Newton-Cartan part 3: gravitating particles

As a simple example of a physical theory on a Galilean manifold, let us consider the physics of a collection of massive particles that do not interact except for their gravitational interaction. In other words, let us consider a collisionless kinetic theory coupled to Newtonian gravity.

Vlasov system
The Vlasov system is a transport equation describing the free flow of collisionless particles. Let $(M,\nabla)$ be a manifold with an affine connection that represents the spacetime. We postulate Newton’s first law:

Physical assumption 1
The motion of a free particle is geodesic.

Therefore the motion of a free particle is described by the following system of equations: let $\tau$ denote proper time as experienced by the particle, and $\gamma(\tau)$ the world-line of the particle (its spacetime trajectory) parametrized by $\tau$, then we have the hyperbolic system of equations.

Equation 2
$(\frac{d}{d\tau}\gamma)(\tau) = V\circ\gamma(\tau) \in T\gamma \subset TM$
and
$\frac{d}{d\tau}(V\circ\gamma) (= \frac{d^2}{d\tau^2}\gamma) = \nabla_VV = 0$

Now let us try to generalize this to a collection of particles. In fact, we’ll take the continuum limit. Instead of looking at a single particle as a single trajectory in phase space (which is equal to $TM$), we’ll consider a entire collection as a distribution on phase space. In symbols, we take $f:TM\to \mathbb{R}_+\cup \{0\}$ to be the phase-space density of particles. In other words, given a point $p\in M$ and a vector $V\in T_pM$, $f(p,V)$ is the density of particles sitting at the point $p$ and traveling with velocity $V$. The geodesic equation 2 tells us that the physical system must obey

Equation 3 (conservation of particles on geodesic motion)
$f(\gamma(\tau), \dot\gamma(\tau)) = f(\gamma(\tau'), \dot\gamma(\tau'))$

for any fixed geodesic $\gamma$ affinely parametrized by $\tau$. To write this as a differential equation, we can use the language of a geodesic spray. Given our affine connection $\nabla$, let $p$ be a point in $M$ and $V$ a tangent vector at $p$, there exists (locally) a unique geodesic $\gamma: (-\epsilon,\epsilon)\to M$ such $\gamma(0) = p$ and $\dot\gamma(0) = V$. Now, this curve naturally lifts to a curve $\tilde\gamma$ in $TM$ by requiring $\tilde\gamma(s) = (\gamma(s), \dot\gamma(s))$. From this we can define a vector $X \in T_{(p,V)}TM$ given by $X = \frac{d}{ds}\tilde\gamma |_{s = 0}$. Now if we do this for every point $(p,V)\in TM$, we obtain a vector field $X: TM\to TTM$. This vector field is the geodesic spray associated to the connection $\nabla$. Equation 3 then can be written in differential form

Equation 4
$Xf = 0$

Equation 4 is the free Vlasov equation on the fixed spacetime $(M,\nabla)$. It maybe of interest to express Equation 4 locally. Here we’ll give the expression in a local frame, the expression in a coordinate system can be established analogously. Let $\{e_i\}$ be a local frame, and let $\omega^i_j$ be the rotation one-form

$\nabla e_i = \omega_i^j e_j$

With the frame, we get a coordinate system on $TM$ by representing a point $p$ and a tangent vector $V\in T_pM$ by $(p,V^i)$, where $\sum V^i e_i = V$. The double tangent space $T_{p,V}TM$ then can be spanned by $(W,\partial^{(2)}_i)$ where $W\in T_pM$ and $\partial^{(2)}_i$ are the coordinate derivatives on $T_pM$ in the above representation. In this coordinates, we can write $f(p,V^i) = f(p,V)$. It remains to compute what $X$ is. Let $\gamma(\tau)$ be a geodesic. Its tangent vector can be written as $\sum_{i} x^i(\tau) e_i(\gamma(\tau))$. The geodesic equation implies that

$\displaystyle \frac{d}{d\tau}\left(\sum_{i} x^i(\tau) e_i(\gamma(\tau))\right) = 0$

Applying the Leibniz rule we have

$\sum_i\dot{x}^i e_i + \sum_i x^i \nabla_{\dot{\gamma}}e_i = \sum_i\dot{x}^i e_i + \sum_{i,j}x^i \omega(\dot{\gamma})^j_i e_j$

Which implies, after evaluating at $\tau = 0$, and setting the initial velocity to $V$

Equation 5
$\frac{d}{d\tau}x^i |_{\tau = 0} = - \sum_j V^j\omega^i_j(V)$

Which implies that in the coordinates given,

Equation 5′
$T_{p,V}TM \ni X = (V, - \sum_j V^j\omega^i_j(V) \partial^{(2)}_i)$

and Equation 4 becomes

Equation 4′
$\sum V^ie_i(f(p,V^k)) - V^j\omega^i_j(V)\cdot \partial^{(2)}_if(p,V^k) = 0$

Where the $e_i$ only acts on the $p$ coordinate.

Coupling to Galilean geometry
Now we make another physical assumption, assuming we are now working with a Galilean manifold $(M,\epsilon_t,\sigma,\nabla)$

Physical assumption 6
Time flows equally for all particles. In other words, the only allowed velocities are those for which $\epsilon_t(V) = +1$.

(N.b. In Einstein-Vlasov theory, where the equations are coupled to general relativity, the assumption is that particle velocity is described by a future-pointing, time-like vector with unit norm, which says that in the rest frame times flow the same way.)

The set of valid velocities we will denote by $\mathcal{V}\subset TM$, which we call the mass shell. Now, whereas the general Vlasov system as described in the previous section is not typed, it is clear that the Vlasov equation on a Galilean manifold with velocities restricted to the mass shell is hyperbolic relative to the spatial hypersurfaces $\Sigma_\tau$: Let $\pi: TM\to M$ be the canonical projection map, and $\iota: \mathcal{V}\hookrightarrow TM$ the canonical inclusion, and take $\tilde\Sigma_\tau := (\iota\circ\pi)^{-1}\Sigma_\tau$. Now, given an arbitrary admissible velocity $V$ at some point $p$, we develop the geodesic associated to it $\gamma$. But now observe that due to the structure $\nabla\epsilon_t = 0$, we have that $\epsilon_t(\dot\gamma) = \epsilon_t(V) = 1$ along $\gamma$. Therefore the geodesic flow carries $\mathcal{V}$ to itself. And this implies that $X$ is tangent to $\mathcal{V}$. (N.b. in the Einstein-Vlasov case, the analogous mass shell is also preserved under geodesic flow.) So by an abuse of notation we will use $X$ to also denote the vector field on $\mathcal{V}$ that generates the Vlasov evolution. Now, it is clear that by definition of $\mathcal{V}$ that $X$ cannot be tangent to $\tilde\Sigma_\tau$ (this is somewhat related to Lemma 9 of this post), therefore the first-order transport equation $Xf$ is locally well-posed, and the Vlasov system on the mass shell is thus a hyperbolic system.

In fact, the Vlasov system on $\mathcal{V}$ is integrable, as evident by Equation 3.

Gravitational coupling
Now we need to fully couple to the Newton-Cartan theory. In view of Definition 10 of my previous post, to define how the particles produce gravity, it suffices that we write down a stress-energy tensor $T[f]$. For this I’ll just appeal to physical intuition and not explain too deeply why this tensor is the appropriate one. (See the Wikipedia article on stress-energy tensor for what each component of the tensor represents. I hope that the definition becomes self-evident with that in mind.) In particular, we’ll take the stress-energy tensor associated to a point-particle of mass $m$ and space-time velocity $V$ to be $\frac12 m V\otimes V$.

Now, our distribution function $f$ is a function on $\mathcal{V}$, but we need to extract from it a tensor on $M$, to do so we need to somehow “integrate out the fibre”. Let $\mathcal{V}_p$ denote the fibre of $\mathcal{V}$ over a point $p\in M$ (uh… did I mention that $\mathcal{V}$ has a natural structure as a fibre bundle over $M$ with the projection map $\iota\circ\pi$?). While $\mathcal{V}$ is not a vector bundle, $\mathcal{V}_p$ has the structure of a projective space. But more importantly, we observe that, given any vector $Y\in T_p\Sigma_\tau$ and $V\in \mathcal{V}_p$, we have that $\epsilon_t(Y+V) = 0 + 1 = 1$ and so $Y+V$ is also in $\mathcal{V}_p$. Now, using that $\sigma$ induces on $\Sigma_\tau$ a Riemannian structure, we see that $\mathcal{V}_p$ admits a flat, Euclidean metric. This metric induces a volume form $dvol(\mathcal{V}_p)$. The key point is that this volume form is invariant under local symmetry transformations. And hence we can use this to define integrals on $\mathcal{V}_p$. (N.b. in the Einstein-Vlasov case, the mass shell acquires the structure of hyperbolic space.) So now we can write down the stress-energy tensor

Equation 7 (stress-energy)
$\displaystyle T[f](p) = \frac12 \int_{\mathcal{V}_p} V\otimes V f(p,V) dvol(\mathcal{V}_p)$

If we take an orthonormal basis $e_i$ of $T_p\Sigma_\tau$ and fix an element $V\in \mathcal{V}_p$, we can rewrite the integral as

Equation 7′
$\displaystyle T[f](p) = \frac12 \int_{\mathbb{R}^n} (V+ x^ie_i)\otimes (V+x^ie_i) f(p,V+x^ie_i)dx$

where $M$ has dimension (n+1). A simple computation shows that this tensor is indeed divergence free.

Now, using the fact that each $\Sigma_\tau$ is flat, we can pick a geodesic coordinate system around a point $p$ in the following way: locally since $\epsilon_t$ is closed, we can lift to a time function $\tau$ such that $\Sigma_\tau$ are level sets. Now, pick $V\in \mathcal{V}_p$, and let $\gamma_0$ denote its associated geodesic. (Note that $\nabla_{\dot\gamma}\tau = 1$ by construction.) We’ll call the points where $\gamma_0\cap \Sigma_\tau$ the “origin” for each slice. Now, at $p$ choose $e_i$ an orthonormal basis of $\Sigma_\tau$. Extend $e_i$ to vector fields along $\gamma_0$ by parallel transport. Now using each spatial slice is flat, we can locally build a coordinate system $(\tau = x^0, x^1, \ldots, x^n)$ such that the induced metric on $\Sigma$ is Euclidean, and such that $\partial_i |_{\gamma_0} = e_i$.

Now, recalling the Christoffel symbols $\Gamma_{\alpha\beta}^\gamma \partial_\gamma = \nabla_{\partial_\alpha} \partial_\beta$. The flatness and totally geodesy of $\Sigma$ implies $\Gamma_{ij}^\gamma = 0$. Using that $\epsilon_t(\partial_0) = 0$ we have $\Gamma_{\alpha 0}^0 = 0$. A direct computation then yields

Equation 8
$Ric_{00} = \partial_i\Gamma^i_{00} - \partial_0\Gamma^i_{i0} - \Gamma_{i0}^j\Gamma_{0j}^i$
$Ric_{i0} = \partial_j\Gamma^j_{i0} - \partial_i\Gamma^j_{j0}$

Now, evaluating at point $p$, where by construction we have $\Gamma^i_{j0} = \partial_0\Gamma^i_{j0} = 0$, the last two terms of the first equation drops out. Now, using also that $\Gamma_{00}^\alpha = 0$ along $\gamma_0$, we have that $\partial_i\Gamma^{j}_{00} = \partial_j\Gamma^i_{00}$. So writing $E^i = \Gamma^i_{00}$, we see that the gravitational coupling formally reduces to the equations

Equation 9
$\mathop{curl}_{\Sigma} E = 0~,\quad \mathop{div}_{\Sigma} E = 2\pi \int_{\mathcal{V}_p} f(p,V)dvol - \Lambda$

which, in the case where there is no cosmological constant, is in the same form as the Vlasov-Poisson equations. (In the special case where our initial data admits a vector field $v_0$ along $\Sigma_0$ with $\epsilon_t(v_0) = 1$ and such that for any $w\in T\Sigma_0$ we have $\nabla_w v_0 = 0$, we can actually make this precise.)