# Lorentz Variant Adapter Family ## Status `BEAUTIFUL_PROVISIONAL` This document defines the Lorentz family as a bounded transformation and invariance adapter family for OTOM. The goal is not to treat “Lorentz” as one loose symbol. The goal is to separate the mathematically distinct Lorentz-family objects so they can be used without semantic collision. --- ## One-sentence definition ```text Lorentz-family adapters are transformations, invariance laws, and local-frame structures that preserve a metric, causal cone, or relativistic force/field relation under a declared signature, domain, and group action. ``` --- ## Core invariant In special relativity, a Lorentz transformation `Lambda` preserves the Minkowski quadratic form: ```math \Lambda^T\eta\Lambda=\eta ``` where the Minkowski metric may use either convention: ```math \eta=\operatorname{diag}(-1,+1,+1,+1) ``` or: ```math \eta=\operatorname{diag}(+1,-1,-1,-1) ``` The interval is invariant: ```math s^2=\eta_{\mu\nu}x^\mu x^\nu ``` Admissibility requires the signature convention to be explicit. --- ## Variant taxonomy | Variant | Symbol / form | Preserved structure | OTOM role | |---|---|---|---| | full Lorentz group | `O(1,3)` | Minkowski metric | full symmetry family | | proper Lorentz group | `SO(1,3)` | metric + orientation | orientation-preserving transformations | | proper orthochronous Lorentz group | `SO^+(1,3)` | metric + orientation + time orientation | physical connected component | | spatial rotations | `SO(3)` subgroup | spatial norm within frame | frame reorientation | | boosts | `B(v)` or `B(\varphi)` | interval and causal cone | inertial-frame translation in velocity/rapidity space | | parity | `P` | metric, flips spatial orientation | discrete spatial inversion | | time reversal | `T` | metric, flips time orientation | discrete temporal inversion | | PT | `PT` | metric, flips both | combined discrete transformation | | infinitesimal Lorentz algebra | `so(1,3)` | tangent generators | local linearized transformation | | spinor cover | `SL(2,C)` | double cover of `SO^+(1,3)` | spinor/lifted representation | | tensor transformation | index law | covariance of tensors | object transport law | | field tensor transformation | `F' = Lambda F Lambda^T` | Maxwell covariance | electromagnetic field adapter | | four-vector force | `dp^mu/dtau` | covariant dynamics | relativistic dynamics adapter | | Lorentz force | `q(E+v×B)` / covariant form | charged-particle dynamics | force-law adapter | | local Lorentz frame | tetrad/vierbein | tangent-space metric | curved-spacetime local adapter | | Lorentzian manifold | `(M,g)` with signature `(1,n-1)` | causal structure | geometric domain adapter | | conformal Lorentz relation | `g -> Omega^2 g` | null cone | causal/null-structure adapter | --- ## 1. Standard boost in one spatial direction For a boost along `x` using `c=1`: ```math \begin{aligned} t' &= \gamma(t-vx) \\ x' &= \gamma(x-vt) \\ y' &= y \\ z' &= z \end{aligned} ``` where: ```math \gamma=\frac{1}{\sqrt{1-v^2}} ``` With units restored: ```math \gamma=\frac{1}{\sqrt{1-v^2/c^2}} ``` --- ## 2. Rapidity form Define rapidity: ```math \varphi=\operatorname{artanh}(v/c) ``` Then: ```math \beta=\tanh\varphi ``` ```math \gamma=\cosh\varphi ``` ```math \gamma\beta=\sinh\varphi ``` The boost becomes a hyperbolic rotation: ```math \begin{bmatrix} ct' \\ x' \end{bmatrix} = \begin{bmatrix} \cosh\varphi & -\sinh\varphi \\ -\sinh\varphi & \cosh\varphi \end{bmatrix} \begin{bmatrix} ct \\ x \end{bmatrix} ``` Rapidity adds linearly for collinear boosts: ```math \varphi_{total}=\varphi_1+\varphi_2 ``` This makes rapidity the clean coordinate for boost composition. --- ## 3. Arbitrary-direction boost Let `n` be a unit vector in the boost direction and decompose: ```math \vec{x}=\vec{x}_{\parallel}+\vec{x}_{\perp} ``` where: ```math \vec{x}_{\parallel}=(\vec{x}\cdot\vec{n})\vec{n} ``` Then: ```math t'=\gamma\left(t-\frac{\vec{v}\cdot\vec{x}}{c^2}\right) ``` ```math \vec{x}'_{\parallel}=\gamma(\vec{x}_{\parallel}-\vec{v}t) ``` ```math \vec{x}'_{\perp}=\vec{x}_{\perp} ``` --- ## 4. Rotations as Lorentz subgroup Spatial rotations preserve time and rotate space: ```math \Lambda_R= \begin{bmatrix} 1 & 0 \\ 0 & R \end{bmatrix} ``` where: ```math R\in SO(3) ``` These preserve the Minkowski metric and sit inside the Lorentz group. --- ## 5. Discrete Lorentz transformations Parity: ```math P=\operatorname{diag}(1,-1,-1,-1) ``` Time reversal: ```math T=\operatorname{diag}(-1,1,1,1) ``` Combined PT: ```math PT=\operatorname{diag}(-1,-1,-1,-1) ``` These preserve the metric but change orientation and/or time orientation. They must not be silently merged with the proper orthochronous component. --- ## 6. Lorentz algebra The Lie algebra condition is: ```math X^T\eta+\eta X=0 ``` In four dimensions, there are six generators: ```text 3 rotations + 3 boosts ``` Commutation relations: ```math [J_i,J_j]=\epsilon_{ijk}J_k ``` ```math [J_i,K_j]=\epsilon_{ijk}K_k ``` ```math [K_i,K_j]=-\epsilon_{ijk}J_k ``` The minus sign in the boost-boost commutator is the signature mark of Lorentzian geometry. --- ## 7. Four-vector adapter A four-vector transforms as: ```math V'^\mu=\Lambda^\mu{}_\nu V^\nu ``` Scalar contraction is invariant: ```math V_\mu V^\mu = \eta_{\mu\nu}V^\mu V^\nu ``` OTOM adapter: ```math \alpha_{4vec}:(V,\Lambda,\eta)\rightarrow V' ``` Admissibility: ```math \Lambda^T\eta\Lambda=\eta ``` --- ## 8. Tensor adapter A rank `(r,s)` tensor transforms by applying `Lambda` to each contravariant index and inverse/dual transformation to each covariant index. For a rank-2 contravariant tensor: ```math T'^{\mu\nu}=\Lambda^\mu{}_{\alpha}\Lambda^\nu{}_{\beta}T^{\alpha\beta} ``` For a covariant tensor: ```math T'_{\mu\nu}=\Lambda^{\alpha}{}_{\mu}\Lambda^{\beta}{}_{\nu}T_{\alpha\beta} ``` Index placement is part of the adapter domain. Dropping it causes semantic collision. --- ## 9. Electromagnetic field tensor adapter The electromagnetic field tensor transforms as: ```math F'^{\mu\nu}=\Lambda^\mu{}_{\alpha}\Lambda^\nu{}_{\beta}F^{\alpha\beta} ``` This mixes electric and magnetic fields under boosts. Bounded claim: ```text Electric and magnetic fields are frame-dependent components of a single antisymmetric tensor. ``` Forbidden overclaim: ```text Every field-mixing phenomenon is Lorentzian. ``` --- ## 10. Lorentz force adapter Three-vector form: ```math \vec{F}=q(\vec{E}+\vec{v}\times\vec{B}) ``` Covariant form: ```math \frac{dp^\mu}{d\tau}=qF^{\mu\nu}u_\nu ``` This is a dynamics/force adapter, not the same object as a Lorentz transformation. Guardrail: ```text Lorentz transformation != Lorentz force ``` They share historical naming and relativistic compatibility, but they are different adapter classes. --- ## 11. Spinor / double-cover adapter The proper orthochronous Lorentz group has a double cover: ```math SL(2,\mathbb{C}) \rightarrow SO^+(1,3) ``` A Minkowski vector can be represented as a Hermitian matrix: ```math X=x^\mu\sigma_\mu ``` with transformation: ```math X' = A X A^\dagger ``` where: ```math A\in SL(2,\mathbb{C}) ``` This adapter is necessary for spinor-bearing systems. It should not be collapsed into ordinary vector transformation. --- ## 12. Lorentzian manifold adapter A Lorentzian manifold is: ```math (M,g) ``` where `g` has Lorentzian signature, commonly: ```math (-,+,+,+) ``` or: ```math (+,-,-,-) ``` The metric defines: ```text timelike / null / spacelike ``` separation and causal cones. OTOM role: ```text Lorentzian manifold = domain where causal structure is part of the geometry. ``` --- ## 13. Tetrad / local Lorentz adapter In curved spacetime, local inertial frames use a tetrad/vierbein: ```math g_{\mu\nu}=e^a{}_{\mu}e^b{}_{\nu}\eta_{ab} ``` Local Lorentz transformations act on the internal frame index: ```math e'^a{}_{\mu}=\Lambda^a{}_b e^b{}_{\mu} ``` This is a local gauge/frame adapter, not a global inertial-frame transformation. --- ## 14. Velocity addition adapter Collinear velocity addition: ```math u=\frac{u+v}{1+uv/c^2} ``` Rapidity version: ```math \varphi_u+\varphi_v=\varphi_{total} ``` Use rapidity for composition whenever possible to avoid algebraic drift. --- ## 15. Doppler and aberration adapters Relativistic Doppler shift: ```math f'=f\sqrt{\frac{1-\beta}{1+\beta}} ``` for recession along the line of sight under the chosen convention. Aberration relation: ```math \cos\theta'=\frac{\cos\theta-\beta}{1-\beta\cos\theta} ``` These are observational adapters derived from Lorentz transformations. They should be bounded to signal/light propagation contexts. --- ## 16. Conformal Lorentz / null-cone adapter A conformal transformation preserves the metric up to scale: ```math g'_{\mu\nu}=\Omega^2 g_{\mu\nu} ``` This preserves null cones but not lengths. Bounded role: ```text conformal-Lorentz structure preserves causal/null geometry, not full metric scale. ``` --- ## 17. OTOM mapping | Lorentz object | Preserved witness | Adapter role | |---|---|---| | `Lambda` | `Lambda^T eta Lambda = eta` | metric-preserving transform | | boost | interval + causal cone | inertial-frame velocity transform | | rapidity | additive boost coordinate | composition-safe boost parameter | | rotation | spatial metric in frame | frame reorientation | | parity/time reversal | metric but not orientation/time-orientation | discrete symmetry branch | | four-vector law | scalar contraction | covariant object transport | | tensor law | index-aware covariance | structured field transport | | `F^{mu nu}` | EM covariance | field-mixing adapter | | Lorentz force | covariant charged-particle dynamics | dynamics adapter | | spinor cover | double-cover representation | spinor/quantum representation adapter | | tetrad | local tangent metric | curved-spacetime local-frame adapter | | Lorentzian manifold | causal cone | geometric domain adapter | --- ## Relation to Cramer's-rule adapter Cramer's rule taught the rule: ```text hold a reference interface fixed, replace one direction, measure signed response. ``` Lorentz adapters teach a complementary rule: ```text change frame, preserve the metric witness. ``` The invariant is no longer a shared determinant face; it is the interval/metric form. ```text Cramer: preserve reference face. Lorentz: preserve causal metric. ``` --- ## Relation to SCW-8192 SCW-8192 uses salt, schema, adapter class, and evidence state to preserve interpretation context. Bounded analogy: ```text Lorentz transformation preserves metric context across frames. SCW-8192 preserves causal interpretation context across artifact lineages. ``` Forbidden overclaim: ```text SCW-8192 is physically Lorentzian. ``` Allowed claim: ```text Lorentz invariance is a clean model of context-preserving transformation: the coordinates change, but the declared invariant remains stable. ``` --- ## Failure modes | Failure | Meaning | |---|---| | metric signature omitted | sign errors and invalid invariance checks | | boost and rotation merged | group structure lost | | proper/improper components conflated | orientation/time-orientation erased | | Lorentz force confused with Lorentz transform | adapter-class collision | | global Lorentz transform used in curved spacetime | local/global domain error | | spinor cover collapsed into vector rep | representation error | | units omitted | `c=1` convention misapplied | | tensor indices ignored | covariance law corrupted | | analogy overextended | physical invariance used as semantic proof | --- ## Claim ladder ### `BEAUTIFUL_PROVISIONAL` - Use Lorentz variants as a taxonomy of context-preserving transformation families. - Use Lorentz invariance as analogy for adapter-bound transformation. ### `CALIBRATED_ENGINEERING_DELTA` - Implement explicit matrix checks for `Lambda^T eta Lambda = eta`. - Add rapidity/boost composition tests. - Add tensor-index transformation tests. ### `REVIEWED` - Formalize Lorentz group invariance in Lean. - Prove interval preservation for declared metric signature. - Prove group closure for selected variant component. --- ## Minimal implementation checklist - [ ] Define metric signature enum. - [ ] Define Lorentz matrix admissibility check. - [ ] Define variant enum: boost, rotation, parity, time reversal, PT, tensor, spinor, local-frame, force-law. - [ ] Add determinant/orientation checks. - [ ] Add rapidity composition test. - [ ] Add four-vector interval-preservation test. - [ ] Add tensor transformation test. - [ ] Add Lorentz force as separate adapter class. - [ ] Add local Lorentz/tetrad guardrail. - [ ] Add forbidden-overclaim tests in documentation. --- ## Summary ```text Lorentz is not one adapter. It is a family of metric-, causal-, representation-, and force-preserving adapters that must be separated by domain, signature, group component, and representation. ``` The core lawful pattern is: ```text coordinates may change; the declared invariant must not. ```