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Physics: Relativistic Fluid Dynamics

This tutorial explores Relativistic Fluid Dynamics using XAct.jl. We will define the energy-momentum tensor for a perfect fluid, verify its conservation laws, and derive the relativistic Euler equations.

1. Setup

If running on Google Colab or a fresh environment, install the required packages first.

# Uncomment the lines below if running on Google Colab:
# using Pkg
# Pkg.add(url="https://github.com/sashakile/XAct.jl.git")
# Pkg.add("Plots")

2. Setup

Load the required modules.

using XAct
using Plots
using LinearAlgebra

# Headless plotting for build compatibility
ENV["GKSwstype"] = "100"
"100"
Project Profile for AI Agents (LLM TL;DR)
  • Goal: Implement perfect fluid $T_{ab}$ and verify conservation laws.
  • Key Symbols: Velocity $u^a$, Density $\rho$, Pressure $p$.
  • Physics: $T_{ab} = (\rho + p)u_a u_b + p g_{ab}$, verify $\nabla_a T^{ab} = 0$.

2. Define the Manifold and Metric

reset_state!()
M = def_manifold!(:M4, 4, [:alpha, :beta, :gamma, :delta, :mu, :nu])
@indices M4 alpha beta gamma delta mu nu

# General metric g_ab
g = def_metric!(-1, "g[-mu,-nu]", :CD)
MetricObj(:g, :M4, :CD, -1)

3. The Perfect Fluid Energy-Momentum Tensor

A perfect fluid is characterized by its energy density $\rho$, pressure $p$, and 4-velocity $u^\mu$. The energy-momentum tensor is: $T_{\mu\nu} = (\rho + p) u_\mu u_\nu + p g_{\mu\nu}$

# Define scalar fields rho and p
def_tensor!(:rho, String[], :M4)
def_tensor!(:p, String[], :M4)
rho = tensor(:rho)
p_tensor = tensor(:p)

# Define 4-velocity vector u^mu
def_tensor!(:u, ["mu"], :M4)
u = tensor(:u)

# Define T_mu_nu
# T_mu_nu = (rho + p) u_mu u_nu + p g_mu_nu
T_expr = ToCanonical((rho[] + p_tensor[]) * u[-mu] * u[-nu] + p_tensor[] * tensor(:g)[-mu, -nu])

println("Energy-Momentum Tensor T_{μν}:")
T_expr

1 + \g{\mu}{\nu} \p

4. Conservation Laws

The physical evolution of the fluid is governed by the conservation of energy and momentum: $\nabla_\mu T^{\mu\nu} = 0$

# Compute the divergence: g^{αμ} ∇_α T_{μν}
div_T = ToCanonical(
    Contract(tensor(:g)[alpha, mu] * covd(:CD)[-alpha](rho[] * u[-mu] * u[-nu]))
    + Contract(tensor(:g)[alpha, mu] * covd(:CD)[-alpha](p_tensor[] * u[-mu] * u[-nu]))
    + Contract(tensor(:g)[alpha, mu] * covd(:CD)[-alpha](p_tensor[] * tensor(:g)[-mu, -nu]))
)

5. Deriving Continuity and Euler Equations

The conservation equations $\nabla_\mu T^{\mu\nu} = 0$ contain both the energy conservation (continuity) and momentum conservation (Euler) equations.

Energy Conservation

Projecting along the 4-velocity ($u_\nu \nabla_\mu T^{\mu\nu} = 0$) yields the continuity equation: $u^\mu \nabla_\mu \rho + (\rho + p) \nabla_\mu u^\mu = 0$

# Project divergence onto u^nu
energy_cons = ToCanonical(Contract(u[nu] * div_T))

6. Summary

This tutorial demonstrated:

  1. Constructing complex physical tensors from fundamental fields.
  2. Symbolically deriving conservation laws in curved spacetime.
  3. Projecting tensor equations to extract specific physical components (like energy conservation).

Next Steps