FR06 · L1 · Chapter 1 of 9

Find the rest frame of expanding matter

A small piece of matter can have a rest frame even when the whole collision is flying apart. In that frame its net energy flux vanishes. Its pressures can still differ along the beam and across it. This distinction gives us a useful description before local thermal equilibrium has been reached.

Before you beginYou need energy, momentum and the idea that velocity depends on the observer. Matrix notation is introduced below; no prior fluid mechanics is required.

Sandbox Physics · 2026-10-27

Start with a slice through the collision

Take the beam direction as the longitudinal direction. The plane perpendicular to the beams is the transverse plane. Two nuclei that miss a perfectly central impact leave an almond-shaped overlap in that plane. This is a geometrical starting point; a pressure field and an emitted-particle distribution require further physics.

Use $z$ along the beams and $x,y$ across them. The distance between the incoming nuclear centres in the transverse plane is the impact parameter $b$ . Each collision has its own fluctuating density. Even an ensemble with circular average geometry can contain individual triangular or elliptical shapes.

We first remove those transverse details and follow a uniform transverse patch. That reduction isolates the rapid longitudinal dilution. Later chapters restore transverse shape as an input to a limited response model.

Energy density belongs to a chosen observer

The stress–energy tensor $T^{\mu\nu}$ is a compact table: its time–time entry is energy per volume, its time–space entries are energy flux, and its spatial entries describe momentum flow. Greek indices run over time and three spatial directions. Throughout the course the metric is $g_{\mu\nu}=\operatorname{diag}(-1,1,1,1)$ ; this fixes the signs in the equations.

Choose a unit future-directed four-velocity $u^\mu$ , with $u_\mu u^\mu=-1$ . The Landau rest frame is the frame moving with this velocity in which energy has no net spatial flux. Multiplying the tensor by the observer’s four-velocity produces another four-vector. An eigenvector is a direction that this multiplication leaves parallel to itself; the proportionality factor here gives minus the rest-frame energy density.

T^{\mu\nu}u_\nu=-e\,u^\mu

The eigenvalue gives the local rest-frame energy density. This definition uses energy flow and does not assume an equilibrium momentum distribution.

A four-velocity is the relativistic version of the velocity of the moving patch. Finding its rest frame removes bulk motion from the local energy density, just as subtracting the motion of a box helps separate its contents’ random motion from the box’s overall motion.

Three pressures need not be the same

In a longitudinally symmetric rest frame, write the two equal transverse pressures as $P_T$ and the longitudinal pressure as $P_L$ . The conformal approximation used in these Labs assumes zero trace. Its spatial pressure sum therefore equals the energy density.

T^{\mu\nu}_{\mathrm{rest}}=\operatorname{diag}(e,P_T,P_T,P_L),\qquad 2P_T+P_L=e

Conformality is an assumption about this model. The QCD equation of state, especially near the crossover, needs a more general relation.

A convenient shear variable $\phi$ measures the reduction of longitudinal pressure: $P_L=e/3-\phi$ and $P_T=e/3+\phi/2$ . Positive shear here means weaker longitudinal pressure. This sign convention will stay fixed when the patch begins to expand.

At equilibrium the pressures are equal. Away from equilibrium, one can still define an energy-matched temperature: the temperature an equilibrium conformal gas would have at the same energy density. It labels energy density. A full nonequilibrium description also needs the directional momentum distribution.

Worked example

One moving patch, two energy densities

Suppose an isotropic patch has $e=12$ and pressure $p=4$ in consistent energy-density units. An observer sees it move along the beam with speed $v=0.6$ in units where light speed is one. The Lorentz factor squared is $\gamma^2=1/(1-v^2)=1.5625$ .

T^{00}=\gamma^2(e+p)-p=21,\qquad T^{0z}=\gamma^2(e+p)v=15

The moving observer measures more energy per coordinate volume and a nonzero energy flux. Returning to the Landau frame recovers the rest-frame energy density of 12.

The difference is not heating. The observer has included bulk motion. Keeping these two energies separate prevents us from assigning a temperature to motion of the entire patch.

Predict, then try the model

Predict which pressure decreases when the initial shear is increased at fixed initial energy.

Open Bjorken expansion and pause. Focus the time handle and press Home to return to the initial time, then move Initial MIS shear. Compare the longitudinal-to-transverse pressure ratio and the pressure glyph.

Bjorken expansion ↗

Compare your explanation

At the initial time, the two transverse pressures plus the longitudinal pressure sum to the fixed energy density. The ratio and glyph show the same directional imbalance.

Exercises with explanations

1. A conformal patch has $e=12$ and $P_L=1$ . Find its transverse pressure and say whether its Landau rest frame is also an isotropic state.

Read the reasoning

The trace condition gives $P_T=(12-1)/2=5.5$ . Energy flux can vanish with these unequal pressures. A rest frame specifies bulk motion; isotropy specifies the directional stress.

2. Why is the collision’s centre-of-momentum frame generally insufficient as a rest frame for every fluid patch?

Read the reasoning

Patches at different positions can move apart with different velocities. A single global frame removes only the net momentum of the entire system; each patch needs its own local energy-flow velocity.

Sources and reading scope

J. D. Bjorken, Highly relativistic nucleus–nucleus collisions: The central rapidity region (1983)
Original boost-invariant spacetime construction; proper time and spacetime rapidity. The course derives the pressure-work equation explicitly.
Baier et al., Relativistic viscous hydrodynamics, conformal invariance, and holography, arXiv:0712.2451v3
Version 3 (15 July 2008), §§3.5 and 6: Bjorken reduction, relaxation and causality. The implemented closure sets the nonlinear lambda-one coefficient to zero.
Florkowski, Heller & Spaliński, New theories of relativistic hydrodynamics in the LHC era, arXiv:1707.02282v3
Version 3 (19 October 2017), §§5.1, 5.3, 6 and 7.4; notation and limits of kinetic and hydrodynamic descriptions. This is a review, not an independent experimental dataset.