ANISOTROPIC SPACE DYNAMICS · CANONICAL THEORY

Length Vector Effect · Full Derivation

The necessary steps of Chapter 1, Eqs. (1.1)–(1.19), with explicit measurement assumptions, motion geometry, directional projection and consistency conditions.

Based on the final manuscript · Chapter 1

TheoryTheory · DERIVEDReviewMathematical review · CONDITIONALEvidenceEvidence · UNTESTEDCanonicalYesStatus, review and sources

01Starting Assumptions

ASD expresses the motion-associated scale as a directional physical relation: $r'=r\gamma(1-\beta\cos\phi)$ in the specified measurement configuration, with $L'_\perp=\gamma L_\perp$ transversely. The derivation identifies the measured objects and their pairing, solves for the scale factor, projects onto the measurement direction, and returns to comparable lengths and durations. The forward and reverse construction is not imported from a standard Lorentz transformation.

This page reconstructs the necessary derivation in the order of Chapter 1, Section II. A1–A5 are website identifiers for assumptions in the manuscript, not separately numbered axioms in the book. Assumption, Mathematical Result and Physical Interpretation denote roles in the argument; they do not replace the theory-maturity or evidence-status systems.

A1 · Assumption

Metrological standards and vacuum propagation

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).

A2 · Assumption

Length carries a propagation and measurement direction

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.

A3 · Assumption

Two measurement configurations in uniform relative motion

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.

A4 · Assumption

A common scale factor in the forward and reverse constructions

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).

A5 · Assumption

Path pairing used to determine the scale factor

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.

02Measurement Relations

ASD-WEB-LVE-003

r=ct,\qquad \mathbf r=\mathbf c\,t

r denotes the effective path in the specified propagation direction. Vector notation follows the manuscript; r is not automatically an arbitrary three-dimensional Euclidean norm.
Chapter reference: Final consolidated manuscript · Chapter 1, Section II, Eqs. (1.1)–(1.2) · ASD-BOOK-ddbade997b45

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For the two measurements separately:

ASD-WEB-LVE-004

\begin{aligned}\mathbf r&=x\mathbf i+y\mathbf j+z\mathbf k,&r&=ct\\\mathbf r^{\prime}&=x^{\prime}\mathbf i+y^{\prime}\mathbf j+z^{\prime}\mathbf k,&r^{\prime}&=ct^{\prime}\end{aligned}

The same c is used in each measurement. The primed object, endpoints and propagation direction must be specified together with its configuration.
Chapter reference: Final consolidated manuscript · Chapter 1, Section II, Eqs. (1.3)–(1.6) · ASD-BOOK-ddbade997b45

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$t,t'$ 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 for measurement history and unit realization.

03Geometric Construction

ASD-WEB-LVE-005

\begin{aligned}\mathbf r^{\prime}&=\gamma(\mathbf r-\mathbf v t)\\\mathbf v&=v_x\mathbf i+v_y\mathbf j+v_z\mathbf k\\\mathbf r&=\gamma(\mathbf r^{\prime}+\mathbf v t^{\prime})\end{aligned}

Both measurements use the same scale factor. Event pairing and inverse-mapping conditions are addressed in the consistency review. This expression is not assumed to be a Lorentz transformation.
Chapter reference: Final consolidated manuscript · Chapter 1, Section II, Eqs. (1.7)–(1.9) · ASD-BOOK-ddbade997b45

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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.

04Intermediate 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}.

Mathematical Result

The positive root of the scale factor

Choose the positive root by requiring $\gamma=1$ at $v=0$ and a continuous correspondence between positive lengths:

ASD-WEB-LVE-006

\gamma(v)=\frac{1}{\sqrt{1-v^2/c^2}}=\frac{1}{\sqrt{1-\beta^2}},\qquad\beta=\frac vc

The positive root is taken under the manuscript’s forward/reverse pairing and normalization conditions; 0 ≤ v < c. Geometric sufficiency of the normalization remains under formal review.
Chapter reference: Final consolidated manuscript · Chapter 1, Section II, Eq. (1.10) · ASD-BOOK-ddbade997b45

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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:

ASD-WEB-LVE-007

t^{\prime}=t\,\gamma(1-\beta\cos\phi)

φ is the angle between observer relative velocity and the specified propagation or measurement direction. This projection result is not a general coordinate-time transformation without a synchronization rule.
Chapter reference: Final consolidated manuscript · Chapter 1, Section II, Eqs. (1.11)–(1.12) · ASD-BOOK-ddbade997b45

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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 $c$ :

ASD-WEB-LVE-008

r^{\prime}=r\,\gamma(1-\beta\cos\phi)

r and r′ denote length scalars for the selected direction. The manuscript’s arrow notation and directional meaning are explained in the derivation.
Chapter reference: Final consolidated manuscript · Chapter 1, Section II, Eq. (1.13) · ASD-BOOK-ddbade997b45

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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.

05Final 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:

ASD-WEB-LVE-009

\begin{aligned}x^{\prime}&=\gamma(x-vt)\\y^{\prime}&=\gamma y\\z^{\prime}&=\gamma z\end{aligned}

Set v = (v,0,0) and expand the manuscript’s vector construction. The transverse components retain γ and must not be replaced by standard Lorentz transverse-coordinate relations.
Chapter reference: Final consolidated manuscript · Chapter 1, Section II, Eqs. (1.15)–(1.18) · ASD-BOOK-ddbade997b45

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Because $v_y=v_z=0$ , the translational terms in the transverse components vanish. Those components still carry the manuscript's factor $\gamma$ , giving:

ASD-WEB-LVE-001

L^{\prime}_{\perp}=\gamma L_{\perp},\quad\gamma=\frac{1}{\sqrt{1-\beta^2}}

L⊥ is the reference transverse length; L′⊥ is the true length associated with motion as defined by ASD. The compared scale is perpendicular to motion; 0 ≤ v < c.
Chapter reference: Final consolidated manuscript · Chapter 1, Eqs. (1.17)–(1.19) · ASD-BOOK-ddbade997b45

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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.

06Physical Interpretation

Physical Interpretation

The scale factor acts on real spatial relations

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 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.

07Domain of Validity

The basic construction concerns uniform relative motion, specified directions and specified measurement endpoints; $0\le v<c$ , with $\gamma>0$ .
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.

08Consistency Checks

Directly reproducible checks

Rest limit: at $v=0$ , $\gamma=1$ , and the general directional relation reduces to $r'=r$ , manuscript Eq. (1.14).
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.
Transverse limit: at $\phi=\pi/2$ , the directional and transverse-component expressions give the same factor $\gamma$ .
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.
Low-speed limit: the website supplies the following Taylor expansion of Eq. (1.10):

ASD-WEB-LVE-002

\gamma=1+\frac{\beta^2}{2}+O(\beta^4)

β ≪ 1. An algebraic supplement on this website obtained by expanding Chapter 1, Eq. (1.10); the manuscript assigns no separate number to this expansion.
Chapter reference: Final consolidated manuscript · Website low-speed expansion of Chapter 1, Eq. (1.10); no new manuscript number · ASD-BOOK-ddbade997b45

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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.

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.
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.
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.

09Dependencies

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 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.

10Sources

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. It provides metrological context, not institutional confirmation of the full ASD spatial transformation.

Physical Meaning → · Experimental Relations and Tests →

On this page

STATUS & REVIEW

Theory StatusTheory · DERIVED

Mathematical ReviewMathematical review · CONDITIONAL

Overall Evidence StatusEvidence · UNTESTED

Canonical ASDYes · canonical theory baselineChapter 1

Mathematical 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.

Read the formal review items

Module version 1.1.0Reviewed 2026-09-13ASD-BOOK-ddbade997b45

Adopted source, conditions and previous versions

Chapter 1 §II, Eqs. (1.1)–(1.20)

Equation-by-equation check of scale factor, projection, components and transverse result; A1–A5 review retained

Conditions retained in this adoption

A1–A5; tt′≠0; 0≤v<c
Directed effective length is not an arbitrary Euclidean vector norm

Specific unresolved source issues

FINAL-CH1-PAIRING

Source locators: p0309 · p0311 · p0327 · p0331 · p0333 · p0335 · p0337 · p0348 · p0360 · p0365

Manuscript SHA-256: ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e

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Previous source, retained as history

CH1-caf38018fc5a · Chapter 1, Section II, Eqs. (1.1)–(1.19), PDF pp. 4–7; Eq. (1.20) as context

RELATIONSHIP INDEX

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These links index inputs and research relationships. The completion status of each derivation is specified in its text. Untranslated supporting pages open in Chinese.

Depends on

Metrological Foundations

Used by

Dynamics · Inertia and Momentum

Revision history

2026-09-13

CONTENT REVISION

LVE v1.1 · Scientific status and full English edition

Editorial record (Chinese)

· Chinese original

SOURCE & SCOPE

Chapter 1 · Chapter 1 §II, Eqs. (1.1)–(1.20)

This is the English translation of the Chinese canonical module, based on the same final-manuscript source snapshot. The Chinese final manuscript remains authoritative. Website equation IDs are stable citation identifiers; original manuscript numbers are listed separately.

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CITATION & SOURCE FILE

ASD: Length Vector Effect · Full Derivation. ASD-DOC-THEORY-LVE-DERIVATION-EN. Version 1.1.0; 2026-10-24. https://asd-physics.org/en/theory/lve/derivation. Chapter 1 §II, Eqs. (1.1)–(1.20). Source ASD-BOOK-ddbade997b45.

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01Starting Assumptions

Metrological standards and vacuum propagation

Length carries a propagation and measurement direction

Two measurement configurations in uniform relative motion

A common scale factor in the forward and reverse constructions

Path pairing used to determine the scale factor

02Measurement Relations

03Geometric Construction

04Intermediate Steps

1. Determine the scale factor using the manuscript's pairing

The positive root of the scale factor

2. Project the vector construction onto the measurement direction

3. Return from elapsed time to length

05Final LVE Relation

Component route

Directional route

06Physical Interpretation

The scale factor acts on real spatial relations

07Domain of Validity

08Consistency Checks

Directly reproducible checks

09Dependencies

10Sources

Conditions retained in this adoption

Specific unresolved source issues

Previous source, retained as history

Position within the framework

Depends on

Used by

Related open problems

Revision history

LVE v1.1 · Scientific status and full English edition

Continue with the assumptions, equations and sources.