Nonlinear Constraints on Relativistic Fluids Far From Equilibrium

New constraints are found that must necessarily hold for Israel-Stewart-like theories of fluid dynamics to be causal far away from equilibrium. Conditions that are sufficient to ensure causality, local existence, and uniqueness of solutions in these theories are also presented. Our results hold in the full nonlinear regime, taking into account bulk and shear viscosities (at zero chemical potential), without any simplifying symmetry or near-equilibrium assumptions. Our findings provide fundamental constraints on the magnitude of viscous corrections in fluid dynamics far from equilibrium.

of such theories is also crucial to reliably assess the role of viscous effects in early universe cosmology [62]. Here, we make essential steps towards solving this critical problem by finding conditions that must necessarily hold for IS-like theories to be causal. We also present conditions that are sufficient to ensure causality, local existence, and uniqueness of solutions of IS-like theories. Our results hold in the full nonlinear regime, with bulk and shear viscosities (at zero chemical potential), in three spatial dimensions, without any symmetry or near-equilibrium assumptions. Our conditions are simple algebraic inequalities that can be easily checked in a given problem. This is the first time that such general statements (causality, local existence, uniqueness) are proven for IS-like theories with shear and bulk viscosities in the full nonlinear regime without simplifying dynamical assumptions.
We note that {η, ζ, τ π , τ Π } are the only coefficients that remain after linearization around equilibrium where π µν = 0 and Π = 0. This shows why linearized analyses [56,57] necessarily miss the effects from the other coefficients, {δ ΠΠ , λ Ππ , δ ππ , τ ππ , λ πΠ }, which contribute to the nonlinear evolution. However, other nonlinear terms such as π µν π µν , Π 2 , π µν Π, π µ α π ν α , which appear in [54], could have been trivially added to the equations as they do not contribute to a causality analysis since they do not involve derivatives of the fields. Nevertheless, there are still some other nonlinear terms that can be considered such as π µ α Ω ν α , where Ω µν = (∆ α µ ∇ α u ν − ∆ α ν ∇ α u µ )/2 is the vorticity, and also Ω µ α Ω ν α [3]. The former will be investigated in a separate publication. The latter contributes with derivatives of the fields to the principal part of the system of equations and, thus, a different analysis than presented here would be required.
3. Causality. Causality is the concept in relativity theory asserting that no information propagates faster than the speed of light and no closed timelike curves exist (so the future cannot influence the past). See the Supplemental Material and references [65][66][67][68][69] for a mathematically precise definition of causality. Causality can be investigated by determining the characteristic manifolds associated with a system of PDE's. In fact, the existence of domains of dependence for solutions of a system of PDEs, as well as their corresponding propagation speeds, can be inferred from the system's characteristics [70,71]. Let us write equations (1)-(2) ν . The eigenvalues are such that Λ 0 = 0, since u µ is in the kernel of π µ ν (u µ π µ ν = 0), and Λ 1 +Λ 2 +Λ 3 = 0, so that the trace is kept zero. Without loss of generality, let us take Λ 1 ≤ Λ 2 ≤ Λ 3 with Λ 1 ≤ 0 ≤ Λ 3 . We now state our assumptions, which are the following: (A1) for the transport coefficients and relaxation times, suppose that τ Π , τ π > 0 and η, ζ, τ ππ , δ ΠΠ , λ Ππ , δ ππ , λ πΠ , c 2 s ≥ 0; (A2) for the fluid variables, suppose that ε > 0, P ≥ 0, and ε + P + Π > 0; finally, also assume that (A3) ε + P + Π + Λ a > 0, a = 1, 2, 3. Then, the following conditions are necessary for causality, i.e., if any of the inequalities below is not satisfied then the system is not causal: where (4c)-(4f) must hold for a, d = 1, 2, 3. The proof that (4) are necessary conditions for causality under assumptions (A1)-(A3) is given in the Supplemental Material. Here, we discuss the significance of this result.
We stress that assumptions (A1) and (A2) are standard in heavy-ion collision applications [55], and (A3) is a very natural assumption since P + Π + Λ a for a = 1, 2, 3 may be interpreted as the pressure in each spatial axis in the local rest frame. Furthermore, it is natural to make assumptions that hold close to equilibrium, and since (A2) guarantees ε + P + Π > 0, for small deviations from equilibrium Λ a will be small, giving ε + P + Π + Λ a > 0. That said, we stress that although (A3) is expected to hold near equilibrium, it is itself not a near-equilibrium assumption.
Conditions (4) could never have been found using a linearized analysis as they depend on Π and Λ a both of which vanish in equilibrium. Consequently, if in any fluid dynamic simulation in heavy-ion collisions that employs (1)-(2) the necessary conditions above are not fulfilled, causality is necessarily violated. It is important to point out that this causality violation has nothing to do with the ability of numerical schemes to produce a solution, a point we discuss in the Conclusion.
While the above conditions must hold for the system to be causal, they are not sufficient conditions, i.e., by themselves, conditions (A1)-(A3) and (4) do not assure the system to be causal (see the Supplemental Material). Therefore it is important to have conditions that are sufficient for causality. In this regard, assume again that (A1)-(A3) hold. Then the following conditions are sufficient to ensure that causality holds, i.e., if they are satisfied then the system is causal: where condition (5h) can be dropped if δ ππ = τ ππ = 0. The detailed proof can be found in the Supplemental Material. Since (4) must hold for causality, they must be satisfied for any set of conditions that imply causality, and it is possible to verify that (5) imply (4) under assumptions (A1)-(A3). When shear viscous effects are neglected, (5) reduces to the conditions for the bulk viscosity case found in [73]. Conditions (A1)-(A3)-(5) also ensure the unique local solvability of the initial-value problem in the class of quasi-analytic functions. More precisely, given initial data of sufficient regularity satisfying (5), there exists a unique solution to the nonlinear equations taking the given initial data, defined for a certain time interval (again, we refer to the Supplemental Material for details). Therefore, if (A1)-(A3) and (5) hold, the evolution of the viscous fluid is guaranteed to be well defined and causal even far from equilibrium where the gradients (and, hence, π µν and Π) are large. This is especially relevant for the open question in heavy-ion collisions concerning the properties of hydrodynamic attractors [20] under general flow conditions [27,74] and also for an overall validation of a fluid dynamic description of small systems, such as proton-proton collisions.
Although here we focus on applications to heavy-ion collisions, where g µν is the Minkowski metric, it is not difficult to see that the methods of [73] can be adapted to show that our conclusions hold when (1)- (2) are coupled to Einstein's equations (see the Supplemental Material). Therefore, our results are also crucial to determine the far from equilibrium behavior of viscous fluids with shear and bulk viscosity in general relativity, which may be directly relevant to neutron star mergers [75]. When we linearize the equations around the equilibrium, terms involving τ ππ , δ ΠΠ , λ Ππ , δ ππ , λ πΠ drop out and, thus, (A1) can be replaced by τ π , τ Π > 0, η, ζ, c 2 s ≥ 0 and (A2) and (A3) can be replaced by ε + P > 0 and P ≥ 0. Then, conditions (5) become necessary and reduce to ε + P > 0, ε + P − η τπ ≥ 0 and s . These conditions coincide with the corresponding well-known results previously found in [56,57] that ensure causality and stability in the linearized regime around equilibrium.
We presented two sets of conditions for causality, namely, conditions that are necessary and conditions that are sufficient. Further studies must be done to discover conditions that are necessary and sufficient, i.e., conditions that ensure the system to be causal if and only if they hold. This is an extremely challenging task given the complexity of the characteristic equation in the nonlinear problem, and would require developing essential new ideas to analyze its roots.
5. Conclusions. In this work, we established for the first time that causality in fact holds for the full set of nonlinear equations in IS-like theories without the need for symmetry assumptions and in the presence of both shear and bulk viscosity. All our conditions are simple algebraic inequalities among the dynamical variables that can be easily checked in a given system or simulation. Previous attempts to go beyond the linear regime were restricted to 1 + 1 dimensions [58] or assumed strong symmetry conditions [59,79]. Without such restrictions, the only other work where nonlinear causality has been showed for IS-like systems is [73]. The latter, however, included only bulk viscosity and, thus, it is more important for applications in cosmology or neutron star mergers than in heavy-ion collisions. We have also studied the Cauchy problem for (1)-(2), establishing that it is well-defined, so that it is meaningful to talk about solutions.
Prior to our work, one could only identify whether a numerical simulation of (1)-(2) violated causality if this caused (a) a breakdown of the simulation, (b) a manifestly spurious solution, or (c) clear non-physical behavior. These constraints are all too weak, as we now explain. For illustration, consider −∂ 2 t ψ + (1 + ψ)∆ψ = 0, where ∆ is the Laplacian. This is a nonlinear wave equation with (nonlinear) speed given by √ 1 + ψ for [90] ψ > −1. Indeed, the characteristics are given by ξ 0 = ± √ 1 + ψ | ξ|. Therefore, the solutions are not causal when ψ > 0, but are causal for −1 < ψ ≤ 0. Nevertheless, the equation remains hyperbolic as long as ψ > −1. Standard hyperbolic theory (see, e.g., [80]) ensures that, given smooth initial data ψ| t=0 and ∂ t ψ| t=0 , there exists a unique smooth solution defined for some time. So any numerical scheme that is able to track the unique solution will produce results in both the acausal and causal cases ψ > 0 and −1 < ψ ≤ 0, respectively. This makes it extremely difficult to infer violations of causality using (a) or (b) as criteria. Exactly the same situation can happen in simulations of (1)- (2). We also note that linearizing the equation about the "equilibrium" ψ = 0 gives −δψ tt + ∆δψ = 0, which is always causal, reinforcing again the idea that causality cannot always be obtained from linearizations.
Criteria (c) also has limited applicability. First, there are different mechanisms that can produce nonphysical solutions. Thus, it is still important to understand if unphysical behavior is being caused by causality violation, or some other mechanism, such as running beyond the limit where the effective description is valid. Second, relativistic fluids in the far from equilibrium regime, such as the QGP, may exhibit unexpected behavior, so one needs to be careful to differentiate genuine exotic features from those that are consequences of running a simulation in a superluminal regime. This may be particularly relevant to heavy-ion simulations where the values of the fields drop extremely rapidly at the edges of the QGP at early times and in the cold/dilute regions of plasma where a rescaling of dissipative tensors has been employed [81][82][83][84]. Third, numerical simulations of relativistic fluids must be based on equations of motion that respect causality, a fundamental physical principle in relativity.
The results we presented here address all these difficulties, as one can check if (A1)-(A3), (4), or (5) hold at any moment in numerical simulations [91] since all the quantities involved in our inequalities can be readily extracted in numerical simulations [3]. We also note that our results apply, in particular, to the initial conditions, so (4) and (5) can be used to rule out initial conditions that violate causality or to select initial conditions for which causality holds. This can be particularly relevant to further constrain the physical assumptions behind the modeling of initial conditions in QGP simulations.
In sum, in this Letter we established, for the first time in the literature, conditions to settle the longstanding questions concerning causality in Israel-Stewart-theories in the nonlinear, far-from-equilibrium regime. As such, our general results provide the most stringent tests to date to determine the validity of relativistic fluid dynamic approaches in heavy-ion collisions, astrophysics, and cosmology.
Acknowledgements. MMD is partially supported by a Sloan Research Fellowship provided by the Alfred P. Sloan foundation, NSF grant DMS-1812826, and a Discovery Grant administered by Vanderbilt University. VH's work on this project was funded (full or in-part) by the University of Texas at San Antonio, Office of the Vice President for Research, Economic Development, and Knowledge Enterprise. VH acknowledges partial support by NSF grants DMS-1614797 and DMS-1810687.

Supplemental material
In this Supplemental Material, in Section II we provide the proof that conditions (4) are necessary for causality, in Section III we provide the proof that conditions (5) are sufficient for causality, and in Section IV we establish local existence and uniqueness of solutions to the initial-value problem for equations (1)- (2). All these results depend on a careful analysis of the roots of the characteristic equation det(A α ξ α ) = 0. Thus, we first present in Section I a suitable factorization of det(A α ξ α ). In Section V we show that conditions (4), albeit necessary, are not sufficient for causality. In Section VI we provide the formal definition of causality and comment on why, in our case, it can be reduced to conditions (C1) and (C2). Since causality is intrinsically tied to concepts of relativity theory, we refer to the standard literature (e.g., [85]) for further background. Throughout this Supplemental Material, we continue to use the notation and definitions of the paper.
The eigenvalues are such that Λ 0 = 0 and Λ 1 + Λ 2 + Λ 3 = 0. Without any loss of generality, let us take Λ 1 ≤ Λ 2 ≤ Λ 3 with Λ 1 ≤ 0 ≤ Λ 3 so that the trace is kept zero (note that if π µ ν = 0, this allows degeneracies to occur with multiplicity up to two). Since {e µ A } is a complete set in R 4 , we may define a tetrad of dual vectors {e A ν } by setting e A ν ≡ η AB (e B ) ν so that [92] δ B A = e ν A e B ν . Also, the following completeness relation holds: δ µ ν = A e µ A e A ν = −u µ u ν + a e µ a (e a ) ν . Therefore, the components of any four-vector z µ relative to the tetrad {e µ A } are defined by z A ≡ z ν e A ν . We can then use this to define v A ≡ e µ A v µ and ξ A ≡ e µ A ξ µ . Given that ξ µ = −bu µ + a v a e µ a (a = 1, 2, 3) one finds that where we used that Λ 0 = 0 and again ξ a = v a (note also that v a = v a since η ab = δ ab ). Using these observations, we can show that the determinant det(M ) needed for the characteristics in (A1) is given by where we defined k ≡ b 2 /v · v and In this notation det(M ) = m 0 (v · v) 3 f (k) because we used the definitionm a = m a /v · v. Note that although G(k) hasm a appearing in denominators, these are canceled by the multiplication of G(k) bym 1m2m3 in the definition of f (k). Thus, f (k) is a polynomial of degree 3 in k (of degree 6 in b) and is defined for all values of k ∈ R. Then, it is possible to factorize f (k) as where k 1 , k 2 , k 3 as the three roots of f (k). Note that for the sake of brevity, we have suppressed the dependence onv in writing G(k) and f (k) (to be more precise, these should have been written as G(k,v), f (k,v)). Conditions (C1) and (C2) for causality demand that all the 22 roots ξ 0 = ξ 0 (ξ i ) of det(A α ξ α ) = 0 are real and satisfy ξ α ξ α = −b 2 + v · v ≥ 0, i.e., 0 ≤ k ≤ 1. The 14 roots b = 0 are causal. Thus, the rest the analysis of necessary conditions in Section II will focus on the remaining roots defined by f (k) = 0. We summarize this in the following important statement: The system is causal if and only if for all for allv on the unit sphere, the roots ofm 0 (k,v) = 0 and f (k,v) = 0 are real and 0 ≤ k ≤ 1. (C3)
Proof of Theorem 1: Our derivation of necessary conditions for causality is via the following reasoning. Causality requires that conditions (C1) and (C2) hold for all ξ i . Thus, in order to violate causality, it suffices to show that for some ξ i , (C1) or (C2) fails. Suppose now that we find a condition, say Z, for which we can exhibit one ξ i such that (C1) or (C2) fail, i.e., we obtain the statement "Z implies non-causality." This statement is logically equivalent to "Causality implies non-Z." In other other, non-Z is a necessary condition for causality: if it is violated, the system is not causal. In our case, conditions like Z will be inequalities among the scalars of the problem (e.g., the relaxation times, eigenvalues Λ a , etc.) of the form A > B, whose negation is then A ≤ B. The latter is then the necessary condition we are looking for: if A ≤ B does not hold, the system is not causal.
Recall that (C1) and (C2) is equivalent to (C3), so in view of the foregoing discussion, we aim to violate (C3). With the choicev a = δ ad , one can write m 0 = ρ(v · v)(k − g d ) = 0. Under our assumptions, the only root is k = g d . Since we need 0 ≤ k ≤ 1, as discussed, and since g 1 ≤ g 2 ≤ g 3 , causality if violated if g 1 < 0, leading to condition (4a), or if g 3 > 1, leading to condition (4b). Observe also that if ρ were allowed to vanish, then the characteristic determinant would also vanish, leading to non-causality. See our discussion of the condition ε + P + Π > 0 in the main text.
As for the roots of f (k), we may note that now in f (k) =m 1m2m3 G(k) we havē and because we have setv a = δ ad . We may therefore rewrite where a = b and a, b = d. Setting each of the factors m a , m b equal to zero, we obtain the roots Causality is violated if k < 0, leading to condition (4c), of if k > 1, leading to condition (4d). The remaining root in (B3) is obtained when the term in brackets vanishes, giving Causality is violated if k < 0, leading to (4e), or if k > 1, leading to (4f). This finishes the proof.
We remark that the diagonalization of π µν was carried out in terms of orthonormal frames which can be defined for any Lorentzian metric. Also, our computations are manifestly covariant. Thus, the result of Theorem 1 remains true in a general globally hyperbolic space-time, as mentioned in the main text. This includes, in particular, the cases where the equations hold in a globally hyperbolic subset of Minkowski space or in I × T 3 with the Minkowski metric, where I ⊆ R is an interval and T 3 is the three-dimensional torus.
Appendix C: III. Derivation of sufficient conditions for causality Here we establish that conditions (5) are sufficient for causality. More precisely, we establish the following Theorem.
Theorem 2. Let Ψ = (ε, u ν , Π, π 0ν , π 1ν , π 2ν , π 3ν ) ν=0,...,3 be a smooth solution to equations (1)- (2) in Minkowski space, with u µ u µ = −1 and π µν satisfying π µ µ = 0 and u µ π µν = 0. Suppose that (A1)-(A3) and (5)  (C1) (A3) together with conditions (5a) and (5b) give 0 ≤ g 1 ≤ g 2 ≤ g 3 ≤ 1. From g 1 ≤ a g av 2 a ≤ g 3 , we see that (C1) is satisfied. Now we analyze the remaining 6 roots of det(M ) = 0 coming from f (k) defined in Eq. (A4) and written explicitly as a polynomial in (A6). We will show further below that the three roots k i in (A6) are real. But let us first show that any real root of f must lie within [0, 1]. Since f is a cubic polynomial, it either has only one real root, say s 1 , or three real roots, in which case we can order them as k 1 ≤ k 2 ≤ k 3 in (A6). Invoking (5a), we see that in the first case f is negative to the left of s 1 and positive to its right, and in the second case that f is a growing cubic polynomial except in the interval between the roots k 1 and k 3 . In either situation, any real root will be between 0 and 1 if and Let us first verify the inequality (C3). For k > 1 for k > 1, hence the condition (5a) lead us tom a (k ≥ 1) > 0. This guarantees that To obtain f (k > 1) > 0 in (C3), we therefore need G(k > 1) > 0. By means of (5c) and (5d), and thus, which follows from the ordering of the eigenvalues Λ a . Hence (5e) implies G(k) > 0 for k > 1. It now remains to verify the inequality (C2). In this case, when k < 0 From condition (5b), one has thatm a (k ≤ 0) < 0. Then, if, and only if, G(k < 0) > 0. Due tom a (k ≤ 0) < 0 together with (5c) and (5d), we obtain that Condition (5f) guarantees that a . . . > 0 in the above inequality. Moreover, where we used (C8) and ( we have G(k < 0) > 0 from condition (5g), finally implying f (k < 0) < 0. It remains to establish the reality of the roots k i in (A6). To do that, let us write G(k) as where Note, in particular, thatr 1 ≤ r a ≤r 3 , wherer 1,3 ≡ 1 2τπ (2η + λ πΠ Π) + τππΛ1,3 2τπ > 0 from (5b). By applying conditions (5) one has that R a , S ab , ρ a , r a ≥ 0. Then, f (k) can be written as where a 0 = − r 1 r 2 r 3 + r 1 r 2 R 3v 2 3 + r 2 r 3 R 1v In view of (5), we have a 3 > 0 and a 2 < 0. Since all coefficients of f (k) are real, then at least one of the roots must be real, say k = s 1 ∈ R is the real root. Then, we know that the other two roots s 2 and s 3 are real or complex conjugate, i.e., s * 3 = s 2 . Let us assume that s 2 and s 3 can be imaginary and set s 2,3 = k R ± ik I , k I = 0. By using Vieta's formula s 1 + s 2 + s 3 = − a2 a3 = |a2| a3 > 0 we obtain that Thus, the following condition holds, because the real root s 1 ∈ [0, 1] when (5a)-(5g) apply, as we have already showed. Since we are assuming s 2,3 = k R ± ik I , where k I = 0, we have thatm a (s 2,3 ) = ρ a k R − r a ± ik I cannot be zero (unless k I = 0 and the roots are real). Consequently, from (C12) we obtain that f (s 2,3 ) = 0 lead us to G(s 2,3 ) = 0, where s 2,3 must obey the above conditions implied by f being a cubic polynomial, in particular the condition on k R in (C24). Thus, let us split G(s To show that the roots are real, if suffices to have G I (s 2,3 ) = 0. We distinguish two cases. If S ab = 0 then G I (s 2,3 ) = 0 because we assumed k I = 0. This means that in this case the roots must all be real. On the other hand, if S ab = 0 and ρ 1 R 1 −r 3 > 0, then Eq. (C25) also gives G I (s 2,3 ) = 0, because then the sum over b in (C25) is > 0. To check that ρ 1 R 1 −r 3 > 0, note first that (5a) guarantees that ρ a > r a . Then, by means of (C24), we obtain that because of condition (5h), and this implies ρ 1 k R −r 3 > 0. Since we have already showed that any real root of f (k) must lie within [0, 1], this finishes our proof.
We remark that the diagonalization of π µν was carried out in terms of orthonormal frames which can be defined for any Lorentzian metric. Also, our computations are manifestly covariant. Thus, the result of Theorem 2 remains true in a general globally hyperbolic space-time, as mentioned in the main text. This includes, in particular, the cases where the equations hold in a globally hyperbolic subset of Minkowski space or in I × T 3 with the Minkowski metric, where I ⊆ R is an interval and T 3 is the three-dimensional torus. In this Section, we establish the local existence and uniqueness of solutions to the Cauchy problem. Below, G is the space of Gevrey functions or quasi-analytic functions.
Proof of Theorem 3: The calculations provided in Section I and in the proof of Theorem 2 imply that, under the assumptions, the characteristic polynomial of the system evaluated at the initial data is a product of strictly hyperbolic polynomials. One also sees that intersection of the interior of the characteristic cones defined by these strictly hyperbolic polynomials has non-empty interior and lies outside the light-cone defined by the metric. Under these circumstances we can apply theorems A.18, A.19, and A.23 of [86] to conclude the result (the remaining assumptions of these theorems are easily verified in our case).
For the sake of brevity, we refer readers to [87] for a definition of G δ , making only the following remarks. The case of δ = 1 corresponds to the space of analytic functions, of which G δ with δ > 1 is a generalization. This is why G is sometimes referred to as the space of quasi-analytic functions. The usefulness of Gevrey functions to the study of hyperbolic problems is at least two-fold. On the one hand, one can prove very general existence and uniqueness theorems for Gevrey data given on a non-characteristic surface that are akin to the Cauchy-Kovalewskaya theorem for analytic data. On the other hand, an advantage of Gevrey maps over analytic ones is that one can construct Gevrey functions that are compactly supported; hence one can appeal to the type of localization arguments that are so useful in the study of hyperbolic equations. This is particularly important when one is considering coupling to Einstein's equations.
While typical evolution problems consider solutions in more general function spaces than G δ , we stress that ours is the very first existence and uniqueness result for equations (1)- (2). In other words, while it is desirable to extend our result to more general function spaces, Theorem 3 is important because it shows, for the very first time in the literature, that the initial value problem for equation (1)-(2) is well-defined, so that it is meaningful to talk about solutions.
We remark that the diagonalization of π µν was carried out in terms of orthonormal frames which can be defined for any Lorentzian metric. Also, our computations are manifestly covariant. Thus, the result of Theorem 3 remains true in a general globally hyperbolic space-time, as mentioned in the main text. This includes, in particular, the cases where the equations hold in a globally hyperbolic subset of Minkowski space or in I × T 3 with the Minkowski metric, where I ⊆ R is an interval and T 3 is the three-dimensional torus. Moreover, as also mentioned in the main text, the result extends to the case when (1)-(2) are coupled to Einstein's equations. This follows by computing the characteristic determinant of the coupled system and observing that it factors into the product of the characteristic determinant of (1)-(2), which we analyzed here, and the characteristic determinant of Einstein's equations. The argument is the same as given in [73].

Appendix E: V. Insufficiency of conditions for causality
In this Section, we show that conditions (4), albeit necessary, are not sufficient for causality. We do this by showing that causality can be violated if we only assume (A1)-(A3) and (4).
Since the notion of causality is central in our work, we find it appropriate to give its precise mathematical definition. We also comment on how it is equivalent, in our context, to conditions (C1) and (C2).
Definition 4. Let (M, g) be the Minkowski space. Consider in M a system of partial differential equations for an unknown ψ, which we write as Pψ = 0, where P is a differential operator (which is allowed to depend on ψ) [94]. Let ϕ be a solution to the system. We say that ϕ is causal if the following holds true: given a Cauchy surface Σ ⊂ M, for any point x in the future of Σ, ϕ(x) depends only on ϕ| J − (x)∩Σ , where J − (x) is the causal past of x.
The case of most interest is when the Cauchy surface is the hypersurface {t = 0} where initial data is prescribed. We also notice that since we are working in Minkowski space, J − (x) is simply the past light-cone with vertex at x. The situation in Definition 4 is illustrated in Fig. 1. In particular, causality implies that ϕ(x) remains unchanged if the the values of ϕ along Σ are altered [95] only outside J − (x) ∩ Σ. Observe that this definition says that ϕ(x) can only be influenced by points in the past of x that are causally connected to x, so no information is allowed to propagate faster than the speed of light.
Definition 4 is for a given solution ϕ to the system. While it would be desirable to state causality as a general property of the system Pψ = 0, i.e., saying that the system is causal if any solution is causal in the sense of Definition 4, this would be too restrictive, as it can be seen from our discussion of the equation −ψ tt + (1 + ψ)∆ψ = 0 in the Conclusion.
The connection between Definition 4 and conditions (C1) and (C2) is via the characteristics of the system Pψ = 0. It is beyond the scope of this Supplemental Material to provide a detailed description of the connections between Definition 4 and the system's characteristics. We refer readers to Appendix A of [86], [72,Chapter VI], and [70]. Here, we restrict ourselves to the following comments. Finite speed of propagation is a property of hyperbolic equations. For such equations, there exist domains of dependence that show precisely how the values of a solution at a point x is determined solely by values within a domain of dependence in the past with "vertex" at x (this is exactly the generalization of the past light-cone). The domain of dependence, in turn, is determined by the system's characteristics. While it is mathematically possible for hyperbolic equations to exhibit domains of dependence where information propagates faster than the speed of light (see, again, discussion in the Conclusion), for solutions to be causal (i.e., to not have faster-than-light signals), the domains of dependence must always lie inside the light-cones. This is equivalent to the statement (C1) and (C2) that we have used. Definition 4 can be generalized to arbitrary globally hyperbolic spaces, which is needed for the aforementioned generalization of our Theorems to this setting. Again, we refer to Appendix A of [86], [72,Chapter VI], and [70].