Newton-Cartan part 3: gravitating particles

by Willie Wong

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.)

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