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Cosmological redshift: the observer-energy ratio of a null geodesic between comoving observers is the inverse scale-factor ratio

Physical scope

Conditional prediction of the adopted homogeneous cosmological model: the ratio of the observer energies of a specified future null geodesic measured by comoving observers at two events is the inverse ratio of the scale factor at those cosmic times, and the redshift parameter so defined satisfies 1+z = a(t_R)/a(t_E). The statement is an exact consequence of the model metric, the null-geodesic light-ray rule and the observer-energy definition; it asserts no observed redshift, no spectral line, no distant-velocity interpretation and no measurement, and it excludes non-geometric-optics propagation, dispersion and any apparatus or analysis model.

Statement

Adopt the homogeneous cosmological model of Adopted homogeneous and isotropic cosmological model with a comoving perfect fluid with an FLRW model datum $(I, Σ, γ, k, a)$ and metric $g = - c^{2} d t^{2} + a (t)^{2} γ$ (FLRW model data: warped product metric, cosmic time, comoving congruence, scale factor and Hubble parameter), let $\nabla$ be its metric-selected connection (Metric compatibility, torsion freedom and the metric-selected connection class), and adopt the light-ray rule that ideal light rays are future null geodesics of $\nabla$ (Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework clause (iii)). Let $η : J \to M$ , $s \mapsto η (s)$ , be such a ray: a $C^{2}$ curve on an interval $J$ with tangent $V : = η^{'}$ , $g (V, V) = 0$ , $V$ future-directed, and $\nabla_{V} V = 0$ (Geodesic of an affine connection, Timelike, null, spacelike and causal vectors of a Lorentz vector space). Let $s_{E}, s_{R} \in J$ with $s_{E} < s_{R}$ , let $t_{E} : = t (η (s_{E}))$ and $t_{R} : = t (η (s_{R}))$ be the cosmic times of the two events, and for each $s \in J$ define the observer energy of the ray at $η (s)$ relative to the comoving observer by

$E (s) : = - c g_{η (s)} (V (s), n (η (s))), n = c^{- 1} \partial_{t} .$

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

(a) $a (t (η (s))) V^{t} (s)$ is constant along $J$ , where $V^{t} = d t (V)$ ; in particular $a (t_{R}) V^{t} (s_{R}) = a (t_{E}) V^{t} (s_{E})$ .

(b) $E (s) > 0$ along the ray and

$\frac{E (s_{R})}{E (s_{E})} = \frac{a (t_{E})}{a (t_{R})} .$

(c) If the redshift parameter of the pair of events is defined by $z : = \frac{E (s_{E})}{E (s_{R})} - 1$ , that is, by $1 + z = E (s_{E}) / E (s_{R})$ , then

$1 + z = \frac{a (t_{R})}{a (t_{E})} .$

The ratio in (b) and (c) is unchanged if $V$ is replaced by $μ V$ with a positive constant $μ > 0$ : a future null ray determines its tangent only up to positive scale, and only the ratio of energies is asserted. No observed wavelength, spectral line, distance, velocity or measurement enters the statement.

Facts & Assumptions

Given: The adopted model with FLRW datum $(I, Σ, γ, k, a)$ , its metric $g$ , its comoving congruence $n = c^{- 1} \partial_{t}$ , its metric-selected connection $\nabla$ , a future null geodesic $η : J \to M$ with tangent $V$ , events $s_{E} < s_{R}$ in $J$ , and the energies $E (s)$ of the statement.

[L1]

In a product chart $(x^{0}, x^{1}, x^{2}, x^{3})$ with $x^{0} = t$ the metric components are $g_{00} = - c^{2}$ , $g_{0 i} = 0$ , $g_{i j} = a (t)^{2} γ_{i j}$ , the comoving congruence is $n = c^{- 1} \partial_{t}$ , and the Christoffel symbols of $\nabla$ are $Γ^{0}_{00} = Γ^{0}_{0 i} = Γ^{i}_{00} = 0$ , $Γ^{0}_{i j} = a \dot{a} γ_{i j} / c^{2}$ and $Γ^{i}_{0 j} = H δ^{i}_{j}$ with $H = \dot{a} / a$ (The FLRW metric is smooth Lorentzian; the comoving field is unit timelike and cosmic time parametrizes comoving curves by proper time; the slices carry the scaled spatial metric, Christoffel symbols of the FLRW metric in product charts).

[L2]

A curve $η$ is a geodesic with affine parameter $s$ when $\nabla_{V} V = 0$ , equivalently in a chart ${\dot{V}}^{a} + \sum_{b, c} Γ^{a}_{b c} V^{b} V^{c} = 0$ for $a = 0, 1, 2, 3$ (Geodesic of an affine connection, Christoffel symbols of an affine connection, Coordinate geodesic equation).

[L3]

A nonzero causal vector $V$ is future-directed when $g (V, N) < 0$ for the time orientation $N = \partial_{t}$ ; with the components of [L1] this is $- c^{2} V^{t} < 0$ , so $V^{t} > 0$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Timelike, null, spacelike and causal vectors of a Lorentz vector space).

[L4]

The observer energy of a tangent vector $P$ relative to an observer $n$ is $E_{n} (P) = - c g (P, n)$ (Observer-relative energy, spatial momentum and rest mass of a tangent vector clause (a)); for a nonzero future causal $P$ it is positive (Pairing of an observer with future causal vectors, Observer-relative energy, spatial momentum and rest mass of a tangent vector). The ideal light ray of the model is a future null geodesic, and its tangent is determined only up to positive scale (Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework clause (iii)).

[L5]

Differentiable functions of the parameter satisfy the chain and product rules (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ). A function continuous on an interval and with zero derivative at its interior points is constant on the whole interval (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant).

Proof

technique · reduce the time-component geodesic equation using nullness, differentiate the conserved product, and read the energies off the metric

1.1L1L2algebra

(Geodesic equation for $V^{t}$ .) In a product chart around a point of the ray, [L2] gives ${\dot{V}}^{0} + \sum_{b, c} Γ^{0}_{b c} V^{b} V^{c} = 0$ ; by [L1] every symbol $Γ^{0}_{b c}$ vanishes except $Γ^{0}_{i j} = a \dot{a} γ_{i j} / c^{2}$ with both lower indices spatial, so ${\dot{V}}^{0} + \frac{a \dot{a}}{c^{2}} \sum_{i, j} γ_{i j} V^{i} V^{j} = 0$ . Since $V^{0} = V^{t} = d t (V)$ , this is ${\dot{V}}^{t} + \frac{a \dot{a}}{c^{2}} γ_{i j} V^{i} V^{j} = 0$ .

2.1L1L5step 1.1algebra

(Nullness gives a conserved product.) Nullness and [L1] give $0 = - c^{2} (V^{t})^{2} + a^{2} γ_{i j} V^{i} V^{j}$ , so $γ_{i j} V^{i} V^{j} = c^{2} (V^{t})^{2} / a^{2}$ . Substitution in step 1.1 yields ${\dot{V}}^{t} + H (V^{t})^{2} = 0$ . By the product and chain rules [L5], $\frac{d}{d s} (a (t (η (s))) V^{t} (s)) = \dot{a} (t (η (s))) (V^{t} (s))^{2} + a (t (η (s))) {\dot{V}}^{t} (s) = a H (V^{t})^{2} - a H (V^{t})^{2} = 0$ . This calculation holds in a product chart near each ray point; the product is the globally defined $C^{1}$ function $a (t \circ η) d t (η^{'})$ , so its derivative vanishes everywhere in the interior of $J$ . It is continuous on $J$ , including any endpoints present. Since $J$ is an interval, [L5] makes this product constant on all of $J$ , proving (a) and its endpoint evaluation.

3.1L1L3L4step 2.1algebra∎

(Observer energies and the redshift ratio.) By [L4], $E (s) = - c g (V (s), n (η (s)))$ ; since $n = c^{- 1} \partial_{t}$ has components $n^{0} = c^{- 1}$ , $n^{i} = 0$ and $n_{0} = g_{00} n^{0} = - c$ , the pairing is $g (V, n) = n_{0} V^{0} = - c V^{t}$ , so $E (s) = c^{2} V^{t} (s) > 0$ along the ray. Hence $\frac{E (s_{R})}{E (s_{E})} = \frac{V^{t} (s_{R})}{V^{t} (s_{E})} = \frac{a (t_{E})}{a (t_{R})}$ by the constant of step 2.1, which is claim (b). Defining $z$ by $1 + z = E (s_{E}) / E (s_{R})$ and inverting the ratio gives $1 + z = a (t_{R}) / a (t_{E})$ , which is claim (c). Finally, replacing $V$ by $μ V$ with $μ > 0$ replaces every $E (s)$ by $μ E (s)$ , leaving the ratios in (b) and (c) unchanged; this is the scale-independence asserted in the statement.

Remarks

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 the comparison is the subject of Scope and empirical status of the homogeneous cosmological model: model predictions, reported observations and statistical inference are distinct and of future experiment items. In particular the theorem does not by itself assign a relative velocity to the emitter and does not establish that an observed redshift is produced by expansion.
Where each assumption entered. The metric form and its Christoffel symbols produced the conservation law of claim (a); the null-geodesic light rule of the framework entered through the identification of the ideal ray with a future null geodesic; the observer-energy definition entered through $E_{n} (P) = - c g (P, n)$ . No field equation, no equation of state and no matter model was used, so the result holds in every FLRW model datum, vacuum or fluid-filled, irrespective of the spatial curvature parameter $k$ .

Cosmological redshift: the observer-energy ratio of a null geodesic between comoving observers is the inverse scale-factor ratio

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources