Gravitational Redshift · Scale, Wavelength and Frequency

01The question raised by comparing clocks at different positions

The same atomic transition at different gravitational potentials shows a systematic frequency difference when compared in a common reference. ASD begins its explanation with the standard process itself: an atomic clock records the periodic process sustained by the transition; the spatial state in which matter is located changes the process's characteristic scale, and its reading changes accordingly. Gravitational redshift thus becomes a metrological route into the study of static spatial structure.

The manuscript's physical sequence is that matter stretches space; spatial stretching increases the associated wavelength and characteristic phase length; with the local vacuum propagation speed $c$ unchanged, frequency decreases and the period increases. The change concerns the scale relationship between space and material processes. Time records this change as a counting parameter in a common reference.

The problem becomes clearer when realizing a standard locally is distinguished from comparing standards at different positions. Each clock can stably repeat the same kind of process at its location, yet comparing two clocks still requires reference conditions, a transfer link, motion and environment to be addressed. The manuscript uses temperature correction of gauge blocks as an analogy: correction makes quantities under different conditions comparable, while physical theory explains why the standard responds. This separates calibration from mechanism; it does not interpret gravitational frequency shifts as thermal effects.

02How a standard process responds to spatial stretching

Let $L_0$ and $f_0$ be the characteristic length and frequency of a standard process under far-field reference conditions, and $L(r)$ and $f(r)$ the corresponding quantities at position $r$. All ratios of lengths and frequencies at different positions on this page are expressed in a common reference. With $\Omega(r)$ describing linear spatial expansion, Chapter 2, Eq. (2.2), adopts

The first relation specifies how the process's characteristic scale changes with the spatial state; the second expresses the frequency response to that same scale change. This is ASD's process-coupling input. SI unit definitions establish relations between quantities, but how a particular atomic or other standard obeys this scale law requires its physical response model. Here $L$ is the characteristic length of the specified process, not automatically every geometric dimension of an arbitrary device.

For a standard process whose phase advances at a characteristic speed $c$, the characteristic phase length of one cycle can be written $S_{\rm cyc}=c/f$, giving the period $\tau=S_{\rm cyc}/c=1/f$. This connects scale and period. For an atomic transition, it does not require a classical closed light path inside the atom. $S_{\rm cyc}$ is a phase scale in a specified process model and should be distinguished from an actual propagation path.

03The scale law in a spherical background

Now specify a calculable gravitational background: an isolated, static, spherical, uncharged source of mass $M$, with background gravitational coupling $G_0$, neglecting the standard's backreaction on the source. Define the areal radius $r$ through $A=4\pi r^2$. In the exterior region $r>2G_0M/c^2$, adopt

Here $\Omega$ is the linear expansion coefficient and $\Phi=\Omega^{-1}$ is the dimensionless line-density coefficient. Neither is the spatial medium's three-dimensional volume density; $\Phi$ on this page also does not denote a wave field from another module. Far from the source, $\Omega$ and $\Phi$ both approach 1. Closer to the source, $\Omega$ increases and $\Phi$ decreases. Under the preceding process coupling, the corresponding characteristic length increases and the reference frequency decreases.

These complete exterior functions are the constitutive relations adopted here. Weak-field redshift constrains their low-order behavior; weak-field data alone do not uniquely determine the full strong-field form. Making this input explicit allows the frequency ratio and weak-field expansion below to be checked step by step, without concealing the microscopic constitutive choice within algebra.

04Why wavelength and frequency vary inversely

First follow the physical sequence. With the standard-process scale coupling stated above, the associated wavelength grows with the spatial linear scale: $\lambda(r)=\Omega(r)\lambda_0$, where $\lambda_0=c/f_0$. Using the vacuum relation $c=\lambda f$ under the same calibration conditions gives

Thus spatial scale first enters the associated wavelength through the declared coupling, then enters frequency and period with $c$ fixed. The same relations can also be rearranged in reverse as $\lambda=c/f=\Omega\lambda_0$ or $\Omega=f_0/f$. This checks their mutual consistency; it does not prove inevitable spatial stretching from $c=\lambda f$ or unit definitions alone. Taking ratios between two positions in a common reference gives

This equation retains $\nu$ for frequency, with the same meaning as $f$ elsewhere on the page. A deeper potential corresponds to a longer wavelength and a lower reference frequency, connected by the same scale factor. The relation $c=\lambda f$ itself connects quantities; ASD's physical interpretation additionally assigns the wavelength change to the spatial stretching produced by matter.

05Connecting frequency to actual counts

Frequency enters measurement through counts of a process. Establish the parameter $t$ with a far-field reference. The common-reference frequency of the local standard process is $f(r)=f_0\Phi(r)$, so during $dt$ the accumulated number of cycles is $dN=f_0\Phi\,dt$. If a segment of motion has true spatial length $d\ell_s$, calibrating its speed with the same local standard process gives

Here $N$ is the cycle count, $t$ the common-reference counting parameter, and $v_*$ the speed calibrated by local length and cycles. At a static position, $\Phi$ does not change during the counting interval, so the number of cycles accumulated within a specified reference interval is determined directly by the local $\Phi$. A clock's nominal reading additionally requires a reference frequency: $t_{\rm read}=N/f_{\rm ref}$. Changing $f_{\rm ref}$ changes how the reading is expressed without removing the comparison between the actual processes at the two positions.

The quantity $f(r_2)/f(r_1)$ calculated here is the common-reference frequency ratio of standard processes of the same kind at two positions. Another experimental quantity is the shift of light emitted at one position and received elsewhere, relative to the receiver's local standard. That requires emission, propagation and reception to be included in a common link model. The compared objects differ, so the direction or sign of one ratio cannot simply be assigned to the other. Further definitions of counting and calibration are given in clocks and frequency.

06From the frequency ratio to the weak-field relation

Compare standard processes of the same kind at positions $r_1$ and $r_2$. Both use the same far-field reference $f_0$, which cancels on division:

This result uses both the standard-process scale response and the spherical exterior scale law. To connect it to weak-field clock comparisons, define $X=\Phi^2$ and the gravitational potential $\Psi=c^2(X-1)/2=-G_0M/r$. The potential $\Psi$ has dimensions of speed squared and is negative near an attractive source. Set $\epsilon_i=\Psi(r_i)/c^2$. For $|\epsilon_i|\ll1$,

Expanding both numerator and denominator, rather than only one of them, gives

Therefore,

The notation $O(\Psi^2/c^4)$ in the last expression abbreviates second-order terms in the small potentials at both positions. Near Earth's surface, over a sufficiently small height difference, take $\Delta h=h_2-h_1$ and let $g>0$ be the local gravitational acceleration magnitude. Then $\Psi(r_2)-\Psi(r_1)\simeq g\Delta h$, giving $\Delta f/f\simeq g\Delta h/c^2$. Here $\Delta f/f$ is defined as $f(r_2)/f(r_1)-1$.

When $r_2>r_1$, the potential $\Psi(r_2)$ is less negative and $\Omega(r_2)$ is smaller. Raising the clock shortens its associated wavelength and increases its reference frequency, with the sign given by the expansion. For $r_2=r_1$ the ratio returns to 1; for $M\to0$ both positions recover the uniform far-field state. These limiting checks show the consistency of scale, wavelength and counting relations within the adopted model.

07Two physical constructions of the same ratio

ASD constructs this frequency ratio through spatial stretching, wavelength change and actual periodic processes. General relativity constructs clock comparisons through proper time and static spacetime geometry. Under the spherical exterior functions and common calibration used here, the two expressions have the same mathematical form and their weak-field expansions give the same potential-difference relation.

Measuring the ratio therefore tests its numerical relation, but does not by itself select ASD's spatial interpretation. If different clock types or apparatus configurations have distinguishable responses, the difference must be calculated from their respective process-coupling models. It cannot be declared directly from unit definitions or an existing identical formula.

08What an experimental comparison measures

Gravitational frequency-shift experiments convert frequencies at different positions into comparable quantities and relate them to a potential-difference model. The experiment by Bothwell and colleagues cited in Chapter 2 resolved a frequency gradient consistent with gravitational redshift within a millimetre-scale strontium sample, providing an observational reference for the weak-field relation. See the original paper. It constrains this frequency gradient without independently distinguishing its physical interpretation.

An actual clock comparison must include transition type, positions and potential model, signal transfer, relative motion and environmental frequency shifts in its measurement equation. Connecting the theoretical $f(r_i)$ to the ratio finally reported by the experiment completes the link from scale relations to device observables. Available material and comparison scope are recorded in gravitational-redshift test GRV-EXP-001. Citing the experiment here is not a claim to have refitted its raw data.

09Scope and microscopic coupling

The complete ratio applies to a static spherical exterior with $\Phi>0$, comparing standard processes of the same kind under the same scale response. The potential-difference formula additionally requires weak fields; the small-height approximation also requires $g$ to be approximately constant over the relevant range. Propagation, motion and environmental corrections belong in the experimental link model. One static scale factor cannot replace an entire apparatus model.

At this level ASD has connected spatial stretching, wavelength and process counts quantitatively. Further progress requires specific material structures and spatial-field coupling to explain why a standard process obeys the adopted scale law and where it applies. The complete strong-field background also requires deeper dynamics. The corresponding tasks concern clock response (Chinese) · Chinese original and strong-field closure. These microscopic conditions remain distinct from the algebraic results developed under this page's constitutive model.

10Derivation sources and related research

This page uses Chapter 2, §I, Eqs. (2.1)–(2.4), and the counting relations in §IV.1. The gravity overview connects this process response to true spatial length, reference optical path and orbital action. The radial length law and universal energy coupling are stated separately where each enters the argument. Metrological and reference conditions are developed further in metrology and measurement and processes.