Gravitational Time Dilation

Two identical clocks held stationary at different radii outside a static, spherical mass accumulate different amounts of proper time over the same reference interval. The closer clock accumulates less. General relativity describes this difference through spacetime geometry. Measurement Quantization (MQ) asks how discrete count relationships can be represented in that geometry.

From Fundamental Measure to Gravitational Geometry

MQ starts from one axiom. Measure is discrete. The Internal Frame is the discrete domain of count configurations. The Frames mapping relates these configurations to continuous observable relationships in the System Frame. MQ uses this geometry to anchor an expanding frame of information that describes a universe.

Four physical inputs select and normalize the modeled universe in MQ’s reconstruction of fundamental measures. The fine-structure constant selects the dimensionless coefficient for the modeled universe, the speed of light anchors the length–time scale, and the electron mass and Bohr radius jointly normalize mass relative to length and time. The numerical value of the speed of light is exact in SI by definition. The measured cosmic microwave background temperature serves separately as an epoch selector.

The fundamental measures of length, mass, and time are $l_{\mathit{f}}$, $m_{\mathit{f}}$, and $t_{\mathit{f}}$. With $n_{\mathit{L}}$, $n_{\mathit{M}}$, and $n_{\mathit{T}}$ denoting their respective counts, resolved length, mass, and time are $l=n_{\mathit{L}}{l}_{\mathit{f}}$, $m=n_{\mathit{M}}{m}_{\mathit{f}}$, and $t=n_{\mathit{T}}{t}_{\mathit{f}}$. Here $t$ denotes a generic resolved duration. For light, equal length and time counts give $c=\frac{l_{\mathit{f}}}{t_{\mathit{f}}}$, where $c$ is the speed of light. The MQ treatment of frames of reference describes how these quantities enter observable geometry.

The Count Relation Behind the Gravitational Effect

Let $m$ be the source mass, $r$ the exterior radial separation represented by the length count, and $\mathrm{G}$ the gravitational constant at the upper count limit. Using that coupling, MQ writes the classical exterior escape-speed expression as $v_{\mathit{e}}=c\sqrt{\frac{2n_{\mathit{M}}}{n_{\mathit{L}}}}$, where $v_{\mathit{e}}$ denotes escape speed with the zero-energy reference at infinity. Squaring the dimensional and count forms gives the identity used in the clock comparison.

The Gravitational Clock Rate in Count Form

In the Schwarzschild exterior, let $\tau$ denote the proper time of a stationary clock. For this comparison $t$ denotes Schwarzschild coordinate time, normalized to the proper time of a stationary clock at infinity. The symbol $\mathrm{d}$ denotes a differential. The time component of the Schwarzschild exterior metric gives^[3] $\frac{\mathrm{d}\tau }{\mathrm{d}t}=\sqrt{1-\frac{2\mathrm{G}m}{r{c}^2}}$. Substituting the upper-count identity gives the clock rate in terms of mass and length counts.

At a fixed radial length count, increasing the source mass count reduces the clock-rate factor. At a fixed source mass count, increasing the radial length count raises it. The factor approaches one as radial separation increases without bound.

Comparing Two Clocks

For two stationary clocks at different radii about the same source, the subscripts 1 and 2 distinguish the two proper times and their respective radial length counts. Dividing the two clock rates against the same coordinate-time reference gives $\frac{\mathrm{d}{\tau }_1}{\mathrm{d}{\tau }_2}=\sqrt{\frac{1-\frac{2n_{\mathit{M}}}{n_{\mathit{L}1}}}{1-\frac{2n_{\mathit{M}}}{n_{\mathit{L}2}}}}$. The clock at smaller radial length count accumulates less proper time over that shared interval.

The Boundary in the Exterior Expression

The classical escape-speed expression reaches $c$ at the following count relation. With the upper-count identity, it represents the Schwarzschild horizon condition when the source has a horizon.

The stationary exterior clock formula applies for $n_{\mathit{L}}>2n_{\mathit{M}}$ and outside the material surface of the source, if present. As the Schwarzschild horizon is approached from outside, the rate relative to the asymptotic reference tends to zero. The zero describes the limit of the stationary exterior expression. A stationary clock at the horizon is outside its domain.

Relation to Gravitational Contraction

The Schwarzschild geometry governs spatial measurements as well as the rate of stationary clocks. The related discussions of contraction and dilation with gravity and the equivalence principle develop MQ’s interpretation of those relationships.

MQ separately defines the Informativity differential as a count-dependent, non-Lorentz length contraction associated with count normalization in the Frames mapping. The Diophantine realization residual $Q_{\mathit{L}}$ is the amount by which the realized hypotenuse count exceeds the radial length count in the unit-leg construction. The geometry gives $Q_{\mathit{L}}=\sqrt{1+n_{\mathit{L}}^2}-n_{\mathit{L}}$, with normalization departure $1-2Q_{\mathit{L}}{n}_{\mathit{L}}=Q_{\mathit{L}}^2$. These are finite-count relations. The clock comparison above uses the Schwarzschild factor with the upper-count gravitational coupling.

Experimental Reality and MQ Interpretation

In their 1960 paper “Apparent Weight of Photons,” Pound and Rebka measured gravitational redshift using nuclear resonance.^[1] Bothwell and colleagues later resolved a gravitational frequency gradient across a millimetre-scale sample of strontium atoms in an optical lattice clock.^[2]

These results test the predicted gravitational frequency shift. The count form above gives the same stationary Schwarzschild rate under the stated assumptions, so experiments that test that common factor leave the proposed discrete mechanism open.

For the broader MQ treatment, see Gravity, Gravitational Constant, Fundamental Measures, Frames of Reference, and Contraction and Dilation with Gravity.