Shapiro Delay · Excess Optical Path and Propagation Counts

01The question posed by signal arrival counts

When a light or radio signal is sent to a distant location and its reception is recorded, the measured propagation quantity does not depend only on the straight-line distance drawn between the endpoints. If the link passes near a mass source such as the Sun, the effect of nonuniform space on the accumulated optical path must also be calculated. Shapiro delay asks how many additional propagation counts a signal accumulates relative to propagation without gravity between the same endpoints and in the same reference.

ASD interprets this increment as an excess reference optical path through a spatially stretched region. Matter changes the true spatial length of the path and the scale of the standard underlying the recording process. Converting the whole process to a common reference gives additional arrival counts. The local propagation speed of light remains $c$. Light deflection measures the transverse gradient of the same optical path; this page calculates its accumulation along the path.

The measured quantity must first be specified. One-way propagation involves emission and reception endpoints and the connection between their references. A round-trip measurement records the total interval for the outgoing and returning signal. Both use the same background, but their events and paths cannot be interchanged without conditions. The calculation below first treats a fixed-endpoint vacuum link in a static background, then connects it to actual measurement.

02How the same spatial state enters propagation

Let the central source have mass $M$ and gravitational coupling coefficient $G_0$, and define $\ell_g=G_0M/c^2$. The area radius $r$ is defined by the sphere area $4\pi r^2$. In the static spherically symmetric region outside the matter source with $r>2\ell_g$, the manuscript adopts $X=1-2\ell_g/r$, linear expansion coefficient $\Omega=X^{-1/2}$, and dimensionless line-density coefficient $\Phi=\Omega^{-1}$. Here $\Phi$ is not a quantum wave field, and $\rho$ will denote only a coordinate radius, not a volume density.

The true spatial line element in the orbital plane is $d\ell_s^2=X^{-1}dr^2+r^2d\varphi^2$, where $\varphi$ is the azimuthal angle. A far-field standard establishes counting parameter $t$. The cycles accumulated by a local standard process satisfy $dN=f_0\Phi\,dt$, where $f_0$ is the far-field reference frequency. The locally calibrated speed is therefore $v_*=\Phi^{-1}d\ell_s/dt$. Setting $v_*=c$ for light gives the reference optical path $d\ell_{\rm opt}=\Phi^{-1}d\ell_s$. True length and reference optical path have different roles and cannot be conflated with one Euclidean geometrical path.

For convenient integration, introduce an isotropic radial coordinate $\rho$, related to the area radius by $r=\rho(1+\ell_g/2\rho)^2$. In this representation, $d\ell_{\rm opt}=n_{\rm iso}\,ds_0$, where $ds_0$ is the reference Euclidean line element and $n_{\rm iso}$ weights each reference-length segment to give its optical path. This weight comes from the combined conversion of radial length and cycle reference. The coordinate change itself adds no physical effect.

This response of space and standard processes is a constitutive input to the model. Once it is specified, path integration gives the delay. Constancy of local $c$ alone does not fix the amount of excess optical path.

03Establishing the propagation reference without gravity

Let $\Gamma$ be the actual stationary light path connecting two fixed endpoints. Dividing each reference-optical-path segment by $c$ and summing gives

The functional $T[\Gamma]$ is the propagation quantity for the entire path. To define an excess, the same pair of endpoints must be used in the same coordinates. Let their Euclidean straight line without gravity be $\Gamma_0$, with length $R_{12}$. The baseline propagation quantity is then $R_{12}/c$, and the one-way gravitational delay is defined by

This subtraction removes the propagation quantity associated with ordinary distance from the total, leaving the increment due to the gravitational background. It does not also remove every other link contribution from a measured total travel time. Endpoint motion, clock reference, and instrument response must still be treated in a specific measurement.

The two modes in the illustration retain the same endpoints, with color showing the relative along-path weight of $n_{\rm iso}-1$. This is a conceptual illustration and first-order model calculation, not experimental data or a fit to an actual spacecraft link. It uses an arbitrary reference length $L$ rather than assumed Earth–Sun or spacecraft distances. Retaining the straight path $\Gamma_0$ corresponds to the first-order method below; the appearance of the illustration does not give a physical higher-order bending magnitude.

04Why the first-order integral can use a straight line

The actual light path bends, so why can the delay still be calculated along $\Gamma_0$? The essential distinction is between changing the path and changing its optical-path weight, expanding both to the same order. Write $n=1+\delta n$ and $\Gamma=\Gamma_0+\delta\Gamma$, where $\delta n$ and the path perturbation are both first order in the weak field.

A straight line between fixed endpoints already makes Euclidean length stationary, so $\delta\int ds_0=0$. The purely geometrical length increment from a small deflection begins at higher order. Multiplying the first-order weight $\delta n$ by the first-order path change also exceeds the present accuracy. Thus the first-order delay needs only

This is the construction of manuscript Eq. (2.53). The true spatial path and the reference conversion of the standard process have already entered $n_{\rm iso}$; integration along a reference straight line does not discard them. The excess optical path is read primarily through the along-path weight. Equating it entirely with the extra Euclidean distance caused by a bent ray would omit the first-order term here.

05From the endpoint integral to a logarithmic relation

In the reference coordinate plane, place the central source at the origin and take the unperturbed path at $y=b>0$, parametrized by $x$. The quantity $b$ is the minimum reference distance from this straight line to the center: the impact parameter of the present weak-field calculation. Place emission and reception on opposite sides of closest approach, at $x=-x_1$ and $x=x_2$, with $x_1,x_2>0$. Their coordinate radii are $\rho_i=\sqrt{x_i^2+b^2}$, while along the path $\rho=\sqrt{x^2+b^2}$.

Substituting $n_{\rm iso}-1\simeq2\ell_g/\rho$ into the preceding expression and using $ds_0=dx$ gives

An antiderivative is

Normalizing lengths in the logarithm by $b$ makes its argument dimensionless. Substituting both endpoints and denoting the integral by $I$ gives

Since $(\rho_1-x_1)(\rho_1+x_1)=b^2$, the contribution from the negative endpoint can be rewritten using a product of positive lengths:

For the same reference straight line, $R_{12}=x_1+x_2$. These endpoint relations let the same result be written using only the two endpoint radii and their separation:

Final equation (2.57) explicitly supplies the exact algebraic identity behind this rewriting:

This identity is exact for the stated endpoint geometry. The gravitational delay that multiplies its logarithm remains a first-order weak-field calculation.

The latter expression is an algebraic rearrangement of the former, with no additional propagation effect. Equalities in both are understood at the first-order weak-field accuracy of this section. They retain the integration endpoints but are not exact closed forms for complete strong-field propagation. The quantities $\rho_1$, $\rho_2$, and $R_{12}$ must all be given in the same reference coordinates.

06Distant endpoints and a round-trip link

When both endpoints lie far from the closest-approach region, $\rho_1,\rho_2\gg b$ and $x_i\simeq\rho_i$. The endpoint expression reduces to

This approximation highlights two physical ingredients: source mass sets the scale of the excess optical path, while the closest-approach distance and endpoint positions determine its accumulation. The quantity $G_0M/c^3$ has dimensions of time, the logarithm has a dimensionless argument, and the increment vanishes for $M\to0$.

Under this condition, the round-trip delay is

The registered expression now retains the full endpoint ratio in final (2.58); it does not impose $\rho_i\gg b$. Only after that additional limit does it reduce to $4G_0M\ln(4\rho_1\rho_2/b^2)/c^3$.

If the ground station or spacecraft has moved appreciably by the time the signal is retransmitted at the spacecraft, outgoing and returning propagation must each use their corresponding events and paths before being combined in a common counting reference. The static doubling relation explains the connection between one-way and round-trip quantities; it does not replace a raw ranging model for a moving spacecraft.

07Connecting to actual experimental comparisons

A Shapiro measurement connects propagation theory to emission, retransmission, and reception events. In addition to the gravitational term, the measurement equation must include endpoint orbits, station clocks and frequency references, propagation media, and instrument effects. Radio links near the Sun particularly require treatment of plasma and related effects. In the Cassini radio experiment, for example, link frequency variations are related to corresponding changes in round-trip light time; the quantity compared is the signal after instrument and propagation treatment. See the NASA/JPL experiment report.

This page derives a static first-order delay from the ASD spatial optical path. With endpoints, coordinates, and approximation order matched, it has the same mathematical form as the comparison-theory result. The result itself therefore does not discriminate between ASD and general relativity. No independent refit of Cassini data has been performed here, and the interactive illustration is not treated as an apparatus model. Observational references and the scope of calculation are recorded in gravity test entry GRV-EXP-003.

08Scope and connections to further calculations

The integrals above apply to a static spherically symmetric vacuum exterior, with the entire path in the weak field and the small-deflection expansion valid. The explicit endpoint integral uses endpoints on opposite sides of closest approach; distant-endpoint and round-trip relations add their own approximation conditions. Higher precision requires a consistent expansion of the complete optical path and endpoint conditions, including source multipoles, rotation, path bending, and actual link terms. Replacing a coordinate radius by the area radius in just one formula does not provide that expansion.

ASD connects two propagation phenomena through the same spatial state: transverse optical-path variation changes ray direction, while accumulation along the path changes arrival counts. The common background is described in the gravity overview, the optical-path coordinate conversion in light deflection, and the scale and counting relations of standard processes in gravitational redshift. Strong-field propagation requires the complete exterior optical model and its scope as described under static compact objects.