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Stationary RN observer frequency ratio

Physical scope

Conditional phase-frequency ratio for a local future null eikonal phase crossing two static observers in a source-free RN model region. Ideal-clock and geometric-optics readings are adopted assumptions; no emitted or received measurement, apparatus model, or empirical agreement is asserted.

Statement

Let $m_{*} > 0$ , $Q_{e}, Φ_{B} \in R$ , and use SI and geometric solution parameter identifications and Dyonic Reissner-Nordström metric and field domains for the dyonic RN metric on $M_{J} = R_{x^{0}} \times J_{r} \times S^{2}$ , where $J$ is a connected open interval with $f (r) = 1 - 2 M / r + (q_{e}^{2} + q_{m}^{2}) / r^{2} > 0$ . The coordinate $x^{0} = c t$ and $r$ have units of metres. Put $ξ = \partial_{x^{0}}$ ; it is the future static Killing field on this region. Let $I_{γ} \subseteq R$ be a nonempty open interval, let $γ : I_{γ} \to M_{J}$ be a $C^{2}$ curve, and let $W$ be an open neighborhood of $γ (I_{γ})$ carrying a dimensionless $C^{2}$ phase $θ : W \to R$ . Set $k = d θ$ and $K = k^{♯}$ . Assume the eikonal equation $g^{- 1} (d θ, d θ) = 0$ holds throughout $W$ , and that $K$ is nowhere zero and future directed along $γ (I_{γ})$ . Parameterize the phase ray by $γ^{'} (λ) = K (γ (λ))$ , which fixes its affine parameter normalization. For each spatial point $(r, ω) \in J \times S^{2}$ and nonempty open interval $I \subset R$ , define the static observer curve $σ_{r, ω} : I \to M_{J}$ by $σ_{r, ω} (s) = (s, r, ω)$ , where $s = x^{0}$ has units of metres. It is smooth and regular with tangent $ξ$ , and it is future timelike because $g (ξ, ξ) = - f (r) < 0$ . Its proper-time parameter satisfies $d τ / d s = \sqrt{f (r)} / c$ under the metric proper-time definition, so its normalized four-velocity is $U (r) : = \frac{c}{\sqrt{f (r)}} ξ, g (U, U) = - c^{2}, ν_{U} : = - \frac{k (U)}{2 π} .$ Here $U = d σ_{r, ω} / d τ$ is normalized by the SI proper-time convention of SI unit and coordinate conventions of the classical gravitational framework, and $ν_{U}$ is the phase frequency in cycles per unit ideal-clock time under Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework. If $γ$ intersects static observer curves $σ_{r_{E}, ω_{E}}$ and $σ_{r_{R}, ω_{R}}$ at events $E$ and $R$ , respectively, with $r_{E}, r_{R} \in J$ , then the conditional model prediction is $\frac{ν_{R}}{ν_{E}} = \sqrt{\frac{f (r_{E})}{f (r_{R})}} .$ The phase data and source-free RN model are conditional inputs; the formula is not a reported observation or a claim of measured precision.

Facts & Assumptions

Given: Positive $G_{N}, c, μ_{0}$ , the SI RN parameters and metric on a connected $f > 0$ interval, the adopted classical clock/light-ray postulate, and a local $C^{2}$ dimensionless phase satisfying $g^{- 1} (d θ, d θ) = 0$ on its open domain $W$ , with nowhere-zero future-directed differential along the ray.

[F1]

SI and geometric solution parameter identifications and Einstein–Maxwell primitive quantities and SI conventions: $M = G_{N} m_{*} / c^{2}$ , $q_{e} = \sqrt{G_{N} / (4 π ε_{0} c^{4})} Q_{e}$ , $q_{m} = α Φ_{B} / (4 π)$ , $ε_{0} μ_{0} c^{2} = 1$ , and $x^{0} = c t$ is a length coordinate.

[F2]

Dyonic Reissner-Nordström metric and field domains: on $f > 0$ , $g = - f (d x^{0})^{2} + f^{- 1} d r^{2} + r^{2} γ_{S^{2}}$ and the chosen future time orientation is $ξ = \partial_{x^{0}}$ .

[F3]

Dyonic RN as adopted Einstein–Maxwell model: the exact dyonic RN datum on each stated positive-radius static domain with $f \neq 0$ is a conditional solution of the adopted zero-source Einstein–Maxwell model; the carrier's proof remains draft and is not independently accepted.

[A1]

Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework: ideal clocks read proper time, and in the geometric-optics approximation ideal light rays are future null geodesics of the metric-selected connection. These are adopted physical assumptions, not deductions from the Einstein–Maxwell equations.

[F4]

Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields and The inverse metric, musical isomorphisms and index raising: the selected connection is torsion-free and metric-compatible, its tensor extension obeys the product rule with $\nabla g^{- 1} = 0$ , and $K = k^{♯}$ is characterized by $g (K, V) = k (V)$ .

[F8]

Clairaut--Schwarz theorem for continuous second partial derivatives: the mixed coordinate partials of a $C^{2}$ scalar field on an open Euclidean chart commute.

[F5]

Killing fields of a Lorentzian metric and Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor: a field is Killing when $L_{ξ} g = 0$ ; for a Killing field and an affinely parametrized geodesic $γ$ , $g (ξ, γ^{'})$ is constant along $γ$ .

[F6]

Classification of future and past causal vectors by a timelike field: a nonzero future null vector has strictly negative metric pairing with every future timelike vector.

[F7]

SI unit and coordinate conventions of the classical gravitational framework: in length coordinates $x^{0} = c t$ , $U = d γ / d τ$ has SI velocity components and the proper-time parameter $τ$ is measured in seconds.

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): the inherited geometric RN and metric-connection conventions use only ${A C}_{ω}$ . No selection from a family is made by this local calculation; full AC is not used.

Proof

1.1F1F2F3F5F7givenalgebra

The SI map and length-coordinate convention [F1] identify $x^{0} = c t$ and the RN length parameters used in [F2]. The metric in [F2] is independent of $x^{0}$ , so its Lie derivative along $ξ = \partial_{x^{0}}$ vanishes and $ξ$ is Killing by [F5]. For any curve $σ_{r, ω} (s) = (s, r, ω)$ in the Statement, $σ_{r, ω}^{'} = ξ$ and $- g (ξ, ξ) = f (r)$ is constant along the curve, so the metric proper-time definition gives $d τ / d s = \sqrt{f (r)} / c$ and $d σ_{r, ω} / d τ = c ξ / \sqrt{f (r)} = U (r)$ . Thus $g (U, U) = - c^{2}$ and $U$ is future timelike on $J$ . The solution is the conditional physical RN model of [F3]; no observational premise is used.

1.2F4F8A1givenalgebra

The metric dual $K = k^{♯}$ is nonzero and null because $k \neq 0$ and $g^{- 1} (k, k) = 0$ . In coordinates, $k_{a} = \partial_{a} θ$ ; [F8] makes the mixed second partials commute, and torsion freedom in [F4] makes the Christoffel terms symmetric, so $\nabla_{b} k_{a} = \nabla_{a} k_{b}$ . Metric compatibility and the tensor product rule in [F4] then give $K^{b} \nabla_{b} K_{a} = k^{b} \nabla_{b} k_{a} = k^{b} \nabla_{a} k_{b} = \frac{1}{2} \nabla_{a} (g^{b c} k_{b} k_{c}) = 0.$ Thus the integral curves of $K$ are affinely parametrized null geodesics on the phase domain by the geodesic definition; the phase ray $γ^{'} = K$ is one such curve. This is the local eikonal input used for the geometric-optics ray.

2.1F5F6step 1.1step 1.2

Apply the Killing contraction identity [F5] to the affinely parametrized geodesic tangent $K$ from step 1.2. It makes $g (ξ, K) = k (ξ)$ constant along that ray. Since $K$ is future null and $ξ$ is future timelike, [F6] gives $C : = - k (ξ) > 0$ .

3.1A1F7step 2.1algebra

At radius $r$ , the phase frequency is $ν_{U} = - \frac{k (c ξ / \sqrt{f})}{2 π} = \frac{c C}{2 π \sqrt{f}} .$ The positive value is a phase rate; the ideal-clock postulate [A1] supplies its clock-time interpretation. Evaluating at the two events and dividing cancels the common positive constant $c C / (2 π)$ , yielding $ν_{R} / ν_{E} = \sqrt{f (r_{E}) / f (r_{R})}$ .

4.1A1F3step 3.1A2∎

The ratio in step 3.1 is a conditional phase-frequency prediction for the chosen RN domain and local ray. The model postulates and phase assumptions remain explicit; no clock apparatus, data, uncertainty, or empirical confirmation has been supplied.

Stationary RN observer frequency ratio

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources