13 KiB
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
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:
\Lambda^T\eta\Lambda=\eta
where the Minkowski metric may use either convention:
\eta=\operatorname{diag}(-1,+1,+1,+1)
or:
\eta=\operatorname{diag}(+1,-1,-1,-1)
The interval is invariant:
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:
\begin{aligned}
t' &= \gamma(t-vx) \\
x' &= \gamma(x-vt) \\
y' &= y \\
z' &= z
\end{aligned}
where:
\gamma=\frac{1}{\sqrt{1-v^2}}
With units restored:
\gamma=\frac{1}{\sqrt{1-v^2/c^2}}
2. Rapidity form
Define rapidity:
\varphi=\operatorname{artanh}(v/c)
Then:
\beta=\tanh\varphi
\gamma=\cosh\varphi
\gamma\beta=\sinh\varphi
The boost becomes a hyperbolic rotation:
\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:
\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:
\vec{x}=\vec{x}_{\parallel}+\vec{x}_{\perp}
where:
\vec{x}_{\parallel}=(\vec{x}\cdot\vec{n})\vec{n}
Then:
t'=\gamma\left(t-\frac{\vec{v}\cdot\vec{x}}{c^2}\right)
\vec{x}'_{\parallel}=\gamma(\vec{x}_{\parallel}-\vec{v}t)
\vec{x}'_{\perp}=\vec{x}_{\perp}
4. Rotations as Lorentz subgroup
Spatial rotations preserve time and rotate space:
\Lambda_R=
\begin{bmatrix}
1 & 0 \\
0 & R
\end{bmatrix}
where:
R\in SO(3)
These preserve the Minkowski metric and sit inside the Lorentz group.
5. Discrete Lorentz transformations
Parity:
P=\operatorname{diag}(1,-1,-1,-1)
Time reversal:
T=\operatorname{diag}(-1,1,1,1)
Combined PT:
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:
X^T\eta+\eta X=0
In four dimensions, there are six generators:
3 rotations + 3 boosts
Commutation relations:
[J_i,J_j]=\epsilon_{ijk}J_k
[J_i,K_j]=\epsilon_{ijk}K_k
[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:
V'^\mu=\Lambda^\mu{}_\nu V^\nu
Scalar contraction is invariant:
V_\mu V^\mu = \eta_{\mu\nu}V^\mu V^\nu
OTOM adapter:
\alpha_{4vec}:(V,\Lambda,\eta)\rightarrow V'
Admissibility:
\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:
T'^{\mu\nu}=\Lambda^\mu{}_{\alpha}\Lambda^\nu{}_{\beta}T^{\alpha\beta}
For a covariant tensor:
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:
F'^{\mu\nu}=\Lambda^\mu{}_{\alpha}\Lambda^\nu{}_{\beta}F^{\alpha\beta}
This mixes electric and magnetic fields under boosts.
Bounded claim:
Electric and magnetic fields are frame-dependent components of a single antisymmetric tensor.
Forbidden overclaim:
Every field-mixing phenomenon is Lorentzian.
10. Lorentz force adapter
Three-vector form:
\vec{F}=q(\vec{E}+\vec{v}\times\vec{B})
Covariant form:
\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:
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:
SL(2,\mathbb{C}) \rightarrow SO^+(1,3)
A Minkowski vector can be represented as a Hermitian matrix:
X=x^\mu\sigma_\mu
with transformation:
X' = A X A^\dagger
where:
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:
(M,g)
where g has Lorentzian signature, commonly:
(-,+,+,+)
or:
(+,-,-,-)
The metric defines:
timelike / null / spacelike
separation and causal cones.
OTOM role:
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:
g_{\mu\nu}=e^a{}_{\mu}e^b{}_{\nu}\eta_{ab}
Local Lorentz transformations act on the internal frame index:
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:
u=\frac{u+v}{1+uv/c^2}
Rapidity version:
\varphi_u+\varphi_v=\varphi_{total}
Use rapidity for composition whenever possible to avoid algebraic drift.
15. Doppler and aberration adapters
Relativistic Doppler shift:
f'=f\sqrt{\frac{1-\beta}{1+\beta}}
for recession along the line of sight under the chosen convention.
Aberration relation:
\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:
g'_{\mu\nu}=\Omega^2 g_{\mu\nu}
This preserves null cones but not lengths.
Bounded role:
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:
hold a reference interface fixed, replace one direction, measure signed response.
Lorentz adapters teach a complementary rule:
change frame, preserve the metric witness.
The invariant is no longer a shared determinant face; it is the interval/metric form.
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:
Lorentz transformation preserves metric context across frames.
SCW-8192 preserves causal interpretation context across artifact lineages.
Forbidden overclaim:
SCW-8192 is physically Lorentzian.
Allowed claim:
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
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:
coordinates may change; the declared invariant must not.