--- { "category": "Metrology & Kinematics", "chapter": "Chapter 1", "template": "theory", "version": "1.1.0", "updated": "2026-10-24", "canonical": true, "source_tier": "canonical", "theory_status": "DERIVED", "module_key": "lve", "module_navigation": [ { "path": "/en/theory/lve", "label": "Overview", "en": "Overview" }, { "path": "/en/theory/lve/derivation", "label": "Derivation", "en": "Derivation" }, { "path": "/en/theory/lve/physical-meaning", "label": "Physical Meaning", "en": "Physical Meaning" }, { "path": "/en/theory/lve/tests", "label": "Tests", "en": "Tests" } ], "source_chapter": "Chapter 1", "source_version": "ASD-BOOK-ddbade997b45", "source_sha256": "dace8ff491b4e4ae8a59b2b98e2a6c9c10e689079021ce0bc117985baba68e86", "last_reviewed": "2026-09-13", "open_problems": [ "/open-problems/master-equation", "/open-problems/lve-muon-decay" ], "sources": [ { "label": "Final combined manuscript — adopted scope documented on this page", "kind": "final-combined-manuscript", "url": "/downloads/ASD_Final_Manuscript_Source_Manifest.json" }, { "label": "Previous source: Anisotropic Space Dynamics, final Chinese Chapter 1 (supplied manuscript, 23 pages)", "kind": "historical-source" } ], "id": "ASD-DOC-THEORY-LVE-DERIVATION-EN", "path": "/en/theory/lve/derivation", "title": "Length Vector Effect · Full Derivation", "title_en": "Length Vector Effect · Full Derivation", "summary": "The necessary steps of Chapter 1, Eqs. (1.1)–(1.19), with explicit measurement assumptions, motion geometry, directional projection and consistency conditions.", "evidence_status": "UNTESTED", "equation": "lve", "depends_on": [ "/theory/metrology" ], "related": [ "/en/theory/lve", "/en/theory/lve/physical-meaning", "/en/theory/lve/tests" ], "equations": [ "lve-measurement", "lve-paths", "lve-geometry", "lve-gamma", "lve-time-direction", "lve-direction", "lve-components", "lve", "lve-expansion" ], "source_reference": "Chapter 1 §II, Eqs. (1.1)–(1.20)", "outline": [ "Starting Assumptions", "Measurement Relations", "Geometric Construction", "Intermediate Steps", "Final LVE Relation", "Physical Interpretation", "Domain of Validity", "Consistency Checks", "Dependencies", "Sources" ], "seo_title": "Length Vector Effect (LVE) — Derivation | Anisotropic Space Dynamics", "canonical_url": "https://asd-physics.org/en/theory/lve/derivation", "locale": "en", "mathematical_review": "CONDITIONAL", "review_path": "/en/theory/lve/derivation#review-notes", "translation_path": "/theory/lve/derivation", "review_note": "The solution for γ depends on the forward/reverse path pairing specified in A5. The geometric sufficiency of that pairing and the inverse interpretation of the forward and reverse mappings remain under formal mathematical review.", "translation_of": "/theory/lve/derivation", "translation_version": "1.1.0", "final_source": { "source_id": "ASD-BOOK-ddbade997b45", "chapter": "Chapter 1", "reference": "Chapter 1 §II, Eqs. (1.1)–(1.20)", "locators": [ "p0309", "p0311", "p0327", "p0331", "p0333", "p0335", "p0337", "p0348", "p0360", "p0365" ], "adopted_scope": "Equation-by-equation check of scale factor, projection, components and transverse result; A1–A5 review retained", "conditions": [ "A1–A5; tt′≠0; 0≤v Length and time use calibrated, consistent measurement standards. Vacuum propagation uses the same $c$, with $r=ct$ for a specified light-propagation process. Chapter 1 begins with measurement relations without presupposing the Lorentz transformation, relativistic energy or invariant spacetime intervals. Source: Chapter 1, Sections I–II, Eq. (1.1). The manuscript writes the directional relation as $\mathbf r=\mathbf c\,t$, where $r$ is the effective path length along the actual propagation direction. Without a specified geometric correspondence, it cannot be replaced by the Euclidean norm of an arbitrary three-dimensional vector. A unit vector $\hat{\mathbf n}$ identifies the propagation or measurement direction under discussion. Source: Eq. (1.2) and the directional explanation in Section II. First use $K$ as reference, with $K'$ moving at constant $\mathbf v$; then consider the measurement with reversed relative motion. $\mathbf r,t$ and $\mathbf r',t'$ must be used together with their emission and reception events and their respective configurations. ASD does not interpret a choice of reference frame itself as a change in the internal properties of calibrated clocks. Source: final manuscript, Chapter 1 §II, p0317–p0331. The manuscript sets $\mathbf r'=\gamma(\mathbf r-\mathbf v t)$ and writes $\mathbf r=\gamma(\mathbf r'+\mathbf v t')$ for the reverse measurement. $\gamma$ depends on speed magnitude and is the same in both directions. This construction is a theoretical input to the derivation, not a conclusion established by the unit definition alone. Source: Eqs. (1.7)–(1.9). Chapter 1 normalizes the product of forward and reverse paths to $c^2tt'$ and writes the cross terms as the cancelling pair $\mathbf c\cdot\mathbf v\,tt'$ and $\mathbf v\cdot\mathbf c\,tt'$. This page retains that step as the manuscript's pairing condition. It is not treated as an identity for arbitrary Euclidean vectors in arbitrary directions. Its geometric sufficiency is listed among the review items below. ## Measurement Relations For the two measurements separately: $t,t'$ are the elapsed times of their respective specified propagation processes; $\mathbf i,\mathbf j,\mathbf k$ denote the component directions used in the manuscript. The same $c$ appears in $r=ct$ and $r'=ct'$. These relations do not yet provide a complete mapping between the two sets of events. Chapter 1 subsequently summarizes a process relation as $S=vt$ (Eq. 1.20, Chapter 1 §II). This explains the length–rate–elapsed-time connection. The present LVE derivation uses its vacuum-propagation instance. See [Metrological Foundations](/en/theory/metrology) for measurement history and unit realization. ## Geometric Construction The first expression accounts for relative displacement $\mathbf r-\mathbf v t$, followed by the scale response $\gamma$ of the moving configuration. Relative velocity reverses in the second measurement, hence the plus sign in the manuscript's third expression. The forward and reverse formulas are displayed as its two measurement constructions. No further assumption is made that they already form mutually inverse transformations acting on the same event quadruple. Only the velocity-component expansion, the light-propagation measurement relation and the manuscript's scale-pairing rule enter the calculation. The standard Lorentz transformation is not used as an intermediate step. ## Intermediate Steps ### 1. Determine the scale factor using the manuscript's pairing Chapter 1 first multiplies Eqs. (1.7) and (1.9), writing the following inner-product form. The next two displayed expressions are not separately numbered in the original chapter. $$ \mathbf r'\cdot\mathbf r =\gamma^2(\mathbf r-\mathbf v t)\cdot(\mathbf r'+\mathbf v t'). $$ Using the path pairing and normalization in A5, the manuscript continues: $$ c^2tt' =\gamma^2\left( c^2tt'+\mathbf c\cdot\mathbf v\,tt' -\mathbf v\cdot\mathbf c\,tt'-v^2tt' \right). $$ The cross terms cancel. For nonzero propagation times $tt'\ne0$, this yields successively: $$ c^2=\gamma^2(c^2-v^2), \qquad \gamma^2=\frac{1}{1-v^2/c^2}. $$ Choose the positive root by requiring $\gamma=1$ at $v=0$ and a continuous correspondence between positive lengths: This algebraic result depends on A5. The conditions relating inner-product normalization to direction must remain attached to the result. ### 2. Project the vector construction onto the measurement direction Substituting the propagation relation into Eq. (1.7), manuscript Eq. (1.11) reads: $$ \mathbf c\,t'=\gamma(\mathbf c\,t-\mathbf v t). $$ Following the directional interpretation in Chapter 1, project onto the specified $\hat{\mathbf n}$ and use $\mathbf v\cdot\hat{\mathbf n}=v\cos\phi$: $$ ct'=\gamma(ct-vt\cos\phi). $$ Dividing by $c$ gives: This preserves the manuscript's use of the same $\mathbf c$ symbol for the two propagation processes. Whether a mapping between their directions is required forms part of A5 and the consistency review. The calculation does not directly take the norm of a general three-dimensional expression. ### 3. Return from elapsed time to length Substitute $r'=ct'$ and $r=ct$ into the preceding relation and cancel $c$: Manuscript Eq. (1.13) retains arrows on its length symbols to emphasize direction dependence. In the scalar ratio displayed here, $r,r'$ denote lengths along the selected direction, with $\phi$ retaining the directional information. The physical claim has not been converted into a coordinate effect. ## Final LVE Relation ### Component route Manuscript Eq. (1.15) restates $\mathbf r'=\gamma(\mathbf r-\mathbf v t)$. Set $\mathbf v=v\mathbf i$ and compare components: Because $v_y=v_z=0$, the translational terms in the transverse components vanish. Those components still carry the manuscript's factor $\gamma$, giving: ### Directional route Set $\phi=\pi/2$ in the general directional relation. Then $\cos\phi=0$, so $r'=\gamma r$, manuscript Eq. (1.19). This is the same transverse factor as in Eqs. (1.17) and (1.18). The website notation $L_\perp$ explicitly identifies the transverse scale being compared. ## Physical Interpretation ASD interprets $L'_\perp=\gamma L_\perp$ as a transverse expansion of real spatial length associated with motion. The mathematical form of the factor is the same as $\gamma$ in SR; this does not imply identical interpretations of space or time. When $r$ denotes light propagation over one period, Chapter 1 applies the same directional relation to wavelength and then uses $\lambda f=c$ to obtain the frequency relation. That application is not part of the preceding derivation of $\gamma$. See [Physical Meaning](/en/theory/lve/physical-meaning) for the distinction between rods, measurements, wavelengths and true lengths, and for the formal comparison with SR. This page preserves the argument and its dependencies; electromagnetic, momentum and gravitational derivations belong to their respective modules. ## Domain of Validity - The basic construction concerns uniform relative motion, specified directions and specified measurement endpoints; $0\le v0$. - The transverse result requires a scale perpendicular to the defined direction of motion. Use Eq. (1.13) for a general direction; multiplication by $\gamma$ alone is not universal. - The basic propagation relation refers to vacuum $c$. Media require optical paths, refractive indices, boundary conditions and conditions for moving media. - Rotation or changing velocity cannot be modeled merely by applying one uniform-motion expression to an entire apparatus. Segments, closed paths and clock-reading procedures must be specified. - $\gamma$ diverges as $v\to c$. Chapter 1 does not establish a superluminal domain or complete a dynamical construction of the microscopic mechanism. ## Consistency Checks ### Directly reproducible checks 1. **Rest limit:** at $v=0$, $\gamma=1$, and the general directional relation reduces to $r'=r$, manuscript Eq. (1.14). 2. **Dimensions:** $\beta,\gamma,\cos\phi$ are dimensionless. Both sides of the length relation have length units; both sides of the elapsed-time relation have time units. 3. **Transverse limit:** at $\phi=\pi/2$, the directional and transverse-component expressions give the same factor $\gamma$. 4. **Opposite-direction scalar factors:** $\gamma^2(1-\beta)(1+\beta)=1$. This checks the product of the co- and counter-directional scalar factors; it does not prove that general vector mappings are inverses. 5. **Low-speed limit:** the website supplies the following Taylor expansion of Eq. (1.10):
Mathematical definitions and application conditions requiring review These are review records from preparation of the website module. They do not replace the original Chapter 1 expressions. 1. **Inner-product normalization, Chapter 1 §II:** a general Euclidean inner product includes the angle between the two directions. Writing $\mathbf r'\cdot\mathbf r$ directly as $c^2tt'$ and representing both cross terms with the same $\mathbf c$ require an explicit path pairing and directional mapping. The original calculation is retained, with its condition identified as A5. 2. **Correspondence of the forward and reverse formulas, Eqs. (1.7) and (1.9):** if both are interpreted as general Euclidean inverse transformations of the same event, transverse projection gives $r_\perp=\gamma r'_\perp=\gamma^2r_\perp$, a conflicting constraint for nonzero transverse length and nonzero speed. The measured objects, bases and endpoints in the two measurements need an explicit correspondence. This page does not assume that inverse consistency has already been established. 3. **Experimental mapping, Section III:** the refractive-index dependence of Fizeau, muon survival distributions, actual transported-clock trajectories, and the area and timing conventions for Sagnac each require further specification. The original expressions and their review status remain on the [tests page](/en/theory/lve/tests).
## Dependencies | Step | Dependency | Role | | --- | --- | --- | | Directional measurement | Chapter 1, Eqs. (1.1)–(1.6), measurement standards | A1–A3; physical and measurement inputs | | Moving construction | Chapter 1, Eqs. (1.7)–(1.9) | A4; ASD construction | | Determining $\gamma$ | Forward/reverse product and path normalization | A5; conditional algebraic result | | General directional length | $\gamma$, directional projection, $r=ct$ | Eq. (1.13) | | Transverse LVE | General directional or component relation | Eqs. (1.17)–(1.19) | | Wavelength and frequency | LVE, $\lambda=r$, $\lambda f=c$ | Downstream relations, not upstream assumptions | See [Metrology](/en/theory/metrology) for context. Later momentum, magnetic-interaction and orbital modules have their own additional inputs; they are not used to prove this scale factor retrospectively. The microscopic generating mechanism is associated with the [master-equation open problem](/en/open-problems/master-equation). ## Sources The main argument comes from Chapter 1, Section II, Eqs. (1.1)–(1.19). Eq. (1.20) is used only to explain process relations. Intermediate calculations that were unnumbered in the manuscript remain unnumbered. A1–A5 and ASD-WEB-LVE IDs are website citation indices. The external SI definition is available from [BIPM](https://www.bipm.org/en/si-base-units/metre). It provides metrological context, not institutional confirmation of the full ASD spatial transformation. [Physical Meaning →](/en/theory/lve/physical-meaning) · [Experimental Relations and Tests →](/en/theory/lve/tests)