Xing’s notes. This article demonstrates a method to construct the energy–momentum tensor of a perfect fluid from “dust”, and aims to give an intuition of this tensor's meaning. It builds upon the lectures of Professor Ori Ganor.

Background

(This section is included as a brief review for those familiar with the basics of special relativity. For a much more thorough introduction, I highly recommend Eigenchris’s YouTube series on relativity and tensor algebra, as well as the first part of Sean Carroll’s lecture notes on general relativity. Conversely, feel free to skip this section if you want to get right to the derivations.)

A key postulate of special relativity is

1. The speed of light in vacuum is constant in every inertial reference frame.

In other words, $c$ is invariant. This motivates the definition of invariant spacetime intervals: \[ (\d s)^2 \equiv -(c\,\d t)^2 + (\d x)^2 + (\d y)^2 + (\d z)^2. \] Just like how the lengths of everyday objects do not change under spatial rotations, the spacetime interval between events do not change under Lorentz boosts. Even as strange effects like length–contraction and time–dilation become visible when objects are moving relative to each other near the speed of light, the spacetime interval stays constant. It’s like having a new and improved, relativistic ruler.

To produce the invariant spacetime interval with this ruler, it must be able to measure time alongside the spatial lengths. Geometrically, this motivates adding a zeroth $t$ element to our vectors’ existing three $x$, $y$, and $z$. For example, a position vector is now

\[\vec r = r^\alpha \vec e_\alpha.\]

(To simplify the writing, Greek-letter indices like $\alpha, \beta, \mu, \nu$ implicitly range over 0, 1, 2, 3; Repeated upper–lower index pairs like ${^\beta}{_\beta}$ are implicitly summed over. Do not confuse them with exponents, which will always be outside parentheses! This notation is known as the Einstein summation convention.)

This way, a spacetime displacement

\[\d\vec s = s^\alpha \vec e_\alpha \equiv \d t\,\vec e_0 + \d x\,\vec e_1 + \d y\,\vec e_2 + \d z\,\vec e_3\]

with components

\[s^\alpha \to \begin{pmatrix} s^0 \\ s^1 \\ s^2 \\ s^3 \end{pmatrix} \equiv \begin{pmatrix} \d t \\ \d x \\ \d y \\ \d z \end{pmatrix}\]

has the spacetime interval (again, summing over repeated index pairs)

\[(\d\vec s)^2 = \eta_{\alpha\beta} s^\alpha s^\beta,\]

where

\[\vec\eta = \eta_{\alpha\beta} \ (\vec e^\alpha \otimes \vec e^\beta)\]

is the Minkowskian metric with components

\[\eta_{\alpha\beta} \to \begin{pmatrix} -1 \\ & 1 \\ & & 1 \\ & & & 1 \end{pmatrix}.\]

(When dealing with objects having two indices, the left index corresponds to the row, and the right index the column. For instance, because the metric is symmetric, $\eta_{\alpha\beta} = \eta_{\beta\alpha}$.)

The metric is useful for a lot of things (see the links above for more details), but most fundamentally, it underscores an interweaving of space and time that is thematic throughout special and general relativity. This means

2. To gain full intuition, we must treat space and time on an equal footing.

I strongly believe that to fully understand special relativity, it must feel natural to mix quantities derived from space and those derived from time. For example:

\[u^\alpha \to \begin{pmatrix}1\\0\\0\\0\end{pmatrix}.\] \[p^\alpha = mu^\alpha \to \begin{pmatrix}m\\0\\0\\0\end{pmatrix}.\]

(We can create new four–vectors by multiplying them with scalar invariants. Here, $m$ always refers to “the mass as measured in the rest frame,” which is by-definition invariant.)

These intertwined interpretations are key to understanding the stress–energy–momentum tensor. More on that in a bit.

For now, notice that we have only defined these quantities in the rest frames of their respective objects. With all the length–contraction and time–dilation stuff, how do we calculate how these quantities might look when we are moving relative to these objects near the speed of light? Luckily,

3. We can transform between any inertial reference frames via Lorentz boosts

Suppose another inertial reference frame $K'$ is traveling with (ordinary) velocity $v\vec e_1$ as measured in $K$. If we know that the four–velocity $\vec u$ has components $u^\alpha$ in $K$, then it will have the components

\[u^{\alpha^\prime} = \Lambda{^{\alpha^\prime}}_\alpha u^\alpha\]

in $K'$, where the Lorentz boost in $v\vec e_1$ is

\[\begin{align*} \Lambda{^{\alpha^\prime}}_\alpha \equiv \frac{\d x^{\alpha^\prime}}{\d x^\alpha} \to \begin{pmatrix} \gamma_v & -\gamma_v v\\\ -\gamma_v v & \gamma\\\ & & 1\\\ & & & 1 \end{pmatrix},\quad \gamma_v \equiv \frac1{\sqrt{1 - v^2}}. \end{align*}\]

(This subsection is unfinished because I gotta study for midterms lol)

Summary of the notation

With these things in mind, let’s get dusty!

Intuitions

A fluid is a macroscopic phenomenon composed of particles — packets of mass–energy — traveling at significant speeds relative to neighboring particles. It has emergent properties, such as density and pressure, which are not well-defined for its individual constituents. Examples include water, neutron star interiors, and photons scattering in a sparse medium (but not, for instance, photons concentrated in a laser beam).

In particular, we are interested in the behavior of perfect fluids, which are isotropic in their rest frame, and have negligible interactions (except some scattering to keep them isotropic).

Fluids are locally modeled by a stress–energy–momentum tensor, which qualitatively means

\[T^{\alpha\beta} \to \begin{pmatrix} \text{energy density} & \text{momentum density in }\beta \\ \text{energy flux flowing}\to\alpha & \text{momentum flux flowing }\alpha\to\beta \end{pmatrix}.\]

At first glance, this looks rather asymmetric. But

(This section is unfinished)

Derivations

We want to find the stress–energy–momentum tensor for a small volume $\Delta V$ of a perfect fluid.

Because tensors components of the same inertial reference frame are additive, our method of attack is as follows:

  1. Find the stress–energy–momentum tensor of each constituent particle in their respective rest frames.
  2. Lorentz boost every constituent tensor to the rest frame of the fluid volume.
  3. Add their components up.

Let’s do this!

1. Dust

As noted in the previous section, fluids are composed of many particles.

\[\begin{align*} T^{\alpha\beta} &= \sum_\# T_\#^{\alpha\beta} \\ &= \sum_\# \rho_\# U_\#^{\alpha} U_\#^{\beta} \\ &= \sum_\# \f{m_\#}{\Delta V_\#} U_\#^{\alpha} U_\#^{\beta} \\ &= \f1{\Delta V} \sum_\# \f{m_\#}{\gamma_\#} U_\#^{\alpha} U_\#^{\beta} \\ &= \f1{\Delta V} \sum_\# \f{m_\#}{\gamma_\#} \begin{bmatrix} \gamma_\# \\ \gamma_\# u^1_\# \\ \gamma_\# u^2_\# \\ \gamma_\# u^3_\# \end{bmatrix}^{\!\alpha} \begin{bmatrix} \gamma_\# \\ \gamma_\# u^1_\# \\ \gamma_\# u^2_\# \\ \gamma_\# u^3_\# \end{bmatrix}^{\!\beta} \\ &= \f1{\Delta V} \sum_\# m_\# \gamma_\# \begin{bmatrix} 1 \\ u^1_\# \\ u^2_\# \\ u^3_\# \end{bmatrix}^{\!\alpha} \begin{bmatrix} 1 \\ u^1_\# \\ u^2_\# \\ u^3_\# \end{bmatrix}^{\!\beta} \\ &= \f1{\Delta V} \sum_\# m_\# \gamma_\# \begin{bmatrix} 1 & u^1_\# & u^2_\# & u^3_\# \\ u^1_\# & u^1_\# u^1_\# & u^2_\# u^1_\# & u^3_\# u^1_\# \\ u^2_\# & u^1_\# u^2_\# & u^2_\# u^2_\# & u^3_\# u^2_\# \\ u^3_\# & u^1_\# u^3_\# & u^2_\# u^3_\# & u^3_\# u^3_\# \\ \end{bmatrix}^{\!\alpha\beta} \\ \end{align*}\]

Suppose we have a fluid volume $\Delta V$.