# Physics 616

• Prof. Andrew W. Steiner
(or Andrew or "Dr. Steiner")
• Office hour: 103 South College, Thursday 11am
• Email: awsteiner@utk.edu
• Homework: Electronically as .pdf
• You may work with each other on the homework, but you must write the solution in your own words

Use down and up arrows to proceed to the next or previous slide.

## Outline

• Ch. 4
• Stress-energy tensor, conservation laws, surface one-form, force equation

## Perfect Fluids

• Hydrodynamics:
• Local thermodynamic equilibrium
• Mean free path is small compared to all other scales
• Perfect fluids: no viscosity and no heat conduction
• Define a new set of quantities to describe fluids which transform in a regular way under Lorentz transformations
• Define new scalars, vectors, one-forms, tensors
• Use term "dust" to define a fluid which is a collection of particles at rest in some Lorentz frame

## Number density

• Number of particles doesn't change under a Lorentz transformation, but volume contracts
• We define a new vector $\vec{N} = n \vec{U}$ $$\left( n \gamma, n v^{x} \gamma, n v^{y} \gamma, n v^{z} \gamma \right)$$
• The norm, $\vec{N} \cdot \vec{N} = -n^2$. The quantity $n$, the "rest number density", is frame invariant

## A one-form defines a surface

• A surface is given by $$\phi(t,x,y,z) = \mathrm{constant}$$
• The gradient defines the one-form normal to the surface $\tilde{d}\phi$
• Unit normal vector is $\tilde{n} \equiv \tilde{d}\phi / | \tilde{d}\phi |$ where $$|\tilde{d}\phi| = \left| \eta^{\alpha \beta} \phi_{,\alpha} \phi_{,\beta} \right|^{1/2}$$

## Energy and Momentum

• Define $\rho = n m$ as the energy density in the MCRF
• Stress energy tensor is a $\left( \begin{array}{c} 2 \\ 0 \end{array} \right)$ tensor, $T^{\alpha \beta}$
• $T^{00}$ is the energy density
• $T^{0i}$ is the energy flux across the surface defined by $x^{i}$
• $T^{i0}$ is the ith component of the momentum density
• $T^{ij}$ is the ith component of the momentum across the surface defined by $x^{j}$
• For dust, $\mathbf{T} = \rho \vec{U} \otimes \vec{U}$
• If forces are perpendicular to surfaces, then $T^{ij}$ is zero when $i \neq j$
• Stress energy tensor is symmetric

## Thermodynamics

• Thermodynamic identity (when $N$ is constant) $$d \rho - \left( \rho + p \right) \frac{dn}{n} = n T dS$$ where $S$ is the specific entropy, i.e. the entropy per particle
• This relation is valid in nonrelativistic thermodynamics as well
• Compare this with the Gibbs-Duhem relation, $$d p = n d \mu + s dT$$
• The appearance of $\rho + p$ is not accidental, and is related to the stress energy tensor

## Conservation Laws

• In terms of $T^{\alpha \beta}$, $$\frac{\partial T^{\alpha \beta}}{\partial x^{\beta}} = {T^{\alpha \beta}}_{,\beta} = 0$$
• Number conservation $${N^{\alpha}}_{,\alpha} = \frac{\partial N^{\alpha}}{\partial x^{\alpha}} = \left( n U^{\alpha} \right)_{,\alpha} = 0$$

## Stress Energy Tensor in Perfect Fluids

• "No heat conduction", constant specific entropy, "adiabatic"
• No viscosity, therefore $T$ is diagonal, $$T^{\alpha \beta} = \left( \begin{array}{cccc} \rho & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \\ \end{array} \right)$$ or $T^{\alpha \beta} = \left( \rho + p \right) U^{\alpha} U^{\beta} + p \eta^{\alpha \beta}$
• Or, $$\mathbf{T} = \left( \rho + p \right) \vec{U} \otimes \vec{U} + p \mathbf{g}^{-1}$$
• One can derive the result that $$\frac{d S}{d \tau} = 0$$

## Force as the Gradient of Pressure

• One can also derive $$\left( \rho + p\right) a_i + p_{,i} = 0$$ where $a_i$ is the acceleration one form. This is the analog of $$\rho \left[ \dot{\mathbf{v}} + \left( \mathbf{v} \cdot \nabla \right) \mathbf{v} \right] + \nabla p = 0$$
• Note again the presence of $\rho + p$, without strong gravity, $p \ll \rho$
• Compare the ideal gas case, where $\rho = \frac{3}{2} n k T$ and $p = n k T$?
• In strong GR systems, $p \sim \rho$
• But it is never the case that $p > \rho$ . Why?

## Surfaces and Gauss' Law

• One-forms imply surfaces
• Given a surface defined by $\phi(t,x,y,z) = \mathrm{constant}$, one can define a normal one-form
• The unit-normal one-form is $$\tilde{n} \equiv \tilde{d} \phi / \left| \tilde{d} \phi \right|$$ where $$\left| \tilde{d} \phi \right| = \left| \eta^{\alpha \beta} \phi_{,\alpha} \phi_{,\beta} \right|^{1/2}$$
• In this notation $$\int {V^{\alpha}}_{,\alpha}~d^4 x = \oint V^{\alpha} n_{\alpha}~d^{3} S$$ where $n_{\alpha}$ is the unit-normal one-form

## Group Work

• The inverse of the metric, $\eta_{\alpha \beta}$, is $\eta^{\alpha \beta}$, and the inverse of a Lorentz transformation ${\Lambda^{\mu}}_{\nu}$ is ${\Lambda_{\mu}}^{\nu}$. Is there a simple expression for the inverse of $T^{\alpha \beta}$?
• What is the inverse of $T^{\alpha \beta}$ in the case of dust?
• Define $\tilde{S}$ as the entropy per particle. Prove Eq. 4.25 backwards, by recasting $d\rho$, $dn$ and $d \tilde{S}$ in terms of $E,P,V,T,S,\mu$ and $N$ and then showing that one obtains $dE = -P dV + T dS$ (which is just the Euler relation with $dN=0$)