Skip to main content

Solvers API

Overview​

The solvers module provides numerical methods for computing Einstein's field equations and related quantities.

Energy Tensor Solver​

get_energy_tensor​

Computes the stress-energy tensor from a metric using Einstein's field equations.

Parameters:

  • metric (Metric): Input metric object
  • order (int): Finite difference order (default: 4)

Returns:

  • EnergyTensor: Object containing T_μν components and derived quantities

Example:

from core.solver.energy import get_energy_tensor

energy_tensor = get_energy_tensor(metric)
print(f"Energy density: {energy_tensor.T_00}")

Field Equation Solvers​

get_ricci_tensor​

Computes the Ricci curvature tensor from metric derivatives.

Parameters:

  • metric (Metric): Input metric object
  • order (int): Finite difference order (default: 4)

Returns:

  • torch.Tensor: Ricci tensor R_μν

get_scalar_curvature​

Computes the Ricci scalar curvature.

Parameters:

  • metric (Metric): Input metric object
  • ricci_tensor (torch.Tensor): Pre-computed Ricci tensor

Returns:

  • torch.Tensor: Ricci scalar R

get_einstein_tensor​

Computes the Einstein tensor G_μν = R_μν - ½Rg_μν.

Parameters:

  • metric (Metric): Input metric object
  • ricci_tensor (torch.Tensor): Pre-computed Ricci tensor
  • ricci_scalar (torch.Tensor): Pre-computed Ricci scalar

Returns:

  • torch.Tensor: Einstein tensor G_μν

Derivative Solvers​

get_christoffel_symbols​

Computes Christoffel symbols Γ^λ_μν from metric derivatives.

Parameters:

  • metric (Metric): Input metric object

Returns:

  • torch.Tensor: Christoffel symbols (4×4×4×grid_size)

get_metric_derivatives​

Computes first and second derivatives of the metric.

Parameters:

  • metric (Metric): Input metric object
  • order (int): Finite difference order (default: 4)

Returns:

  • dict: Dictionary containing ∂g and ∂²g components

Performance Notes​

  • Finite Difference Order: Higher orders (4-6) provide better accuracy but slower computation
  • GPU Acceleration: All solvers are fully vectorized for GPU acceleration
  • Memory Usage: 4D grids can be memory-intensive for large simulations