In special relativity only the two-way (round-trip) speed of light is measurable without a prior synchronization convention; the one-way speed depends on Reichenbach’s parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varepsilon \in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, whose value is conventional. We develop the corresponding framework in general relativity, in the threading decomposition of an arbitrary stationary frame. Resynchronizations of the frame’s clocks act as gauge transformations <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$g_i \mapsto g_i + \partial _i\phi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>g</mi> <mi>i</mi> </msub> <mo>↦</mo> <msub> <mi>g</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mi>∂</mi> <mi>i</mi> </msub> <mi>ϕ</mi> </mrow> </math></EquationSource> </InlineEquation> on the gravitomagnetic potential, leaving the redshift factor h and the radar metric <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\gamma _{ij}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mrow> <mi mathvariant="italic">ij</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> invariant; the one-way speed obeys the exact Reichenbach identity <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$1/c_+(\textbf{n}) + 1/c_+(-\textbf{n}) = 2/c$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <msub> <mi>c</mi> <mo>+</mo> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">n</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mn>1</mn> <mo stretchy="false">/</mo> <msub> <mi>c</mi> <mo>+</mo> </msub> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi mathvariant="bold">n</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mo stretchy="false">/</mo> <mi>c</mi> </mrow> </math></EquationSource> </InlineEquation>, with synchrony field <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varepsilon (x,\textbf{n}) = \tfrac{1}{2}\bigl (1 + \sqrt{h}\,g_i n^i\bigr )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi mathvariant="bold">n</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mi>h</mi> </msqrt> <mspace width="0.166667em" /> <msub> <mi>g</mi> <mi>i</mi> </msub> <msup> <mi>n</mi> <mi>i</mi> </msup> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>; the obstruction to a global Einstein synchronization is the gauge-invariant holonomy <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\oint g_i\,\textrm{d}x^i$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>∮</mo> <msub> <mi>g</mi> <mi>i</mi> </msub> <mspace width="0.166667em" /> <mtext>d</mtext> <msup> <mi>x</mi> <mi>i</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, realized physically as the Sagnac effect and equivalent to the vorticity of the frame; and slow clock transport carries the same holonomy. Hence the conventional content of one-way light propagation is exactly a gradient, and the triple <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(h,\gamma _{ij},\textrm{d}g)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>h</mi> <mo>,</mo> <msub> <mi>γ</mi> <mrow> <mi mathvariant="italic">ij</mi> </mrow> </msub> <mo>,</mo> <mtext>d</mtext> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> exhausts the invariant chronometric content of the frame. We then give the general-relativistic dynamics of the invariants: <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\textrm{d}g$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>d</mtext> <mi>g</mi> </mrow> </math></EquationSource> </InlineEquation> is the twist of the frame’s Killing congruence; matter and light probe it — never the gauge sector — through an exact gravitomagnetic Lorentz-force law and Fermat’s principle, light rays being the magnetic geodesics of the Randers optical geometry <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\gamma _{ij}/h$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>γ</mi> <mrow> <mi mathvariant="italic">ij</mi> </mrow> </msub> <mo stretchy="false">/</mo> <mi>h</mi> </mrow> </math></EquationSource> </InlineEquation>; and the Einstein equations govern it: in vacuo the pair formed by h and the twist potential <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\chi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>χ</mi> </math></EquationSource> </InlineEquation> obeys the Ernst sigma-model on the radar geometry (for Kerr, <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\chi = 2ma\cos \theta /\rho ^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>χ</mi> <mo>=</mo> <mn>2</mn> <mi>m</mi> <mi>a</mi> <mo>cos</mo> <mi>θ</mi> <mo stretchy="false">/</mo> <msup> <mi>ρ</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>), while in the weak field mass currents source it. Exact case studies (rotating frames, Schwarzschild, Kerr, the GPS) include the Kerr synchronization holonomy <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\Delta \tau = -8\pi ma/\bigl (c\sqrt{r(r-2m)}\bigr )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Δ</mi> <mi>τ</mi> <mo>=</mo> <mo>-</mo> <mn>8</mn> <mi>π</mi> <mi>m</mi> <mi>a</mi> <mo stretchy="false">/</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>c</mi> <msqrt> <mrow> <mi>r</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>-</mo> <mn>2</mn> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </msqrt> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> per revolution, of weak-field magnitude <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$8\pi GJ/(c^4 r) \approx 1.9\times 10^{-16}\,\textrm{s}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>8</mn> <mi>π</mi> <mi>G</mi> <mi>J</mi> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <msup> <mi>c</mi> <mn>4</mn> </msup> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>≈</mo> <mn>1.9</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>16</mn> </mrow> </msup> <mspace width="0.166667em" /> <mtext>s</mtext> </mrow> </math></EquationSource> </InlineEquation> for Earth-girdling light paths, nine orders of magnitude below the kinematic Sagnac delay corrected for daily by the GPS. Consequences for one-way-speed test theories in gravitational environments are drawn.

The two-way speed of light in general relativity: conventional and physical content of synchronization

期刊论文
发表日期：

2026年7月27日

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Paulo Guilherme Santos

摘要

In special relativity only the two-way (round-trip) speed of light is measurable without a prior synchronization convention; the one-way speed depends on Reichenbach’s parameter $\varepsilon \in (0,1)$ $ε \in (0, 1)$ , whose value is conventional. We develop the corresponding framework in general relativity, in the threading decomposition of an arbitrary stationary frame. Resynchronizations of the frame’s clocks act as gauge transformations $g_i \mapsto g_i + \partial _i\phi $ $g_{i} \mapsto g_{i} + \partial_{i} ϕ$ on the gravitomagnetic potential, leaving the redshift factor h and the radar metric $\gamma _{ij}$ $γ_{ij}$ invariant; the one-way speed obeys the exact Reichenbach identity $1/c_+(\textbf{n}) + 1/c_+(-\textbf{n}) = 2/c$ $1 / c_{+} (n) + 1 / c_{+} (- n) = 2 / c$ , with synchrony field $\varepsilon (x,\textbf{n}) = \tfrac{1}{2}\bigl (1 + \sqrt{h}\,g_i n^i\bigr )$ $ε (x, n) = \frac{1}{2} (1 + \sqrt{h} g_{i} n^{i})$ ; the obstruction to a global Einstein synchronization is the gauge-invariant holonomy $\oint g_i\,\textrm{d}x^i$ $\oint g_{i} d x^{i}$ , realized physically as the Sagnac effect and equivalent to the vorticity of the frame; and slow clock transport carries the same holonomy. Hence the conventional content of one-way light propagation is exactly a gradient, and the triple $(h,\gamma _{ij},\textrm{d}g)$ $(h, γ_{ij}, d g)$ exhausts the invariant chronometric content of the frame. We then give the general-relativistic dynamics of the invariants: $\textrm{d}g$ $d g$ is the twist of the frame’s Killing congruence; matter and light probe it — never the gauge sector — through an exact gravitomagnetic Lorentz-force law and Fermat’s principle, light rays being the magnetic geodesics of the Randers optical geometry $\gamma _{ij}/h$ $γ_{ij} / h$ ; and the Einstein equations govern it: in vacuo the pair formed by h and the twist potential $\chi $ $χ$ obeys the Ernst sigma-model on the radar geometry (for Kerr, $\chi = 2ma\cos \theta /\rho ^2$ $χ = 2 m a cos θ / ρ^{2}$ ), while in the weak field mass currents source it. Exact case studies (rotating frames, Schwarzschild, Kerr, the GPS) include the Kerr synchronization holonomy $\Delta \tau = -8\pi ma/\bigl (c\sqrt{r(r-2m)}\bigr )$ $Δ τ = - 8 π m a / (c \sqrt{r (r - 2 m)})$ per revolution, of weak-field magnitude $8\pi GJ/(c^4 r) \approx 1.9\times 10^{-16}\,\textrm{s}$ $8 π G J / (c^{4} r) \approx 1.9 \times 10^{- 16} s$ for Earth-girdling light paths, nine orders of magnitude below the kinematic Sagnac delay corrected for daily by the GPS. Consequences for one-way-speed test theories in gravitational environments are drawn.