Restricted redshift prediction of the Schwarzschild vacuum model: the observer-energy ratio between static observers

Adopt the Schwarzschild exterior vacuum model of The Schwarzschild exterior vacuum model: field law, static clock readings and the mass parameter: the exterior datum $(M,g,N)$ of mass parameter $m_{\ast }>0$ with $s=2G_N{m}_{\ast }\mathrm{/}{c}^2$ and $f(r)=1-s\mathrm{/}r$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation), its metric-selected connection $\mathrm{\nabla }$ (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), its static congruence $n=f^{-1\mathrm{/}2}{\mathrm{\partial }}_0$ (The static observer congruence of the Schwarzschild exterior model: unit field, integral curves, acceleration and clock-rate ratios), and the light-ray rule that ideal light rays are future null geodesics of $\mathrm{\nabla }$ (Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework clause (iii)). Let $\eta :J\to M$ be the specified ray, a smooth affinely parametrized geodesic of $\mathrm{\nabla }$ on an interval $J$, with tangent $W={\eta }^{\mathrm{\prime }}$, $g(W,W)=0$ and $W$ future-directed (Geodesic of an affine connection, Timelike, null, spacelike and causal vectors of a Lorentz vector space). Let ${\lambda }_E,{\lambda }_R\in J$ with ${\lambda }_E<{\lambda }_R$ and suppose $\eta ({\lambda }_E)$ and $\eta ({\lambda }_R)$ lie on static observer worldlines of radii $r_E=r(\eta ({\lambda }_E))$ and $r_R=r(\eta ({\lambda }_R))$ with $r_E,r_R\in (s,\mathrm{\infty })$. For each parameter value define the observer energy of the ray relative to the static observers by

$E(\lambda ):=-c{g}_{\eta (\lambda )}(W(\lambda ),n(\eta (\lambda ))),$

the value of the observer-energy definition of Observer-relative energy, spatial momentum and rest mass of a tangent vector clause (a) for the unit observer $n$. Then:

(a) Constancy of the Killing pairing. $K={\mathrm{\partial }}_0$ is a Killing field of $g$ and the function $C(\lambda ):=-g_{\eta (\lambda )}(W(\lambda ),K(\eta (\lambda )))$ is constant along $J$; moreover $C>0$.

(b) The observer-energy ratio. $E(\lambda )>0$ along the ray and

$\frac{E({\lambda }_R)}{E({\lambda }_E)}=\sqrt{\frac{f(r_E)}{f(r_R)}}=\sqrt{\frac{1-s\mathrm{/}{r}_E}{1-s\mathrm{/}{r}_R}}\mathrm{.}$

(c) Redshift parameter. If the redshift parameter of the pair of events is defined by $z:={\displaystyle \frac{E({\lambda }_E)}{E({\lambda }_R)}}-1$, equivalently by $1+z=E({\lambda }_E)\mathrm{/}E({\lambda }_R)$, then

$1+z=\sqrt{\frac{f(r_R)}{f(r_E)}}\mathrm{.}$

The ratio in (b) and (c) is unchanged if $W$ is replaced by $\mu W$ with a positive constant $\mu$: a future null ray determines its future tangent only up to positive scale, and only the ratio of energies is asserted. No observed wavelength, spectral line, calibration or measurement enters the statement; the identification of the ratio with a frequency ratio of an ideal ray is part of the framework's geometric-optics reading, and moving observers require additional factors.

(d) Scope. The statement is a model prediction for the specified ideal ray and the specified static observers; it asserts no redshift measurement and no property of any real source, and it excludes non-geometric-optics propagation, dispersion and any apparatus or analysis model.

• Model prediction, not an observation. The theorem compares two ideal observer energies of one ideal null geodesic in the adopted model. A reported redshift is a measured quantity with an emission line, a spectrograph, calibration, resolution and uncertainty; identifying the two requires an apparatus and analysis model, and this is a subject for experiment items and for the experimental-interface page, not a consequence of the mathematics proved here.

• Where each assumption entered. The metric form gave the pairing $g(W,{\mathrm{\partial }}_0)$ and the factor $f^{-1\mathrm{/}2}$; the Killing property of ${\mathrm{\partial }}_0$ gave constancy; the light-ray rule gave the identification of the ideal ray with a future null geodesic; the observer-energy definition entered through $E=-cg(W,n)$. No field equation and no matter model was used: the result holds for every Schwarzschild exterior datum with $s>0$ irrespective of the field law, and it is the physical reading of the model that attaches the frequency interpretation.