Coordinates in the relativistic Boltzmann theory

It is often the case in mathematical analysis that solving an open problem can be facilitated by finding a new set of coordinates which may illumniate the known difficulties. In this article, we illustrate how to derive an assortment coordinates in which to represent the relativistic Boltzmann collision operator. We show the equivalence between some known representations, and others which seem to be new. One of these representations has been used recently to solve several open problems.


Introduction and main result
The relativistic Boltzmann equation can be written as p µ ∂ µ F = C(F, F ).
In this expression the collision operator [3,15] is given by The transition rate, W (p, q|p ′ , q ′ ), from [3] in n-dimensions (n ≥ 2) is denoted (1) W (p, q|p ′ , q ′ ) = 1 2 where σ(̺, θ) is the differential cross-section which is a measure of the interactions between particles. This is an important model for fast moving particles. Standard references in relativistic Kinetic theory include [13,15,28,40,41]. In this paper we give a complete reduction of the collision integrals for the operator C(f, h), deriving several sets of coordinates for the particles momentum, some old and some new. The rest of our notation is given after the following historical discussion. [49] proves the smoothing effects for relativistic Landau-Maxwell system. And [48] proves time decay rates in the whole space for the relativistic Boltzmann equation (with hard potentials) and the relativistic Landau equation as well.
1.2. Notation. In this section we define several notations which will be used throughout the article. A relativistic particle has momentum p = (p 1 , . . . , p n ) ∈ R n , with its energy defined by p 0 = c 2 + |p| 2 where |p| 2 def = p · p. Here c denotes the speed of light. We use the standard Euclidean dot product: p · q def = n i=1 p i q i . As is customary we write p µ = (p 0 , p) where p µ also denotes the µ-th element of (p 0 , p). In general, Latin (spatial) indices i, j, etc., take values in {1, . . . , n}, while Greek indices κ, λ, µ, ν, etc., take on the values {0, 1, . . . , n}. Indices are raised and lowered as usual with the Minkowski metric g µν and its inverse g µν , where (g µν ) def = diag(−1 1 · · · 1) is an (1 + n) × (1 + n) matrix. In other words p µ = g µν p ν . We furthermore use the Einstein convention of implicit summation over repeated indices. The Lorentz inner product is then given by where ∇ x is the spatial gradient. Conservation of momentum and energy for elastic collisions is expressed as These conservation laws are enforced by the 1 + n delta functions in (1). The angle θ in the Boltzmann collision operator (1) is then defined by Note that this angle is well defined under (2), see [28,Lemma 3.15.3]. Here the relative momentum, ̺ = ̺(p µ , q µ ), is denoted Furthermore the quantity s = s(p µ , q µ ) is defined as This is in other words s = 2(p 0 q 0 − p · q + c 2 ). Notice that s = ̺ 2 + 4c 2 . To proceed, we will quickly review the Lorentz transformations.
The Boost matrix. The most common Lorentz transformation is probably the Boost matrix. Given v = (v 1 , . . . , v n ) ∈ R n , the (1 + n) × (1 + n) Boost matrix is where ρ = (1 − |v| 2 ) −1/2 and 1 n is the n × n identity matrix. Notice that Λ b has only n free parameters. Our goal is to choose v such that Λ b satisfies (7).
Plugging these choices into Λ b above, we obtain that By a direct calculation this example satisfies (7).

Main results.
We will now state our main results in the language of the notation just introduced. For this we use the operator Q(f, h) . The main point is to carry out the reduction of the number of integrations in this expression by evaluating the 1 + n delta functions from (1). In the literature, there are two approaches to performing this goal [15,31]. One [31] uses algebraic manipulation of polynomials; this results in a representation (Theorem 4) that is well known in the mathematics literature and has been widely used (at least in dimension n = 3). The other representation [15] (from Corollary 3 and Theorem 2), which starts by using the change of variables as in (7), is sketched in physics texts but seems to be hard to locate in the mathematics literature on the relativistic Boltzmann equation; this approach was developed (as in Corollary 3 below in dimension n = 3) in the authors thesis [43]. Additionally the representations of the collision operator (10) from Corollary 3 have only recently been used to solve several open problems on the relativistic Boltzmann equation [34,39,44,45].
For this reason, we are motivated at this time to write down a complete mathematical proof of these different representations. Our results improve upon those given previously [15,31] in the following ways. We prove these representations in n dimensions with n ≥ 2. Note that the representation in Theorem 4 is dimensionally dependent; there is an additional term (19) which is not present when n = 3. Furthermore it is discussed in the physics paper [3] that there are physical situations in which the Boltzmann equation may be of interest in dimensions other than n = 3. We also prove the exact formula for the post-collisional energies as in (21). We give the precise expressions for the angles in (13), (14), and (22) which do not seem to have been previously computed. What the author finds most interesting is that we can show that there are several alternative expressions for the post-collisional momentum and energy, as in (15) and (16), one for each Lorentz transformation satisfying (7). We hope that the availability of these additional alternative representations may be useful to future investigations in the relativistic Boltzmann theory; in particular we observe in Corollary 5 the equivalence of these different representations. This equivalence has been crucial to our recent proof (joint with Yan Guo) of the global in time stability for the relativistic Vlasov-Maxwell-Boltzmann system [34] with near Maxwellian initial conditions.
To begin we state the center of momentum (7) reduction.
Theorem 2. (Center of momentum reduction). Recall (10) and (1). For any suitable integrable function G : R n × R n × R n × R n → R, it holds that where ω = (ω 1 , . . . , ω n ) ∈ S n−1 and v ø = v ø (p, q) is the Møller velocity given by The angle θ in this expression is defined by (13) and (14). The post-collisional momentum and energy above are defined by (15) and (16) respectively.
The angle in the reduced expression in Theorem 2 is defined by (13) cos where k ∈ R n is given as Moreover the post-collisional momentum satisfy (i = 1, . . . , n) Here we use the tensor L µκ def = −g µλ Λ κ λ . Furthermore, the energies are As usual, in each of these expressions, we implicitly sum over j ∈ {1, . . . , n}. Note that the formulae above hold for any Lorentz transformation Λ satisfying only (7). Now (1), (10) and (11) together imply that where the relevant quantities are defined above as in (13), (14), (15) and (16).
If we use the Lorentz boost (9) satisfying (7) we have the following simplification.
Corollary 3. In the particular case of the Boost matrix (9), the post-collisional momentum (15) in the expressions above can be written precisely as where ρ def = (p 0 + q 0 )/ √ s. Furthermore, the post-collisional energies are given by Additionally in the angle (13), the vector (14) can be simplified to In these formula we use s from (5).
Now we turn to the expression given by Glassey-Strauss in [31].
where B n (p, q, ω) is given by (19) and (p ′ , q ′ ) on the right are given by (20). The angle θ is also defined by (22).
In the above reduction, we consider the expression Then the kernel in Theorem 4 is given by Notice that B 3 (p, q, ω) = B(p, q, ω) when the dimension is n = 3. For the reduction in Theorem 4, the post-collisional momentum are given by where a(p, q, ω) And the energies can be expressed as These formula clearly satisfy the collisional conservations (2). The angle (3) in Theorem 4, can then be reduced to (22) Evidently, in this case θ is not simply a dot product of a unit vector with ω.

Corollary 5. Combining Theorem 2 and Theorem 4 yields
where the angle and post-collisional momentum on the left are defined as in Theorem 2, and the similar expressions on the right are given as in Theorem 4. (20), it was computed in [30, Theorem 1] that the mapping (p, q) → (p ′ , q ′ ) has the following Jacobian

Now in the variables
Note also that when (p, q) → (p ′ , q ′ ) then additionally (p ′ , q ′ ) → (p, q). We conclude This holds with the variables (20) which are used in Theorem 4. However iterating this formula and using Corollary 5 we additionally observe that The variables in these integrals are those from (15) which are used in Theorem 2. This article is organized as follows. In the next Section 2 we will prove Theorem 2, from which we conclude Corollary 3. Then in Section 3 we will prove Theorem 4. We remark that Corollary 5 will be an important part of [34].

Center of Momentum reduction of the Collision Integrals
In this section we will prove Theorem 2 and in particular (11). To this end we consider the following integral where θ is defined by (3). Above and in the following, for convenience, we write G(p µ , q µ , p ′µ , q ′µ ) = G(p, q, p ′ , q ′ ) when there is no opportunity for confusion.
This holds because dq ′ q ′0 and dp ′ p ′0 are Lorentz invariant measures and . But notice that the claim is not true unless the angle θ from (3) is redefined as where k is defined in (14). We have also employed the following calculation For that we used (8). Note also, as a result of (7), we can deduce (from the delta function in I) that p ′ + q ′ = 0 which further yields p ′0 = q ′0 . Then the integration over q ′ can be carried out immediately and we obtain Here now, with p ′ = −q ′ , the angle is Next change to polar coordinates as p ′ = |p ′ |ω with ω ∈ S n−1 and dp ′ = |p ′ | n−1 d|p ′ |dω.

Glassey-Strauss reduction of the Collision Integrals
The goal of this section is to prove Theorem 4. This reduction was given by Glassey & Strauss [31] in dimension n = 3 and without the presence of the arguments, (̺, θ), in the differential cross section, σ(̺, θ). The reduction below is essentially similar to [31] in that we perform the examination of the roots of polynomials; it is also however slightly different from [31] in that we integrate the radial variable, r, below over the entire real line R (rather than r ≥ 0). We also observe a dimensionally dependent factor A(p, q, ω) as in (19) which is not present when n = 3. We are further able to compute the formula for the angle θ in (22) and establish the equivalence of the representations as in Corollary 5.
To reduce the number of integrals in (24), we split I = 1 2 I + 1 2 I. Letting q ′ = p + q − p ′ we can immediately remove n of the delta functions in the first copy of I.
We combine the last two splittings to conclude that where p ′ = p + rω and q ′ = q − rω. Now the angle (3) satisfies We will return to this expression below. We now focus on the argument of the delta function. For λ i > 0 (i = 1, 2), we use the identity δ(λ 1 − λ 2 ) = 2λ 1 δ(λ 2 1 − λ 2 2 ) and (2) to see that then the delta function is zero. Now we write the argument of the last delta function above as Plugging in p ′ = p + rω and q ′ = q − rω we observe that This means that p(r) is quadratic in r. Moreover, We conclude that p(r) = 4D 1 r 2 − 8D 2 r for some D 1 , D 2 ∈ R.
Remark. We point out that there is unfortunately a misprint in our recent paper [44] if the dimension n ≥ 2 is not n = 3. In particular the transition rate W at the top of the paper [44], should be replaced by the transition rate from (1) (and [3]). The main difference between the two is the factor ̺ 3−n , which is unity when n = 3. Furthermore, the expression (1.9) in [44] for the collision operator is correct when n = N = 3 (N is the notation for the dimension used in [44]), but otherwise the kernel B(p, q, ω) in [44, (1.9)] needs to be replaced by B n (p, q, ω) in (19). In other words the factor (A(p, q, ω)) n−3 from (19) is missing from [44, (1.9)]. We point out that this factor in the expression [44, (1.9)] does not affect the main theorems of [44]. Furthermore the condition in [44, (2.7)] on the collisional cross section is written as 0 ≤ γ < −3 (which is empty); this [44, (2.7)] should be 0 ≤ γ < N .