Outside special relativity, here are mathematical frameworks for FTL travel from alternative theories and speculative physics.

 

Outside special relativity, here are mathematical frameworks for FTL travel from alternative theories and speculative physics:

1. Variable Speed of Light (VSL) TheoriesProposed by João Magueijo and others, where c varies with time/space:Modified Lorentz transformation:

x' = γ(λx - vt)
t' = γ(λt - vx/c²)

Where λ = c₀/c(t) varies, allowing superluminal propagation in early universe.VSL metric:

ds² = -c(t)²dt² + dx² + dy² + dz²

2. Krasnikov TubesCreates a permanent "superluminal tunnel" through spacetime:Krasnikov metric:

ds² = -dt² + dx² + dy² + dz² + 2K(x,y,z,t)dxdt

Where K creates a preferred direction for FTL travel without causality violation in one direction.Travel time reduction:

Δτ = ∫√(g_{\mu\nu}dx^μdx^ν) < distance/c

3. Horava-Lifshitz GravityHigher-dimensional gravity theory with anisotropic scaling:Action:

S = ∫d³xdt [N√g (K_{ij}K^{ij} - λR + μR² + ...)]

Allows superluminal graviton propagation at high energies while preserving causality.Dispersion relation:

E² = p² + βp⁴/M² + ...

β > 0 permits superluminal modes at Planck scale.4. bimetric Gravity TheoriesTwo metrics: one for matter (g_{\mu\nu}), one for gravity (f_{\mu\nu}):Action:

S = ∫√(-g)m²gR[g] + √(-f)M²fR[f] + √(-g)Vm(g⁻¹f)

FTL solutions: Gravitational waves on f_{\mu\nu} can exceed c defined by g_{\mu\nu}.5. Superluminal Quantum TunnelingIn quantum mechanics, apparent FTL through barriers:Transmission probability:

T(E) ≈ exp[-2∫κ(x)dx] = exp[-2∫√(2m(V-E))/ℏ dx]

Group delay:

τ_g = dφ/dω < L/c

Where phase time suggests FTL, though information transfer limited by uncertainty.Wigner phase time:

τ_W = ℏdφ/dE

6. Non-commutative GeometrySpacetime coordinates don't commute: [x^μ, x^ν] = iθ^{μν}Modified dispersion:

p_μp^μ = m² + θ²p⁴

Leads to superluminal propagation for high-energy particles.FTL velocity:

v = c/√(1 - θ²E²)

7. MOND-like Modified GravityModified Newtonian dynamics extended to relativistic regime:Aether-like field:

∇_μ(√-g g^{μν}∇_νφ) = 0

With c_eff = c/√(1 - φ) allowing effective superluminal speeds in weak field limits.8. Torsion-based Theories (Einstein-Cartan)Spacetime torsion enables superluminal spinor propagation:Torsion tensor:

T^λ_{μν} = Γ^λ_{[μν]} + K^λ_{μν}

Superluminal Dirac equation:

(iγ^μ∇_μ - m + γ^5γ^μT_{μν}σ^{νρ})ψ = 0

9. Brane World ScenariosOur universe as 3-brane in higher-dimensional bulk:Randall-Sundrum metric:

ds² = e^{-2k|y|}η_{μν}dx^μdx^ν + dy²

Graviton propagation in bulk can appear superluminal on brane:

v_grav = c/√(1 - e^{-2kL})

10. Pregeometric TheoriesSpacetime emerges from quantum entanglement network:Causal set theory:

N(x,y) = ∑_{p∈C} μ(p,x)μ(p,y)

Where μ is measure, allowing FTL signaling through discrete causal structure.Discrete Lorentz violations:

Δx = √(N)ℓ_p > cΔt

Key Features of These Approaches:

1. Modified causality - Redefines light cones or preferred frames
2. Higher dimensions - Extra dimensions allow bulk shortcuts
3. Scale dependence - FTL only at specific energy scales
4. Anisotropic propagation - Direction-dependent speed limits
5. Emergent spacetime - Geometry arises from quantum structure

None of these are experimentally confirmed, but they represent mathematically consistent alternatives to special relativity's universal speed limit.