🎯 Learning Objectives
Fiber Bundles as Dependent Types
See how E → M corresponds to a dependent type E : M → Type.
Connections as Transport
Understand connections as specifying how to parallel transport along paths.
Synthetic Differential Geometry
Learn the topos-theoretic approach where infinitesimals are genuine objects.
Type-Safe Einstein Equations
Formulate GR so that coordinate-dependent expressions are type errors.
📖 Key Concepts
The Coordinate Covariance Problem
General relativity is about geometry, but physicists compute with coordinates. This creates a type-safety problem:
- Coordinate-dependent quantities (Christoffel symbols Γ) have different types than
- Coordinate-independent observables (proper time, curvature invariants)
Traditional notation obscures this distinction. Type-safe GR makes it explicit!
Fiber Bundles as Dependent Types
-- A fiber bundle E → M with fiber F corresponds to:
E : M → Type
-- where E(p) ≅ F for each point p : M
-- Sections of the bundle are dependent functions:
Γ(E) ≡ Π(p : M). E(p)
-- Examples:
TangentBundle : M → Type -- Tangent vectors at each point
CotangentBundle : M → Type -- Covectors at each point
TensorBundle(r,s) : M → Type -- (r,s)-tensors at each pointType checking prevents mixing tensors of different ranks or at different points!
Connections as Parallel Transport
A connection on E → M specifies, for each path γ: [0,1] → M, a transport map:
transport_γ : E(γ(0)) → E(γ(1))The curvature measures failure of transport around closed loops to be trivial—a higher categorical structure!
Einstein's Equations (Type-Safe)
-- All terms must have the same type: symmetric (0,2)-tensor fields
einstein_equation :
(g : Metric M) →
(Λ : Scalar) →
(T : Sym²(T*M)) →
Type
einstein_equation g Λ T =
(G g + Λ • g ≡ 8π • T)
-- G g : Sym²(T*M) (Einstein tensor, derived from g)
-- Λ • g : Sym²(T*M) (cosmological term)
-- T : Sym²(T*M) (stress-energy)
-- Type-checker verifies all terms match!🤖 AI Learning Prompts
"Explain fiber bundles as dependent types. How does a bundle E → M correspond to a family of types E : M → Type? What do sections, local trivializations, and transition functions look like in this language?"
"How can type systems enforce coordinate independence in general relativity? What types should Christoffel symbols have vs curvature tensors? How do we prevent accidentally treating a coordinate-dependent quantity as observable?"
"Explain synthetic differential geometry (SDG). How does it treat infinitesimals as genuine objects in a topos? What is the 'Kock-Lawvere axiom'? How does this approach make differential geometry more algebraic and type-safe?"
"Explain connections on fiber bundles from a type-theoretic perspective. How is parallel transport a family of isomorphisms indexed by paths? How does curvature arise as the failure of transport around loops to be trivial? Connect this to higher category theory."
"Explain Penrose's abstract index notation for tensors. How does distinguishing abstract indices from concrete indices create a kind of type system? How does this prevent common errors in tensor calculations?"
"How should we think about spacetime singularities type-theoretically? A singularity is where geodesics can't be extended—a failure of 'totality'. How do partial types and partiality monads help formalize this?"
"How does homotopy type theory (HoTT) relate to general relativity? In HoTT, types are spaces and equalities are paths. How might this connect to the geometry of spacetime? What would a 'univalent' formulation of GR look like?"
"Explain gauge theories from a type-theoretic perspective. How are gauge fields connections on principal bundles? How does the type system distinguish gauge-invariant observables from gauge-dependent quantities?"
✏️ Exercises
Exercise 1: Tangent Bundle Types
Write out the type signature for a vector field on a manifold M. What is the type of the Lie bracket [X, Y] of two vector fields?
Exercise 2: Covariant Derivative
The covariant derivative ∇ takes a vector field and produces a (1,1)-tensor. Write this as a type signature. Why can't we just use ordinary derivatives?
Exercise 3: Einstein Equations
Verify that each term in G_μν + Λg_μν = 8πT_μν has the same type. What happens if you try to add G_μν + R (Ricci scalar)?