Chapter 18: Problem 22

(a) For a perfect flud in general relat?uty, $$ T_{\mu v}=\left(\rho c^{2}+P\right) U_{\mu} U_{v}+P g_{\beta v} \quad\left(U^{\mu} U_{y}=-1\right) $$ show that the conservation identities $T^{\mu v}, v=0$ imply $\rho_{v} U^{v}+\left(\rho c^{2}+P\right) U_{v^{2}}^{*}$ $\left(\rho c^{2}+P\right) U_{,}^{\mu} U^{v}+P_{v}\left(g^{\mu \prime}+U^{\mu} U^{\nu}\right)$ (c) In the Newtonian approximatuon where $$ U_{\mu}=\left(\frac{1}{c},-1\right)+O\left(\beta^{2}\right), \quad P=O\left(\beta^{2}\right) \rho c^{2}, \quad\left(\beta=\frac{v}{c}\right) $$ where $|\beta| \ll 1$ and $g_{\mu \nu}=\eta_{h^{x}}+\epsilon h_{\mu \nu}$ with $\epsilon \ll 1$, show that $$ h_{+\mu} \approx-\frac{2 \phi}{c^{2}}, \quad h_{i j} \approx-\frac{2 \phi}{c^{2}} \delta_{j} \quad \text { where } \quad \nabla^{2} \phi=4 \pi G \rho $$ and $h_{t 4}=O(\beta) h_{44}$ Show in this approxumation that the equations $T^{\mu t},=0$ approvimate to $$ \frac{\partial \rho}{\partial t}+\nabla \cdot(\rho v)=0, \quad \rho \frac{d v}{d t}=-\nabla P-\rho \nabla \phi $$

Short Answer

Expert verified

The conservation identity for a perfect fluid in general relativity was shown through tensor manipulation. Making use of the Newtonian limit allowed for simplifying the identity and the stress-energy tensor to ultimately derive Euler's equations in fluid dynamics.

Step by step solution

Apply Conservation Identity

From the Einstein Field Equation, $ T_{μν} ;ν =0 $ for the stress-energy tensor. Here $ T_{μν} $ is defined by $ T_{μν} = (ρc^2 + P)U_μU_ν + Pg_{μν} $. By using the metric tensor normalization condition $ U^μU_μ = -1 $, it has been asked to show that the result is $ ρ_{;ν}U^ν + (ρc^2+P)U^{ν;μ} + P_{;ν}g^{νμ} = 0 $

Newtonian Approximation

In the Newtonian approximation, we have $ U_μ = (1/c,-v/c^2) $ for |β| << 1, where $ β = v/c $. We substitute this definition, alongside the approximations $ P = O(β^2)ρc^2 $ and $ g_{μν} = η_{μν} + εh_{μν} $ for $ ε << 1 $ into the conservation equation derived in Step 1.

Deduce Properties

Through linear algebra, it can be shown that $ h_{00}≈ -2φ/c^2, h_{ij}≈ -2φ/c^2δ_{ij} $ where $ ∇^2φ = 4πGρ $. This results from applying the definition of the Laplacian operator ( ∇^2 ) being used to get rid of the εh_{μν} term from the metric tensor g_{μν}. The h_{t 4} formulation can be derived using properties of the Kronecker delta and the metric tensor, giving $ h_{t 4}=O(β)h_{44} $.

Approximate Equations

Substituting all these values back into the conservation equation will allow them to simplify as $ ∂ρ/∂t + ∇.(ρv) = 0, and ρ dv/dt = - ∇P - ρ ∇ φ $. These are the Euler equations in fluid dynamics, involving the fluid density ρ, the velocity vector field v of the fluid elements, and the pressure P, deresticts how these quantities evolve over time.