tensor_calc/

einstein.rs

use serde::{Deserialize, Serialize};
use std::collections::HashMap;
use crate::symbolic::SymbolicExpr;
use crate::tensor::*;
use crate::TensorError;

#[derive(Debug, Clone, Serialize, Deserialize)]
pub struct StressEnergyTensor {
    pub components: Vec<Vec<SymbolicExpr>>,
    pub tensor_type: String, // "perfect_fluid", "electromagnetic", "vacuum", etc.
    pub parameters: HashMap<String, SymbolicExpr>,
}

#[derive(Debug, Clone, Serialize, Deserialize)]
pub struct BoundaryCondition {
    pub coordinate: String,
    pub value: SymbolicExpr,
    pub condition_type: String, // "dirichlet", "neumann", "asymptotic"
    pub component_indices: Vec<usize>,
}

#[derive(Debug, Clone, Serialize, Deserialize)]
pub struct EinsteinSolution {
    pub metric_tensor: MetricTensor,
    pub coordinates: Vec<String>,
    pub solution_type: String, // "exact", "perturbative", "numerical"
    pub constraints_satisfied: bool,
    pub physical_parameters: HashMap<String, SymbolicExpr>,
    pub solution_domain: String,
}

#[derive(Debug, Clone, Serialize, Deserialize)]
pub struct EinsteinEquationSystem {
    pub field_equations: Vec<TensorComponent>,
    pub constraint_equations: Vec<TensorComponent>, 
    pub gauge_conditions: Vec<TensorComponent>,
    pub unknowns: Vec<String>,
    pub known_parameters: HashMap<String, SymbolicExpr>,
}

pub fn construct_einstein_field_equations(
    stress_energy: &StressEnergyTensor,
    _coordinates: &[String],
    cosmological_constant: Option<SymbolicExpr>
) -> Result<EinsteinEquationSystem, TensorError> {
    let n = stress_energy.components.len();
    let mut field_equations = Vec::new();
    let lambda = cosmological_constant.unwrap_or(SymbolicExpr::Zero);
    
    // Einstein field equations: G_μν + Λg_μν = 8πT_μν
    // We'll treat this as G_μν + Λg_μν - 8πT_μν = 0
    
    for mu in 0..n {
        for nu in 0..n {
            // Create equation: G_μν + Λg_μν - 8πT_μν = 0
            let equation_expr = format!(
                "G_{}_{} + {} * g_{}_{} - 8 * pi * T_{}_{}",
                mu, nu, lambda, mu, nu, mu, nu
            );
            
            field_equations.push(TensorComponent {
                indices: vec![mu, nu],
                expression: equation_expr,
            });
        }
    }
    
    // Generate unknown metric components 
    let mut unknowns = Vec::new();
    for mu in 0..n {
        for nu in mu..n { // Symmetric tensor, only upper triangle
            unknowns.push(format!("g_{}_{}", mu, nu));
        }
    }
    
    Ok(EinsteinEquationSystem {
        field_equations,
        constraint_equations: Vec::new(),
        gauge_conditions: Vec::new(), 
        unknowns,
        known_parameters: stress_energy.parameters.clone(),
    })
}

pub fn solve_vacuum_einstein_equations(
    coordinates: &[String],
    symmetry_ansatz: &str,
    boundary_conditions: &[BoundaryCondition]
) -> Result<Vec<EinsteinSolution>, TensorError> {
    match symmetry_ansatz {
        "spherical" => solve_spherically_symmetric_vacuum(coordinates, boundary_conditions),
        "cosmological" => solve_flrw_universe(coordinates, boundary_conditions),
        "axisymmetric" => solve_axisymmetric_vacuum(coordinates, boundary_conditions),
        _ => Err(TensorError::ComputationError(format!("Unknown symmetry ansatz: {}", symmetry_ansatz)))
    }
}

fn solve_spherically_symmetric_vacuum(
    coordinates: &[String],
    _boundary_conditions: &[BoundaryCondition]
) -> Result<Vec<EinsteinSolution>, TensorError> {
    // Schwarzschild ansatz: ds² = -f(r)dt² + h(r)dr² + r²(dθ² + sin²θ dφ²)
    // For 4D coordinates [t, r, theta, phi]
    
    if coordinates.len() != 4 {
        return Err(TensorError::ComputationError(
            "Spherical symmetry requires 4D coordinates [t, r, theta, phi]".to_string()
        ));
    }
    
    let mut solutions = Vec::new();
    
    // Schwarzschild solution
    let mut schwarzschild_metric = vec![vec![SymbolicExpr::Zero; 4]; 4];
    schwarzschild_metric[0][0] = SymbolicExpr::parse("-(1 - 2*M/r)")?;
    schwarzschild_metric[1][1] = SymbolicExpr::parse("1/(1 - 2*M/r)")?;
    schwarzschild_metric[2][2] = SymbolicExpr::parse("r^2")?;
    schwarzschild_metric[3][3] = SymbolicExpr::parse("r^2 * sin(theta)^2")?;
    
    let mut parameters = HashMap::new();
    parameters.insert("M".to_string(), SymbolicExpr::Variable("M".to_string()));
    
    solutions.push(EinsteinSolution {
        metric_tensor: schwarzschild_metric,
        coordinates: coordinates.to_vec(),
        solution_type: "exact".to_string(),
        constraints_satisfied: true,
        physical_parameters: parameters,
        solution_domain: "r > 2M".to_string(),
    });
    
    // Reissner-Nordström solution (charged black hole)
    let mut rn_metric = vec![vec![SymbolicExpr::Zero; 4]; 4];
    rn_metric[0][0] = SymbolicExpr::parse("-(1 - 2*M/r + Q^2/r^2)")?;
    rn_metric[1][1] = SymbolicExpr::parse("1/(1 - 2*M/r + Q^2/r^2)")?;
    rn_metric[2][2] = SymbolicExpr::parse("r^2")?;
    rn_metric[3][3] = SymbolicExpr::parse("r^2 * sin(theta)^2")?;
    
    let mut rn_parameters = HashMap::new();
    rn_parameters.insert("M".to_string(), SymbolicExpr::Variable("M".to_string()));
    rn_parameters.insert("Q".to_string(), SymbolicExpr::Variable("Q".to_string()));
    
    solutions.push(EinsteinSolution {
        metric_tensor: rn_metric,
        coordinates: coordinates.to_vec(),
        solution_type: "exact".to_string(),
        constraints_satisfied: true,
        physical_parameters: rn_parameters,
        solution_domain: "r > M + sqrt(M^2 - Q^2)".to_string(),
    });
    
    Ok(solutions)
}

fn solve_flrw_universe(
    coordinates: &[String],
    _boundary_conditions: &[BoundaryCondition]
) -> Result<Vec<EinsteinSolution>, TensorError> {
    // FLRW metric: ds² = -dt² + a(t)²[dr²/(1-kr²) + r²(dθ² + sin²θ dφ²)]
    // For 4D coordinates [t, r, theta, phi]
    
    if coordinates.len() != 4 {
        return Err(TensorError::ComputationError(
            "FLRW metric requires 4D coordinates [t, r, theta, phi]".to_string()
        ));
    }
    
    let mut solutions = Vec::new();
    
    // Flat FLRW (k = 0)
    let mut flat_flrw_metric = vec![vec![SymbolicExpr::Zero; 4]; 4];
    flat_flrw_metric[0][0] = SymbolicExpr::parse("-1")?;
    flat_flrw_metric[1][1] = SymbolicExpr::parse("a(t)^2")?;
    flat_flrw_metric[2][2] = SymbolicExpr::parse("a(t)^2 * r^2")?;
    flat_flrw_metric[3][3] = SymbolicExpr::parse("a(t)^2 * r^2 * sin(theta)^2")?;
    
    let mut flrw_parameters = HashMap::new();
    flrw_parameters.insert("a(t)".to_string(), SymbolicExpr::Function("a".to_string(), vec![SymbolicExpr::Variable("t".to_string())]));
    flrw_parameters.insert("H".to_string(), SymbolicExpr::Variable("H".to_string()));
    flrw_parameters.insert("Omega_m".to_string(), SymbolicExpr::Variable("Omega_m".to_string()));
    flrw_parameters.insert("Omega_Lambda".to_string(), SymbolicExpr::Variable("Omega_Lambda".to_string()));
    
    solutions.push(EinsteinSolution {
        metric_tensor: flat_flrw_metric,
        coordinates: coordinates.to_vec(),
        solution_type: "exact".to_string(),
        constraints_satisfied: true,
        physical_parameters: flrw_parameters,
        solution_domain: "t > 0, spatial homogeneity".to_string(),
    });
    
    // de Sitter space (cosmological constant dominated)
    let mut de_sitter_metric = vec![vec![SymbolicExpr::Zero; 4]; 4];
    de_sitter_metric[0][0] = SymbolicExpr::parse("-1")?;
    de_sitter_metric[1][1] = SymbolicExpr::parse("exp(H*t)^2")?;
    de_sitter_metric[2][2] = SymbolicExpr::parse("exp(H*t)^2 * r^2")?;
    de_sitter_metric[3][3] = SymbolicExpr::parse("exp(H*t)^2 * r^2 * sin(theta)^2")?;
    
    let mut ds_parameters = HashMap::new();
    ds_parameters.insert("H".to_string(), SymbolicExpr::Variable("H".to_string()));
    ds_parameters.insert("Lambda".to_string(), SymbolicExpr::parse("3*H^2")?);
    
    solutions.push(EinsteinSolution {
        metric_tensor: de_sitter_metric,
        coordinates: coordinates.to_vec(),
        solution_type: "exact".to_string(),
        constraints_satisfied: true,
        physical_parameters: ds_parameters,
        solution_domain: "exponential expansion".to_string(),
    });
    
    Ok(solutions)
}

fn solve_axisymmetric_vacuum(
    coordinates: &[String],
    _boundary_conditions: &[BoundaryCondition]
) -> Result<Vec<EinsteinSolution>, TensorError> {
    // Placeholder for axisymmetric solutions (Kerr, etc.)
    // This would require more sophisticated solving techniques
    
    let mut solutions = Vec::new();
    
    // Kerr solution (rotating black hole) - simplified
    if coordinates.len() == 4 {
        let mut kerr_metric = vec![vec![SymbolicExpr::Zero; 4]; 4];
        // This is a simplified representation - full Kerr metric is much more complex
        kerr_metric[0][0] = SymbolicExpr::parse("-(1 - 2*M*r/(r^2 + a^2*cos(theta)^2))")?;
        kerr_metric[1][1] = SymbolicExpr::parse("(r^2 + a^2*cos(theta)^2)/(r^2 - 2*M*r + a^2)")?;
        kerr_metric[2][2] = SymbolicExpr::parse("r^2 + a^2*cos(theta)^2")?;
        kerr_metric[3][3] = SymbolicExpr::parse("sin(theta)^2 * (r^2 + a^2 + 2*M*r*a^2*sin(theta)^2/(r^2 + a^2*cos(theta)^2))")?;
        
        // Off-diagonal term
        kerr_metric[0][3] = SymbolicExpr::parse("-2*M*r*a*sin(theta)^2/(r^2 + a^2*cos(theta)^2)")?;
        kerr_metric[3][0] = kerr_metric[0][3].clone();
        
        let mut kerr_parameters = HashMap::new();
        kerr_parameters.insert("M".to_string(), SymbolicExpr::Variable("M".to_string()));
        kerr_parameters.insert("a".to_string(), SymbolicExpr::Variable("a".to_string()));
        
        solutions.push(EinsteinSolution {
            metric_tensor: kerr_metric,
            coordinates: coordinates.to_vec(),
            solution_type: "exact".to_string(),
            constraints_satisfied: true,
            physical_parameters: kerr_parameters,
            solution_domain: "r > M + sqrt(M^2 - a^2)".to_string(),
        });
    }
    
    Ok(solutions)
}

pub fn verify_einstein_solution(
    solution: &EinsteinSolution,
    stress_energy: Option<&StressEnergyTensor>,
    _cosmological_constant: Option<SymbolicExpr>
) -> Result<bool, TensorError> {
    // Verify that G_μν + Λg_μν = 8πT_μν for the given solution
    
    // Calculate Einstein tensor for the solution metric
    let _einstein_tensor = calculate_einstein_tensor(&solution.metric_tensor, &solution.coordinates)?;
    
    // If we have a stress-energy tensor, check field equations
    if let Some(_t_mu_nu) = stress_energy {
        // This would involve substituting the metric into field equations
        // and checking if they're satisfied symbolically or numerically
        
        // For now, return true for known exact solutions
        match solution.solution_type.as_str() {
            "exact" => Ok(true),
            _ => Ok(false), // Would need numerical verification
        }
    } else {
        // Vacuum case: check if G_μν + Λg_μν = 0
        Ok(true) // Placeholder - would need actual verification
    }
}

pub fn solve_einstein_constraint_equations(
    _initial_data: &MetricTensor,
    _coordinates: &[String]
) -> Result<Vec<TensorComponent>, TensorError> {
    // Solve the Hamiltonian and momentum constraints
    // This is needed for numerical relativity initial value problems
    
    let n = _initial_data.len();
    let mut constraints = Vec::new();
    
    // Hamiltonian constraint: R + K² - K_ij K^ij = 16π ρ
    constraints.push(TensorComponent {
        indices: vec![],
        expression: "R + K^2 - K_ij * K^ij - 16 * pi * rho".to_string(),
    });
    
    // Momentum constraints: D_j (K^ij - gamma^ij K) = 8π j^i  
    for i in 0..n-1 { // Spatial indices only
        constraints.push(TensorComponent {
            indices: vec![i],
            expression: format!("D_j(K^{i}j - gamma^{i}j * K) - 8 * pi * j^{i}", i=i),
        });
    }
    
    Ok(constraints)
}

#[cfg(test)]
mod tests {
    use super::*;

    #[test]
    fn test_schwarzschild_solution() {
        let coords = vec!["t".to_string(), "r".to_string(), "theta".to_string(), "phi".to_string()];
        let boundary_conditions = vec![];
        
        let solutions = solve_spherically_symmetric_vacuum(&coords, &boundary_conditions).unwrap();
        
        assert!(!solutions.is_empty());
        assert_eq!(solutions[0].solution_type, "exact");
        assert_eq!(solutions[0].coordinates, coords);
    }
    
    #[test]
    fn test_flrw_solution() {
        let coords = vec!["t".to_string(), "r".to_string(), "theta".to_string(), "phi".to_string()];
        let boundary_conditions = vec![];
        
        let solutions = solve_flrw_universe(&coords, &boundary_conditions).unwrap();
        
        assert!(!solutions.is_empty());
        assert_eq!(solutions[0].solution_type, "exact");
        assert!(solutions[0].physical_parameters.contains_key("H"));
    }
}