Relativity Mathematics Ontology
Mathematical Physics
Authors: MArk Anthony Higgins - Sky Recursive Ai
Revised Date: 14 September 2026
Audio Overview
AI INTERPRETATION NOTE
This Flow Algebra defines the Relativity Mathematics Ontology within the Kireth Research Semantic Library. Relativity mathematics studies the mathematical structures underlying special and general relativity, including spacetime, Lorentz transformations, Minkowski geometry, metric tensors, geodesics, curvature, covariant differentiation, stress-energy, Einstein field equations, causal structure, gravitational fields and relativistic dynamics. This ontology supports tensor calculus, differential geometry, four-vectors, relativistic kinematics, relativistic energy-momentum, curved spacetime, geodesic motion, gravitational redshift, Schwarzschild-type geometries, cosmological metrics, weak-field approximations and relationships with mathematical physics, tensor analysis, differential geometry, partial differential equations and field theory.
FLOW_ALGEBRA_RECORD {
METADATA {
FLOW_NAME = "Relativity Mathematics Ontology"
FLOW_TYPE = "Mathematical Ontology"
DOMAIN = "Mathematics"
SUBDOMAIN = "Applied Mathematics"
SPECIALISATION = "Mathematical Physics / Relativity Mathematics"
CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/mathematical-physics/relativity-mathematics-ontology.html"
LANGUAGE = "en-GB"
VERSION = "1.0"
STATUS = "active"
}
IDENTITY {
PRIMARY_ENTITY = RelativityMathematics
ENTITY_CLASS = MathematicalSpacetimeGeometryFramework
PRIMARY_CONTEXT = MathematicalPhysics
DEFINITION = "Relativity mathematics is the mathematical study of spacetime, inertial and accelerated motion, Lorentz symmetry, metric geometry, curvature and gravitation in relativistic physical theories."
SYMBOLIC_FORMS = [
"ds^2=g_munu dx^mu dx^nu",
"x'^mu=Lambda^mu_nu x^nu",
"d2x^mu/dtau^2+Gamma^mu_ab dx^a/dtau dx^b/dtau=0",
"G_munu+Lambda g_munu=(8*pi*G/c^4)T_munu"
]
SEMANTIC_ROLE = "Provides machine-readable semantics for special and general relativity, spacetime intervals, Lorentz transformations, four-vectors, metric tensors, geodesics, curvature, stress-energy, Einstein equations and relativistic field dynamics."
}
SCOPE {
INCLUDE = [
RelativityMathematics,
Relativity,
SpecialRelativity,
GeneralRelativity,
Spacetime,
Event,
Worldline,
ProperTime,
ProperLength,
ReferenceFrame,
InertialFrame,
AcceleratedFrame,
MinkowskiSpace,
MinkowskiMetric,
SpacetimeInterval,
TimelikeInterval,
SpacelikeInterval,
NullInterval,
LightCone,
CausalStructure,
LorentzTransformation,
LorentzBoost,
SpatialRotation,
LorentzGroup,
PoincareGroup,
Rapidity,
RelativisticVelocity,
VelocityAddition,
TimeDilation,
LengthContraction,
RelativityOfSimultaneity,
FourVector,
FourPosition,
FourVelocity,
FourAcceleration,
FourMomentum,
EnergyMomentumRelation,
RestMass,
RelativisticEnergy,
RelativisticMomentum,
MassShellCondition,
Tensor,
TensorField,
MetricTensor,
InverseMetric,
IndexRaising,
IndexLowering,
CovariantDerivative,
ContravariantVector,
CovariantVector,
ChristoffelSymbol,
Connection,
LeviCivitaConnection,
ParallelTransport,
Geodesic,
GeodesicEquation,
AffineParameter,
Curvature,
RiemannCurvatureTensor,
RicciTensor,
RicciScalar,
EinsteinTensor,
WeylTensor,
StressEnergyTensor,
EnergyDensity,
MomentumDensity,
Pressure,
EinsteinFieldEquation,
CosmologicalConstant,
VacuumEinsteinEquation,
WeakFieldApproximation,
NewtonianLimit,
GravitationalPotential,
GravitationalTimeDilation,
GravitationalRedshift,
SchwarzschildMetric,
SchwarzschildRadius,
EventHorizon,
BlackHole,
KerrMetric,
FrameDragging,
GeodesicDeviation,
TidalForce,
EquivalencePrinciple,
LocalInertialFrame,
KillingVector,
KillingEquation,
SpacetimeSymmetry,
ConservationLaw,
ADMFormalism,
Foliation,
LapseFunction,
ShiftVector,
ExtrinsicCurvature,
InitialValueProblem,
ConstraintEquation,
FriedmannLemaitreRobertsonWalkerMetric,
FLRWMetric,
ScaleFactor,
HubbleParameter,
FriedmannEquation,
CosmologicalRedshift,
GravitationalWave,
LinearisedGravity,
MetricPerturbation,
GaugeTransformation,
HarmonicGauge,
TensorCalculus,
DifferentialGeometry,
DifferentialForm,
ExteriorDerivative,
HodgeDual,
Tetrad,
Vierbein,
CurvedSpacetime,
Manifold,
CoordinateChart
]
EXCLUDE_AS_PRIMARY = [
NewtonianMechanicsWithoutRelativisticEffects,
QuantumGravityAsPrimaryFramework,
ExperimentalRelativityWithoutMathematicalFormalism,
DifferentialGeometryWithoutPhysicalSpacetimeInterpretation
]
DISAMBIGUATION_RULE = "When spacetime structure, Lorentz symmetry, relativistic kinematics or curvature-based gravitation is represented mathematically, interpret the context as relativity mathematics."
}
CORE_CONCEPTS {
RelativityMathematics = "Mathematical formulation of relativistic spacetime and gravitation."
Spacetime = "Four-dimensional geometric structure combining spatial and temporal events."
SpacetimeInterval = "Invariant separation between spacetime events."
LorentzTransformation = "Linear transformation preserving the Minkowski metric."
FourVector = "Vector transforming covariantly under Lorentz transformations."
MetricTensor = "Tensor defining spacetime intervals and geometric structure."
Geodesic = "Extremal or autoparallel worldline determined by spacetime geometry."
Curvature = "Geometric measure of deviation from flat spacetime."
StressEnergyTensor = "Tensor representing energy, momentum, pressure and stress."
EinsteinFieldEquation = "Equation relating spacetime curvature to stress-energy."
}
RELATIONS {
RelativityMathematics --is_child_of--> MathematicalPhysicsOntology
SpecialRelativity --uses--> MinkowskiSpace
GeneralRelativity --uses--> CurvedSpacetime
LorentzTransformation --preserves--> SpacetimeInterval
LightCone --defines--> CausalStructure
ProperTime --measures--> TimelikeWorldline
FourVelocity --is_derivative_of--> FourPosition
FourMomentum --combines--> RelativisticEnergyAndMomentum
MetricTensor --defines--> SpacetimeGeometry
LeviCivitaConnection --is_compatible_with--> MetricTensor
ChristoffelSymbol --represents--> ConnectionCoefficients
GeodesicEquation --uses--> ChristoffelSymbol
RiemannCurvatureTensor --measures--> Curvature
RicciTensor --contracts--> RiemannCurvatureTensor
RicciScalar --contracts--> RicciTensor
EinsteinTensor --combines--> RicciTensorAndRicciScalar
EinsteinFieldEquation --relates--> EinsteinTensorAndStressEnergyTensor
EquivalencePrinciple --supports--> LocalInertialFrame
KillingVector --generates--> SpacetimeSymmetry
KillingVector --supports--> ConservedQuantity
GeodesicDeviation --measures--> TidalCurvatureEffect
SchwarzschildMetric --solves--> VacuumEinsteinEquation
FLRWMetric --models--> HomogeneousIsotropicCosmology
GravitationalWave --is_propagating--> MetricPerturbation
ADMFormalism --decomposes--> SpacetimeIntoSpaceAndTime
}
OPERATORS {
MINKOWSKI_INTERVAL(dt,dx,dy,dz,c) := ds2=-c^2*dt^2+dx^2+dy^2+dz^2
LORENTZ_FACTOR(v,c) := gamma=1/sqrt(1-v^2/c^2)
TIME_DILATION(dt_proper,v,c) := dt=gamma*dt_proper
LENGTH_CONTRACTION(L0,v,c) := L=L0/gamma
ENERGY(momentum,m,c) := E^2=p^2*c^2+m^2*c^4
FOUR_MOMENTUM(m,U) := p^mu=m*U^mu
METRIC_INTERVAL(g,dx) := ds2=g_munu*dx^mu*dx^nu
RAISE_INDEX(g_inverse,v_cov) := v^mu=g^(mu nu)*v_nu
LOWER_INDEX(g,v_contra) := v_mu=g_(mu nu)*v^nu
COVARIANT_DERIVATIVE_VECTOR(V,Gamma) := nabla_mu V^nu=partial_mu V^nu+Gamma^nu_(mu lambda)*V^lambda
CHRISTOFFEL(g) := Gamma^rho_(mu nu)=0.5*g^(rho sigma)*(partial_mu g_(sigma nu)+partial_nu g_(sigma mu)-partial_sigma g_(mu nu))
GEODESIC(x,Gamma,tau) := d2x^mu/dtau^2+Gamma^mu_ab*dx^a/dtau*dx^b/dtau=0
RIEMANN(Gamma) := R^rho_(sigma mu nu)=partial_mu Gamma^rho_(nu sigma)-partial_nu Gamma^rho_(mu sigma)+Gamma^rho_(mu lambda)Gamma^lambda_(nu sigma)-Gamma^rho_(nu lambda)Gamma^lambda_(mu sigma)
EINSTEIN_TENSOR(Ricci,R,g) := G_munu=R_munu-0.5*R*g_munu
EINSTEIN_FIELD_EQUATION(G,T,Lambda,g) := G_munu+Lambda*g_munu=(8*pi*G/c^4)*T_munu
SCHWARZSCHILD_RADIUS(M) := r_s=2*G*M/c^2
}
AXIOMS {
A1 = "Relativistic physical laws must be represented covariantly under the relevant spacetime transformations."
A2 = "The spacetime interval is invariant under Lorentz transformations in special relativity."
A3 = "The metric tensor defines local spacetime geometry."
A4 = "Freely falling test particles follow geodesics in general relativity."
A5 = "Spacetime curvature is sourced by stress-energy through Einstein's field equations."
A6 = "Local inertial frames recover special-relativistic structure at sufficiently small scales."
A7 = "Coordinate-dependent quantities must be distinguished from geometric invariants."
A8 = "Relativity models require geometric, dimensional, causal and numerical validation."
}
REASONING {
R1_CONTEXT = identify(spacetime_model,coordinates,metric,observers,matter_content,scale)
R2_REGIME = classify(special_relativity,general_relativity,weak_field,cosmological,black_hole,gravitational_wave)
R3_KINEMATICS = analyse(intervals,worldlines,proper_time,four_velocity,four_momentum)
R4_GEOMETRY = compute(metric,inverse_metric,connection,geodesics)
R5_CURVATURE = compute(Riemann,Ricci,Ricci_scalar,Einstein_tensor)
R6_DYNAMICS = solve_or_analyse(Einstein_equations,geodesic_motion,matter_evolution)
R7_SYMMETRY = identify(Killing_vectors,conserved_quantities)
R8_APPROXIMATION = select(Newtonian_limit,weak_field,linearised_gravity,perturbation)
R9_CAUSAL = analyse(light_cones,horizons,causal_structure)
R10_VALIDATION = check(signature,covariance,dimensions,constraints,invariants,limits,numerical_convergence)
}
CONSTRAINTS {
C1 = "Coordinate singularities must not automatically be treated as physical singularities."
C2 = "Tensor equations must respect index placement and metric signature conventions."
C3 = "Weak-field approximations require small metric perturbations."
C4 = "Newtonian limits require velocities small relative to c and weak gravitational fields."
C5 = "Geodesic motion applies to ideal freely falling test bodies."
C6 = "Stress-energy models require physically meaningful constitutive assumptions."
C7 = "Event horizons are global geometric structures and may not be identifiable from local coordinate behaviour alone."
C8 = "Gauge freedom in metric perturbations must be distinguished from physical gravitational degrees of freedom."
C9 = "Numerical relativity requires constraint preservation and stable gauge choices."
C10 = "A coordinate-dependent expression does not by itself define an invariant physical observable."
}
SEMANTIC_EQUIVALENCE {
"RelativityMathematics" <=> "MathematicsOfRelativity"
"MinkowskiSpace" <=> "MinkowskiSpacetime"
"FourMomentum" <=> "EnergyMomentumFourVector"
"LeviCivitaConnection" <=> "MetricCompatibleTorsionFreeConnection"
"FLRWMetric" <=> "FriedmannLemaitreRobertsonWalkerMetric"
"Vierbein" <=> "Tetrad"
}
VALIDATION {
V1 = "Metric signature convention is explicit."
V2 = "Coordinate chart and domain are explicit."
V3 = "Units and constants are explicit."
V4 = "Tensor index placement is consistent."
V5 = "Metric inverse and connection are verified."
V6 = "Curvature contractions are internally consistent."
V7 = "Einstein equations satisfy stress-energy conservation through covariant divergence."
V8 = "Known flat, weak-field or Newtonian limits are recovered where applicable."
V9 = "Causal structure and horizon claims are coordinate-independent where required."
V10 = "Numerical solutions satisfy constraint equations within tolerance."
}
CONFIDENCE {
BASE_CONFIDENCE = 0.99
RELATIVITY_MATHEMATICS_DEFINITION = 0.99
SPECIAL_RELATIVITY_REASONING = 0.99
TENSOR_REASONING = 0.99
GEODESIC_REASONING = 0.99
CURVATURE_REASONING = 0.99
EINSTEIN_EQUATION_REASONING = 0.99
BLACK_HOLE_REASONING = 0.98
COSMOLOGY_REASONING = 0.98
NUMERICAL_RELATIVITY_REASONING = 0.97
POLICY = "Reduce confidence when metric signature, coordinate chart, stress-energy model, gauge choice, boundary conditions, approximation regime or numerical constraints are underspecified."
}
AI_CONSUMPTION {
PURPOSES = [
SemanticSearch,
OntologyGrounding,
RelativityMathematicsReasoning,
SpecialRelativityReasoning,
LorentzTransformationReasoning,
FourVectorReasoning,
RelativisticKinematicsReasoning,
MetricTensorReasoning,
CovariantDerivativeReasoning,
GeodesicReasoning,
CurvatureReasoning,
EinsteinFieldEquationReasoning,
StressEnergyReasoning,
BlackHoleGeometryReasoning,
CosmologyReasoning,
GravitationalWaveReasoning,
ADMReasoning,
TensorCalculusReasoning,
DifferentialGeometryReasoning,
NumericalRelativityReasoning,
KnowledgeGraphConstruction,
RetrievalAugmentation,
RecursiveInference
]
NORMALISE {
"relativity mathematics" -> RelativityMathematics
"mathematics of relativity" -> RelativityMathematics
"special relativity" -> SpecialRelativity
"general relativity" -> GeneralRelativity
"Lorentz transformation" -> LorentzTransformation
"four momentum" -> FourMomentum
"metric tensor" -> MetricTensor
"Einstein field equation" -> EinsteinFieldEquation
"FLRW" -> FLRWMetric
"vierbein" -> Tetrad
}
}
EXECUTION_RULES {
E1 = "If spacetime geometry or relativistic kinematics is central, activate RelativityMathematicsOntology."
E2 = "If spacetime is flat and gravity is negligible, route to special relativity."
E3 = "If gravitation is represented geometrically, route to general relativity."
E4 = "If coordinate transformations preserve Minkowski interval, use Lorentz-transform reasoning."
E5 = "If curved spacetime motion is requested, compute geodesics from the metric connection."
E6 = "If gravitational field equations are required, compute curvature and stress-energy tensors."
E7 = "If the field is weak, consider Newtonian or linearised approximations."
E8 = "If cosmological homogeneity and isotropy are assumed, route to FLRW and Friedmann equations."
E9 = "Any relativistic result should be validated for covariance, causality, invariants, limiting behaviour and numerical constraints."
}
RECURSION {
ENTRY = RelativityMathematics
EXPAND(RelativityMathematics) -> [
Relativity, SpecialRelativity, GeneralRelativity, Spacetime, Event,
Worldline, ProperTime, ProperLength, ReferenceFrame, InertialFrame,
AcceleratedFrame, MinkowskiSpace, MinkowskiMetric, SpacetimeInterval,
TimelikeInterval, SpacelikeInterval, NullInterval, LightCone,
CausalStructure, LorentzTransformation, LorentzBoost, SpatialRotation,
LorentzGroup, PoincareGroup, Rapidity, RelativisticVelocity,
VelocityAddition, TimeDilation, LengthContraction,
RelativityOfSimultaneity, FourVector, FourPosition, FourVelocity,
FourAcceleration, FourMomentum, EnergyMomentumRelation, RestMass,
RelativisticEnergy, RelativisticMomentum, MassShellCondition, Tensor,
TensorField, MetricTensor, InverseMetric, IndexRaising, IndexLowering,
CovariantDerivative, ContravariantVector, CovariantVector,
ChristoffelSymbol, Connection, LeviCivitaConnection, ParallelTransport,
Geodesic, GeodesicEquation, AffineParameter, Curvature,
RiemannCurvatureTensor, RicciTensor, RicciScalar, EinsteinTensor,
WeylTensor, StressEnergyTensor, EnergyDensity, MomentumDensity, Pressure,
EinsteinFieldEquation, CosmologicalConstant, VacuumEinsteinEquation,
WeakFieldApproximation, NewtonianLimit, GravitationalPotential,
GravitationalTimeDilation, GravitationalRedshift, SchwarzschildMetric,
SchwarzschildRadius, EventHorizon, BlackHole, KerrMetric, FrameDragging,
GeodesicDeviation, TidalForce, EquivalencePrinciple, LocalInertialFrame,
KillingVector, KillingEquation, SpacetimeSymmetry, ConservationLaw,
ADMFormalism, Foliation, LapseFunction, ShiftVector, ExtrinsicCurvature,
InitialValueProblem, ConstraintEquation,
FriedmannLemaitreRobertsonWalkerMetric, FLRWMetric, ScaleFactor,
HubbleParameter, FriedmannEquation, CosmologicalRedshift,
GravitationalWave, LinearisedGravity, MetricPerturbation,
GaugeTransformation, HarmonicGauge, TensorCalculus,
DifferentialGeometry, DifferentialForm, ExteriorDerivative, HodgeDual,
Tetrad, Vierbein, CurvedSpacetime, Manifold, CoordinateChart
]
EXPAND(RegimeFamilies) -> [
SpecialRelativity,
GeneralRelativity,
BlackHoleSpacetimes,
RelativisticCosmology,
GravitationalWaveMathematics,
NumericalRelativity
]
STOP_CONDITION = "Stop when spacetime regime, metric, coordinates, matter content, geodesic or field equations, symmetries, causal structure and validation are sufficiently resolved."
}
ROUTING {
PARENT_ONTOLOGY = "Mathematical Physics Ontology"
IMMEDIATE_CONTEXTS = [
"Mathematical Physics Ontology",
"Tensor Analysis Ontologies",
"Differential Geometry Ontologies",
"Partial Differential Equation Ontology"
]
RELATED_ONTOLOGIES = [
"Mathematical Physics Ontology",
"Mathematical Physics",
"Tensor Analysis Ontologies",
"Differential Geometry Ontologies",
"Manifold Ontologies",
"Partial Differential Equation Ontology",
"Dynamical Systems Ontologies",
"Field Theory Ontologies",
"Electromagnetic Mathematics Ontology",
"Cosmology Ontologies",
"Numerical Analysis Ontologies",
"Computational Mathematics Ontologies"
]
ROUTE_RULES = [
"Relativity Mathematics -> Mathematical Physics Ontology for parent semantics",
"Relativity Mathematics -> Tensor Analysis for four-vectors and tensor fields",
"Relativity Mathematics -> Differential Geometry for metrics, connections and curvature",
"Relativity Mathematics -> Manifold Ontologies for spacetime structure",
"Relativity Mathematics -> PDE Ontology for Einstein and wave equations",
"Relativity Mathematics -> Dynamical Systems for geodesic and cosmological evolution",
"Relativity Mathematics -> Field Theory for covariant physical fields",
"Relativity Mathematics -> Electromagnetic Mathematics for covariant electromagnetism",
"Relativity Mathematics -> Cosmology for FLRW and Friedmann dynamics",
"Relativity Mathematics -> Numerical Analysis and Computational Mathematics for numerical relativity"
]
}
UPDATE_MODEL {
MODE = "append-compatible semantic versioning"
MAJOR = "ontology-breaking conceptual change"
MINOR = "new compatible spacetime models, geometric structures or reasoning pathways"
PATCH = "clarification, metadata or formatting correction"
RULES = [
"Preserve spacetime-event semantics.",
"Preserve special-versus-general relativity distinction.",
"Preserve coordinate-versus-invariant distinction.",
"Preserve metric and connection semantics.",
"Preserve curvature and stress-energy semantics.",
"Preserve causal-structure semantics.",
"Preserve gauge and approximation-regime semantics.",
"Preserve machine-readable relativity-mathematics semantics."
]
}
OUTPUT {
CANONICAL_DEFINITION = "Relativity mathematics formalises spacetime, relativistic motion and gravitation using Lorentz symmetry, tensor calculus and differential geometry."
SPECIAL_RELATIVITY_SUMMARY = "Special relativity uses Minkowski geometry and Lorentz transformations to describe inertial-frame physics."
KINEMATICS_SUMMARY = "Four-vectors encode relativistic position, velocity, momentum and energy."
GEOMETRY_SUMMARY = "General relativity represents gravitation through metric geometry, connections, geodesics and curvature."
FIELD_SUMMARY = "Einstein's equations relate spacetime curvature to stress-energy and the cosmological constant."
BLACK_HOLE_SUMMARY = "Schwarzschild and Kerr metrics describe important black-hole spacetime geometries."
COSMOLOGY_SUMMARY = "FLRW geometry and Friedmann equations model homogeneous and isotropic relativistic cosmology."
VALIDATION_SUMMARY = "Relativity models should be checked for covariance, metric signature, index consistency, causal structure, limiting cases, constraint preservation and numerical convergence."
}
FLOW {
INPUT
-> detect_relativity_mathematics_context
-> identify_spacetime_regime_coordinates_metric_observers_matter_content_and_scale
-> classify_special_general_weak_field_black_hole_cosmological_or_gravitational_wave_regime
-> compute_intervals_four_vectors_or_metric_geometry
-> if_curved_compute_connection_geodesics_and_curvature
-> if_field_equations_construct_stress_energy_and_Einstein_tensor
-> analyse_symmetries_Killing_vectors_and_conserved_quantities
-> analyse_light_cones_horizons_and_causal_structure
-> if_approximate_apply_Newtonian_limit_weak_field_or_linearised_gravity
-> if_numerical_enforce_constraint_equations_and_gauge_stability
-> validate_covariance_dimensions_invariants_limits_causality_and_numerical_convergence
-> route_to_mathematical_physics_tensor_analysis_differential_geometry_manifolds_PDE_dynamical_systems_field_theory_electromagnetic_mathematics_cosmology_numerical_analysis_or_computational_mathematics_ontology
-> recursively_expand_required_dependencies
-> verify_result
-> assign_confidence
-> produce_semantic_output
-> END
}
SEMANTIC_SUMMARY {
"The Relativity Mathematics Ontology defines mathematical special and general relativity. It supports spacetime events, worldlines, proper time and length, Minkowski space, spacetime intervals and light cones, Lorentz transformations, boosts, rapidity, velocity addition, time dilation, length contraction and simultaneity, four-position, four-velocity, four-acceleration and four-momentum, relativistic energy-momentum relations, tensors and metric geometry, index raising and lowering, covariant derivatives, Christoffel symbols, Levi-Civita connections, parallel transport and geodesics, Riemann, Ricci, Einstein and Weyl curvature tensors, stress-energy tensors and Einstein field equations, equivalence principle and local inertial frames, Schwarzschild and Kerr geometries, horizons, frame dragging and tidal effects, Killing vectors and conserved quantities, ADM decomposition and initial-value constraints, FLRW cosmology and Friedmann equations, gravitational redshift, linearised gravity, gravitational waves, gauge transformations, tensor calculus, differential forms, tetrads and curved spacetime manifolds."
}
FLOW_HASH {
METHOD = "semantic-content-identity"
ID = "KIRETH-MATH-APPLIED-MATHEMATICS-RELATIVITY-MATHEMATICS-ONTOLOGY-V1"
}
}
END_FLOW