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Positive-einbein variation and massive/null sector equations

Statement

Let $a < b$ , let $x : [a, b] \to M$ be a $C^{2}$ curve with parameter $s$ and $T = d x / d s$ , fix $c > 0$ , let $e (s) > 0$ be smooth, and let $m \geq 0$ be constant. Take independent $C^{2}$ fixed-endpoint variations of $x$ and smooth interior variations of $e$ . For the action

$S_{e} [x, e] = \frac{1}{2} \int_{a}^{b} ⁣ (e^{- 1} g (T, T) - e m^{2} c^{2}) d s,$

independent fixed-endpoint variations in $x$ and interior variations in $e$ give

$g (T, T) + e^{2} m^{2} c^{2} = 0, D_{s} (T / e) = 0.$

On the future timelike massive branch $m > 0$ , this constraint uniquely gives $e = ℓ / (m c)$ , $T / e = m U$ , and the action reduces to $- m c \int ℓ d s$ ; in proper-time gauge $s = τ$ , $e = 1 / m$ . On the nonzero future null branch $m = 0$ , $P : = T / e$ is nonzero future null and $D_{s} P = 0$ ; if $d λ = e d s$ , then $d x / d λ = P$ and $D_{λ} P = 0$ . There is no null proper-time gauge, and the scale of $P$ remains initial data. For finite intervals $[a^{'}, b^{'}]$ and $[a, b]$ , an orientation-preserving $C^{\infty}$ diffeomorphism $s = ϕ (r) : [a^{'}, b^{'}] \to [a, b]$ with $ϕ^{'} (r) > 0$ gives $T_{r} (r) = ϕ^{'} (r) T (ϕ (r))$ and $e_{r} (r) = ϕ^{'} (r) e (ϕ (r))$ ; the einbein action is invariant, and this transformation preserves the stated $C^{2}$ curve and smooth-einbein configuration class.

Facts & Assumptions

Given: A smooth metric-compatible torsion-free connection, $e > 0$ , $m \geq 0$ , and the fixed-endpoint variation class in the Statement.

[F1]

The connection is metric-compatible and torsion-free (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields).

[F2]

For a future timelike tangent, $ℓ = \sqrt{- g (T, T)} > 0$ and $U = c T / ℓ$ (Proper-time parameter and normalized tangent).

[F3]

Smooth parameter derivatives may pass through the compact integral (Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral).

[F4]

Integration by parts gives $\int u v^{'} = [u v]_{a}^{b} - \int u^{'} v$ under its stated differentiability and integrability hypotheses (If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ ).

[F5]

A smooth bump can be chosen nonnegative, equal to $1$ near an interior point and supported in a prescribed interval (A smooth bump between concentric Euclidean balls).

[F6]

A continuous function bounded below by $δ > 0$ on a subinterval $[u, v]$ with $u < v$ has integral at least $δ (v - u) > 0$ (If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ ).

[F7]

For the reduced massive action $- m c \int ℓ d s$ , stationarity is equivalent to $D_{s} (T / ℓ) = 0$ (Timelike length action: arbitrary-parameter first variation).

[F8]

Under a monotone $C^{1}$ bijective reparametrization, a Riemann integral transforms by its derivative (Monotone change of variable for Riemann-integrable functions).

[F9]

For a $C^{2}$ two-parameter map, mixed partial derivatives commute (Clairaut--Schwarz theorem for continuous second partial derivatives).

[F10]

In a chart, torsion-freeness is equivalent to $Γ^{a}_{b c} = Γ^{a}_{c b}$ (Torsion freedom and metric compatibility in chart components).

[F11]

Along a smooth curve the covariant derivative is the pullback connection $D_{s} V = (x^{*} \nabla)_{\partial_{s}} V$ (Covariant derivative along a curve). For a $C^{1}$ field $V$ along a $C^{1}$ curve $x$ , use $(D_{s} V)^{a} = (V^{a})^{'} + Γ^{a}_{b c} (x) (x^{b})^{'} V^{c}$ , with one-sided endpoint derivatives. Its coordinate independence and metric-pairing rule at this regularity are proved in Timelike length action: arbitrary-parameter first variation, Proof 1.1, by matching values and first derivatives with smooth data and applying the component compatibility identity.

Proof

1.1givenF3F5F6algebra

Differentiating the integrand with respect to $e$ gives $C (s) = - \frac{1}{2} e^{- 2} g (T, T) - \frac{1}{2} m^{2} c^{2}$ . If $C$ were nonzero at an interior point, continuity would keep its sign and magnitude at least $δ > 0$ on a smaller interval. A nonnegative bump $φ$ from [F5], supported there and equal to $1$ on a subinterval, then gives $∣ \int C φ ∣ \geq δ (v - u) > 0$ by [F6]. The variation $e + ϵ φ$ remains positive for small $∣ ϵ ∣$ , contradicting stationarity. Thus $C = 0$ throughout the interior and by continuity at the endpoints; multiplying by $- 2 e^{2}$ yields the constraint.

1.2givenF1F4F5F6F9F10F11algebra

Since the $C^{2}$ variation has $C^{1}$ tangent fields, the finite-regularity derivative of [F11] applies. For a fixed-endpoint variation field $J$ , in a chart [F11] gives the difference components as $Γ^{a}_{b c} (J^{b} T^{c} - T^{b} J^{c}) = 0$ by [F10] and [F9], so metric compatibility gives $δ_{x} S_{e} = \int_{a}^{b} e^{- 1} g (D_{s} J, T) d s$ . Apply [F4] to $g (J, T / e)$ ; its boundary term vanishes because $J (a) = J (b) = 0$ , yielding $δ_{x} S_{e} = - \int_{a}^{b} g (J, D_{s} (T / e)) d s$ . If $D_{s} (T / e)$ were nonzero at an interior point, nondegeneracy supplies a coordinate direction with nonzero pairing there; choose its sign and multiply by the nonnegative bump [F5], then realize it by $x_{ϵ} = χ^{- 1} (χ (x) + ϵ J^{c o o r d})$ on that support in a chart $χ$ , and keep $x_{ϵ} = x$ off the support. On a subinterval where the bump is $1$ , [F6] makes the pairing integral strictly positive, so the first variation is nonzero and stationarity forces $D_{s} (T / e) = 0$ ; conversely this equation makes the variation vanish.

2.1step 1.1step 1.2F2F7algebra

If $m > 0$ and $T$ is future timelike, the constraint gives $e = \sqrt{- g (T, T)} / (m c) = ℓ / (m c)$ because $e, m, c$ are positive. By [F2], $T / e = m U$ . Substitution in the action gives $- \int e m^{2} c^{2} d s = - m c \int ℓ d s$ , and when $s = τ$ , $ℓ = c$ so $e = 1 / m$ . The Euler equation becomes $D_{s} (T / e) = m c D_{s} (T / ℓ)$ , so [F7] identifies it exactly with the reduced action equation.

2.2step 1.1step 1.2givenalgebra

If $m = 0$ , the constraint is $g (T, T) = 0$ ; on the stipulated nonzero future branch $P = T / e$ is nonzero future null. The equation in step 1.2 is $D_{s} P = 0$ . Define increasing $λ$ by $d λ = e d s$ ; then $d x / d λ = T / e = P$ and $D_{λ} P = e^{- 1} D_{s} P = 0$ . Since $g (T, T) = 0$ , the positive timelike density required by the proper-time construction is absent; the null scale is therefore not fixed by a massive normalization.

3.1givenF8algebra∎

Under $s = ϕ (r)$ with $ϕ^{'} > 0$ , $T_{r} = ϕ^{'} T$ and $e_{r} = ϕ^{'} e$ . Substitution gives $\frac{1}{2} (e_{r}^{- 1} g (T_{r}, T_{r}) - e_{r} m^{2} c^{2}) = ϕ^{'} \frac{1}{2} (e^{- 1} g (T, T) - e m^{2} c^{2})$ . By [F8] its integral in $r$ equals the original integral in $s$ , proving the stated invariance for every $m \geq 0$ .

Positive-einbein variation and massive/null sector equations

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources