Conserved reference energy

Differentiate final (2.27) with respect to both generalized velocities to obtain (2.28), then apply the Legendre transform; D>0.

Theory · DERIVEDEquation version 1.0.0

Based on the final manuscript · Chapter 2

ASD-WEB-GRV-028

\begin{aligned}p_r&=\frac{m_0\dot r}{X\sqrt D},\qquad p_\varphi=\frac{m_0r^2\dot\varphi}{\sqrt D},\\\mathcal E&=\frac{E_\infty}{m_0c^2}=\frac{p_r\dot r+p_\varphi\dot\varphi-\mathcal L}{m_0c^2}=\frac X{\sqrt D}.\end{aligned}

Differentiate final (2.27) with respect to both generalized velocities to obtain (2.28), then apply the Legendre transform; D>0.
Chapter reference: Final collected manuscript · Chapter 2 §IV.2 (2.28)—(2.31) · ASD-BOOK-ddbade997b45

Assumptions and derivation Registry and dependencies Research this equation Discuss this equation

Source and ID

ASD-WEB-GRV-028 is the stable website equation ID. It retains existing links and does not replace the manuscript reference. Use the copy button to obtain LaTeX.

Return to premises and derivation

Adopted source, conditions and previous versions

Final collected manuscript · Chapter 2 §IV.2 (2.28)—(2.31)

Differentiate final (2.27) with respect to both generalized velocities to obtain (2.28), then apply the Legendre transform; D>0.

Conditions retained in this adoption

Fixed G₀; isolated, static, spherically symmetric, uncharged exterior.
Areal radius, actual spatial length and reference optical path are defined separately.
Weak-field perturbation, slow-rotation far field and boundary statistics each have separate conditions.

Specific unresolved source issues

Microscopic source matching, dynamical strong fields and full boundary-state dynamics remain open.

Source locators: p0682 · p0694 · p0698 · p0701 · p0703 · p0705 · p0715 · p0717 · p0719 · p0730 · p0732 · p0779 · 最终合订稿 · 第二章 §四.2 (2.28)—(2.31)

Manuscript SHA-256: ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e

Download source identity manifest

The manifest identifies the source; it is not the complete manuscript. Adopting this source does not upgrade evidence or grant a new licence.

Previous source, retained as history

FND-3dde8d8e081d · 第二章 / Chapter 2 · (2.31) and unnumbered radial momentum, p. 13

DEPENDS ON

Dependencies

USED BY

Uses of this equation

Discuss this page Browse all equations