Compact Objects · Static Optics and Strong-field Boundaries

01From small deflections to critical propagation

Far from a central source, spatial stretching produces small deflections of light and an accumulated excess optical path. When the source is sufficiently concentrated and a ray enters a more strongly stretched region, the same propagation law also produces critical orbiting and capture conditions. This page first examines phenomena calculable from the exterior optical path, then considers how ASD interprets the limit of the exterior spatial continuum and its boundary.

The model uses a static, spherical, uncharged source, keeps the gravitational coupling $G_0$ fixed, and neglects the probe light's effect on the background. Let $M$ be the mass calibrated by the far-field longitudinal flux, $c$ the local vacuum light speed, and $\ell_g=G_0M/c^2$ the gravitational length. Areal radius $r$ is defined by the sphere's area $4\pi r^2$, and $\varphi$ is azimuth. Adopt the exterior constitutive laws $X=1-2\ell_g/r$, $\Omega=X^{-1/2}$ and $\Phi=\sqrt X$. Here $\Omega$ denotes linear expansion and $\Phi$ its inverse, not a quantum field variable.

Extending these relations into the strong-field exterior $r>2\ell_g$ gives the static optical model below. It uses the complete exterior function, without treating $\ell_g/r$ as small everywhere. This strong-field continuation remains a constitutive input. The derivation determines the motion of light in a specified background; formation of the boundary is a separate dynamical question.

02Describing a light path in a strong field

The actual spatial line element remains $d\ell_s^2=X^{-1}dr^2+r^2d\varphi^2$. Because $r$ already specifies the sphere's actual area, radial stretching and tangential arc length enter separately; no additional factor of $\Omega$ is applied to the tangential term. After conversion through the standard process to a common reference, the optical path is $d\ell_{\rm opt}=\Phi^{-1}d\ell_s$, and the reference propagation count satisfies $dt=d\ell_{\rm opt}/c$.

To distinguish the roles of radial and circumferential propagation, write this same optical path as

The radial optical factor $\mathcal A$ is dimensionless. The quantity $\mathcal B$ has dimensions of length and represents circumferential optical path per unit azimuth. Nearer the source, the sphere's geometric scale $r$ decreases while the scale-conversion factor $X^{-1/2}$ increases. The expression $\mathcal B=rX^{-1/2}$ combines these changes in a single quantity. Their competition determines the critical ray.

A light ray makes the optical-path functional stationary. Let $s_{\rm opt}$ be optical path accumulated along the ray. Its unit-tangent condition is

Axial symmetry conserves the quantity conjugate to azimuthal change, giving the optical impact parameter

Here $\psi_{\rm opt}$ is the angle between propagation and the radial direction, measured using the optical line element. In the far field, $\mathcal B\to r$, so $b$ becomes the transverse offset of a distant incoming straight line from the center. This page uses its nonnegative magnitude to label different incident rays. Substitute $d\varphi/ds_{\rm opt}=b/\mathcal B^2$ into the unit-tangent condition to obtain radial propagation:

When the right-hand side vanishes, radial advance momentarily stops at a closest-approach point $r_0$, so $b=\mathcal B(r_0)$. The requirement that the right-hand side be nonnegative also determines which radii a ray with a given impact parameter can enter.

03The photon sphere and unstable circular rays

Zero radial speed alone is insufficient for a ray to remain in a circular orbit. A critical circular ray also requires stationarity of the turning condition with radius, namely $\mathcal B'(r)=0$. A prime in this section denotes differentiation with respect to $r$. Differentiating the circumferential optical-path function gives

The denominator is positive in the exterior region. The zero of the numerator $1-3\ell_g/r$ gives

The radius $r_{\rm ph}$ is called the photon-sphere radius. In a spherical background, critical circular rays in different orbital planes lie at this same radius and together define a sphere. It is a critical propagation condition, not a luminous shell of material.

At this radius, $\mathcal B$ is minimal and hence $1/\mathcal B^2=X/r^2$ is maximal. The critical ray is unstable to radial perturbations: a slight outward displacement may allow escape, while a slight inward displacement directs the path toward more strongly stretched space. Continuous weak-field deflection and critical orbiting follow from the same optical-path law, without adding a separate change in local light speed.

04From the capture scale to an observable image

Substituting the critical radius into the impact parameter gives

The capture interpretation additionally requires an absorbing boundary and geometric optics: propagation is approximated by rays when the wavelength is small relative to the scale over which the background varies. This page adopts the manuscript's boundary $r_h=2\ell_g<r_{\rm ph}$, with the exterior constitutive model covering the critical ray and the region beyond it. The absorbing surface must lie inside the critical orbit. If a material surface lies outside $r_{\rm ph}$, light encounters that surface first; capture must then be calculated using its boundary conditions, rather than directly applying the value of $b_c$ above. Under this page's boundary conditions, distant incident rays with $b<b_c$ have no exterior turning point and enter the capture region. Rays with $b>b_c$ have an exterior closest-approach point and can return to the far field. The case $b=b_c$ separates these behaviors. For an observer at a distance $D_{\rm obs}$ much larger than the gravitational scale, the critical angular radius is approximately $b_c/D_{\rm obs}$.

This result determines the angular scale projected by the capture region at large distance. It does not specify the brightness in each direction. An actual image also depends on where emitting material is located, how light is absorbed or scattered, how the material moves, the viewing inclination and the instrument response. Comparison with an observation requires these processes in a single forward model that calculates the recorded detector signal from the source and its propagation.

The photon-sphere radius $r_{\rm ph}$, distant capture scale $b_c$ and bright ring in an image therefore have distinct meanings. A complete imaging model can connect them, but a measured bright ring alone does not confirm the physical structure of the boundary. See GRV-EXP-005 for the relevant observations and modeling requirements.

05Which lengths diverge at the exterior limit

The adopted background function $X$ approaches zero at a finite areal radius:

Approaching $r_h$ from outside gives $\Phi\to0$ and $\Omega\to\infty$. The associated wavelength of a local standard process lengthens, and its common-reference frequency decreases. An increasing expansion factor does not mean that all accumulated lengths diverge in the same way. Actual radial length accumulates $dr/\sqrt X$, while reference optical path accumulates $dr/X$; these must be integrated separately.

Both integrals can be evaluated analytically in the adopted exterior model:

The first expression uses the boundary as its lower limit, with $r>r_h$; the second uses $r_2\geq r_1>r_h$. Differentiating the first expression gives $1/\sqrt{1-r_h/r}$ and its boundary value is zero. The optical antiderivative is $r+r_h\ln(r-r_h)$, so the arbitrary logarithmic scale cancels between endpoints. Near the boundary, $r=r_h+\epsilon$ gives $\ell_s(r_h,r)=2\sqrt{r_h\epsilon}+O(\epsilon^{3/2}/\sqrt{r_h})$, while the optical path from $r_h+\epsilon$ to a fixed outer endpoint diverges logarithmically. The finite actual length and divergent reference optical path are therefore consequences of two exact integrals, not an absence of analytic length results.

The quantities $r_h$, $r_{\rm ph}=3r_h/2$ and $b_c=3\sqrt3\,r_h/2$ mark the exterior limiting position, critical orbiting radius and distant capture scale, respectively. All have dimensions of length, but their physical roles are not interchangeable.

06Far-field gradient and local hovering force

How much support is needed to keep a body stationary near the boundary? This question involves the spatial gradient, actual displacement and energy conversion together. First define the exterior longitudinal gradient normalized to the common reference:

The quantity $\kappa_X$ has dimensions of acceleration and represents the boundary-gradient scale in the far-field reference. For fixed $M$, it remains finite in the exterior limit. Local support force must instead be calculated from the test body's energy in the same background. Use the stationary energy $E_\infty=m_0c^2\Phi$ from the orbital action and set $dW_\infty=dE_\infty$ for an outward virtual displacement. Then use $dW_\infty=\Phi dW_{\rm loc}$ and $d\ell_s=dr/\sqrt X$ to obtain

Here $m_0$ is the test-body mass and $F_{\rm hold,loc}=dW_{\rm loc}/d\ell_s$ is the magnitude of the local outward support force. It grows as $X^{-1/2}$ and diverges toward the boundary; the converted quantity $\sqrt X F_{\rm hold,loc}/m_0$ approaches the finite value $\kappa_X$. The two results answer different questions—local support and far-field normalization—and are consistent. Free fall and hovering also impose different dynamical conditions: the force needed to prevent infall does not replace the free trajectory.

07Boundary physics and the scope of the static model

The manuscript further proposes a strong-field boundary hypothesis: $X=0$ is the physical termination boundary of the spatial continuum, and $r<r_h$ is not continued as a region of the same continuous medium. In this picture, ordinary matter maintains its identity through a localized topological structure, while a black hole's overall state is sustained by boundary states and the exterior field together. Energy and state information from incoming structures can be transferred to the boundary and exterior field.

This hypothesis identifies the physical object ASD seeks to investigate. The zero of $X$ and exterior optics do not, however, provide a dynamical solution for boundary formation, stability or interior structure. Preservation of information also depends on the reversibility of the joint boundary–field evolution; it does not follow from total energy conservation alone. See the strong-field closure problem.

08How a rotating source establishes an azimuthal accompanying field

Rotation gives different parts of a source differently directed mass currents. Beyond static radial stretching, their spatial response forms an azimuthal structure. The manuscript uses the stationary mass current $\mathbf j_m=\rho_m\mathbf v_m$, with $\nabla\cdot\mathbf j_m=0$, and total angular momentum $\mathbf J=\int\mathbf x'\times\mathbf j_m\,d^3x'$. The effective directional transport $\mathbf u_{\rm rot}$ has velocity units. Given the lowest-order transverse coupling and decay at infinity, the Poisson equation has the Green-function solution

Here $\zeta$ is a dimensionless coupling normalization. For a localized source without bulk translation, $\int\mathbf j_m d^3x'=0$, so the $1/r$ term in the far-field Green-function expansion vanishes. Integrating $\nabla\cdot(x'_ix'_j\mathbf j_m)$ with a vanishing boundary term sets the symmetric part of the first current moment to zero. Its antisymmetric part is fixed by $\mathbf J$, yielding $\int\mathbf j_m(\mathbf x'\cdot\hat{\mathbf r})d^3x'=\mathbf J\times\hat{\mathbf r}/2$ and the far-field dipole term above. Its amplitude is calculated from this source relation.

With the rotation axis along $z$, $u_\varphi=\zeta G_0J\sin\theta/(2c^2r^2)$. Dividing by the distance $r\sin\theta$ to the axis gives the field angular velocity $\omega_{\rm rot}$. Here $\theta$ is the spatial polar angle and $\varphi$ the azimuth; neither is Chapter 1’s motion-measurement angle. The source’s own rotation, the exterior $\omega_{\rm rot}$ and the later boundary thermodynamic conjugate are distinct quantities. The manuscript adopts $\zeta=4$ as an empirical rotation-coupling calibration, not a parameter-free microscopic prediction.

09From directed optical path to opposite-direction propagation

Projecting the azimuthal field onto the propagation direction $\hat{\mathbf k}$ in the LVE-type directional factor adds a term that changes sign when propagation is reversed:

The first line states the adopted directional factor. The second retains first order in rotation for weak fields and $u_{\rm rot}/c\ll1$; these expressions have different approximation roles. The one-way correction is $\delta T_{\rm rot}=-\zeta G_0\int(\mathbf J\times\mathbf r)\cdot d\mathbf x/(2c^4r^3)$. On the same fixed closed path, define the positive orientation along the azimuthal field. Reversal changes the sign of the projection, giving twice its circulation divided by $c^2$.

The final circular-loop expression further requires a loop coaxial with the source, at fixed $r$ and polar angle. Its radius is $R_\perp=r\sin\theta$ and its area $A_C=\pi R_\perp^2$. Then $\omega_{\rm rot}$ is constant along the loop and $\oint\mathbf u_{\rm rot}\cdot d\mathbf x=2\omega_{\rm rot}A_C$, yielding (2.82). These equations already calculate a far-field directional response; they are not a complete rotating strong-field photon-sphere or shadow solution. Apparatus rotation, the source’s azimuthal field and moving endpoints must enter an experimental path separately.

10How boundary microstates give an area entropy

ASD organizes absorption, storage and radiation as transformations between the boundary and exterior field. Denoting a boundary microstate by $\mathcal B$, the process is $(\mathcal B,\mathcal I_{\rm in})\to(\mathcal B',\mathcal R_{\rm out})$. Mass and angular momentum record macroscopic totals, while microstates distinguish incident histories compatible with those totals. Energy conservation controls the total exchange; information preservation additionally requires the joint state map to retain distinctions between inputs.

For a nonrotating, uncharged quasistatic sequence at fixed $G_0$, normalize stored energy as $E_{\rm bdy}=Mc^2$. The powers $P_{\rm in}$ and $P_{\rm out}$ use the same reference. If the boundary contains effective area cells $a_*$, each with $g_*$ accessible states, approximately independent in this counting model, then

Here $N_h$ counts cells, $W$ counts states, and $\sigma_h$ has inverse-area units. Multiplication of the cell counts makes $W$ exponential in cell number; the logarithm makes entropy proportional to area. This area law is established within the stated boundary model. Microscopic cell area, accessible-state count and correlation corrections still require boundary dynamics. Obtaining the area law and independently calculating its coefficient are distinct results.

11Temperature, heat capacity and the boundary first law

At fixed $G_0,c,\sigma_h$, $dA_h=32\pi G_0^2M\,dM/c^4$, while $dE_{\rm bdy}=c^2dM$. Inserting these into $T=dE/dS$ determines the temperature of the area model:

The first line is the thermodynamic result for this fixed-parameter quasistatic sequence. Adopting the Bekenstein–Hawking reference normalization $\sigma_h=c^3/(4G_0\hbar)$ gives the second line’s temperature coefficient. This is explicitly a normalization match, not an independent determination from unknown $a_*$ and $g_*$. Since $T\propto M^{-1}$, the heat capacity is negative. If effective emitting area is proportional to $A_h$, channel counts and greybody factors are approximately fixed, and thermal spectral flux is adopted, then $P_{\rm out}\propto A_hT^4\propto M^{-2}$. Net evaporation additionally accounts for absorption and background input.

The last line is the extension when a differentiable state function $E(S,J,Q)$ exists. The quantities $\omega_{\rm bdy}=(\partial E/\partial J)_{S,Q}$ and $\mathcal V_{\rm bdy}=(\partial E/\partial Q)_{S,J}$ are energy conjugates. Their relation to the exterior azimuthal response requires the joint boundary–exterior model. If $G$, the action scale or cell area varies, the first law needs additional conjugate terms rather than fixed-parameter derivatives.

Under the stated static exterior constitutive relations, the optical path is mathematically equivalent to Schwarzschild optical geometry and gives the same photon-sphere and capture scales. ASD's spatial interpretation must continue to be tested through its dynamical origin and eventual differences in observables. Equivalent strong-field results are not independent experimental confirmation.