Outside special relativity, here are mathematical frameworks for FTL travel from alternative theories and speculative physics: Credit Grok Expert
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:Where λ = c₀/c(t) varies, allowing superluminal propagation in early universe.VSL metric:2. Krasnikov TubesCreates a permanent "superluminal tunnel" through spacetime:Krasnikov metric:Where K creates a preferred direction for FTL travel without causality violation in one direction.Travel time reduction:3. Horava-Lifshitz GravityHigher-dimensional gravity theory with anisotropic scaling:Action:Allows superluminal graviton propagation at high energies while preserving causality.Dispersion relation: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: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:Group delay:Where phase time suggests FTL, though information transfer limited by uncertainty.Wigner phase time:6. Non-commutative GeometrySpacetime coordinates don't commute: [x^μ, x^ν] = iθ^{μν}Modified dispersion:Leads to superluminal propagation for high-energy particles.FTL velocity:7. MOND-like Modified GravityModified Newtonian dynamics extended to relativistic regime:Aether-like field: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:Superluminal Dirac equation:9. Brane World ScenariosOur universe as 3-brane in higher-dimensional bulk:Randall-Sundrum metric:Graviton propagation in bulk can appear superluminal on brane:10. Pregeometric TheoriesSpacetime emerges from quantum entanglement network:Causal set theory:Where μ is measure, allowing FTL signaling through discrete causal structure.Discrete Lorentz violations:Key Features of These Approaches:
x' = γ(λx - vt)
t' = γ(λt - vx/c²)
ds² = -c(t)²dt² + dx² + dy² + dz²
ds² = -dt² + dx² + dy² + dz² + 2K(x,y,z,t)dxdt
Δτ = ∫√(g_{\mu\nu}dx^μdx^ν) < distance/c
S = ∫d³xdt [N√g (K_{ij}K^{ij} - λR + μR² + ...)]
E² = p² + βp⁴/M² + ...
S = ∫√(-g)m²gR[g] + √(-f)M²fR[f] + √(-g)Vm(g⁻¹f)
T(E) ≈ exp[-2∫κ(x)dx] = exp[-2∫√(2m(V-E))/ℏ dx]
τ_g = dφ/dω < L/c
τ_W = ℏdφ/dE
p_μp^μ = m² + θ²p⁴
v = c/√(1 - θ²E²)
∇_μ(√-g g^{μν}∇_νφ) = 0
T^λ_{μν} = Γ^λ_{[μν]} + K^λ_{μν}
(iγ^μ∇_μ - m + γ^5γ^μT_{μν}σ^{νρ})ψ = 0
ds² = e^{-2k|y|}η_{μν}dx^μdx^ν + dy²
v_grav = c/√(1 - e^{-2kL})
N(x,y) = ∑_{p∈C} μ(p,x)μ(p,y)
Δx = √(N)ℓ_p > cΔt
- Modified causality - Redefines light cones or preferred frames
- Higher dimensions - Extra dimensions allow bulk shortcuts
- Scale dependence - FTL only at specific energy scales
- Anisotropic propagation - Direction-dependent speed limits
- Emergent spacetime - Geometry arises from quantum structure
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