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Free-fall congruences, tidal (curvature) relative acceleration and the flat comparison

Physical scope

Smooth test-field regime of the classical general-relativistic framework: a smooth time-oriented Lorentzian four-manifold with metric-selected connection, ideal freely falling structureless test bodies whose worldlines are future timelike geodesics parametrized by proper time, and smooth one-parameter families of candidate geodesic histories. Their physical interpretation applies when body histories realize the family on a common interval; no existence of bodies with arbitrary release data is assumed or deduced. Excludes bodies with non-negligible self-gravity, self-force, spin, finite size or internal structure, non-geodesic motion, non-smooth or distributional matter and fields, quantum effects, and any comparison with measured data. Every conclusion is a conditional model deduction from the adopted postulates, not an observational claim.

Statement

Adopt the postulates Adopted spacetime model and classical domain of the gravitational framework and Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework of the classical gravitational framework, and let $(M, g, N)$ be the model's smooth time-oriented Lorentzian four-manifold with metric-selected connection $\nabla$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, 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 curvature $R$ (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors). All statements below are deductions within the model so specified; they assert nothing about nature.

At an event $q$ , a frame means an ordered basis $(E_{0}, E_{1}, E_{2}, E_{3})$ of $T_{q} M$ . If an orthonormal frame is intended, it means a tetrad satisfying $g_{q} (E_{A}, E_{B}) = diag (- 1, 1, 1, 1)_{A B}$ ; these terms and their distinction from coordinate frames are defined in Local frame fields, coordinate frames and tetrad fields. Such a frame is pointwise basis data, not a chart on an open set or an observer congruence, which is a smooth future unit timelike vector field on an open set (Observer congruences, their observer curves and adapted charts).

(a) Free-fall congruences. A free-fall congruence of the model is a smooth map $γ : (- ε, ε) \times I \to M$ , with $ε > 0$ and $I \subseteq R$ an interval with nonempty interior, whose values we write $γ (r, s)$ . For each $r \in (- ε, ε)$ the curve $γ_{r} : I \to M$ is defined by $γ_{r} (s) : = γ (r, s)$ and is a future timelike geodesic parametrized by proper time, with tangent field $U_{r} (s) : = \partial_{s} γ (r, s)$ satisfying $g (U_{r}, U_{r}) = - c^{2}, D_{s} U_{r} = 0,$ where $D_{s}$ is the covariant derivative along $γ_{r}$ . Here this term denotes a mathematical family of candidate free-fall histories. Clause (ii) of Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework requires any ideal freely falling massive test body's history to be such a geodesic on its actual proper-time interval; it does not guarantee bodies realizing arbitrary release data or an entire geodesic domain. If a smooth family of these body histories is realized on a common interval $I$ , it is a free-fall congruence in the sense just defined. Conversely each individual such body history can be included mathematically in a constant family $γ (r, s) = γ_{0} (s)$ , without asserting the existence of additional bodies. No injectivity or immersion condition is imposed on the family map.

(b) Deviation equation (model deduction). Let $γ$ be a free-fall congruence and set $J : = \partial_{r} γ$ . Then, along each $γ_{r}$ , the separation field satisfies $D_{s}^{2} J = R (U_{r}, J) U_{r},$ where $D_{s}$ is the covariant derivative along $γ_{r}$ . In particular, the relative four-acceleration encoded by the family’s infinitesimal separation field $J$ is exactly the curvature term: at each event, $J \mapsto R (U_{r}, J) U_{r}$ is an $R$ -linear map on the event's tangent space, vanishes at $J = 0$ , is tensorial and independent of the frame or chart used for its components, and contains no rest mass, charge or internal-structure parameter of the bodies. When the family is realized by ideal test-body histories, this is their infinitesimal relative acceleration within the adopted model.

(c) Flat comparison and the vanishing criterion. If the curvature vanishes on an open subset $O \subseteq M$ , then every free-fall congruence with worldlines in $O$ has $D_{s}^{2} J = 0$ there: the model predicts no tidal relative acceleration of freely falling test bodies in that region. Conversely, if at an event $q$ the operator $J \mapsto R (U, J) U$ vanishes for every future timelike $U$ and every $J \in T_{q} M$ , then $R_{q} = 0$ ; hence within the model the tidal response of all free-fall congruences detects exactly the curvature when all mathematical candidate families of (a) are considered, and $R$ is the model's complete tidal field at every event. Every normalized future timelike $U$ and every separation value $J$ are realized by such a local mathematical family. This does not assert the existence of test bodies realizing those families; inference from only physically realized families would additionally require that they supply all these directions and values.

Status. Clauses (a)-(c) are conditional consequences of the stated postulates and the cited mathematics. No observation, apparatus, measured input or statistical inference is used; the item contains no empirical premises, and the word "tidal" denotes the curvature term in the model's deviation equation, not a measured force on any particular body.

Facts & Assumptions

Given: The adopted framework postulates, a model spacetime with metric-selected connection, a free-fall congruence $γ$ , and an event $q$ .

[A1]

The framework model has a smooth time-oriented Lorentzian four-manifold $(M, g, N)$ with metric-selected connection $\nabla$ and curvature $R$ , all mathematics as in the cited definitions (Adopted spacetime model and classical domain of the gravitational framework clauses (i) and (iv), Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors).

[A2]

An ideal freely falling massive test body of the model has as worldline a future timelike geodesic with four-velocity $U$ , $g (U, U) = - c^{2}$ , and proper time an affine parameter. Its history is a restriction of the unique maximal geodesic with its release event and four-velocity; no force, charge or rest mass enters that evolution rule. This constrains a body's history when realized, without guaranteeing a body for each initial datum or requiring its history to have the maximal geodesic's domain (Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework clause (ii), The free-fall worldline of an ideal test body is determined by its initial event and four-velocity, Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent clause 3).

[L1]

For a smooth family of geodesics parametrized by an affine parameter $σ$ with $g (T, T) = - 1$ , the variation field satisfies $D_{σ}^{2} J = R (T, J) T$ ; for a geodesic with $g (T, T) = - 1$ the proper time is $τ = σ / c$ , and the physical four-velocity is $U = c T$ (The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields clause (b), Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent clauses 2-3, Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold clauses (d)-(e)).

[L2]

If an algebraic curvature tensor at a point of a Lorentzian four-manifold satisfies $R (U, J) U = 0$ for every timelike $U$ and every $J$ , then it vanishes at that point (A curvature tensor vanishes if the tidal operator vanishes for every timelike direction).

[L3]

The curvature tensor is a smooth $(1, 3)$ -tensor; its lowered components satisfy the symmetries and the pairing identity $g (R (X, Y) Z, W) = R_{a b c d} W^{a} X^{c} Y^{d} Z^{b}$ (Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clause (a), The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors clauses (a)-(b)).

[L4]

The geodesic evaluation map is smooth on an open neighborhood of each initial position-velocity datum and time zero (The geodesic initial-value problem and the exponential map of the metric-selected connection clause (b)). Metric compatibility gives $\frac{d}{d s} g (γ', γ') = 2 g (D_{s} γ', γ') = 0$ along each geodesic (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields), so a future timelike geodesic with initial squared speed $- c^{2}$ has proper-time parameter $s$ up to an additive constant (Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent clause 3). These are mathematical conclusions, not body-existence assertions.

Proof

technique · direct

1.1L1given

(From proper time to the unit affine parameter.) Let $γ$ be a free-fall congruence and fix $r$ . By definition, $t \mapsto γ_{r} (t)$ is a future timelike geodesic with $U_{r} = \partial_{t} γ$ and $g (U_{r}, U_{r}) = - c^{2}$ ; by [L1] the rescaled map $Γ_{r} (σ) : = γ_{r} (σ / c)$ is a geodesic with unit tangent $T_{r} = U_{r} / c$ and $g (T_{r}, T_{r}) = - 1$ , so $σ$ is an affine parameter for it.

2.1step 1.1L1L3algebra

(Deviation equation in the model.) Put $c I : = {c t : t \in I}$ and define the smooth family $Γ : (- ε, ε) \times c I \to M$ by $Γ (r, σ) : = γ (r, σ / c)$ . For each fixed $r$ , its variation field is ${\tilde{J}}_{r} (σ) : = \partial_{r} Γ (r, σ) = J (r, σ / c)$ , so [L1] gives $D_{σ}^{2} {\tilde{J}}_{r} = R (T_{r}, {\tilde{J}}_{r}) T_{r}$ . Since $D_{t} = c D_{σ}$ along $γ_{r}$ by the chain rule for the covariant derivative along a curve, $D_{t}^{2} J = c^{2} D_{σ}^{2} {\tilde{J}}_{r} = c^{2} R (T_{r}, J) T_{r} = R (c T_{r}, J) (c T_{r}) = R (U_{r}, J) U_{r}$ by $R$ -bilinearity of $R$ . This proves (b).

3.1step 2.1A2given

(No mass or structure parameter.) The right-hand side of the equation of step 2.1 is built from the worldlines (through $U_{r}$ and $J$ ) and the geometry only; no rest mass, charge or internal-structure parameter occurs in it, and by [A2] the worldlines themselves do not depend on the rest mass of the body. Hence two congruences of the model that agree in the worldlines of their test bodies have identical relative-acceleration fields, whether or not the bodies would be assigned different masses; this is a comparison inside the model, not a statement about any real bodies.

3.2step 2.1L2L3algebra

(Flat comparison and vanishing criterion.) If $R = 0$ on an open set $O$ , then step 2.1 gives $D_{t}^{2} J = 0$ for every free-fall congruence with worldlines in $O$ : the model predicts no tidal relative acceleration there. Conversely, suppose at the event $q$ that $R (U, J) U = 0$ for every future timelike $U$ and every $J$ . If $W$ is timelike, then either $W$ is future timelike or $- W$ is future timelike; in the second case $R (W, J) W = R (- W, J) (- W) = 0$ as well. Thus the tidal operator vanishes for every timelike direction at $q$ , and [L2] gives $R_{q} = 0$ .

3.3A2L4step 2.1L3constructalgebra

(Mathematical realization of arbitrary separation.) Fix $q \in M$ , a future timelike $U \in T_{q} M$ with $g (U, U) = - c^{2}$ , and $J \in T_{q} M$ . In a chart $x$ around $q$ choose $q (r) = x^{- 1} (x (q) + r d x_{q} J)$ for sufficiently small $∣ r ∣$ , and let $W (r)$ have the constant chart components of $U$ . By continuity, $W (r)$ remains future timelike after shrinking the interval. Set $V (r) = c W (r) / \sqrt{- g_{q (r)} (W (r), W (r))}$ ; this is smooth, future timelike, has squared norm $- c^{2}$ , and $V (0) = U$ . The open smooth evaluation domain of [L4] supplies a smooth geodesic family $γ (r, s)$ on a common open rectangle with $γ (r, 0) = q (r)$ and $\partial_{s} γ (r, 0) = V (r)$ . Shrink the rectangle so its tangents remain future-directed. By [L4] their squared speed is $- c^{2}$ , making $s$ proper time up to a constant. Thus this is a mathematical family of (a), without any body-existence inference from [A2]. At $(0, 0)$ its separation is $\partial_{r} γ = J$ and its velocity is $U$ , hence step 2.1 realizes every $R (U, J) U$ with this normalization. Every other future timelike $\tilde{U}$ is a positive multiple of such a $U$ , and tensor bilinearity gives $R (\tilde{U}, J) \tilde{U}$ as that positive square multiple. This supplies the mathematical family interpretation of (c); if body histories realize a restriction of the family on a common interval, [A2] and step 2.1 give its physical interpretation there.

4.1step 2.1step 3.1step 3.2step 3.3A1A2given∎

(Conditional status and conclusion.) Steps 2.1, 3.1, 3.2 and 3.3 derive clauses (b) and (c) from [A1]-[A2] and the cited mathematical results, using no observation, measurement or statistical model; the statements are therefore conditional deductions within the adopted model, and the item carries no empirical premises. Together with the definitions of clause (a) this completes the proof.

Remarks

What the deduction does and does not establish. The theorem shows that, if the framework's spacetime, free-fall and clock postulates are adopted, then free-fall relative acceleration is governed by curvature and is detected by the tidal operator for every observer direction. It does not establish the postulates, does not show that any real body is exactly freely falling, and does not compare the prediction with any measured tidal effect; such a comparison would require a specified apparatus model, empirical premises with uncertainties and a statistical analysis, none of which is supplied here.
Frame independence. The operator $J \mapsto R (U, J) U$ is a tensor contraction at an event; changing the ordered basis of that tangent space or changing coordinates alters its components but not the operator, its vanishing, or the geometric content of the equation. Neither this pointwise basis nor a chart is the smooth family of worldlines defined in clause (a). The absence of a zeroth-order relative acceleration term is the model's precise formulation of the statement that free fall is "locally inertial", and the curvature term is the model's tidal field; the equivalence-principle reading of these facts is motivational and is not asserted as a separate claim.

Free-fall congruences, tidal (curvature) relative acceleration and the flat comparison

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources