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.
The Vlasov system is a transport equation describing the free flow of collisionless particles. Let 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 denote proper time as experienced by the particle, and the world-line of the particle (its spacetime trajectory) parametrized by , then we have the hyperbolic system of equations.
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 ), we’ll consider a entire collection as a distribution on phase space. In symbols, we take to be the phase-space density of particles. In other words, given a point and a vector , is the density of particles sitting at the point and traveling with velocity . The geodesic equation 2 tells us that the physical system must obey
Equation 3 (conservation of particles on geodesic motion)
for any fixed geodesic affinely parametrized by . To write this as a differential equation, we can use the language of a geodesic spray. Given our affine connection , let be a point in and a tangent vector at , there exists (locally) a unique geodesic such and . Now, this curve naturally lifts to a curve in by requiring . From this we can define a vector given by . Now if we do this for every point , we obtain a vector field . This vector field is the geodesic spray associated to the connection . Equation 3 then can be written in differential form
Equation 4 is the free Vlasov equation on the fixed spacetime . 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 be a local frame, and let be the rotation one-form
With the frame, we get a coordinate system on by representing a point and a tangent vector $V\in T_pM$ by , where . The double tangent space then can be spanned by where and are the coordinate derivatives on in the above representation. In this coordinates, we can write . It remains to compute what is. Let be a geodesic. Its tangent vector can be written as . The geodesic equation implies that
Applying the Leibniz rule we have
Which implies, after evaluating at , and setting the initial velocity to
Which implies that in the coordinates given,
and Equation 4 becomes
Where the only acts on the coordinate.
Coupling to Galilean geometry
Now we make another physical assumption, assuming we are now working with a Galilean manifold
Physical assumption 6
Time flows equally for all particles. In other words, the only allowed velocities are those for which .
(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 , 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 : Let be the canonical projection map, and the canonical inclusion, and take . Now, given an arbitrary admissible velocity at some point , we develop the geodesic associated to it . But now observe that due to the structure , we have that along . Therefore the geodesic flow carries to itself. And this implies that is tangent to . (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 to also denote the vector field on that generates the Vlasov evolution. Now, it is clear that by definition of that cannot be tangent to (this is somewhat related to Lemma 9 of this post), therefore the first-order transport equation is locally well-posed, and the Vlasov system on the mass shell is thus a hyperbolic system.
In fact, the Vlasov system on is integrable, as evident by Equation 3.
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 . 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 and space-time velocity to be $ \frac12 m V\otimes V$.
Now, our distribution function is a function on , but we need to extract from it a tensor on , to do so we need to somehow “integrate out the fibre”. Let denote the fibre of over a point (uh… did I mention that has a natural structure as a fibre bundle over with the projection map ?). While is not a vector bundle, has the structure of a projective space. But more importantly, we observe that, given any vector and , we have that and so is also in . Now, using that induces on a Riemannian structure, we see that admits a flat, Euclidean metric. This metric induces a volume form . The key point is that this volume form is invariant under local symmetry transformations. And hence we can use this to define integrals on . (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)
If we take an orthonormal basis of and fix an element , we can rewrite the integral as
where has dimension (n+1). A simple computation shows that this tensor is indeed divergence free.
Now, using the fact that each is flat, we can pick a geodesic coordinate system around a point in the following way: locally since is closed, we can lift to a time function such that are level sets. Now, pick , and let denote its associated geodesic. (Note that by construction.) We’ll call the points where the “origin” for each slice. Now, at choose an orthonormal basis of . Extend to vector fields along by parallel transport. Now using each spatial slice is flat, we can locally build a coordinate system such that the induced metric on is Euclidean, and such that .
Now, recalling the Christoffel symbols . The flatness and totally geodesy of implies . Using that we have . A direct computation then yields
Now, evaluating at point , where by construction we have , the last two terms of the first equation drops out. Now, using also that along , we have that . So writing , we see that the gravitational coupling formally reduces to the equations
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 along with and such that for any we have , we can actually make this precise.)