area radius; it is not the true length accumulated radially from the center. The calculation is restricted to the region outside the matter source with $r>2\ell_g$, neglecting the backreaction of light on the source.

Here $\Omega$ is the linear expansion coefficient, $\Phi$ is the dimensionless line-density coefficient, and $X=\Phi^2$. The symbol $\Phi$ denotes neither the medium's three-dimensional volume density nor the wave field used in the quantum module. The quantity $d\ell_s$ is the true spatial line element in the orbital plane, and $\varphi$ is the azimuthal angle. The central-source gradient selects the radial direction, whose true length is $\Omega\,dr$. The tangential arc length is already fixed by the area radius as $r\,d\varphi$ and does not receive the radial factor a second time.

Path length must also be connected to the standard used to record the process. Let $t$ be the common-reference counting parameter established by a far-field standard, and let $f_0$ be its frequency. The manuscript adopts the local process response $dN=f_0\Phi\,dt$, where $N$ counts cycles. The speed calibrated by local length and cycles is $v_*=f_0d\ell_s/dN=\Phi^{-1}d\ell_s/dt$. Setting $v_*=c$ for light gives

The quantity $d\ell_{\rm opt}$ is the reference optical path, and $d\ell_{\rm opt}/c$ is the propagation-count interval in this reference. The radial optical-path factor $\Omega^2$ has two sources: stretching of the true path and the cycle conversion associated with the wavelength stretching of the standard process. The true tangential arc length is already defined through sphere area, so its optical path receives only the additional cycle-conversion factor $\Omega$. Together these components describe propagation in any direction.

03Determining a stationary light path from propagation counts

For given emission and reception points, one can consider multiple connecting paths $\Gamma$. Each has an accumulated optical path and therefore a common-reference propagation quantity $T[\Gamma]$. Calling $T$ a functional means that it depends on the whole path, not just one position.

Geometric optics represents wave propagation by rays and requires the wavelength to be sufficiently short relative to the scale of background variation. In a static background, phase accumulation at fixed frequency is proportional to optical path. Varying the path while holding its endpoints fixed, the actual ray satisfies

The condition $\delta T=0$ means that an infinitesimal change of path produces no first-order change in the propagation quantity. Stationarity does not guarantee a unique path in every configuration or make every path a global minimum; a strong-lensing background can admit several stationary paths. The weak-deflection calculation selects the branch continuously connected to the straight path with no source.

04Isotropic coordinates for the same optical path

The area radius makes the physical roles of radial and tangential lengths explicit. To calculate the transverse gradient directly, the manuscript introduces another radial coordinate, $\rho$, in which optical path is a scalar weight multiplying a reference Euclidean line element. This is another coordinate representation of the same background:

The variable $q=\ell_g/(2\rho)$ is dimensionless, and the exterior coordinate branch uses $0<q<1$. Substitution into the radial and tangential terms of the original optical path gives, respectively,

Both directions now have the same coefficient. With $ds_0=\sqrt{d\rho^2+\rho^2d\varphi^2}$ denoting the reference Euclidean line element in these coordinates,

The quantity $n_{\rm iso}$ is a reference optical-path weight; it does not mean that the local speed of light has become a different constant. In the weak field, $(1+q)^3/(1-q)=1+4q+O(q^2)$, giving $2\ell_g/\rho$. This coefficient follows from converting the complete optical path. The coordinate change neither adds another physical stretching nor removes the distinct physical roles of radial and tangential lengths. Endpoints, impact parameter, and optical path must consistently use the same coordinate representation.

05From an optical-path gradient to a change of direction

In the isotropic coordinate plane, let $\mathbf x$ be the position vector and $\mathbf e=d\mathbf x/ds_0$ the unit tangent to the ray. Fermat variation holds for arbitrary transverse path variations, giving the Euler–Lagrange equation

The left side describes how propagation direction and optical-path weight change along the path; the right side is the spatial gradient of that weight. Place the central source at the origin and take the unperturbed ray to travel in the $x$ direction at $y=b>0$. The parameter $b$ is the far-field-normalized impact parameter: the perpendicular distance from the incoming straight line to the center. On the zeroth-order path, $\rho=\sqrt{x^2+b^2}$.

Let $\theta$ be the small angle of the ray relative to its unperturbed direction. At first order, $ds_0\simeq dx$. The factor $n_{\rm iso}$ multiplying the direction change on the left can take its zeroth-order value 1, while the first-order gradient of $2\ell_g/\rho$ is retained on the right. Thus

This step retains optical-path differences between positions; it does not replace all of $n_{\rm iso}$ by a constant. The minus sign means that an incoming ray on the $y=b>0$ side turns toward negative $y$, hence toward the central mass.

06The deflection integral for distant endpoints

Total deflection is the direction change accumulated along the ray. For emission and reception both far away, let $\alpha_{\rm def}>0$ denote the magnitude of deflection toward the mass source:

The required antiderivative is

Its values at positive and negative infinity are $1/b$ and $-1/b$, respectively. Their difference is $2/b$, so

The infinite integration limits represent the distant-endpoint limit. If the source or observer is not sufficiently far from the closest-approach region, the actual endpoints must be retained and the local reception direction converted to the angle reported by the apparatus. The total angle above cannot simply be applied. Deflection vanishes as $M\to0$ or $b\to\infty$, and $G_0M/(c^2b)$ is dimensionless, as required for an angle.

The manuscript uses the solar gravitational parameter and nominal radius $6.957\times10^8\,\mathrm m$ to estimate a grazing $b$, obtaining about $1.75$ arcseconds. At the present first-order accuracy, the center-to-surface distance can be used for this estimate. This number is a central-field calculation, not new observational data. Parameter sources are the JPL gravitational parameters and IAU nominal constants.

07Connecting the result to observation and comparison theories

The derivation begins with ASD spatial scales, true length, and standard-cycle response, obtains a gradient from the complete optical path, and accumulates the direction change. Comparison with a particular astronomical observation also requires the source mass, impact parameter, emission and reception positions, angular reference, and nongravitational propagation effects. Only after these quantities are matched does the deflection in the formula become the angle reported by an apparatus or observation.

For the same $M$, $b$, distant endpoints, and first-order accuracy, this result agrees with general relativity. Chapter 2 also notes that the adopted complete static optical path has the same mathematical form as the optical geometry of the Schwarzschild exterior. Higher-order expansion of this optical construction does not automatically generate a distinctive difference. See the optical-geometry reference. This page therefore shows how the ASD constitutive framework reproduces the propagation relation; mathematical agreement cannot be presented as independent experimental proof of its physical interpretation.

Gravity test entry GRV-EXP-002 records formula reproduction, observational references, and data refitting separately. The calculation on this page currently yields no discriminating light-deflection observable.

08Further propagation problems and derivation sources

This page uses geometric optics, a static spherically symmetric vacuum exterior, and the weak-deflection approximation. The spatial scale law, radial length law, standard-process response, and Fermat condition are stated where each enters the argument. Microscopic source coupling and the complete dynamical constitutive description require further determination by space dynamics. Finite endpoints, material media, rotation, and strong lensing require their own backgrounds and boundary conditions.

Accumulation along this same optical path gives Shapiro delay. Retaining the full weight instead of its weak-field expansion leads to the critical light paths of static compact objects. Matter trajectories connect the same spatial length to inertia and the action. All three share the background definitions in the gravity overview, addressing direction, propagation counts, and matter motion, respectively.