---
title: Post-Newtonian Boltzmann Equation
url: https://www.emergentmind.com/topics/post-newtonian-boltzmann-equation
type: topic
---

# Post-Newtonian Boltzmann Equation

The post-Newtonian Boltzmann equation is the weak-field, slow-motion expansion of relativistic kinetic theory for a one-particle distribution function in a gravitational field. In collisionless problems it is the post-Newtonian Vlasov equation for self-gravitating systems; in collisional problems it is coupled either to the full Boltzmann collision operator or to model operators such as BGK/Marle. Its characteristic feature is that the Newtonian streaming operator is supplemented by terms of order \(1/c^2\) and higher involving scalar and vector gravitational potentials, corrected phase-space measures, and post-Newtonian source terms for the gravitational field equations. The formalism has been developed at \(1\)PN, \(2\)PN, and \(2.5\)PN orders and applied to hydrodynamics, dissipative transport, Jeans instability, spherical stellar systems, Brownian motion, and multicomponent self-gravitating media [2010.11017, 2112.02419, 2303.03331, 2604.27575].

## 1. Weak-field kinetic framework

In the standard post-Newtonian hierarchy, Newtonian order gives the classical Boltzmann or Vlasov equation, \(1\)PN retains corrections of order \(1/c^2\), \(2\)PN retains corrections of order \(1/c^4\), and \(2.5\)PN introduces the first odd-order, radiative terms, with the \(2.5\)PN Boltzmann equation and equilibrium Maxwell–Jüttner distribution determined up to order \(1/c^7\) [2010.11017, 2604.27575]. The dynamical variable is the one-particle distribution function, written either as \(f(x,p,t)\) in momentum variables or as \(f(\mathbf{x},\mathbf{v},t)\) in three-velocity variables.

A common \(1\)PN metric choice is Chandrasekhar’s form
\[
g_{00}=1-\frac{2U}{c^2}+\frac{2}{c^4}(U^2-2\Phi),\qquad g_{0i}=\frac{\Pi_i}{c^3},\qquad g_{ij}=-\left(1+\frac{2U}{c^2}\right)\delta_{ij},
\]
where \(U\) is the Newtonian gravitational potential, \(\Phi\) is a scalar \(1\)PN potential, and \(\Pi_i\) is a vector \(1\)PN potential [2112.02419]. Some treatments use the notation
\[
g_{00}=1+\frac{2\phi}{c^2}+\frac{2}{c^4}(\phi^2+\psi),\qquad g_{0i}=-\frac{\xi_i}{c^3},\qquad g_{ij}=-\left(1-\frac{2\phi}{c^2}\right)\delta_{ij},
\]
with \(\phi,\psi,\xi_i\) playing the analogous roles [2203.06922]. At \(2\)PN and \(2.5\)PN order, additional tensor and radiative potentials such as \(\Psi_{ij}^{(4)},\Psi_{00}^{(6)},\mathcal Y_{00}^{(5)}\), and \(\mathcal Y_{ij}^{(5)}\) enter the metric and the kinetic equation [2010.11017, 2604.27575].

The post-Newtonian Boltzmann equation is therefore not a single fixed formula but a family of weak-field kinetic equations whose explicit appearance depends on approximation order, variable choice, and whether collisions are retained. A plausible implication is that the term designates a framework rather than one canonical operator.

## 2. Collisionless \(1\)PN equation and corrected conserved quantities

For collisionless self-gravitating systems, the central object is the \(1\)PN collisionless Boltzmann equation obtained by expanding the general relativistic Liouville operator. In Chandrasekhar-type notation one widely used explicit form is
\[
\frac{\partial f}{\partial t} +v_i\frac{\partial f}{\partial x^i} +\frac{\partial U}{\partial x^i}\frac{\partial f}{\partial v^i} +\frac{1}{c^2}\bigg[ \left(v^2-4U\right)\frac{\partial U}{\partial x^i} -4v_iv_j\frac{\partial U}{\partial x^j} -3v_i\frac{\partial U}{\partial t} +2\frac{\partial \Phi}{\partial x^i} +\frac{\partial \Pi_i}{\partial t} +v_j\left(\frac{\partial \Pi_i}{\partial x^j}-\frac{\partial \Pi_j}{\partial x^i}\right) \bigg]\frac{\partial f}{\partial v^i}=0,
\]
which is the equation used in post-Newtonian Jeans analyses [2108.03885]. Equivalent formulations may appear with a prefactor multiplying the Newtonian streaming term, but all retain only terms through order \(1/c^2\).

A major correction to earlier literature was established in “Kinetic Theory of Collisionless Self-Gravitating Gases: Post-Newtonian Polytropes” [1104.5262]. That work argued that the \(1\)PN collisionless equation previously used by Rezania and Sobouti is not consistent with the actual \(1\)PN equations of motion, because the quantities identified there as energy and angular momentum are not true integrals of motion. The corrected \(1\)PN collisionless Boltzmann equation is consistent with the \(1\)PN test-particle dynamics, the correct conserved energy
\[
E=\frac{\mathbf{v}^{2}}{2}+\phi +\frac{3\mathbf{v}^{4}}{8c^2} -\frac{3\mathbf{v}^{2}\phi}{2c^2} +\frac{\phi^2}{2c^2} +\frac{\psi}{c^2},
\]
and, for spherical systems, the corrected angular momentum
\[
l_{i}=\varepsilon_{ijk}x^{j}v^{k}\left(1-\frac{3\phi}{c^{2}}+\frac{\mathbf{v}^{2}}{2c^{2}}\right).
\]
The same paper states that Jeans theorem remains valid at \(1\)PN order: static equilibria are functions of the true integrals of motion [1104.5262].

This correction resolved a common misconception that the post-Newtonian collisionless Boltzmann equation can be constructed by a naive relativistic modification of the Newtonian operator. In the corrected formulation, consistency with the microscopic equations of motion is a necessary criterion.

## 3. Gravitational coupling, invariant measures, and equilibrium distributions

The post-Newtonian Boltzmann equation is coupled to post-Newtonian field equations through velocity moments of the distribution function. In kinetic form, the energy-momentum tensor is written as
\[
T^{\mu\nu}=mc\int u^\mu u^\nu f\,\frac{\sqrt{-g}\,d^3u}{u^0}
\]
or, in equivalent species-dependent notation,
\[
T_a^{\mu\nu}=m_a c \int u_a^\mu u_a^\nu f_a \,\frac{\sqrt{-g}\,d^3u_a}{u_{a0}},
\]
with the invariant measure expanded consistently at \(1\)PN order. One representative measure expansion is
\[
\frac{\sqrt{-g}\,d^3u}{u^0} = \left[ 1+\frac{1}{c^2}\left(2v^2+6U\right) \right]\frac{d^3v}{c},
\]
and the particle four-velocity has the \(1\)PN form
\[
u^0=c\left[1+\frac{1}{c^2}\left(\frac{v^2}{2}+U\right)\right],\qquad u^i=\frac{u^0v^i}{c}
\]
[2112.02419, 2303.03331].

The corresponding Poisson-type field equations couple the kinetic moments to the gravitational potentials. In one common notation,
\[
\nabla^2 U = -4\pi G\rho,\qquad
\nabla^2 \Phi=-4\pi G\rho\left(V^2+U+\varepsilon+\frac{3p}{\rho}\right),\qquad
\nabla^2 \Pi_i=-16\pi G\rho V_i + \frac{\partial^2 U}{\partial t\,\partial x_i},
\]
while in multicomponent collisionless systems the sources are sums over species-specific moments of \(T_{\mu\nu}\) [2112.02419, 2303.03331].

Equilibrium is described by post-Newtonian Maxwell–Jüttner distributions. For a stationary background with vanishing hydrodynamic velocity, one \(1\)PN form is
\[
f^{MJ}=f^0\left\{1-\frac{1}{c^2}\left[\frac{15\sigma^2}{8} +\frac{3V^4}{8\sigma^2} +\frac{2UV^2}{\sigma^2}\right]\right\},
\]
with \(f^0\) the Maxwellian part [2303.03331]. In peculiar-velocity notation \(\mathcal V_i=v_i-V_i\), the \(1\)PN Maxwell–Jüttner distribution acquires corrections involving \(\mathcal V^2,\mathcal V^4,U,V^2\), and \(V_i\mathcal V_i\), and at \(2\)PN the equilibrium distribution contains quartic, sextic, and octic peculiar-velocity terms together with \(\Phi,\Pi_i\), and the \(2\)PN potentials \(\Psi_{ij},\Psi_{0i},\Psi_{00}\) [2010.11017]. For non-perfect fluids, the Grad 14-moment distribution has also been expanded to \(1\)PN order, yielding dissipative corrections linear in the pressure deviator, heat flux, and dynamic pressure [2112.02419].

These kinetic ingredients support macroscopic balance laws. In the \(1\)PN non-equilibrium theory, the conserved mass density
\[
\rho^*=\rho\left(1+\frac{1}{c^2}\left[\frac{V^2}{2}+3U\right]\right)
\]
satisfies
\[
\frac{\partial \rho^*}{\partial t} +\frac{\partial(\rho^*V_i)}{\partial x_i}=0,
\]
and analogous \(1\)PN balance equations exist for mass-energy density and momentum density, with explicit viscous-stress and heat-flux terms [2112.02419].

## 4. Jeans instability and multicomponent self-gravitating media

A central application of the post-Newtonian Boltzmann equation is gravitational instability. In the one-component collisionless case, the perturbative construction uses a homogeneous background, the Jeans swindle, and plane-wave perturbations
\[
f_1=\bar f_1 e^{i(\mathbf{k}\cdot\mathbf{x}-\omega t)},\qquad
U_1=\bar U_1 e^{i(\mathbf{k}\cdot\mathbf{x}-\omega t)},\qquad
\Phi_1=\bar \Phi_1 e^{i(\mathbf{k}\cdot\mathbf{x}-\omega t)},\qquad
\Pi_1^i=\bar \Pi_1^i e^{i(\mathbf{k}\cdot\mathbf{x}-\omega t)}.
\]
After solving the linearized kinetic equation and evaluating Gaussian velocity integrals, the resulting dispersion relation yields a post-Newtonian threshold
\[
\kappa_* = 1+\frac{\sigma^2}{c^2}\left(4+\frac{U_0}{\sigma^2}\right),
\]
so the critical wavelength is reduced and the Jeans mass is smaller than in the Newtonian theory. With the estimate \(U_0\simeq \sigma^2\), the paper gives
\[
\frac{M_J^{PN}}{M_J^N}\approx 1-15\frac{\sigma^2}{c^2},
\]
which makes explicit that the \(1\)PN correction shifts the instability threshold toward collapse at smaller masses [2108.03885].

The multicomponent generalization was developed for a self-gravitating baryon–dark-matter mixture in “A self-gravitating system composed of baryonic and dark matter analysed from the post-Newtonian Boltzmann equations” [2303.03331]. There the kinetic model is built from two one-particle distribution functions,
\[
f_a=f(\mathbf{x},\mathbf{v}_a,t),\qquad a=b,d,
\]
one for baryons and one for dark matter, each satisfying its own collisionless \(1\)PN Boltzmann equation and coupled only through the common gravitational potentials. The perturbed field equations are sourced by moments of \(f_{b1}\) and \(f_{d1}\), and the dimensionless dispersion relation depends on the density ratio \(\rho_{b0}/\rho_{d0}\), the velocity-dispersion ratio \(\sigma_b/\sigma_d\), the post-Newtonian parameter \(\sigma_d^2/c^2\), and the background potential \(U_0\). The paper shows that both dark matter and \(1\)PN corrections reduce the Jeans mass relative to the standard one-component Newtonian criterion [2303.03331].

Dark matter enters that framework in three distinct ways: it has its own distribution function obeying its own collisionless \(1\)PN Boltzmann equation; it contributes separately to the source terms in the post-Newtonian field equations through its own energy-momentum tensor moments; and its equilibrium density \(\rho_{d0}\) and velocity dispersion \(\sigma_d\) define the reference Jeans scale [2303.03331]. The same paper applies the formalism to Bok globules, using \(U_0=\sigma_b^2\), \(\rho_{d0}/\rho_{b0}\approx 5.5\), and inferred values \(\sigma_d^2\sim 10^6\ \mathrm{m}^2/\mathrm{s}^2\), and states that the model correctly reproduces the observational stability data [2303.03331].

## 5. Spherical systems, Jeans equations, and post-Newtonian polytropes

For stationary, spherically symmetric stellar systems, the collisionless \(1\)PN Boltzmann equation can be rewritten in spherical coordinates and reduced by taking velocity moments. In “Post-Newtonian Jeans Equation for Stationary and Spherically Symmetrical Self-Gravitating System” [2203.06922], the radial moment yields the final \(1\)PN Jeans equation
\[
\frac{d\rho\langle v_r^2\rangle}{d r} +2\rho\frac{\beta\langle v_r^2\rangle}{r} +\rho\frac{d\phi}{d r}\left[1+\frac{\langle v_r^2\rangle(5-6\beta)}{2c^2}\right] +\frac{\rho}{c^2}\frac{d\psi}{d r}=0,
\]
with anisotropy parameter
\[
\beta=1-\frac{\langle v_\theta^2\rangle}{\langle v_r^2\rangle}.
\]
The same analysis derives \(\langle v_\theta^2\rangle=\langle v_\varphi^2\rangle\) from the angular moment equation and couples the Jeans equation to the scalar Poisson equations
\[
\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{d\phi}{dr}\right)=4\pi G\rho,\qquad
\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{d\psi}{dr}\right) =4\pi G\rho\left[\frac{3\langle v_r^2\rangle(3-2\beta)}{2}-2\phi\right]
\]
[2203.06922].

The closure used in that stationary problem is a \(1\)PN Maxwell–Jüttner equilibrium distribution, from which the required higher velocity moments are evaluated [2203.06922]. Applied to a Hernquist model with a central black hole, the post-Newtonian scalar potential \(\psi\) is found to be positive and increasing toward small radius, and the paper states that it attenuates the rise of the velocity-dispersion profile relative to the purely Newtonian prediction [2203.06922].

The corrected collisionless \(1\)PN Boltzmann equation has also been used to construct post-Newtonian polytropes. For the ergodic distribution
\[
f(E)=
\begin{cases}
\dfrac{k_n}{2\pi}(-E)^{n-3/2}, & E<0,\\[1ex]
0, & E\ge 0,
\end{cases}
\]
the resulting field equations are
\[
\nabla^{2}\phi=\alpha_{n}(-\phi)^{n}, \qquad
\nabla^{2}\psi=-n\alpha_{n}(-\phi)^{n-1}\psi+ \beta_{n}(-\phi)^{n+1},
\]
which give the Newtonian polytrope equation together with its post-Newtonian correction [1104.5262]. The same work notes that sufficiently strong relativistic corrections can produce negative mass density in outer regions for some parameter choices, and interprets such cases as unphysical and indicative of the limits of the \(1\)PN approximation [1104.5262].

## 6. Collisional, dissipative, and stochastic extensions

The post-Newtonian Boltzmann equation is not restricted to collisionless dynamics. In “Relaxation-Time Model for the Post-Newtonian Boltzmann Equation” [2301.04880], the collision term is replaced by the BGK/Marle model
\[
\mathcal Q(f,f) = -\nu (f-f_{MJ}) = -\nu f_{\mathrm{NE}},
\]
and the Chapman–Enskog method is used to obtain the non-equilibrium distribution function, the hydrodynamic balance equations, and constitutive laws for viscous stress and heat flux. The paper identifies Newtonian transport coefficients
\[
\lambda_N=\frac{5kp}{2m\nu},\qquad \mu_N=\frac{p}{\nu},
\]
and gives the \(1\)PN-corrected coefficients
\[
\lambda = \lambda_N \left[ 1+\frac{1}{c^2}\left(2U+\frac{5kT}{2m}\right) \right], \qquad
\mu = \mu_N \left[ 1+\frac{1}{c^2}\left(2U+\frac{5kT}{2m}\right) \right],
\]
so both shear viscosity and thermal conductivity depend explicitly on the Newtonian gravitational potential \(U\) [2301.04880].

A more general non-equilibrium theory retaining the full Boltzmann collision operator was developed in “Post-Newtonian non-equilibrium kinetic theory” [2112.02419]. That paper combines the \(1\)PN Boltzmann equation, the Eckart decomposition,
\[
T^{\mu\nu} = p^{\langle\mu\nu\rangle} -(p+\varpi)\Delta^{\mu\nu} +\frac{1}{c^2}e\,U^\mu U^\nu +\frac{1}{c^2}\big(U^\mu q^\nu+U^\nu q^\mu\big),
\]
and the Grad 14-moment distribution expanded to post-Newtonian order. It derives a \(1\)PN Maxwell–Enskog transfer equation and the \(1\)PN hydrodynamic equations for conserved mass density, mass-energy density, and momentum density. The dynamic pressure \(\varpi\) is retained in the Grad distribution but drops out of the final \(1\)PN hydrodynamics for a rarefied monatomic gas because its contribution is of order \(1/c^4\) [2112.02419].

The collisional framework has also been reduced to a stochastic kinetic description. “Fokker-Planck equation for the Brownian motion in the post-Newtonian approximation” [2507.11082] starts from a \(1\)PN Boltzmann equation for a dilute binary mixture, assumes \(m_B\gg m_G\) and \(n_B/n_G\ll 1\), and derives a post-Newtonian Fokker–Planck equation for the heavy particles by expanding the collision integral to second order in the small velocity increment. The derivation retains post-Newtonian corrections not only in the streaming operator but also in the invariant measure, invariant collision flux, equilibrium Maxwell–Jüttner distributions, and cross-section dependence on the flux. For hard spheres, the \(1\)PN friction coefficient depends explicitly on \(U\), and the quoted \(2\)PN extension depends on \(U^2\), \(\Phi\), and \(\Psi_{kk}\) [2507.11082]. The same paper develops Newtonian and post-Newtonian linear stability analyses and states that for perturbation wavelengths smaller than the Jeans wavelength two propagating modes and one nonpropagating mode appear, while for wavelengths larger than the Jeans wavelength the perturbation either grows or decays, with the growing branch corresponding to instability [2507.11082].

## 7. Higher-order post-Newtonian structure and regime of validity

Beyond \(1\)PN, “Post-Newtonian Kinetic Theory” [2010.11017] derives the Boltzmann equation in the second post-Newtonian approximation directly from the proper-time evolution of the one-particle distribution function along the particle worldline. The resulting \(2\)PN equation contains higher-order contributions from \(\Phi,\Pi_i,\Psi_{ij},\Psi_{0i}\), and \(\Psi_{00}\), together with the \(2\)PN Maxwell–Jüttner equilibrium distribution and the \(2\)PN moment equations for mass density, mass-energy density, and momentum density [2010.11017]. An important structural point made there is that combining the mass and mass-energy equations yields the internal-energy equation in the first post-Newtonian approximation, so some conservation laws at a given PN order require higher-order kinetic information [2010.11017].

At \(2.5\)PN, “Boltzmann equation in the \(2{\frac12}\)-post-Newtonian approximation” [2604.27575] extends the kinetic theory through order \(1/c^7\) in harmonic coordinates. The collisionless streaming operator then contains, in addition to the Newtonian and conservative even-parity PN terms, odd-order radiative potentials such as \(\mathcal Y_{00}^{(5)}\), \(\mathcal Y_{ij}^{(5)}\), \(\mathcal Y_i^{(6)}\), and \(\mathcal Y_{00}^{(7)}\), which are tied to time derivatives of the inertia tensor and represent radiation-reaction effects [2604.27575]. The same paper shows that these potentials also enter the equilibrium Maxwell–Jüttner distribution and the higher-order moments \(N^\mu\) and \(T^{\mu\nu}\). It further notes that although the Boltzmann and metric expansions include \(2.5\)PN terms, the final total-energy conservation law derived from the available moments does not yet display the full radiative energy-loss term; higher-order metric information would be required [2604.27575].

Across the literature, several assumptions recur: weak gravitational fields, slow bulk and particle motions, truncation at a fixed post-Newtonian order, and either collisionless evolution or a near-equilibrium closure such as BGK or Grad. Many instability calculations also assume a stationary homogeneous background, the Jeans swindle, and plane-wave perturbations [2108.03885, 2303.03331]. A common misconception is that the post-Newtonian Boltzmann equation is only a \(1\)PN collisionless construct. The published developments instead include collisionless and collisional theories, dissipative closures, stochastic reductions, and explicit extensions to \(2\)PN and \(2.5\)PN order [2010.11017, 2112.02419, 2301.04880, 2507.11082, 2604.27575].

Source: https://www.emergentmind.com/topics/post-newtonian-boltzmann-equation