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The free-fall worldline of an ideal test body is determined by its initial event and four-velocity

Physical scope

Model-conditional statement within the classical smooth gravitational framework with its adopted clock, free-fall and light-ray assumptions: under the test-body idealization, two ideal freely falling massive test bodies released at the same event with the same physical four-velocity have the same worldline. It excludes bodies with non-negligible self-field, finite-size or tidal couplings beyond the idealization, self-force and radiation back-reaction, and quantum effects; it makes no empirical claim that all bodies in nature fall alike, and it does not establish the equivalence principle as a law of nature.

Statement

Let a model of the classical gravitational framework be specified by a time-oriented smooth Lorentzian four-manifold $(M, g, N)$ with its metric-selected connection (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 assume the clock, free-fall and light-ray assumptions of Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework. Let $p \in M$ and let $U_{0} \in T_{p} M$ be a future timelike unit vector, $g_{p} (U_{0}, U_{0}) = - c^{2}$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space).

Suppose $B_{1}$ and $B_{2}$ are two ideal freely falling massive test bodies of the model that are released at the event $p$ with physical four-velocity $U_{0}$ : their worldlines $γ_{i} : I_{i} \to M$ pass through $p$ and their four-vectors there equal $U_{0}$ . Let $τ_{i}$ be their proper times, with origins chosen so that $0 \in I_{i}$ and $γ_{i} (0) = p$ . Let $γ_{p, U_{0}} : I_{p, U_{0}} \to M$ be the unique maximal geodesic through $(p, U_{0})$ . Then each physical worldline is a restriction of that maximal geodesic:

$I_{i} \subseteq I_{p, U_{0}}, γ_{i} (τ) = γ_{p, U_{0}} (τ) (τ \in I_{i}), i = 1, 2.$

In particular, $γ_{1}$ and $γ_{2}$ coincide wherever both are defined. The theorem does not assume that either physical worldline has the entire maximal geodesic's domain. The unique geodesic continuation is determined by the release event and the initial four-velocity alone; parameters that distinguish the two bodies, such as their rest masses, do not enter.

Facts & Assumptions

Given: A smooth model as in the statement, an event $p$ , a future timelike unit vector $U_{0} \in T_{p} M$ , and two ideal freely falling massive test bodies $B_{1}, B_{2}$ released at $p$ with four-velocity $U_{0}$ .

[A1]

Clause (ii) of Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework is the adopted physical assumption that an ideal freely falling massive test body has as worldline a future timelike geodesic of the metric-selected connection, traversed with physical four-velocity $U = d γ / d τ$ satisfying $g (U, U) = - c^{2}$ ; clause (i) is the adopted assumption that an ideal clock measures the proper time of its worldline.

[L1]

For the metric-selected connection there is, for every $(p, v) \in T M$ , a unique maximal geodesic with $γ (0) = p$ and $γ^{'} (0) = v$ , and every geodesic with those data is its restriction. Proper time is an affine parameter of a future timelike geodesic, and along it the four-velocity is $d γ / d τ$ , with $D_{τ} (d γ / d τ) = 0$ (The geodesic initial-value problem and the exponential map of the metric-selected connection, Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent, Geodesic of an affine connection).

[L2]

A worldline with four-velocity $U$ satisfies $g (U, U) = - c^{2}$ and is future-directed; proper time is strictly increasing along it, so the proper-time parametrization is a bijection from its parameter interval onto its image (Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold, Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent).

Proof

technique · direct

1.1A1L2given

For $i = 1, 2$ , [A1] makes the worldline of $B_{i}$ a future timelike geodesic traversed with four-velocity $U_{i} = d γ_{i} / d τ_{i}$ satisfying $g (U_{i}, U_{i}) = - c^{2}$ , and the four-vector of $B_{i}$ at $p$ is its tangent with respect to its own proper time at the parameter value where it passes through $p$ . Choosing the proper-time origins so that this value is $0$ , we have $γ_{i} (0) = p$ and, by the hypothesis, $U_{i} (0) = U_{0}$ .

2.1step 1.1L1algebra

By [L1], there is a unique maximal geodesic $γ_{p, U_{0}}$ with $γ_{p, U_{0}} (0) = p$ and $γ_{p, U_{0}}^{'} (0) = U_{0}$ , and every geodesic with these data is its restriction. Applying this to $γ_{1}$ and $γ_{2}$ from step 1.1 gives $I_{i} \subseteq I_{p, U_{0}}$ and $γ_{i} = γ_{p, U_{0}} ∣_{I_{i}}$ for each $i$ . They therefore agree at every common proper-time value, while their own domains need not be maximal.

3.1step 2.1A1L1algebra∎

The initial data of the geodesic initial-value problem are the event $p$ and tangent vector $U_{0}$ ; the geodesic equation $D_{τ} (d γ / d τ) = 0$ and its maximal solution contain no rest mass, charge, or other parameter of the body, and clause (ii) of [A1] requires each physical worldline only to be a future timelike geodesic. Therefore the unique geodesic continuation, and the values of each physical worldline wherever defined, are determined by $(p, U_{0})$ alone; the rest masses or other distinguishing parameters of $B_{1}$ and $B_{2}$ do not affect them.

Remarks

Model statement, not an empirical law. The theorem is a conditional consequence of the adopted free-fall rule and the mathematical initial-value theory. It does not assert that bodies in nature fall alike, does not test the equivalence principle, and carries the test-body exclusions of Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework: negligible self-field, no self-force or radiation back-reaction, internal structure small compared with the variation scale of the geometry, and no quantum effects. Empirical comparisons of free-fall universality are separate reported observations with their own apparatus, calibration and uncertainty qualifications; none is recorded here.
No new empirical premises. The proof uses no measured input and no experiment as a premise, so no empirical_premises entries are attached; the conclusion inherits the model and idealization qualifications of the postulate.

The free-fall worldline of an ideal test body is determined by its initial event and four-velocity

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources