an optimal transport formulation of the einstein …

AN OPTIMAL TRANSPORT FORMULATION OF THE

EINSTEIN EQUATIONS OF GENERAL RELATIVITY

A. MONDINO AND S. SUHR

Abstract. The goal of the paper is to give an optimal transport for-mulation of the full Einstein equations of general relativity, linking the(Ricci) curvature of a space-time with the cosmological constant andthe energy-momentum tensor. Such an optimal transport formulationis in terms of convexity/concavity properties of the Shannon-Bolzmannentropy along curves of probability measures extremizing suitable opti-mal transport costs. The result gives a new connection between generalrelativity and optimal transport; moreover it gives a mathematical re-inforcement of the strong link between general relativity and thermody-namics/information theory that emerged in the physics literature of thelast years.

Contents

1. Introduction 21.1. Statement of the main results 31.2. Outline of the argument 91.3. An Example. FLRW Spacetimes 91.4. Related literature 13Acknowledgement 142. Preliminaries 142.1. Some basics of Lorentzian geometry 142.2. The Lagrangian Lp, the action Ap and the cost cp 152.3. Ricci curvature and Jacobi equation 172.4. The q-gradient of a function 182.5. cp-concave functions and regular cp-optimal dynamical plans 193. Existence, regularity and evolution of Kantorovich potentials 214. Optimal transport formulation of the Einstein equations 284.1. OT-characterization of Ric ≥ T 294.2. OT-characterization of Ric ≤ T 324.3. Optimal transport formulation of the Einstein equations 37Appendix A. A q-Bochner identity in Lorentzian setting 41

Key words and phrases. Ricci curvature, optimal transport, Lorentzian manifold, gen-eral relativity, Einstein equations, strong energy condition.

A. Mondino: University of Oxford, Mathematical Institut, email: [email protected]. Supported by the EPSRC First Grant EP/R004730/1“Optimal transport and geometric analysis” and by the ERC Starting Grant 802689“CURVATURE”.

S. Suhr: University of Bochum, email: [email protected]. Supportedby the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics”,funded by the Deutsche Forschungsgemeinschaft.

1

2 A. MONDINO AND S. SUHR

Appendix B. A synthetic formulation of Einstein’s vacuumequations in a non-smooth setting 45

B.1. The Lorentzian synthetic framework 45Examples entering the class of Lorentzian synthetic spaces 47B.2. The p-Lorentz-Wasserstein distance on a Lorentzian pre-length

space. 48B.3. Synthetic time-like Ricci upper bounds. 49B.4. Synthetic time-like Ricci lower bounds. 54B.5. Synthetic vacuum Einstein’s equations, 55References 57

1. Introduction

In recent years, optimal transport revealed to be a very effective and in-novative tool in several fields of mathematics and applications. By way ofexample, let us mention fluid mechanics (e.g. Brenier [15] and Benamou-Brenier [11]), partial differential equations (e.g. Jordan-Kinderleher-Otto[55] and Otto [68]), random matrices (e.g. Figalli-Guionnet [34]), opti-mization (e.g. Bouchitte-Buttazzo [18]), non-linear σ-models (e.g. Car-fora [22]), geometric and functional inequalities (e.g. Cordero-Erausquin-Nazaret-Villani [26], Figalli-Maggi-Pratelli [35], Klartag [53], Cavalletti- Mon-dino [24]) Ricci curvature in Riemannian geometry (e.g. Otto-Villani [69],Cordero Erausquin-McCann-Schmuckenschlager [27], Sturm-VonRenesse [77])and in metric measure spaces (e.g. Lott-Villani [57], Sturm [74, 75], Ambrosio-Gigli-Savare [4]). For more details about optimal transport and its appli-cations in both pure and applied mathematics, we refer the reader to themany books on the topic, e.g. [1, 3, 73, 81, 82].

Here let us just quote two of the many applications to partial differen-tial equations. In the pioneering work of Jordan-Kinderleher-Otto [55] itwas discovered a new optimal transport formulation of the Fokker-Planckequation (and in particular of the heat equation) as a gradient flow of asuitable functional (roughly, the Boltzmann-Shannon entropy defined belowin (1.5) plus a potential) in the Wasserstein space (i.e. the space of prob-ability measures with finite second moments endowed with the quadraticKantorovich-Wasserstein distance); later, Otto [68] found a related optimaltransport formulation of the porous medium equation. The impact of theseworks in the optimal transport community has been huge, and opened theway to a more general theory of gradient flows (see for instance the mono-graph by Ambrosio-Gigli-Savare [3]).

The goal of the present work is to give a new optimal transport formu-lation of another fundamental class of partial differential equations: theEinstein equations of general relativity. First published by Einstein in 1915,the Einstein equations describe gravitation as a result of space-time beingcurved by mass and energy; more precisely, the space-time (Ricci) curva-ture is related to the local energy and momentum expressed by the energy-momentum tensor. Before entering into the topic, let us first recall that theEinstein equations are hyperbolic evolution equations (for a comprehensive

AN OPTIMAL TRANSPORT FORMULATION OF THE EINSTEIN EQUATIONS 3

treatment see the recent monograph by Klainerman-Nicolo [52]). Insteadof a gradient flow/PDE approach, we will see the evolution from a geomet-ric/thermodynamic/information point of view.

Next we briefly recall the formulation of the Einstein equations. Let Mn

be an n-dimensional manifold (n ≥ 3, the physical dimension being n = 4)endowed with a Lorentzian metric g, i.e. g is a nondegenerate symmetricbilinear form of signature (−+ + . . .+). Denote with Ric and Scal the Ricciand the scalar curvatures of (Mn, g). The Einstein equations read as

(1.1) Ric− 1

2Scal g + Λg = 8πT,

where Λ ∈ R is the cosmological constant, and T is the energy-momentumtensor. Physically, the cosmological constant Λ corresponds to the energydensity of the vacuum; the energy-momentum tensor is a symmetric bilinearform on M representing the density of energy and momentum, acting as thesource of the gravitational field.

1.1. Statement of the main results. Recall that in a Lorentzian manifold(Mn, g), a non-zero tangent vector v ∈ TxM is called time-like if g(v, v) <0. If M admits a continuous no-where vanishing time-like vector field X,then (M, g) is said to be time-oriented and it is called a space-time. Thevector field X induces a partition on the set of time-like vectors, into twoequivalence classes: the future pointing tangent vectors v for which g(X, v) <0 and the past pointing tangent vectors v for which g(X, v) > 0. The closureof the set of future pointing time-like vectors is denoted

C = Cl({v ∈ TM : g(v, v) < 0 and g(X, v) < 0}) ⊂ TM.

A physical particle moving in the space-time (M, g, C) is represented by acausal curve which is an absolutely continuous curve, γ, satisfying

γt ∈ C a.e. t ∈ [0, 1].

If the particle cannot reach the speed of light (e.g. massive particle), thenit is represented by a chronological curve which is an absolutely continuouscurve, γ, satisfying

γt ∈ Int(C) a.e. t ∈ [0, 1],

where Int(C) is the interior of the cone C made of future pointing time-likevectors. The Lorentz length of a causal curve is

Lg(γ) =

ˆ 1

0

√−g(γt, γt) dt.

A point y is in the future of x, denoted y >> x, if there is a future orientedchronological curve from x to y; in this case, the Lorentz distance or propertime between x and y is defined by

sup{Lg(γ) : γ0 = x and γ1 = y, γ chronological} > 0,

which is achieved by a geodesic which is called a maximal geodesic. See forexample [31, 67, 83]


In this paper, we consider the following Lorentzian Lagrangian on TMfor p ∈ (0, 1):

(1.2) Lp(v) :=

{−1p(−g(v, v))

p2 if v ∈ C

+∞ otherwise.

Note that if p were 1 this would be the negative of the integrand for theLorentz length given above. Here we study p ∈ (0, 1) because this LorentzianLagrangian Lp has good convexity properties for such p [see Lemma 2.1].

Let AC([0, 1],M) denote the space of absolutely continuous curves from[0, 1] to M. The Lagrangian action Ap, corresponding to the Lagrangian Lpand defined for any γ ∈ AC([0, 1],M), is given by

(1.3) Ap(γ) :=

ˆ 1

0Lp(γt)dt ∈ (−∞, 0] ∪ {+∞}.

Observe that Ap(γ) ∈ (−∞, 0] if and only if γ is a causal curve. Note that, ifp were 1, this would be the negative of the Lorentz length of γ or the propertime along γ. Thus −Ap(γ) can be seen as a kind of non-linear p-propertime along γ, enjoying better convexity properties. The reader may note theparallel with the theory of Riemannian geodesics, where one often studiesthe energy functional

´|γ|2 in place of the length functional

´|γ|, due to

the analogous advantages.The choice of the minus sign in (1.3) is motivated by optimal transport

theory, in order to have a minimization problem instead of a maximizationone (as in the sup defining the Lorentz distance between points above). Itis readily checked that the critical points of Ap with negative action aretime-like geodesics [see Lemma 2.2]. The advantage of Ap with p ∈ (0, 1) isthat it automatically selects an affine parametrization for its critical pointswith negative action.

The cost function, cp : M×M → (−∞, 0]∪{+∞}, relative to the p-actionAp is defined by

cp(x, y) = inf{Ap(γ) : γ ∈ AC([0, 1],M), γ0 = x, γ1 = y}.

Note that, if p were 1, then this would be the negative of the Lorentz distancebetween x and y.

Consider a relatively compact open subset

E ⊂⊂ Int(C) ⊂ TM and r ∈ (0, injg(E)),

where injg(E) > 0 is the injectivity radius of the exponential map of g re-stricted to E. If we take pTM→M : TM →M to be the canonical projectionmap then

∀x ∈ pTM→M (E) and v ∈ TxM ∩ E with g(v, v) = −r2,

we have a maximal geodesic γx : [0, 1]→M defined by

γx(t) = expx((t− 1/2)v) such that γx(1/2) = x.

The Ricci curvature, Ricx(v, v) at a point x ∈ M in the direction v, is atrace of the curvature tensor so that intuitively it measures the average wayin which geodesics near γx bend towards or away from it. See Section 1.3. In


Riemannian geometry, the Ricci curvature influences the volumes of balls.Here, instead of balls, we define for any x ∈ pTM→M (E)

Bg,Er (x) := {expgx(tw) : w ∈ TxM ∩ E, g(w,w) = −1, t ∈ [0, r]}

precisely to avoid the null directions.Rather than considering individual paths between a given pair of points,

we will consider distributions of paths between a given pair of distributionsof points using the optimal transport approach.

We denote by P(M) the set of Borel probability measures on M . For anyµ1, µ2 ∈ P(M), we say that a Borel probability measure

π ∈ P(M ×M) is a coupling of µ1 and µ2

if (pi)]π = µi, i = 1, 2, where p1, p2 : M × M → M are the projectionsonto the first and second coordinate. Recall that the push-forward (p1)]π isdefined by

(p1)]π(A) := π(p−1

1 (A))

for any Borel subset A ⊂ M . The set of couplings of µ1, µ2 is denoted byCpl(µ1, µ2). The cp-cost of a coupling π is given byˆ

M×Mcp(x, y)dπ(x, y) ∈ [−∞, 0] ∪ {+∞}.

Denote by Cp(µ1, µ2) the minimal cost relative to cp among all couplingsfrom µ1 to µ2, i.e.

Cp(µ1, µ2) := inf

{ˆcpdπ : π ∈ Cpl(µ1, µ2)

}∈ [−∞, 0] ∪ {+∞}.

If Cp(µ1, µ2) ∈ R, a coupling achieving the infimum is said to be cp-optimal.For t ∈ [0, 1] denote by et : AC([0, 1],M)→M the evaluation map

et(γ) := γt.

A cp-optimal dynamical plan is a probability measure Π on AC([0, 1],M)such that (e0, e1)]Π is a cp-optimal coupling from µ0 := (e0)]Π to µ1 :=(e1)]Π. One can naturally associate to Π a curve

(µt := (et)]Π)t∈[0,1] ⊂ P(M)

of probability measures. The condition that Π is a cp-optimal dynamicalplan corresponds to saying that the curve (µt)t∈[0,1] ⊂ P(M) is a lengthminimizing geodesic with respect to Cp, i.e.

Cp(µs, µt) = |t− s|Cp(µ0, µ1) ∀s, t ∈ [0, 1].

We will mainly consider a special class of cp-optimal dynamical plans,that we call regular : roughly, a cp-optimal dynamical plan is said to beregular if it is obtained by exponentiating the gradient (which is assumedto be time-like) of a smooth Kantorovich potential φ:(1.4)

µt = (Ψt1/2)]µ1/2, Ψt

1/2(x) := expgx

(−(t−1/2)|∇gφ|q−2

g ∇gφ(x)),

1

p+

1

q= 1,

and moreover µt � volg for all t ∈ (0, 1), where volg denotes the standardvolume measure of (M, g). For the precise notions, the reader is referred toSection 2.5.


A key role in our optimal transport formulation of the Einstein equationswill be played by the (relative) Boltzmann-Shannon entropy. Denote by volgthe standard volume measure on (M, g). Given an absolutely continuousprobability measure µ = % volg with density % ∈ Cc(M), its Boltzmann-Shannon entropy (relative to volg) is defined as

(1.5) Ent(µ|volg) :=

ˆM% log % dvolg.

We will be proving that the second order derivative of this entropy along a cp-optimal dynamical plan, is equivalent to the Einstein Equation in Theorem4.10. See Figure 1. Throughout the paper we will assume the cosmologicalconstant Λ and the energy momentum tensor T to be given, say from physicsand/or mathematical general relativity. Given g,Λ and T it is convenientto set

(1.6) T :=2Λ

n− 2g + 8πT − 8π

n− 2Trg(T ) g.

so that the Einstein Equation can be written as Ric = T , see Lemma 4.1.

Theorem 1.1 (Theorem 4.10). Let (M, g, C) be a space-time of dimensionn ≥ 3. Then the following assertions are equivalent:

(1) (M, g, C) satisfies the Einstein equations (1.1), which can be rewrit-

ten in terms of T of (1.6) as Ric = T .(2) For every p ∈ (0, 1) and for every relatively compact open subset

E ⊂⊂ Int(C) there exist R = R(E) ∈ (0, 1) and a function

ε = εE : (0,∞)→ (0,∞) with limr↓0

ε(r) = 0 such that

∀x ∈ pTM→M (E) and v ∈ TxM ∩ E with g(v, v) = −R2

the next assertion holds. For every r ∈ (0, R), setting y = expgx(rv),there exists a regular cp-optimal dynamical plan Π = Π(x, v, r) withassociated curve of probability measures

(µt := (et)]Π)t∈[0,1] ⊂ P(M)

such that• µ1/2 = volg(B

g,Er4 (x))−1 volgxB

g,Er4 (x),

• supp(µ1) ⊂ {expgy(r2w) : w ∈ TyM ∩ C, g(w,w) = −1}and which has convex/concave entropy in the following sense:

(1.7) ∣∣∣ 4r2

[Ent(µ1|volg)− 2Ent(µ1/2|volg) + Ent(µ0|volg)

]− T (v, v)

∣∣∣ ≤ ε(r)(3) There exists p ∈ (0, 1) such that the assertion as in (2) holds true.

Remark 1.2 (On the regularity of the space-time). For simplicity, in thepaper we work with a smooth space-time. However, all the statements andproofs would be valid assuming that M is a differentiable manifold endowedwith a C3-atlas and that g is a C2-Lorentzian metric on M .

Remark 1.3 (A heuristic thermodynamic interpretation of Theorem 1.1).A curve (µt)t∈[0,1] ⊂ P(M) associated to a cp-optimal dynamical plan can


Figure 1. The transport in Theorem 1.1

be interpreted as the evolution (a) of a distribution of gas passing through agiven gas distribution µ1/2 (that in Theorem 1.1 is assumed to be concen-trated in the space-time near x). Theorem 1.1 says that the Einstein equa-tions can be equivalently formulated in terms of the convexity propertiesof the Bolzmann-Shannon entropy along such evolutions (µt)t∈[0,1] ⊂ P(M).Extrapolating a bit more, we can say that the second law of thermodynamics(i.e. in a natural thermodynamic process, the sum of the entropies of theinteracting thermodynamic systems decreases, due to our sign convention)concerns the first derivative of the Bolzmann-Shannon entropy; gravitation

(a)strictly speaking t is not the proper time, but only a variable parametrizing theevolution


(under the form of Ricci curvature) is instead related to the second order de-rivative of the Bolzmann-Shannon entropy along a natural thermodynamicprocess.

Remark 1.4 (Disclaimer). In Theorem 1.1 we are not claiming to solve thegeneral Einstein Equations via optimal transport; we are instead propos-ing a novel formulation/characterization of the solutions of the EinsteinEquations based on optimal transport, assuming the cosmological constantΛ and the energy-momentum tensor T being already given (this can be a bitcontroversial for a general T ; however, the characterization is already newand interesting in the vacuum case T ≡ 0 where there is no controversy).The aim is indeed to bridge optimal transport and general relativity, withthe goal of stimulating fruitful connections between these two fascinatingfields. In particular, optimal transport tools have been very successful tostudy Ricci curvature bounds in a (low regularity) Riemannian and metric-measure framework (see later in the introduction for the related literature)and it is thus natural to expect that optimal transport can be useful also ina low-regularity Lorentzian framework, where singularities play an impor-tant part in the theory; for example it is expected that, at least generically,singularities occur in black-hole interiors.

For equivalent formulations of Theorem 1.1, see Remark 4.11 and Remark4.12.

In the vacuum case T ≡ 0 with zero cosmological constant Λ = 0, theEinstein equations read as

(1.8) Ric ≡ 0,

for an n-dimensional space-time (M, g, C). Specializing Theorem 1.1 with

the choice T = 0 (plus a small extra observation to sharpen the lower boundin (1.9) from −ε(r) to 0; moreover the same proof extends to n = 2) givesthe following optimal transport formulation of Einstein vacuum equationswith zero cosmological constant.

Corollary 1.5. Let (M, g, C) be a space-time of dimension n ≥ 2. Then thefollowing assertions are equivalent:

(1) (M, g, C) satisfies the Einstein vacuum equations with zero cosmo-logical constant, i.e. Ric ≡ 0.

(2) For every p ∈ (0, 1) and for every relatively compact open subsetE ⊂⊂ Int(C) there exist R = R(E) ∈ (0, 1) and a function

ε = εE : (0,∞)→ (0,∞) with limr↓0

ε(r)/r2 = 0 such that


the next assertion holds. For every r ∈ (0, R), setting y = expgx(rv),there exists a regular cp-optimal dynamical plan Π = Π(x, v, r) withassociated curve of probability measures

(µt := (et)]Π)t∈[0,1] ⊂ P(M)

such that• µ1/2 = volg(B


g,Er4 (x),

• supp(µ1) ⊂ {expgy(r2w) : w ∈ TyM ∩ C, g(w,w) = −1}


and which has almost affine entropy in the sense that

(1.9) 0 ≤ Ent(µ1|volg)− 2Ent(µ1/2|volg) + Ent(µ0|volg) ≤ ε(r).

(3) There exists p ∈ (0, 1) such that the assertion as in (2) holds true.

1.2. Outline of the argument. As already mentioned, the Einstein Equa-tions can be written as Ric = T where T was defined in (1.6), see Lemma 4.1.The optimal transport formulation of the Einstein equations will consist sep-arately of an optimal transport characterization of the two inequalities

(1.10) Ric ≥ T

and

(1.11) Ric ≤ T ,

respectively. The optimal transport characterization of the lower bound(1.10) will be achieved in Theorem 4.3 and consists in showing that (1.10) isequivalent to a convexity property of the Bolzmann-Shannon entropy alongevery regular cp-optimal dynamical plan. The optimal transport charac-terization of the upper bound (1.11) will be achieved in Theorem 4.7 andconsists in showing that (1.11) is equivalent to the existence of a large familyof regular cp-optimal dynamical plans (roughly the ones given by exponenti-ating the gradient of a smooth Kantorovich potential with Hessian vanishingat a given point) along which the Bolzmann-Shannon entropy satisfies thecorresponding concavity condition.

Important ingredients in the proofs will be the following. In Theorem4.3, for proving that Ricci lower bounds imply convexity properties of theentropy, we will perform Jacobi fields computations relating the Ricci curva-ture with the Jacobian of the change of coordinates of the optimal transportmap (see Proposition 3.4); in order to establish the converse implicationwe will argue by contradiction via constructing cp-optimal dynamical plansvery localized in the space-time (Lemma 3.2).In Theorem 4.3 we will consider the special class of regular cp-optimal dy-namical plans constructed in Lemma 3.2, roughly the ones given by ex-ponentiating the gradient of a smooth Kantorovich potential with Hessianvanishing at a given point x ∈ M . For proving that Ricci upper boundsimply concavity properties of the entropy, we will need to establish theHamilton-Jacobi equation satisfied by the evolved Kantorovich potentials(Proposition 3.1) and a non-linear Bochner formula involving the p-Box op-erator (Proposition A.1), the Lorentzian counterpart of the p-Laplacian. Inorder to show the converse implication we will argue by contradiction usingTheorem 4.3.

1.3. An Example. FLRW Spacetimes. We illustrate Theorem 1.1 forthe class of Friedmann-Lemaıtre-Robertson-Walker spacetimes (short FLRWspacetimes), a group of cosmological models well known in general relativity.See [67, Chapter 12] for a discussion of the geometry in the case n = 4.

FLRW spacetimes are of the form

(M, g) = (I × Σ,−ds2 + a2(s)σ),


where I ⊂ R is an interval, a : I → (0,∞) is smooth, and (Σ, σ) is a Rie-mannian manifold with constant sectional curvature k ∈ {−1, 0, 1}. TheRicci and scalar curvature are given by

Ric = −(n− 1)a

ads2 +

[a

a+ (n− 2)

(a2 + k

a2

)]a2σ

and

Scal = 2(n− 1)a

a+ (n− 1)(n− 2)

a2 + k

a2,

respectively, where a := dads . The stress-energy tensor is thus (assuming

Λ = 0)

8πT = Ric− 1

2Scal g

=(n− 1)(n− 2)

2

a2 + k

a2ds2 −

[(n− 2)

a

a+

(n− 2)(n− 3)

2

a2 + k

a2

]a2σ.

The foliation

O := {s 7→ (s, x)}x∈Σ

is a geodesic foliation by cp-minimal geodesics. The orthogonal complement

∂⊥s = TΣ with respect to g is integrable. Consider the projection

S : M = I × Σ→ I

and for r > 0 the function

φ : M → R, x 7→ rp−1S(x).

It is easy to see that

∇qgφ(x) := −|∇gφ|q−2g ∇gφ(x) = r∂s.

For the cp-transform we have (see Section 2.5)

φcp(s, y) = infx∈M

cp(x, (s, y))− φ(x)

= infs′<s

cp((s′, y), (s, y))− φ(s′, y)

= infs′<s−1

p(s− s′)p − rp−1s′ =

p− 1

prp − rp−1s,

where the second equality follows from the fact that the geodesics in Ominimize cp to the level sets of φ. It follows that

φcp(s, y) + φ(s− r, y) = −1

prp = cp((s− r, y), (s, y)),

i.e. ∂cpφ(x) = {expx(∇qgφ(x))} for all x ∈ M whenever the right hand sideis well defined (see Section 2.5 for the definition).

It now follows by standard transportation theory (see for instance [1,Theorem 1.13]) that for a Borel probability measure µ1/2 on I × Σ the

family(µt := (Ψt

1/2)]µ1/2

)t∈[0,1]

, where

Ψt1/2 : I × Σ→ I × Σ, (s, x) 7→

(s+ r

(t− 1

2

), x

)


defines a cp-optimal dynamical plan as long as it is defined in accordance withthe notation in (1.4) and φ is a smooth Kantorovich potential for (µt)t∈[0,1].

If µ1/2 � volg with density ρ1/2 ∈ Cc(M), we get (compare with the proofof Theorem 4.3)

Ent(µt|volg) =

ˆM

log ρt (y) dµt (y) =

ˆM

log ρt(Ψt1/2(x)) dµ1/2(x)

=

ˆM

log[ρ1/2(x)(Detg(DΨt1/2)(x))−1] dµ1/2(x)

= Ent(µ1/2|volg)−ˆM

log[Detg(DΨt1/2)(x))] dµ1/2(x).

We have Detg(DΨt1/2)((s, x))) = an−1(s+r(t−1/2))

an−1(s)and thus obtain

(1.12)

− d2

dt2log[Detg(DΨt

1/2)(x))] = −(n−1)aa− a2

a2r2 = Ric(r∂s, r∂s)+(n−1)

a2

a2r2.

Neglecting the term a2

a2 ≥ 0 we conclude (compare with the proof of Propo-sition 3.4)

d2

dt2Ent(µt|volg) = − d2

dt2

ˆM

log[Detg(DΨt1/2)(x))] dµ1/2(x)

≥ˆM

Ric(r∂s, r∂s)Ψt1/2

(x) dµ1/2(x),

(1.13)

which implies

4r2


]≥ˆM

Ric(∂s, ∂s)dµ1/2

for r → 0, i.e. one side of (1.7).Note that the gap in (1.13) is

(n− 1)r2

ˆM

a2

a2dµ1/2.

From this we see that the bound from above4r2 [Ent(µ1|volg)− 2Ent(µ1/2|volg)+Ent(µ0|volg)]

≤ˆM

Ric(∂s, ∂s)dµ1/2 + ε(r),(1.14)

in (1.7) for the aforementioned transports holds for r → 0 if µ1/2 is con-centrated on {(s, x)| a(s) = 0} or, more generally, if µ1/2 = µr1/2 satisfies

limr→0

´M

a2

a2 dµr1/2 = 0.

The Levi-Civita connection ∇ of g satisfies

∇X∂s = ∇∂sX =a

aX

for all vector fields X tangent to Σ. Thus the Hessian of φ is given by(1.15)

Hessφ = rp−1HessS = rp−1∇.∇gS = −rp−1∇.∂s = −rp−1 a

a(Id− ∂s ⊗ ds).


It vanishes at (s, x) if and only if a(s) = 0. Thus the inequality (1.14)follows for these transports if the Hessian of φ vanishes on supp(µ1/2) or,

more generally, if limr→0

´M ‖r

1−pHessφ‖2dµr1/2 = 0.

Finally we illustrate the theory laid out in Appendix B via warping func-tions a : I → (0,∞) of regularity below C2. Compare with [41] for relatedresults on metrics of low regularity.

Consider a probability measure ν1/2 � volσ on Σ and set

µ1/2 :=1

s1 − s0L1|[s0,s1] ⊗ ν1/2

for s0, s1 ∈ I with s0 < s1 and L1 the 1-dimensional Lebesgue measure. Forµt := (Ψt

1/2)]µ1/2 we have

4

r2[Ent(µ1|volg)− 2Ent(µ1/2|volg)+Ent(µ0|volg)]

= − 4

r2(s1 − s0)

ˆ s1

s0

log

(an−1(s+ r

2)an−1(s− r2)

a2n−2(s)

)ds

Assuming s1 − s0 � r and r > 0 sufficiently small (i.e. the setting ofTheorem 1.1), since a is continuous, we obtain:

4

r2[Ent(µ1|volg)− 2Ent(µ1/2|volg)+Ent(µ0|volg)]

= − 4

r2log

(an−1(s+ r

2)an−1(s− r2)

a2n−2(s)

)+ε(r)

r2,

for some function ε(r)→ 0 as r → 0. As an example, we discuss the case ofthe following C1,1-warping function

a : I → R, a(s) :=

{λ+s

2 + 1, s ≥ 0

λ−s2 + 1, s < 0

for λ−, λ+ ∈ R.Ignoring terms of higher order near s = 0 we get

4

r2[Ent(µ1|volg)− 2Ent(µ1/2|volg) + Ent(µ0|volg)]

=

{−(n− 1)[λ+(2s

r + 1)2 + λ−(2sr − 1)2 − 2λ+(2s

r )2], s ≥ 0

−(n− 1)[λ+(2sr + 1)2 + λ−(2s

r − 1)2 − 2λ−(2sr )2], s ≤ 0

for r → 0. It is now easy to see that the possible accumulation points ofthe right hand side for r → 0 and s ∈ [− r

2 ,r2 ] lie between −2(n − 1)λ+ =

lims↓0 Ric(∂s, ∂s) and −2(n− 1)λ− = lims↑0 Ric(∂s, ∂s). Furthermore, everyvalue in that interval is an accumulation point for the right hand side, asr → 0. In case the warping function lies in C2, we get from (1.12) theasymptotic formula:(1.16)

4

r2[Ent(µ1|volg)− 2Ent(µ1/2|volg) + Ent(µ0|volg)]− (n− 1)

ˆM

a2

a2dµ1/2

for Ric(∂s, ∂s) when r → 0. The second integral is well defined for a ∈W 1,2(I). Therefore we can use (1.16) as a definition for Ric(∂s, ∂s) in this


case. This is indeed the spirit of the synthetic bounds on the timelike Riccicurvature given in Definition B.5 and Definition B.8.

1.4. Related literature.

1.4.1. Ricci curvature via optimal transport in Riemannian setting. In theRiemannian framework, a line of research pioneered by McCann [61], Cordero-Erausquin-McCann-Schmuckenschlager [27, 28], Otto-Villani [69] and vonRenesse-Sturm, has culminated in a characterization of Ricci-curvature lowerbounds (by a constant K ∈ R) involving only the displacement convexity ofcertain information-theoretic entropies [77]. This in turn led Sturm [74, 75]and independently Lott-Villani [57] to develop a theory for lower Ricci cur-vature bounds in a non-smooth metric-measure space setting. The theoryof such spaces has seen a very fast development in the last years, see e.g.[2, 4, 5, 6, 7, 19, 23, 24, 32, 39, 40, 65]. An approach to the complementaryupper bounds on the Ricci tensor (again by a constant K ′ ∈ R) has beenrecently proposed by Naber [66] (see also Haslhofer-Naber [44]) in terms offunctional inequalities on path spaces and martingales, and by Sturm [76](see also Erbar-Sturm [33]) in terms of contraction/expansion rate estimatesof the heat flow and in terms of displacement concavity of the Shannon-Bolzmann entropy. The Lorentzian time-like Ricci upper bounds of thispaper have been inspired in particular by the work of Sturm [76].

1.4.2. Optimal transport in Lorentzian setting. The optimal transport prob-lem in Lorentzian geometry was first proposed by Brenier [14] and further in-vestigated in [12, 78, 50]. An intriguing physical motivation for studying theoptimal transport problem in Lorentzian setting called the “early universereconstruction problem” [16, 38]. The Lorentzian cost Cp, for p ∈ (0, 1), wasproposed by Eckstein-Miller [29] and thoroughly studied by Mc Cann [62]very recently. In the same paper [62], Mc Cann gave an optimal transportformulation of the strong energy condition Ric ≥ 0 of Penrose-Hawking[71, 45, 46] in terms of displacement convexity of the Shannon-Bolzmannentropy under the assumption that the space time is globally hyperbolic.

We learned of the work of Mc Cann [62] when we were already in thefinal stages of writing the present paper. Though both papers (inspired bythe aforementioned Riemannian setting) are based on the idea of analyzingconvexity properties of entropy functionals on the space of probability mea-sures endowed with the cost Cp, p ∈ (0, 1), the two approaches are largelyindependent: while Mc Cann develops a general theory of optimal trans-portation in globally hyperbolic space times focusing on the strong energycondition Ric ≥ 0, in this paper we decided to take the quickest path inorder to reach our goal of giving an optimal transport formulation of thefull Einstein’s equations. Compared to [62], in the present paper we removethe assumption of global hyperbolicity on the space-time, we extend the op-timal transport formulation to any lower bound of the type Ric ≥ T for anysymmetric bilinear form T , and we also characterize general upper boundsRic ≤ T .

1.4.3. Physics literature. The existence of strong connections between ther-modynamics and general relativity is not new in the physics literature; it has


its origins at least in the work Bekenstein [10] and Hawking with collabora-tors [8] in the mid-1970s about the black hole thermodynamics. These worksinspired a new research field in theoretical physics, called entropic gravity(also known as emergent gravity), asserting that gravity is an entropic forcerather than a fundamental interaction. Let us give a brief account. In 1995Jacobson [43] derived the Einstein equations from the proportionality of en-tropy and horizon area of a black hole, exploiting the fundamental relationδQ = T δS linking heat Q, temperature T and entropy S. Subsequently,other physicists, most notably Padmanabhan (see for instance the recentsurvey [70]), have been exploring links between gravity and entropy.

More recently, in 2011 Verlinde [80] proposed a heuristic argument sug-gesting that (Newtonian) gravity can be identified with an entropic forcecaused by changes in the information associated with the positions of ma-terial bodies. A relativistic generalization of those arguments leads to theEinstein equations.

The optimal transport formulation of Einstein equations obtained in thepresent paper involving the Shannon-Bolzmann entropy can be seen as anadditional strong connection between general relativity and thermodyna-mics/information theory. It would be interesting to explore this relationshipfurther.

Acknowledgement. The authors wish to thank Christina Sormani andthe anonymous referee for several comments that improved the expositionof the paper.

2. Preliminaries

2.1. Some basics of Lorentzian geometry. Let M be a smooth manifoldof dimension n ≥ 2. It is convenient to fix a complete Riemannian metrich on M . The norm | · | on TxM and the distance dist(·, ·) : M ×M → R+

are understood to be induced by h, unless otherwise specified. Recall thath induces a Riemannian metric on TM . Distances on TM are understoodto the induced by such a metric. The metric ball around x ∈M with radiusr, with respect to h, is denoted by Bh

r (x) or simply by Br(x).A Lorentzian metric g on M is a smooth (0, 2)-tensor field such that

g|x : TxM × TxM → Ris symmetric and non-degenerate with signature (−,+, . . . ,+) for all x ∈M . It is well known that, if M is compact, the vanishing of the Eulercharacteristic of M is equivalent to the existence of a Lorentzian metric; onthe other hand, any non-compact manifold admits a Lorentzian metric. Anon-zero tangent vector v ∈ TxM is called

• Time-like: if g(v, v) < 0,• Light-like (or null): if g(v, v) = 0 as well as v 6= 0,• Spacelike: if g(v, v) > 0 or v = 0.

A non-zero tangent vector v ∈ TxM which is either time-like or light-like,i.e. g(v, v) ≤ 0 and v 6= 0, is called causal (or non-spacelike). A Lorentzianmanifold (M, g) is said to be time-oriented if M admits a continuous no-where vanishing time-like vector field X. The vector field X induces apartition on the set of causal vectors, into two equivalence classes:


• The future pointing tangent vectors v for which g(X, v) < 0,• The past pointing tangent vectors v for which g(X, v) > 0.

The closure of the set of future pointing time-like vectors is denoted

C = Cl({v ∈ TM : g(v, v) < 0 and g(X, v) < 0}) ⊂ TM.

Note that the fiber Cx := C ∩ TxM is a closed convex cone and the openinterior Int(C) is a connected component of {v : g(v, v) < 0}. A time-oriented Lorentzian manifold (M, g, C) is called a space-time.

An absolutely continuous curve γ : I → M is called (C)-causal if γt ∈ Cfor every differentiability point t ∈ I. A causal curve γ : I → M is calledtime-like if for every s ∈ I there exist ε, δ > 0 such that dist(γt, ∂C) ≥ ε|γt|for every t ∈ I for which γt exists and |s − t| < δ. In [17, Section 2.2]time-like curves are defined in terms of the Clarke differential of a Lipschitzcurve. Whereas the definition via the Clarke differential is probably moresatisfying from a conceptual point of view, the definition given here is easierto state. All relevant sets and curves used below are independent of thedefinition, see [17, Lemma 2.11] and Proposition 2.4, though.

We denote by J+(x) (resp. J−(x)) the set of points y ∈ M such thatthere exists a causal curve with initial point x (resp. y) and final point y(resp. x), i.e. the causal future (resp. past) of x. The sets I±(x) are definedanalogously by replacing causal curves by time-like ones. The sets I±(p)are always open in any space-time, on the other hand the sets J±(p) are ingeneral neither closed nor open.

For a subset A ⊂M , define J±(A) := ∪x∈AJ±(x), moreover set

(2.1) J+ := {(x, y) ∈M ×M : y ∈ J+(x)}.

2.2. The Lagrangian Lp, the action Ap and the cost cp. On a space-time (M, g, C) consider, for any p ∈ (0, 1), the following Lagrangian on TM :

(2.2) Lp(v) :=

{−1p(−g(v, v))

p2 if v ∈ C

+∞ otherwise.

The following fact appears in [62, Lemma 3.1]. We provide a proof forthe readers convenience.

Lemma 2.1. The function Lp is fiberwise convex, finite (and non-positive)on its domain and positive homogenous of degree p. Moreover Lp is smoothand fiberwise strictly convex on Int(C).

Proof. It is clear from its very definition that the restriction of Lp to Int(C)is smooth. A direct computation gives

∂Lp∂vi

= (−g(v, v))p−2

2 gikvk, i = 1, . . . , n

(2.3)

∂2Lp∂vi∂vj

= (−g(v, v))p−4

2

(−g(v, v)gij + (2− p)gikvkgjlvl

), i, j = 1, . . . , n.

(2.4)


Fix v ∈ Int(C). Decompose w ∈ TxM into w‖ the part parallel to v and w⊥

the part orthogonal to v, all with respect to g. Then we have

D2vvLp(w,w) = (−g(v, v))

p−42

(− g(w⊥, w⊥)g(v, v)− g(w‖, w‖)g(v, v)

(2.5)

+ (2− p)g(v, w‖)2)

(2.6)

= (−g(v, v))p−4

2

(−g(w⊥, w⊥)g(v, v) + (1− p)g(v, w‖)2

).(2.7)

Since g(w⊥, w⊥) ≥ 0 and p < 1 we have

D2vvLp(w,w) > 0

for w 6= 0. �

We define the Lagrangian action Ap associated to Lp as follows:

(2.8) Ap(γ) :=

ˆ 1

0Lp(γt)dt ∈ (−∞, 0] ∪ {+∞}.

Note that if Ap(γ) ∈ R, then γ is causal. A causal curve γ : [0, 1] → M isan Ap-minimizer between its endpoints x, y ∈M if

Ap(γ) = inf{Ap(η) : η ∈ AC([0, 1],M), η0 = x, η1 = y}.

Lemma 2.2. Any Ap-minimizer with finite action is either a future point-ing time-like geodesic of (M, g) or a future pointing light-like pregeodesic of(M, g), i.e. an orientation preserving reparameterization is a future pointinglight-like geodesic of (M, g).

Proof. Let γ : [0, 1]→M be a Ap-minimizer with finite action. Then γ(t) ∈C for a.e. t. By Jensen’s inequality we haveˆ 1

0−1

p(−g(η, η))

p2 dt ≥ −1

p

(ˆ 1

0

√−g(η, η)dt

)p,

for any causal curve η : [0, 1]→M with equality if and only if η is parametrizedproportionally to arclength.

Recall that the restriction of a minimizer to any subinterval of [0, 1] is aminimizer of the restricted action. Since any point in a spacetime admits aglobally hyperbolic neighborhood, see [64, Theorem 2.14], the Avez-SeifertTheorem [67, Proposition 14.19] implies that every minimizer of A1 withfinite action is a causal pregeodesic.

Combining both points we see that if the action of γ is negative, the curveis a time-like pregeodesic parameterized with respect to constant arclength,i.e. a time-like geodesic. If the action of γ vanishes, the curve is a light-likepregeodesic. �

Consider the cost function relative to the p-action Ap:

cp : M ×M → R ∪ {+∞}(x, y) 7→ inf{Ap(η) : η ∈ AC([0, 1],M), η0 = x, η1 = y}.

Remark 2.3. We will always assume that:


(i) The cost function is bounded from below on bounded subsets ofM ×M . By transitivity of the causal relation this follows from theassumption that cp(x, y) > −∞ for all x, y ∈M .

(ii) The cost function is localizable, i.e. every point x ∈ M has aneighborhood U ⊂ M such that the cost function of the space-time(U, g|U , C|U ) coincides with the global cost function.

Note since the main results of this paper are local in nature, the assump-tions can always be satisfied by restricting the space-time to a suitable opensubset.

Proposition 2.4. Fix p ∈ (0, 1) and let (M, g, C) be a space-time. Thenevery point has a neighborhood U such that the following holds for the space-time (U, g|U , C|U ). For every pair of points x, y ∈ U with (x, y) ∈ J+

U , thecausal relation of (U, g|U , C|U ), there exists a curve γ : [0, 1] → U withγ0 = x, γ1 = y, and minimizing Ap among all curves η ∈ AC([0, 1],M) withη0 = x and η1 = y. Moreover γ is a constant speed geodesic for the metricg, γ ∈ C whenever the tangent vector exists, and Ap(γ) ∈ R.

Proof. It is well known that in a space-time every point has a globally hy-perbolic neighborhood. Let U be such a neighborhood. If (x, y) ∈ J+

U thereexists a curve with finite action Ap between x and y. At the same time theaction is bounded from below, e.g. by a steep Lyapunov function, see [17].Therefore any minimizer γ : [0, 1] → U has finite action, i.e. γ(t) ∈ C foralmost all t. By Jensen’s inequality we haveˆ 1

0−1

p(−g(η, η))

p2 dt ≥ −1

p

(ˆ 1

0

√−g(η, η)dt

)p,

for any causal curve η : [0, 1]→ U with equality if and only if η is parametrizedproportionally to arclength. By the Avez-Seifert Theorem [67, Proposition14.19] every minimizer of the right hand side is a causal pregeodesic. Com-bining both it follows that every Ap-minimizer is a causal geodesic. �

2.3. Ricci curvature and Jacobi equation. We now fix the notationregarding curvature for a Lorentzian manifold (M, g) of dimension n ≥ 2.Called∇ the Levi-Civita connection of (M, g), the Riemann curvature tensoris defined by

(2.9) R(X,Y )Z = ∇X∇Y Z −∇Y∇XX −∇[X,Y ]Z,

where X,Y, Z are smooth vector fields on M and [X,Y ] is the Lie bracketof X and Y .For each x ∈ M , the Ricci curvature is a symmetric bilinear form Ricx :TxM × TxM → R defined by

(2.10) Ricx(v, w) :=n∑i=1

g(ei, ei)g(R(ei, w)v, ei),

where {ei}i=1,...,n is an orthonormal basis of TxM , i.e. |g(ei, ej)| = δij forall i, j = 1 . . . , n.

Given a endomorphism U : TxM → TxM and a g-orthonormal basis{ei}i=1,...,n of TxM , we associate to U the matrix

(2.11) (Uij)i,j=1,...,n, Uij := g(ei, ej) g(Uei, ej).


The trace Trg(U) and the determinant Detg(U) of the endomorphism Uwith respect to the Lorentzian metric g are by definition the trace tr(Uij)and the determinant det(Uij)) of the matrix (Uij)i,j=1,...,n, respectively. Itis standard to check that such a definition is independent of the chosenorthonormal basis of TxM . Note that Ricx(v, w) is the trace of curvatureendomorphism R(·, w)v : TxM → TxM .

A smooth curve γ : I → M is called a geodesic if ∇γ γ = 0. A vectorfield J along a geodesic γ is said to be a Jacobi field if it satisfies the Jacobiequation:

(2.12) ∇γ(∇γJ) +R(J, γ)γ = 0.

2.4. The q-gradient of a function. Finally let us recall the definition ofgradient and hessian. Given a smooth function f : M → R, the gradient off denoted by ∇gf is defined by the identity

g(∇gf, Y ) = df(Y ), ∀Y ∈ TM,

where df is the differential of f . The Hessian of f , denoted by Hessf isdefined to be the covariant derivative of df :

Hessf := ∇(df).

It is related to the gradient through the formula

Hessf (X,Y ) = g(∇X∇gf, Y ), ∀X,Y ∈ TM,

and satisfies the symmetry

(2.13) Hessf (X,Y ) = Hessf (Y,X) ∀X,Y ∈ TM.

Next we recall some notions for the causal character of functions.

• A function f : M → R ∪ {±∞} is a causal function if f(x) ≤ f(y)for all (x, y) ∈ J+;• it is a time function if f(x) < f(y) for all (x, y) ∈ J+ \∆, where ∆

denotes the diagonal in M ×M .• Following [17] we call a differentiable (Ck-) function f : M → R

(k ∈ N ∪ {∞}) a (Ck-) Lyapunov or (Ck-) temporal function ifdfx|Cx\{0} > 0 for all x ∈M .

Let q be the conjugate exponent to p, i.e.

1

p+

1

q= 1, or equivalently (p− 1)(q − 1) = 1.

Notice that, since p ranges in (0, 1) then q ranges in (−∞, 0). In orderto describe the optimal transport maps later in the paper, it is useful tointroduce the q-gradient (cf. [49])

(2.14) ∇qgφ := −|g(∇gφ,∇gφ)|q−2

2 ∇gφ

for differentiable Lyapunov functions φ : M → R; in particular, ∇qgφ(x) ∈Cx \ {0}. Notice that,

For v ∈ Cx \ {0}, ∇gφ(x) = −|g(v, v)|p−2

2 v if and only if ∇qgφ(x) = v.

Moreover

x 7→ ∇qgφ(x) is continuous (resp. Ck, k ≥ 1) on U ⊂ {|g(∇qgφ,∇qgφ)| > 0}


if and only if

x 7→ ∇gφ(x) is continuous (resp. Ck, k ≥ 1) on U ⊂ {|g(∇gφ,∇gφ)| > 0}.The motivation for the use of the q-gradient comes from the Hamiltonian

formulation of the dynamics; let us briefly mention a few key facts that willplay a role later in the paper. For α ∈ T ∗xM , let

(2.15) Hp(α) = supv∈TxM

[α(v)− Lp(v)]

be the Legendre transform of Lp. Denote with g∗ the dual Lorentzian metricon T ∗M and C∗ ⊂ T ∗M the dual cone field to C. Then Hp satisfies

(2.16) Hp(α) :=

{−1q (−g∗(α, α))

q2 if α ∈ C∗ \ T ∗,0M

+∞ otherwise,

for (p−1)(q−1) = 1. By analogous computations as performed in the proofof Lemma 2.1, one can check that

(2.17) ∇qgφ(x) = DHp(−dφ(x)).

By well known properties of the Legendre transform (see for instance [21,Theorem A.2.5]) it follows that DHp is invertible on Int(C∗) with inversegiven by DLp. Thus (2.17) is equivalent to

(2.18) DLp(∇qgφ(x)) = −dφ(x).

2.5. cp-concave functions and regular cp-optimal dynamical plans.We denote by P(M) the set of Borel probability measures on M . For anyµ1, µ2 ∈ P(M), we say that a Borel probability measure

π ∈ P(M ×M) is a coupling of µ1 and µ2

if (pi)]π = µi, i = 1, 2, where p1, p2 : M × M → M are the projectionsonto the first and second coordinate. Recall that the push-forward (p1)]π isdefined by

(p1)]π(A) := π(p−1

1 (A))

for any Borel subset A ⊂ M . The set of couplings of µ1, µ2 is denoted byCpl(µ1, µ2). The cp-cost of a coupling π is given byˆ

M×Mcp(x, y)dπ(x, y) ∈ [−∞, 0] ∪ {+∞}.

Denote by Cp(µ1, µ2) the minimal cost relative to cp among all couplingsfrom µ1 to µ2, i.e.

Cp(µ1, µ2) := inf

{ˆcpdπ : π ∈ Cpl(µ1, µ2)

}∈ [−∞, 0] ∪ {+∞}.

If Cp(µ1, µ2) ∈ R, a coupling achieving the infimum is said to be cp-optimal.We next define the notion of cp-optimal dynamical plan. To this aim, it is

convenient to consider the set of Ap-minimizing curves, denoted by Γp. Theset Γp is endowed with the sup metric induced by the auxiliary Riemannianmetric h. It will be useful to consider the maps for t ∈ [0, 1]:

et : Γp →M, et(γ) := γt

∂et : Γp → TM, ∂et(γ) := γt ∈ TγtM.


A cp-optimal dynamical plan is a probability measure Π on Γp such that(e0, e1)]Π is a cp-optimal coupling from µ0 := (e0)]Π to µ1 := (e1)]Π.We will mostly be interested in cp-optimal dynamical plans obtained by“exponentiating the q-gradient of a cp-concave function”, what we will callregular cp-optimal dynamical plans. In order to define them precisely, let usfirst recall some basics of Kantorovich duality (we adopt the convention of[1]).

Fix two subsets X,Y ⊂ M . A function φ : X → R ∪ {−∞} is said cp-concave (with respect to (X,Y )) if it is not identically −∞ and there existsu : Y → R ∪ {−∞} such that

φ(x) = infy∈Y

cp(x, y)− u(y), for every x ∈ X.

Then, its cp-transform is the function φcp : Y → R ∪ {−∞} defined by

(2.19) φcp(y) := infx∈X

cp(x, y)− φ(x),

and its cp-superdifferential ∂cpφ(x) at a point x ∈ X is defined by

(2.20) ∂cpφ(x) := {y ∈ Y : φ(x) + φcp(y) = cp(x, y)}.Note that

(2.21)

{φ(x) = cp(x, y)− φcp(y), for all x ∈ X, y ∈ ∂cpφ(x)

φ(x) ≤ cp(x, y)− φcp(y), for all x ∈ X, y ∈ Y.

From the definition it follows readily that if φ is cp-concave, then for (x, z) ∈J+ ∩ (X ×X) we have

φ(z) = infy∈Y

cp(z, y)− u(y) ≥ infy∈Y

cp(x, y)− u(y) = φ(x),

i.e. φ is a causal function. The same argument gives that −φcp is a causalfunction as well.

Definition 2.5 (Regular cp-optimal dynamical plan). A cp-optimal dynam-ical plan Π ∈ P(Γp) is regular if the following holds.There exists U, V ⊂ M relatively compact open subsets and a smooth cp-concave (with respect to (U, V )) function φ1/2 : U → R such that

(1) ∇qgφ1/2(x) ∈ K ⊂⊂ Int(C) for every x ∈ U and(− g(∇qgφ1/2,∇qgφ1/2)

)1/2< Injg(U),

where Injg(U) is the injectivity radius of g on U ;(2) Setting

Ψt1/2(x) = expgx((t− 1/2)∇qgφ1/2(x))

and µt := (et)]Π for every t ∈ [0, 1], it holds that

supp(µ1/2) ⊂ U, µt = (Ψt1/2)]µ1/2 and µt � volg ∀t ∈ [0, 1].

Roughly, the above notion of regularity asks that the Ap-minimizingcurves performing the optimal transport from µ0 := (e0)]Π to µ1 := (e1)]Πhave velocities contained in K, i.e. they are all “uniformly” time-like futurepointing. Moreover it also implies that ∪t∈[0,1]supp(µt) ⊂M is compact; inaddition the optimal transport is assumed to be driven by a smooth poten-tial φ1/2. Even if these conditions may appear a bit strong, we will prove in


Lemma 3.2 that there are a lot of such regular plans; moreover in the paperwe will show that it is enough to consider such particular optimal trans-ports in order to characterize upper and lower bounds on the (causal-)Riccicurvature and thus characterize the solutions of Einstein equations.

3. Existence, regularity and evolution of Kantorovichpotentials

In order to characterize Lorentzian Ricci curvature upper bounds, it willbe useful the next proposition concerning the evolution of Kantorovich po-tentials along a regularAp-minimizing curve of probability measures (µt)t∈[0,1]

given by exponentiating the q-gradient of a smooth cp-concave function withtime-like gradient. To this aim it is convenient to consider, for 0 ≤ s < t ≤ 1,the restricted minimal action

cs,tp (x, y) := inf

{ˆ t

sLp(γ(τ))dτ

∣∣∣∣ γ ∈ AC([s, t],M), γ(s) = x, γ(t) = y

}.

Proposition 3.1. Let (M, g, C) be a space-time, fix p ∈ (0, 1) and let q ∈(−∞, 0) be the Holder conjugate exponent, i.e. 1

p + 1q = 1 or equivalently

(p − 1)(q − 1) = 1. Let U, V ⊂ M be relatively compact open subsets andφ1/2 be a smooth cp-concave function relative to (U, V ) such that

• φ1/2 is a smooth Lyapunov function on U ,

•(− g(∇qgφ1/2,∇

qgφ1/2)

)1/2< Injg(U).

For t ∈ [0, 1], let

Ψt1/2 : U →M, Ψt

1/2(x) := expx((t− 1/2)∇qgφ1/2(x)).

For every x ∈ U , define(3.1)

φt(Ψt1/2(x)) = φ(t,Ψt

1/2(x)) :=

{φ1/2(x)− c1/2,t

p (x,Ψt1/2(x)) for t ∈ [1/2, 1]

φ1/2(x) + ct,1/2p (Ψt

1/2(x), x) for t ∈ [0, 1/2).

Then the map (t, y) 7→ φ(t, y) defined on⋃t∈[0,1] {t} × Ψt

1/2(U) is C∞ and

satisfies the Hamilton-Jacobi equation(3.2)

∂tφt(t, y) +1

q(−g(∇gφt(y),∇gφt(y)))q/2 = 0, ∀(t, y) ∈

⋃t∈[0,1]

{t} ×Ψt1/2(U)

with

(3.3)d

dtΨt

1/2(x) = ∇qgφt(Ψt1/2(x)), ∀(t, x) ∈ [0, 1]× U.

Proof. Step 1: smoothness of φ.The fact that t 7→ Ψt

1/2 is a smooth 1-parameter family of maps performing

cp-optimal transport gives that φ defined in (3.19) satisfies (cf. [21, Theorem6.4.6])(3.4)φt(Ψ

t1/2(x)) = φs(Ψ

s1/2(x))−cs,tp (Ψs

1/2(x),Ψt1/2(x)), ∀x ∈ U, 0 ≤ s < t ≤ 1.


In particular it holds

φt(Ψt1/2(x)) = φ0(Ψ0

1/2(x))− c0,tp (Ψ0

1/2(x),Ψt1/2(x)) ∀x ∈ U, t ∈ [0, 1],

(3.5)

φt(Ψt1/2(x)) = φ1(Ψ1

1/2(x)) + ct,1p (Ψt1/2(x),Ψ1

1/2(x)) ∀x ∈ U, t ∈ [0, 1].

(3.6)

Since by construction everything is defined inside the injectivity radius andall the transport rays are non-constant, from (3.5) (respectively (3.6)) it ismanifest that the map (t, y) 7→ φ(t, y) is C∞ on

⋃t∈(0,1]{t}×Ψt

1/2(U) (resp.⋃t∈[0,1){t}×Ψt

1/2(U)). The smoothness of φ on⋃t∈[0,1]{t}×Ψt

1/2(U) follows.

Step 2: validity of the Hamilton-Jacobi equation (3.2).We consider t ∈ (1/2, 1], the case t ∈ [0, 1/2] being analogous. Fix y =Ψt

1/2(x) for some arbitrary x ∈ U and t ∈ (1/2, 1], and let γ : [0, s]→M be

a smooth curve with γ(0) = v ∈ TyM . From (3.19) we have

φ(t+ s, γs) ≥ −ˆ s

0Lp(γτ ) dτ + φ(t, γ0),

with equality for γ(τ) = Ψt+τ (x) for all τ ∈ [0, s]. Dividing by s and takingthe limit for s→ 0, we obtain

lims→0

φ(t+ s, γs)− φ(t, γ0)

s≥ −Lp(v),

which in turn implies

∂tφt(y) ≥ −dφt(v)− Lp(v), for every v ∈ TyM.

Note that equality holds for v = ∇qgφ1/2(x). For α ∈ T ∗yM , let

Hp(α) = supv∈TyM

[α(v)− Lp(v)]

denote the Legendre transform of Lp. Thus we get

(3.7) ∂tφt(y) = Hp(−d(φt)y).

Recalling that Hp has the representation (2.16), we have

Hp(−d(φt)y) = −1

q(−g(∇gφt(y),∇gφt(y)))q/2 ,

which, together with (3.7), implies (3.2).Step 3: validity of (3.3).

Since Ψt1/2 is a smooth 1-parameter family of maps performing cp-optimal

transport and the function φ defined in (3.19) is smooth, it coincides withthe viscosity solution (resp. backward solution)

(3.8) φt(y) =

{supz∈Ψs

1/2(U) φs(z)− c

s,tp (z, y) for t ∈ [s, 1]

infz∈Ψs1/2

(U) φs(z) + ct,sp (y, z) for t ∈ [0, s),

for every s ∈ (0, 1), y ∈ Ψt1/2(U).

Let us discuss the case t ∈ (s, 1], the other is analogous. From (3.4) itfollows that Ψs

1/2(x) is a maximum point in the right hand side of (3.8)


corresponding to y = Ψt1/2(x). Thus

dφs(Ψs1/2(x)) = d

[cs,tp (·,Ψt

1/2(x))]

(Ψs1/2(x)) = −DLp

(d

dsΨs

1/2(x)

).

By construction ddsΨ

s1/2(x) ∈ Int(C) and, as already observed, DLp is invert-

ible on Int(C) with inverse given by DHp. We conclude that

d

dsΨs

1/2(x) = DHp(−dφs(Ψs

1/2(x)))

= ∇qgφs(Ψs1/2(x)).

�

We next show that for every point x ∈M and every v ∈ Cx “small enough”we can find a smooth cp-concave function φ defined on a neighbourhood ofx, such that ∇qgφ = v and the hessian of φ vanishes at x. This is well knownin the Riemannian setting (e.g. [81, Theorem 13.5]) and should be comparedwith the recent paper by Mc Cann [62] in the Lorentzian framework. Thesecond part of the next lemma shows that the class of regular cp-optimaldynamical plans is non-empty, and actually rather rich.

Lemma 3.2. Let (M, g, C) be a space-time, fix x ∈ M and v ∈ Cx withg(v, v) < 0. Then there exists ε = ε(x, v) > 0 with the following property:

(1) For every s ∈ (0, ε), for every C2 function φ : M → R satisfying

(3.9) ∇qgφ(x) = sv, Hessφ(x) = 0,

there exists a neighbourhood Ux of x and a neighbourhood Uy of y :=expgx(sv) such that φ is cp-concave relatively to (Ux, Uy).

(2) Let

Ψt1/2(x) := expx((t− 1/2)∇qgφ(x)), ∀x ∈ Ux

Ψ : Ux → AC([0, 1],M), x 7→ Ψ(·)1/2(x).

Then, for every µ1/2 ∈ P(M) with supp(µ1/2) ⊂ Ux, the measure

Π := (Ψ)]µ1/2 is a cp-optimal dynamical plan.

Proof. (1) Calling y = y(sv) := expgx(sv), notice that ∇qgφ(x) = sv is equiv-alent to

(3.10) dφ(x) = Dxcp(x, y),

where Dxcp(x, y) denotes the differential at x of the function x 7→ cp(x, y).Indeed, a computation shows that Dxcp(x, y) = −DLp(sv) and thus theclaim follows from (2.18).

Let φ : M → R be any smooth function satisfying

(3.11) ∇qgφ(x) = sv, Hessφ(x) = 0.

In what follows we denote with Hessx,cp(x, y)(resp. Hessv,Lp(sv) the Hessianof the function x 7→ cp(x, y) evaluated at x = x (resp. the Hessian of thefunction TxM 3 w 7→ Lp(sv + w)). By taking normal coordinates centredat x one can check that the operator norm

‖Hessx,cp(x, y)−Hessv,Lp(sv)‖ → 0 as t→ 0.


Recalling that from (2.7) there exists Cp,v > 0 such that Hessv,Lp(sv) ≥Cp,vs

−2+p as quadratic forms, we infer(3.12)

Hessx,cp(x, y)−Hessφ(x) > 0 as quadratic forms, for every s ∈ (0, ε),

for some ε = ε(x, v) > 0 small enough. Since by construction we haveDxcp(x, y) − dφ(x) = 0, by the Implicit Function Theorem there exists aneighbourhood Ux×Uy of (x, y) ∈M ×M and a smooth function F : Uy →Ux such that F (y) = x and

Dxcp(F (y), y)− dφ(F (y)) = 0, for every y ∈ Uy.

Differentiating the last equation in y at y and using that Hessφ(x) = 0, weobtain

(3.13) D2yxcp(x, y) + Hessx,cp(x, y)DF (y) = 0.

Using normal coordinates centred at x and (2.4) one can check that theoperator norm∥∥∥D2

yxcp(x, y)− (−g(sv, sv))p−4

2 (−g(sv, sv)g + (2− p)(sv)∗ ⊗ (sv)∗)∥∥∥→ 0

as s→ 0, where (sv)∗ = g(sv, ·) is the covector associated to sv.Since by assumption g(v, v) < 0 and p ∈ (0, 1), it follows from the reverse

Cauchy-Schwartz inequality that det[D2yxcp(x, y)] > 0 for s ∈ (0, ε). Recall-

ing that det[Hessx,cp(x, y)] 6= 0, from (3.13) we infer that det(DF (y)) 6= 0.By the Inverse Function Theorem, up to reducing the neighbourhoods, weget that F : Uy → Ux is a smooth diffeomorphism. Define now

u : Uy → R, u(y) := cp(F (y), y)− φ(F (y)).

For every fixed y ∈ Uy, the function Ux 3 x 7→ cp(x, y) − φ(x) − u(y)vanishes at x = F (y); moreover, from (3.12), it follows that x = F (y) isthe strict global minimum of such a function on Ux, up to further reducingUx and Uy, possibly. In other words, the function Ux × Uy 3 (x, y) 7→cp(x, y) − φ(x) − u(y) is always non-negative and vanishes exactly on thegraph of F . It follows that

(3.14) φ(x) = infy∈Uy

cp(x, y)− u(y), for every x ∈ Ux,

i.e. φ : Ux → R is a smooth cp-concave function relative to (Ux, Uy) satisfy-ing (3.9).

Proof of (2).Step 1. We first show that (e1/2, et)]Π is a cp-optimal coupling for (µ1/2, µt)and every t ∈ [1/2, 1]. To keep notation short, it is convenient to define

Ψ′t(x) := expx(t∇qgφ(x)), ∀x ∈ UxΨ′ : Ux → AC([0, 1],M), x 7→ Ψ′(·)(x).

Setting Π′ := (Ψ′)]µ1/2, we have (et)]Π′ = (et+1/2)]Π for every t ∈ [0, 1/2].

If we show that

(3.15) (e0, e1)]Π′ is a cp-optimal coupling for

((e0)]Π

′, (e1)]Π′),


then, by the triangle inequality, it will follow that (e0, et)]Π′ is a cp-optimal

coupling for((e0)]Π

′, (et)]Π′) for every t ∈ [0, 1]; in particular our claim

that (e1/2, et)]Π is a cp-optimal coupling for (µ1/2, µt), t ∈ [1/2, 1], will beproved. Thus, the rest of step 1 will be devoted to establish (3.15).

Since by construction cp : Ux × Uy → R is smooth, by classical optimaltransport theory it is well know that the cp-superdifferential ∂cpφ ⊂ Uy iscp-cyclically monotone (see for instance [1, Theorem 1.13]). Therefore, inorder to have (3.15), it is enough to prove that

(3.16) ∂cpφ(x) = {expx(∇qgφ(x))} for every x ∈ Ux.Let us first show that ∂cpφ(x) 6= ∅, for every x ∈ Ux. From the proof of part(1), there exists a smooth diffeomorphism F : Uy → Ux such that

(3.17)

{φ(F (y)) = cp(F (y), y)− u(y), for all y ∈ Uyφ(x) ≤ cp(x, y)− u(y), for all x ∈ Ux, y ∈ Uy.

From the definition of φcp in (2.19), it is readily seen that φcp = u onUy. Thus (3.17) combined with (2.21) gives that y ∈ ∂cpφ(F (y)) for everyy ∈ Uy or, equivalently, F−1(x) ∈ ∂cpφ(x) for every x ∈ Ux. In particular,∂cpφ(x) 6= ∅, for every x ∈ Ux.

Now fix x ∈ Ux and pick y ∈ ∂cpφ(x) ⊂ Uy. Since z 7→ cp(z, y) isdifferentiable on Ux, we get(3.18)

cp(z, y) = cp(x, y)+(Dxcp(x, y))[(exphx)−1(z)]+o(dh(z, x)), for every z ∈ Ux.From y ∈ ∂cpφ(x), we have

φ(z)− φ(x)(2.21)

≤ cp(z, y)− cp(x, y)

(3.18)= (Dxcp(x, y))[(exphx)−1(z)] + o(dh(z, x)), for every z ∈ Ux.

Since φ is differentiable at x ∈ Ux, it follows that

dφ(x) = Dxcp(x, y) = −DLp(w),

where w ∈ Int(Cx) is such that y = expgx(w), which by (2.17) is equivalentto

w = DHp (−dφ(x)) = ∇qgφ(x),

which yields y = expgx(w) = expgx(∇qgφ(x)), concluding the proof of (3.16).

Step 2. Up to further reducing the open set Ux and the scale parameters > 0 in the definition of φ, we can assume that φ satisfies the assumptionsof Proposition 3.1 and that φ : Ψ′1/2(Ux)→ R defined by

(3.19) φ(Ψ′1/2(x)) := φ(x)− c0,1/2p (x,Ψ′1/2(x)), ∀x ∈ Ψ′1/2(Ux),

is still a Lyapunov function satisfying

(3.20)(− g(∇qgφ,∇qgφ)

)1/2< Injg(Ψ

′1/2(Ux)).

It is easily seen that

x = expz

(−1

2∇qgφ(z)

), ∀x ∈ Ux, z := Ψ′1/2(x).


Moreover, thanks to (3.20), the curve

[0, 1] 3 t 7→ expz((t− 1)∇qgφ(z)

), ∀x ∈ Ux, z := Ψ′1/2(x),

is still a g-geodesic, solving the method of characteristics associated to theoptimal transport problem (see for instance [21, Ch. 5.1] and [81, Ch. 7];this is actually a variation of step 1 and of the proof of Proposition 3.1). Itfollows that the map

Ξ : Ψ′1/2(Ux)→M, Ξ(z) := expz(−∇qgφ(z)

), ∀x ∈ Ux, z := Ψ′1/2(x)

is a cp-optimal transport map. In other words, for every µ ∈ P(M) withsupp µ ⊂ Ψ′1/2(Ux),

π := (Ξ, Id)] µ ∈ P(M ×M)

is a cp-optimal coupling for its marginals. We conclude that

(e0, e1)]Π = (Ξ, Id)]((e1)]Π

)is a cp-optimal coupling for its marginals and thus Π is a cp-optimal dynam-ical plan. �

We next establish some basic properties of cp-optimal dynamical planswhich will turn out to be useful for the OT-characterization of LorentzianRicci curvature upper and lower bounds.

Lemma 3.3. Let (M, g, C) be a space-time and let Π be a regular cp-optimaldynamical plan with

(et)]Π = µt = (Ψt1/2)]µ1/2 � volg,

Ψt1/2(x) = expgx((t− 1/2)∇qgφ(x)).

Then

(1) ∇∇qgφ(x) : TxM → TxM is a symmetric endomorphism, i.e.

g(∇X∇qgφ, Y ) = g(∇Y∇qgφ,X), ∀X,Y ∈ TxM, ∀x ∈ supp(µs).

(2) Ψt1/2 : suppµ1/2 → M is a diffeomorphism on its image for all

t ∈ [0, 1].(3) Calling µt = ρtvolg, the following Monge-Ampere equation holds

true:

(3.21) ρ1/2(x) = Detg[DΨt1/2(x)]ρt(Ψ

t1/2(x)), µ1/2-a.e. x, ∀t ∈ [0, 1].

Proof. (1) By construction, φ is smooth on U and g(∇gφ,∇gφ) < 0. Thusalso ∇qgφ : M → TM is a smooth section of the tangent bundle and the sym-metry of the endomorphism ∇∇qgφ(x) : TxM → TxM follows by Schwartz’sLemma.

(2) is a straightforward consequence of assumption (1) in Definition 2.5.(3) is a straightforward consequence of the change of variable formula.

�

It will be convenient to consider the matrix of Jacobi fields

(3.22) Bt(x) := DΨt1/2(x) : TxM → TΨt

1/2(x)M, for all x ∈ suppµ1/2,


along the geodesic t 7→ γt := Ψt1/2(x); recalling (2.12), Bt(x) satisfies the

Jacobi equation

(3.23) ∇t∇tBt(x) +R(Bt(x), γt)γt = 0,

where we denoted ∇t := ∇γt for short.Since by Lemma 3.3 we know that Bt is non-singular for all x ∈ suppµ1/2,

we can define

(3.24) Ut(x) := ∇tBt ◦ B−1t : TγtM → TγtM, for all x ∈ suppµ1/2.

The next proposition will be key in the proof of the lower bounds on causalRicci curvature. It is well known in Riemannian and Lorentzian geometry,see for instance [28, Lemma 3.1] and [30]; in any case we report a proof forthe reader’s convenience.

Proposition 3.4. Let Ut be defined in (3.24). Then Ut is a symmetric en-domorphism of TγtM (i.e. the matrix (Ut)ij with respect to an orthonormalbasis is symmetric) and it holds

∇tUt + U2t +R(·, γt)γt = 0.

Taking the trace with respect to g yields

(3.25) Trg(∇tUt) + Trg(U2t ) + Ric(γt, γt) = 0.

Setting y(t) := log DetgBt, it holds

(3.26) y′′(t) +1

n(y′(t))2 + Ric(γt, γt) ≤ 0.

Proof. Using (3.23) we get

∇tUt = (∇t∇tBt)B−1t +∇tBt∇t(B−1

t ) = −R(·, γt)γt − (∇tBt)B−1t (∇tBt)B−1

t

= −R(·, γt)γt − U2t .

Taking the trace with respect to g yields the second identity.The rest of the proof is devoted to show (3.26). Let (ei(t))i=1,...,n be anorthonormal basis of TγtM parallel along γ. Setting y(t) = log detBt, wehave that

y′(t0) =d

dt

∣∣∣∣t=t0

log Detg(BtB−1

t0

)=

d

dt

∣∣∣∣t=t0

log det[(g(ei(t), ej(t))g(BtB−1

t0ei(t), ej(t))

)i,j

]= Trg

[(∇tBt)B−1

t0

]|t=t0 = Trg(Ut0).(3.27)

We next show that Ut is a symmetric endomorphism of TγtM , i.e. the matrix(Ut)ij is symmetric. To this aim, calling U∗t the adjoint, we observe that

(3.28) U∗t − Ut = (B∗t )−1 [(∇tB∗t )Bt − B∗t (∇tBt)]B−1t ,

and that

(3.29) ∇t [(∇tB∗t )Bt − B∗t (∇tBt)] = (∇t∇tB∗t )Bt − B∗t (∇t∇tBt).Now the Jacobi equation (3.23) reads

(3.30) ∇t∇tBt = −R(Bt, γt)γt = −R(t)Bt,


where

R(t) : TγtM → TγtM, R(t)[v] := R(v, γt)γt

is symmetric; indeed, in the orthonormal basis (ei(t))i=1,...,n, it is representedby the symmetric matrix(

g(ei(t), ej(t)) g(R(ei(t), γt)γt, ej(t)))i,j=1,...,n

.

Plugging (3.30) into (3.29), we obtain that

(∇tB∗t )Bt − B∗t (∇tBt)is constant in t. But B1/2 = IdTγ1/2

M and∇tBt|t=1/2 = −∇∇qgφ is symmetric

by assertion (1) in Lemma 3.3. Taking into account (3.28), we conclude thatUt is symmetric for every t ∈ [0, 1].Using that Ut is symmetric, by Cauchy-Schwartz inequality, we have that

(3.31) Trg[U2t

]= tr

[(U2)ij

]≥ 1

n

(tr[Uij])2

=1

n

(Trg[Uij])2

.

The desired estimate (3.26) then follows from the combination of (3.25),(3.27) and (3.31). �

4. Optimal transport formulation of the Einstein equations

The Einstein equations of General Relativity for an n-dimensional space-time (Mn, g, C), n ≥ 3, read as

(4.1) Ric− 1

2Scal g + Λg = 8πT,

where Scal is the scalar curvature, Λ ∈ R is the cosmological constant, andT is the energy-momentum tensor.

Lemma 4.1. The space-time (Mn, g, C), n ≥ 3, satisfies the Einstein Equa-tion (4.1) if and only if

(4.2) Ric =2Λ

n− 2g + 8πT − 8π

n− 2Trg(T ) g.

Proof. Taking the trace of (4.1), one can express the scalar curvature as

(4.3) Scal =2nΛ

n− 2− 16π

n− 2Trg(T ).

Plugging (4.3) into 4.1 gives the equivalent formulation (4.2) of Einsteinequations just in terms of the metric, the Ricci and the energy-momentumtensors. �

The optimal transport formulation of the Einstein equations will consistseparately of an optimal transport characterization of the two inequalities

Ric ≥ T and Ric ≤ T ,where

(4.4) T :=2Λ

n− 2g + 8πT − 8π

n− 2Trg(T ) g.

Subsection 4.1 will be devoted to the lower bound and Subsection 4.2 tothe upper bound on the Ricci tensor.


A key role in such an optimal transport formulation will be played by the(relative) Boltzmann-Shannon entropy defined below. Denote by volg thestandard volume measure on (M, g). Given an absolutely continuous prob-ability measure µ = % volg with density % ∈ Cc(M), define its Boltzmann-Shannon entropy (relative to volg) as

(4.5) Ent(µ|volg) :=

ˆM% log % dvolg.

4.1. OT-characterization of Ric ≥ T . In establishing the Ricci curvaturelower bounds, the next elementary lemma will be key (for the proof see forinstance [81, Chapter 16] or [6, Lemma 9.1])

Lemma 4.2. Define the function G : [0, 1]× [0, 1]→ [0, 1] by

(4.6) G(s, t) :=

{(1− t)s if s ∈ [0, t],

t(1− s) if s ∈ [t, 1],

so that for all t ∈ (0, 1) one has

(4.7) − ∂2

∂s2G(s, t) = δt in D ′(0, 1), G(0, t) = G(1, t) = 0.

If u ∈ C([0, 1],R) satisfies u′′ ≥ f in D ′(0, 1) for some f ∈ L1(0, 1) then

u(t) ≤ (1− t)u(0) + tu(1)−ˆ 1

0G(s, t) f(s) ds, ∀t ∈ [0, 1].(4.8)

In particular, if f ≡ c ∈ R then

u(t) ≤ (1− t)u(0) + tu(1)− ct(1− t)2

, ∀t ∈ [0, 1].(4.9)

The characterization of Ricci curvature lower bounds (i.e. Ric ≥ Kg forsome constant K ∈ R) via displacement convexity of the entropy is by nowclassical in the Riemannian setting, let us briefly recall the key contributions.Otto & Villani [69] gave a nice heuristic argument for the implication “Ric ≥Kg ⇒K-convexity of the entropy”; this implication was proved forK = 0 byCordero-Erausquin, McCann & Schmuckenschlager [27]; the equivalence foreveryK ∈ R was then established by Sturm & von Renesse [77]. Our optimaltransport characterization of Ric ≥ 2Λ

n−2g + 8πT − 8πn−2Trg(T ) g is inspired

by such fundamental papers (compare also with [51] for the implication(3)⇒(1)). Let us also mention that the characterization of Ric ≥ Kg forK ≥ 0 via displacement convexity in the globally hyperbolic Lorentziansetting has recently been obtained independently by Mc Cann [62]. Notethat Corollary 4.4 extends such a result to any lower bounds K ∈ R and tothe case of general (possibly non globally hyperbolic) space times.

The next general result will be applied with n ≥ 3 and T be as in (4.4).

Theorem 4.3 (OT-characterization of Ric ≥ T ). Let (M, g, C) be a space-

time of dimension n ≥ 2 and let T be a quadratic form on M . Then thefollowing are equivalent:

(1) Ric(v, v) ≥ T (v, v) for every causal vector v ∈ C.


(2) For every p ∈ (0, 1), for every regular dynamical cp-optimal plan Πit holds

(4.10)

Ent(µt|volg) ≤ (1−t)Ent(µ0|volg)+tEnt(µ1|volg)−ˆ ˆ 1

0G(s, t)T (γs, γs)ds dΠ(γ),

where we denoted µt := (et)]Π, t ∈ [0, 1], the curve of probabilitymeasures associated to Π.

(3) There exists p ∈ (0, 1) such that for every regular dynamical cp-optimal plan Π the convexity property (4.10) holds.

Proof. (1)⇒ (2)Fix p ∈ (0, 1). Let Π be a regular dynamical cp-optimal coupling and let(µt := (et)]Π)t∈[0,1] be the corresponding curve of probability measures withµt = ρt volg � volg compactly supported. By definition of regular dynamicalcp-optimal coupling there exists a smooth function φ1/2 such that, calling

Ψt1/2(x) = expgx((t− 1/2)∇qgφ1/2(x)),

it holds µt = (Ψt1/2)]µ1/2 for every t ∈ [0, 1]. Moreover the Jacobian DΨt

1/2

is non-singular for every t ∈ [0, 1] on supp(µ1/2). Recall the definition of Btand Ut along the geodesic t 7→ γt := Ψt

1/2(x).

Bt(x) := DΨt1/2(x) : TxM → TΨt

1/2(x)M, for µ1/2-a.e. x,

Ut(x) := ∇tBt ◦ B−1t : TγtM → TγtM, for µ1/2-a.e. x.

Calling yx(t) := log DetgBt(x) and γxt := Ψt1/2(x), from Proposition 3.4 we

get

(4.11) y′′x(t) + T (γxt , γxt ) ≤ y′′x(t) + Ric(γxt , γ

xt ) ≤ 0, µ1/2-a.e. x.

Now, for t ∈ [0, 1] we have

Ent(µt|volg) =

ˆlog ρt(y) dµt(y) =

ˆM

log ρt(Ψt1/2(x)) dµ1/2(x)

=

ˆlog[ρ1/2(x)(Detg(DΨt

1/2)(x))−1] dµ1/2(x)

= Ent(µ1/2|volg)−ˆyx(t) dµ1/2(x),(4.12)

where the second to last equality follows from Lemma 3.3(3). Using (4.11)we obtain(4.13)d2

dt2Ent(µt|volg) ≥

ˆT (γxt , γ

xt )dµ1/2(x) =

ˆT (γt, γt)dΠ(γ), ∀t ∈ (0, 1).

Using Lemma 4.2, we get (4.10).(2)⇒ (3): trivial.(3)⇒ (1)

We argue by contradiction. Assume there exist x0 ∈ M and v ∈ Tx0M ∩ Cwith g(v, v) < 0 such that the Ricci curvature at x0 in the direction of v ∈ Cxsatisfies

(4.14) Ric(v, v) ≤ (T + 3εg)(v, v),


for some ε > 0. Thanks to Lemma 3.2, for η ∈ (0, η(x0, v)] small enough,there exists δ > 0 and a cp-convex function φ1/2, smooth on Bδ(x0) andsatisfying

(4.15) ∇qgφ1/2(x0) = ηv 6= 0 and Hessφ1/2(x0) = 0.

From now on we fix η ∈(0,min(η(x0, v), injg(Bδ(x0)))

], where injg(Bδ(x0))

is the injectivity radius of Bδ(x0) with respect to the metric g. It is easilychecked that, for δ ∈ (0, δ), small enough the map

x 7→ Ψt1/2(x) = expgx((t− 1/2)∇qgφ1/2(x))

is a diffeomorphism from Bδ(x0) onto its image for any t ∈ [0, 1]. Moreover,since ∇qgφ1/2(x0) ∈ Int(C) and arguing by continuity and by parallel trans-

port along the geodesics t 7→ Ψt1/2(x), x ∈ Bδ(x0), for δ > 0 small enough

we have that

(4.16)⋃

t∈[0,1]

⋃x∈Bδ(x0)

d

dtΨt

1/2(x) ⊂⊂ Int(C).

Define µ 12

:= volg(Bδ(x0))−1 volgxBδ(x0). Let Π be the cp-optimal dynami-

cal plan representing the curve of probability measures(µt := (Ψt

1/2)]µ 12

)t∈[0,1]

.

Note that (4.16) together with (4.15) ensures that Π is regular, for δ > 0small enough.Calling γxt := Ψt

1/2(x) = expgx((t − 1/2)∇qgφ1/2(x)

)for x ∈ Bδ(x0) the ge-

odesic performing the transport, note that by continuity there exists δ > 0small enough such that

(4.17) Ric(γxt , γxt ) < (T + 2εg)(γxt , γ

xt ), ∀x ∈ Bδ(x0), ∀t ∈ [0, 1] .

The identity (3.25) proved in Proposition 3.4 reads as(4.18)

[Trg(Uxt )]′ + Trg[(Uxt )2] + Ric(γxt , γxt ) = 0, ∀x ∈ Bδ(x0), ∀t ∈ [0, 1].

Since by construction Ux0

1/2 := ∇tBx0

1/2(Bx0

1/2)−1 = ∇∇qgφ1/2(x0) = 0 and

g(∇qgφ1/2(x0),∇qgφ1/2(x0)) < 0, again by continuity we can choose δ > 0even smaller so that

Trg[(Uxt )2] < −εg(γxt , γxt ), ∀x ∈ Bδ(x0), ∀t ∈ [0, 1].(4.19)

The combination of (4.17), (4.18) and (4.19) yields

[Trg(Uxt )]′ + (T + εg)(γxt , γxt ) > 0, ∀x ∈ Bδ(x0), ∀t ∈ [0, 1] .

Recalling (3.27), the last inequality can be rewritten as

yx(t)′′ + (T + εg)(γxt , γxt ) > 0, ∀x ∈ Bδ(x0), ∀t ∈ [0, 1] .

The combination of the last inequality with (4.12) gives

(4.20)d2

dt2Ent(µt|volg) <

ˆ(T + εg)(γxt , γ

xt ) dµ1/2(x), ∀t ∈ (0, 1).


By applying Lemma 4.2 we get that

Ent(µt|volg) ≥ (1− t) Ent(µ0|volg) + tEnt(µ1|volg)

−ˆ ˆ 1

0G(s, t)(T + εg)(γs, γs)ds dΠ(γ),

= (1− t) Ent(µ0|volg) + tEnt(µ1|volg)−ˆ ˆ 1

0G(s, t)T (γs, γs)ds dΠ(γ)

− εt(1− t)2

ˆg(γ, γ)dΠ(γ),

where, in the equality we used that for every fixed x ∈ Bδ(x0) the functiont 7→ g(γxt , γ

xt ) is constant (as t 7→ γxt is by construction a g-geodesic).

This clearly contradicts (4.10), as´g(γ, γ)dΠ(γ) < 0. �

In the vacuum case, i.e. T ≡ 0, the inequality Ric ≥ T with T as in(4.4) reads as Ric ≥ Kg with K = 2Λ

n−2 ∈ R. Note that for v ∈ C it

holds g(v, v) ≤ 0 so, when comparing the next result with its Riemanniancounterparts [69, 27, 77], the sign of the lower bound K is reversed.

Corollary 4.4 (The vacuum case T ≡ 0). Let (M, g, C) be a space-time ofdimension n ≥ 2 and let K ∈ R. Then the following are equivalent:

(1) Ric(v, v) ≥ Kg(v, v) for every causal vector v ∈ C.(2) For every p ∈ (0, 1), for every regular dynamical cp-optimal plan Π

it holds

Ent(µt|volg) ≤ (1− t)Ent(µ0|volg) + tEnt(µ1|volg)−Kt(1− t)

2

ˆg(γ, γ)dΠ(γ)

= (1− t)Ent(µ0|volg) + tEnt(µ1|volg)−Kt(1− t)ˆA2(γ)dΠ(γ),(4.21)

where we denoted µi := (ei)]Π, i = 0, 1, the endpoints of the curveof probability measures associated to Π.

(3) There exists p ∈ (0, 1) such that for every regular dynamical cp-optimal plan Π the convexity property (4.21) holds.

Remark 4.5 (The strong energy condition). The strong energy conditionasserts that, called T the energy-momentum tensor, it holds T (v, v) ≥12Trg(T ) for every time-like vector v ∈ TM satisfying g(v, v) = −1. As-suming that the space-time (M, g, C) satisfies the Einstein equations (4.1)with zero cosmological constant Λ = 0, the strong energy condition is equiv-alent to Ric(v, v) ≥ 0 for every time-like vector v ∈ TM . This correspondsto the case K = 0 in Corollary 4.4.

The strong energy condition, proposed by Hawking and Penrose [71, 45,46], plays a key role in general relativity. For instance, in the presence oftrapped surfaces, it implies that the space-time has singularities (e.g. blackholes) [31, 83].

4.2. OT-characterization of Ric ≤ T . The goal of the present section isto provide an optimal transport formulation of upper bounds on time-likeRicci curvature in the Lorentzian setting. More precisely, given a quadraticform T (which will later be chosen to be equal to the right hand side of


Einstein equations, i.e. as in (4.4)), we aim to find an optimal transportformulation of the condition

“Ric(v, v) ≤ T (v, v) for every time-like vector v ∈ C”.The Riemannian counterpart, in the special case of Ric ≤ Kg for someconstant K ∈ R, has been recently established by Sturm [76].

In order to state the result, let us fix some notation. Given a relativelycompact open subset E ⊂⊂ Int(C) let pTM→M : TM →M be the canonicalprojection map and injg(E) > 0 be the injectivity radius of the exponentialmap of g restricted to E. For x ∈ pTM→M (E) and r ∈ (0, injg(E)) we denote

Bg,Er (x) := {expgx(tw) : w ∈ TxM ∩ E, g(w,w) = −1, t ∈ [0, r]}.

Definition 4.6 (r-concentrated regular cp-optimal dynamical plan). Fix arelatively compact open subset E ⊂⊂ Int(C), x ∈ pTM→M (E), v ∈ TxM∩E,and r > 0. A regular cp-optimal dynamical plan Π is called r-concentratedin the direction of v (with respect to E) if µt := (et)]Π for t ∈ [0, 1] satisfies:

(1) µ1/2 = volg(Bg,Er4 (x))−1 volgxB

g,Er4 (x);

(2) supp(µ1) ⊂ {expgy(r2w) : w ∈ TyM ∩ C, g(w,w) = −1}, where y :=expgx(rv).

The next general result will be applied with n ≥ 3 and T as in (4.4).

Theorem 4.7 (OT-characterization of Ric ≤ T ). Let (M, g, C) be a space-

time of dimension n ≥ 2 and let T be a quadratic form on M . Then thefollowing assertions are equivalent:

(1) Ric(v, v) ≤ T (v, v) for every causal vector v ∈ C.(2) For every p ∈ (0, 1) and for every relatively compact open subset


ε = εE : (0,∞)→ (0,∞) with limr↓0

ε(r) = 0 such that


the next assertion holds. For every r ∈ (0, R), there exists an r-concentrated regular cp-optimal dynamical plan Π = Π(x, v, r) in the

direction of v (with respect to E) which has T (v, v)-concave entropyin the sense that

(4.22)4

r2


]≤ T (v, v)+ε(r).

(3) There exists p ∈ (0, 1) such that the analogous assertion as in (2)holds true.

Remark 4.8. Given an auxiliary Riemannian metric h on M , with analo-gous arguments as in the proof below, the condition (2) in Theorem 4.7 canbe replaced by

(2)’ For every p ∈ (0, 1) and for every x ∈ M there exist R = R(x) >0 and a function ε = εx : (0,∞) → (0,∞) with limr↓0 ε(r) = 0such that for all v ∈ Int(Cx) with h(v, v) ≤ R2 the next assertionholds. For every r ∈ (0, R), there exists an r-concentrated regularcp-optimal dynamical plan Π = Π(x, v, r) in the direction of v (withrespect to E) satisfying (4.22).


Moreover, both in (2) and (2’) one can replace Bg,Er4 (x) (resp. {expgy(r2w) :

w ∈ TyM ∩ C, g(w,w) = −1}) by Bhr4(x) (resp. Bh

r2(y)).

Proof. (1)⇒ (2)Let (M, g, C) be a space time and let h be an auxiliary Riemannian metricon M such that

1

4h(w,w) ≤ |g(w,w)| ≤ 4h(w,w), ∀w ∈ E.

We denote with dTMh the distance on TM induced by the auxiliary Rie-mannian metric h. Once the compact subset E ⊂⊂ Int(C) is fixed, thanksto Lemma 3.2 there exist a constant

R = R(E) ∈ (0,min(1, injg(E)))

and a function

ε = εE : (0,∞)→ (0,∞) with limr↓0

ε(r) = 0

such that

∀r ∈ (0, R/10), x ∈ pTM→M (E), v ∈ TxM ∩ E with g(v, v) = −R2

we can find a cp-convex function φ : M → R with the following properties:

(1) φ is smooth on Bh100r(x), ∇qgφ(x) = v, ∇qgφ ∈ E on Bh

10r(x),dTMh (∇qgφ, v) ≤ ε(r) on Bh

10r(x);

(2) |Hessφ|h ≤ ε(r) on Bh10r(x).

For t ∈ [0, 1], consider the map

Ψt1/2 : z 7→ expz(r(t− 1/2)∇qgφ(z)).

Notice thatΨt

1/2(Bg,Er4 (x)) ⊂ Bh

10r(x), ∀t ∈ [0, 1].

Letµ1/2 = volg(B


g,Er4 (x)

and defineµt := (Ψt

1/2)](µ1/2) ∀t ∈ [0, 1].

By the properties of φ, the plan Π representing the curve of probabilitymeasures (µt)t∈[0,1] is a regular cp-optimal dynamical plan and

supp(µ1) ⊂ {expgy(r2w) : w ∈ TyM ∩ C, g(w,w) = −1}.

By Proposition 3.1 we can find a smooth family of functions (φt)t∈[0,1] definedon⋃t∈[0,1]{t} × supp(µt) with φ1/2 = φ satisfying

(∂tφt)(γt) +r

q(−g(∇gφt(γt),∇gφt(γt)))q/2 = 0 for Π-a.e. γ, for all t ∈ [0, 1],

(4.23)

r∇qgφt(γt)− γt = 0 for Π-a.e. γ, for all t ∈ [0, 1].(4.24)

Moreover, using the properties of φ1/2 = φ and the smoothness of the family(φt)t∈[0,1], we have(4.25)

dTMh (∇qgφt(γt), v) ≤ ε(r), |Hessφt(γt)|h ≤ ε(r), Π-a.e. γ, for all t ∈ [0, 1],


up to renaming ε(r) with a suitable function

ε = εE : (0,∞)→ (0,∞)

withlimr↓0

ε(r) = 0.

The curve [0, 1] 3 t 7→ Ent(µt|volg) ∈ R is smooth and, in virtue of (4.24),it satisfies(4.26)d

dtEnt(µt|volg) = r

ˆMg(∇qgφt,∇gρt)dvolg = r

ˆM

2qgφt dµt, for all t ∈ [0, 1],

where

ρt :=dµtdvolg

is the density of µt,2qgφt := div(−∇qgφt)

is the q-Box of φt (the Lorentzian analog of the q-Laplacian), and where weused the continuity equation

d

dtρt + r div(ρt∇qgφt) = 0.

For what follows it is useful to consider the linearization of the q-Box ata smooth function f , denoted by Lqf and defined by the following relation:

(4.27)d

dt

∣∣∣∣t=0

2qg(f + tu) = Lqfu, ∀u ∈ C∞c (M).

The map [0, 1] 3 t 7→´M 2

qgφt dµt ∈ R is smooth and, in virtue of (4.23)

and (4.24), it satisfies

d

dtr

ˆM

2qgφt dµt = −r2

ˆMLqφ

(1

q(−g(∇gφt,∇gφt))q/2

)+g(∇g2qgφt,−∇qgφt) dµt,

for every t ∈ [0, 1]. Using the q-Bochner identity (A.2) together with the

assumption Ric(w,w) ≤ T (w,w) for any w ∈ C and the estimates (4.25) onφt, we can rewrite the last formula as

d

dt

1

r

ˆM

2qgφt dµt =

ˆM|g(∇gφt,∇gφt)|q−2

[Ric(∇gφt,∇gφt) + g(Hessφt ,Hessφt)

+

((q − 2)

Hessφt(∇gφt,∇gφt)|g(∇gφt,∇gφt)|

)2

−2(q − 2)Hessφt

((∇gφt,Hessφt(∇gφt)

)|g(∇gφt,∇gφt)|

]dµt

≤ˆM

(T (v, v) + ε(r)

)dµt = T (v, v) + ε(r) for all t ∈ [0, 1],

up to renaming ε(r) with a suitable function

ε = εE : (0,∞)→ (0,∞) with limr↓0

ε(r) = 0.


Thus

Ent(µ1|volg)− 2Ent(µ1/2|volg) + Ent(µ0|volg)

=

ˆ 1/2

0

(d

dtEnt(µt|volg)|t=s+1/2 −

d

dtEnt(µt|volg)|t=s

)ds

=

ˆ 1/2

0

ˆ s+1/2

s

d2

dt2Ent(µt|volg)dt ds

≤ T (v, v) + ε(r)

4r2.

(2)⇒ (3): trivial.

(3)⇒ (1)Fix p ∈ (0, 1) given by (3) and assume by contradiction that there existsx ∈M , ε > 0 and v ∈ TxM ∩ C with −g(v, v) = 1 such that

Ric(v, v) ≥ (T − 2εg)(v, v).

Then, by continuity, we can find a relatively compact neighbourhood E ⊂⊂Int(C) of v in TM such that

(4.28) Ric(w,w) ≥ (T − εg)(w,w), ∀w ∈ E.By Lemma 3.2 we can construct a cp-convex function φ : M → R such thatφ is smooth on a neighbourhood of x and

∇qgφ(x) = Rv.

For t ∈ [0, 1], define

Ψt1/2(z) := expgz(2r(t− 1/2)∇qgφ(z)).

By continuity, for r ∈ (0, R) small enough, we have that(4.29)1

r

d

dtΨt

1/2(z) ∈ E, dTMh

(d

dtΨt

1/2(z), rv

)≤ εr, ∀z ∈ Bg,E

r4 (x), ∀t ∈ [0, 1].

Moreover Ψ11/2(Bg,E

r4 (x)) ⊂ Bg,Er2 (y).

Set µ1/2 := volg(Bg,Er4 (x))−1 volgxB

g,Er4 (x) and consider µt := (Ψt

1/2)]µ1/2.

Notice that

supp(µ1) ⊂ Bg,Er2 (y) ⊂ {expgy(r

2w) : w ∈ TyM ∩ C, g(w,w) = −1}.By the above construction, we get that (µt)t∈[0,1] can be represented by aregular cp-optimal dynamical plan Π such that

supp((∂e)]Π) ⊂ E.Therefore (4.28) together with Theorem 4.3 yields

Ent(µ1/2|volg) ≤1

2Ent(µ0|volg) +

1

2Ent(µ1|volg)

−ˆ ˆ 1

0G(s, 1/2)(T − εg)(γs, γs)ds dΠ(γ),

≤ 1

2Ent(µ0|volg) +

1

2Ent(µ1|volg)−

r2(T (v, v)− εg(v, v) + Cεr)

8,(4.30)


where in the second inequality we used (4.29) and that C > 0 is a constantindependent of r and ε. Note that ε > 0 in (4.30) is fixed independently ofr > 0. Clearly (4.30) contradicts the existence of εE(r) → 0 as r → 0 sothat (4.22) holds. �

In the vacuum case when T ≡ 0, the inequality Ric ≤ T with T as in(4.4) reads as

Ric ≤ Kg with K =2Λ

n− 2∈ R.

Note that for v ∈ C it holds g(v, v) ≤ 0 so, when comparing the next resultwith its Riemannian counterpart [76], the sign of the lower bound K isreversed.

Corollary 4.9. Let (M, g, C) be a space-time of dimension n ≥ 2 and letK ∈ R. Then the following assertions are equivalent:

(1) Ric(v, v) ≤ Kg(v, v) for every causal vector v ∈ C.(2) For every p ∈ (0, 1) and for every relatively compact open subset


ε = εE : (0,∞)→ (0,∞) with limr↓0

ε(r) = 0 such that


the next assertion holds. For every r ∈ (0, R), there exists an r-concentrated regular cp-optimal dynamical plan Π = Π(x, v, r) in thedirection of v (with respect to E) which has −K-concave entropy inthe sense that

(4.31)4

r2


]≤ −K + ε(r).


4.3. Optimal transport formulation of the Einstein equations. Re-call that Einstein equations of general relativity, with cosmological constantequal to Λ ∈ R and energy-momentum tensor T , read as

(4.32) Ric =2Λ

n− 2g + 8πT − 8π

n− 2Trg(T ) g,

for an n-dimensional space-time (M, g, C). Combining Theorem 4.3 withTheorem 4.7, both with the choice

T =2Λ

n− 2g + 8πT − 8π

n− 2Trg(T ) g,

we obtain the following optimal transport formulation of (4.32).

Theorem 4.10. Let (M, g, C) be a space-time of dimension n ≥ 2 and let

T be a quadratic form on M . Then the following assertions are equivalent:

(1) Ric(v, v) = T (v, v) for every v ∈ TxM(2) Ric(v, v) = T (v, v) for every causal vector v ∈ C.



ε = εE : (0,∞)→ (0,∞) with limr↓0

ε(r) = 0 such that


the next assertion holds. For every r ∈ (0, R), there exists an r-concentrated regular cp-optimal dynamical plan Π = Π(x, v, r) in thedirection of v (with respect to E) satisfying

(4.33)

T (v, v)−ε(r) ≤ 4

r2


]≤ T (v, v)+ε(r).


Remark 4.11 (µ1/2 can be chosen more general). From the proof of Theo-rem 4.10 it follows that one can replace (2) (and analogously (3)) with thefollowing (a priori stronger, but a fortiori equivalent) statement. For everyp ∈ (0, 1) the following holds. For every relatively compact open subsetE ⊂⊂ Int(C) there exist R = R(E) ∈ (0, 1) and a function

ε = εE : (0,∞)→ (0,∞) with limr↓0

ε(r) = 0

such that

∀x ∈ pTM→M (E),∀v ∈ TxM ∩ E with g(v, v) = −R2

the next assertion holds. For every r ∈ (0, R) and every

µ1/2 ∈ P(M) with µ1/2 � volg and supp(µ1/2) ⊂ Bg,Er4 (x),

setting y = expgx(rv), there exists a regular cp-optimal dynamical plan Π =Π(µ1/2, v, r) with associated curve of probability measures

(µt := (et)]Π)t∈[0,1] ⊂ P(M)

such that

supp(µ1) ⊂ {expgy(r2w) : w ∈ TyM ∩ C, g(w,w) = −1}

and satisfying (4.33).

Remark 4.12 (An equivalent statement via an auxiliary Riemannian met-ric h). Given an auxiliary Riemannian metric h on M , with analogous ar-guments as in the proof below, the condition (3) in Theorem 4.10 can bereplaced by

(3)’ For every p ∈ (0, 1) the following holds. For every x ∈M there existR = R(x) > 0 and a function

ε = εx : (0,∞)→ (0,∞) with limr↓0

ε(r) = 0 such that

∀v ∈ Int(Cx) with h(v, v) ≤ R2 the next assertion holds.

For every r ∈ (0, R), there exists an r-concentrated regular cp-optimal dynamical plan Π = Π(x, v, r) in the direction of v satisfying(4.33).


Moreover, both in (3) and (3’) one can replace Bg,Er4 (x) by Bh

r4(x) and

{expgy(r2w) : w ∈ TyM ∩ C, g(w,w) = −1}

by Bhr2(y).

Remark 4.13 (The tensor T ). As mentioned above, we will assume thecosmological constant Λ and the energy momentum tensor T to be given,say from physics and/or mathematical general relativity. Given g,Λ and T ,

for convenience of notation we set T to be defined in (1.6).

Let us stress that not any symmetric bilinear form T would correspond toa physically meaningful situation; in order to be physically relevant, it iscrucial that T is given by (1.6) where T is a physical energy-momentumtensor (in particular T has to satisfy ∇aTab = 0, i.e. be “freely gravitating”,it has to satisfy some suitable energy condition like the “dominant energycondition”, etc.).

Proof of Theorem 4.10. (1)⇒ (2): trivial.

(2)⇒ (1): Follows by the identity theorem for polynomials and the factthat C has non-empty open interior.

(2)⇒ (3): From the implication (1)⇒ (2) in Theorem 4.7, we get aregular cp-optimal dynamical plan Π = Π(x, v, r) as in (3) such that theupper bound in (4.33) holds. Moreover, from (4.25) it holds that

(4.34) dTMh (γt, rv) ≤ rε(r), for Π-a.e. γ, for all t ∈ [0, 1].

Recalling that the implication (1)⇒ (2) in Theorem 4.3 gives the convex-ity property (4.10) of the entropy along every regular cp-optimal dynamicalplan, and using (4.34), we conclude that also the lower bound in (4.33) holds.

(3)⇒ (4): trivial.

(4)⇒ (2).The fact that

Ric(v, v) ≤ T (v, v) ∀v ∈ Cfollows directly from Theorem 4.7. The fact that

Ric(v, v) ≥ T (v, v) ∀v ∈ Ccan be showed following arguments already used in the paper, let us brieflydiscuss it. Fix p ∈ (0, 1) given by (4) and assume by contradiction that

∃x ∈M, δ > 0 and v ∈ TxM ∩ C with − g(v, v) > 0

such thatRic(v, v) ≤ (T + 3δg)(v, v).

Thanks to Lemma 3.2, up to replacing v with sv for some s ∈ (0, 1) smallenough, we know that we can construct a cp-convex function φ : M → R,smooth in a neighbourhood of x and satisfying

∇qgφ(x) = v Hessφ(x) = 0.

Then, by continuity, we can find


• a relatively compact open neighbourhood E ⊂⊂ Int(C) of v in TMsuch that

(4.35) Ric(w,w) ≤ (T + 2δg)(w,w), ∀w ∈ E;

• φ is smooth on Bh100r(x),

∇qgφ ∈ E on Bh10r(x),

dTMh (∇qgφ, v) ≤ ε(r) on Bh10r(x);

• |Hessφ|h ≤ ε(r) on Bh10r(x);

where

ε(r)→ 0 as r → 0.

For t ∈ [0, 1], consider the map

Ψt1/2 : z 7→ expz(r(t− 1/2)∇qgφ(z)).

Notice that

Ψt1/2(Bg,E

r4 (x)) ⊂ Bh10r(x) ∀t ∈ [0, 1].

Call

µ1/2 = volg(Bg,Er4 (x))−1 volgxB

g,Er4 (x)

and define

µt := (Ψt1/2)](µ1/2) for t ∈ [0, 1].

By the properties of φ, the plan Π representing the curve of probabilitymeasures (µt)t∈[0,1] is a regular cp-optimal dynamical plan and

supp(µ1) ⊂ {expgy(r2w) : w ∈ TyM ∩ C, g(w,w) = −1}.

Moreover

(4.36)1

r

d

dtΨt

1/2(z) ∈ E ∀z ∈ Bhr (x), ∀t ∈ [0, 1].

We can now follow verbatim the arguments in (1)⇒(2) of Theorem 4.7 byusing (4.35) and (4.36), obtaining a function ε(r)→ 0 as r → 0 such that

Ent(µ1|volg)− 2Ent(µ1/2|volg) + Ent(µ0|volg) ≤(T + δg)(v, v) + ε(r)

4r2.

The last inequality clearly contradicts the lower bound in (4.33).�

In the vacuum case T ≡ 0 with cosmological constant Λ ∈ R, the Einsteinequations read as

(4.37) Ric ≡ Λn2 − 1

g,

for an n-dimensional space-time (M, g, C). Specializing Theorem 4.10 withthe choice

T =Λ

n2 − 1

g

and using Corollary 4.4 to sharpen the lower bound in (4.38) for the con-stant case, we obtain the following optimal transport formulation of Einsteinvacuum equations.


Theorem 4.14. Let (M, g, C) be a space-time of dimension n ≥ 3 and letΛ ∈ R. Then the following assertions are equivalent:

(1) The space-time (M, g, C) satisfies Einstein vacuum equations of Gen-eral Relativity (4.37) corresponding to cosmological constant equal toΛ.


ε = εE : (0,∞)→ (0,∞) with limr↓0

ε(r) = 0 such that



(4.38)

− Λn2 − 1

≤ 4

r2


]≤ − Λ

n2 − 1

+ε(r).


It is worth to isolate the case of zero cosmological constant.

Corollary 4.15. Let (M, g, C) be a space-time of dimension n ≥ 2. Thenthe following assertions are equivalent:

(1) The space-time (M, g, C) satisfies Einstein vacuum equations of Gen-eral Relativity with zero cosmological constant, i.e. Ric ≡ 0.


ε = εE : (0,∞)→ (0,∞) with limr↓0

ε(r)/r2 = 0 such that



(4.39) 0 ≤ Ent(µ1|volg)− 2Ent(µ1/2|volg) + Ent(µ0|volg) ≤ ε(r).(3) There exists p ∈ (0, 1) such that the analogous assertion as in (2)

holds true.

Appendix A. A q-Bochner identity in Lorentzian setting

In this section we prove a Bochner type identity in Lorentzian settingfor the linearization of the q-Box operator, the Lorentzian analog of theq-Laplacian; let us mention that related results have been obtained in theRiemannian [59, 79] and Finsler settings [84, 85] but at best our knowledgethis section is original in the Lorentzian Lp framework.

Throughout the section, (M, g, C) is a space-time, U ⊂ M is an opensubset and q ∈ (−∞, 0) is fixed. Let φ ∈ C3(U) satisfy

−∇gφ ∈ Int(C) on U.


Denote by

∇qgφ := −|g(∇gφ,∇gφ)|q−2

2 ∇gφthe q-gradient, by

2qgφ := div(−∇qgφ)

the q-Box operator of φ and by Lqφ the linearization of the q-Box operator

at φ defined by the following relation:

(A.1)d

dt

∣∣∣∣t=0

2qg(φ+ tu) = Lqφu, ∀u ∈ C∞c (M).

The ultimate goal of the section is to prove the following result.

Proposition A.1. Under the above notation, the following q-Bochner iden-tity holds:

−Lqφ

((−g(∇gφ,∇gφ))

q2

q

)= g(∇g2qgφ,−∇qgφ)

+(q − 2)2|g(∇gφ,∇gφ)|q−2

(Hessφ(∇gφ,∇gφ)

|g(∇gφ,∇gφ)|

)2

+|g(∇gφ,∇gφ)|q−2(Ric(∇gφ,∇gφ) + g(Hessφ,Hessφ)

)−2(q − 2) |g(∇gφ,∇gφ)|q−3Hessφ (∇gφ,Hessφ(∇gφ)) .(A.2)

The proof of Proposition A.1 requires some preliminary lemmas. First ofall we derive an explicit expression for the operator Lqφ.

Lemma A.2. Under the above notation, it holds

Lqφu =(− g(∇gφ,∇gφ)

) q−22

(2gu− (q − 2)

Hessu(∇gφ,∇gφ)

−g(∇gφ,∇gφ)

)+ (2− q)

(− g(∇gφ,∇gφ)

)−1g(∇gφ,∇gu)2qgφ

+ 2(2− q)(− g(∇gφ,∇gφ)

) q−42 Hessφ

(∇gφ,∇gu+

g(∇gφ,∇gu)

−g(∇gφ,∇gφ)∇gφ

).

(A.3)

Proof. By the very definitions of Lqφu and 2qgφ, we have

Lqφu = div

(d

dt

∣∣∣∣t=0

(−g(∇g(φ+ tu),∇g(φ+ tu)))q−2

2 ∇g(φ+ tu)

)= div

((2− q)

(− g(∇gφ,∇gφ)

) q−42 g(∇gφ,∇gu)∇gφ

+(− g(∇gφ,∇gφ)

) q−22 ∇gu

).(A.4)

In order to explicit the last formula, compute

∇g(− g(∇gφ,∇gφ)

)α= −2α

(− g(∇gφ,∇gφ)

)α−1Hessφ(∇gφ)(A.5)

∇g(g(∇gφ,∇gu)

)= Hessu(∇gφ) + Hessφ(∇gu).(A.6)

Plugging (A.5) and (A.6) into (A.4) gives (A.3). �


We next show a q-Bochner identity for the operator Aqφ defined as

(A.7) Aqφ(u) :=(− g(∇gφ,∇gφ)

) q−22

(2gu− (q − 2)

Hessu(∇gφ,∇gφ)

−g(∇gφ,∇gφ)

).

Lemma A.3. Under the above notation, the following identity holds:

− Lqφ

((−g(∇gφ,∇gφ))

q2

q

)+ g(∇g2qgφ,∇qgφ) =

= (q − 2)|g(∇gφ,∇gφ)|q−4

2 2qgφHessφ(∇gφ,∇gφ)

+ |g(∇gφ,∇gφ)|q−2(q(q − 2)


|g(∇gφ,∇gφ)|

)2

+ Ric(∇gφ,∇gφ) + g(Hessφ,Hessφ)).

(A.8)

Proof. We perform the computation at an arbitrary point x0 ∈ U . In or-der to simplify the computations, we consider normal coordinates (xi) in aneighbourhood of x0 with ∂

∂x1 ∈ C. It holds

− 2g(−g(∇gφ,∇gφ))q2

q= gij∂i

((−g(∇gφ,∇gφ))

q−22 gkl∂jkφ∂lφ

)= (−g(∇gφ,∇gφ))

q−22

(− q − 2

−g(∇gφ,∇gφ)gijgmn∂imφ∂nφg

kl∂jkφ∂lφ

+ gijgkl∂ijkφ∂lφ+ gijgkl∂jkφ∂liφ).

(A.9)

Now, from the symmetry of second order derivatives and the very definitionof the the Riemann tensor (2.9), we have

(A.10) ∂ijkφ = ∂ikjφ = g(R(∂xi , ∂xk)∇gφ, ∂xj ) + ∂kijφ.

Thus

gijgkl∂ijkφ∂lφ = g(∇g2gφ,∇gφ) + Ric(∇gφ,∇gφ),

and we can rewrite (A.9) as

−2g(−g(∇gφ,∇gφ))q2

q=(−g(∇gφ,∇gφ))

q−22

((2− q)

g(Hessφ(∇gφ),Hessφ(∇gφ))

−g(∇gφ,∇gφ)

+ g(∇g2gφ,∇gφ) + Ric(∇gφ,∇gφ)

+ g(Hessφ,Hessφ)).(A.11)


We now compute the second part of −Lqφ

((−g(∇gφ,∇gφ))

q2

q

). To this aim

observe that

∇g

((− g(∇gφ,∇gφ)

) q2

q

)= −

(− g(∇gφ,∇gφ)

) q−22 Hessφ(∇gφ)

(A.12)

Hess(−g(∇gφ,∇gφ))

q2

q

(∇gφ,∇gφ) = (q − 2)(− g(∇gφ,∇gφ)

) q−42 [Hessφ(∇gφ,∇gφ)]2

−(− g(∇gφ,∇gφ)

) q−22 g(∇∇gφ∇∇gφ∇gφ,∇gφ).(A.13)

It is useful to express 2qg in terms of 2g:

2qgφ := div((− g(∇gφ,∇gφ)

) q−22 ∇gφ

)=(− g(∇gφ,∇gφ)

) q−22

(2gφ− (q − 2)

Hessφ(∇gφ,∇gφ)

−g(∇gφ,∇gφ)

).(A.14)

Using (A.14), we can write

g(∇g2qgφ,∇gφ) =

= g

(∇g((− g(∇gφ,∇gφ)

) q−22

(2gφ− (q − 2)


−g(∇gφ,∇gφ)

)),∇gφ

)=(− g(∇gφ,∇gφ)

) q−22 g(∇g2gφ,∇gφ)− (q − 2)


−g(∇gφ,∇gφ)2qgφ

− (q − 2)(− g(∇gφ,∇gφ)

) q−42

(g(∇∇gφ∇∇gφ∇gφ,∇gφ)

+ g(Hessφ(∇gφ),Hessφ(∇gφ))

− 2(q − 2)(− g(∇gφ,∇gφ)

) q−62 [Hessφ(∇gφ,∇gφ)]2.

(A.15)

Plugging (A.11) and (A.13) into (A.7), and simplifying using (A.15), givesthe desired (A.8). �

Proof of Proposition A.1 Combining the expression of Lqφu as in (A.3)

with the definition of Aqφu as in (A.7) we can write

Lqφu =Aqφu+ (2− q)(− g(∇gφ,∇gφ)

)−1g(∇gφ,∇gu)2qgφ

+ 2(2− q)(− g(∇gφ,∇gφ)

) q−42 Hessφ

(∇gφ,∇gu+

g(∇gφ,∇gu)

−g(∇gφ,∇gφ)∇gφ

).

(A.16)


Specializing (A.16) with u =(−g(∇gφ,∇gφ))

q2

q and using (A.12) gives

− Lqφ

((−g(∇gφ,∇gφ))

q2

q

)= −Aqφ

((−g(∇gφ,∇gφ))

q2

q

)

− (q − 2)(− g(∇gφ,∇gφ)

) q−22


−g(∇gφ,∇gφ)2qgφ

− 2(q − 2)(− g(∇gφ,∇gφ)

)q−3Hessφ (∇gφ,Hessφ(∇gφ))

− 2(q − 2)(− g(∇gφ,∇gφ)

)q−2(

Hessφ (∇gφ,∇gφ)

−g(∇gφ,∇gφ)

)2

.

(A.17)

Now, the combination of (A.17) and (A.8) yields

− Lqφ

((−g(∇gφ,∇gφ))

q2

q

)= −g(∇g2qgφ,∇qgφ)

+ (q − 2)2|g(∇gφ,∇gφ)|q−2


|g(∇gφ,∇gφ)|

)2

+ |g(∇gφ,∇gφ)|q−2(Ric(∇gφ,∇gφ) + g(Hessφ,Hessφ)

)− 2(q − 2) |g(∇gφ,∇gφ)|q−3Hessφ (∇gφ,Hessφ(∇gφ)) .(A.18)

2

Appendix B. A synthetic formulation of Einstein’s vacuumequations in a non-smooth setting

B.1. The Lorentzian synthetic framework. The goal of this appendixin to give a synthetic formulation of Einstein’s vacuum equations (i.e. zerostress-energy tensor T ≡ 0 but possibly non-zero cosmological constant Λ)and show its stability under a natural adaptation for Lorentzian syntheticspaces of the measured Gromov-Hausdorff convergence (classically designedas a notion of convergence for metric measure spaces).This appendix was written in early 2021, more than two years after the restof the paper was posted in arXiv (in October 2018). One of the reasons isthat we will build on top of the synthetic time-like Ricci curvature lowerbounds in a non-smooth setting developed by the first author in collabora-tion with Cavalletti in [25] (posted on arXiv in 2020).

The general framework is given by Lorentzian pre-length/geodesic spaces.Let us start by recalling some basics of theory, following the approach ofKunziger-Samann [54].A causal space (X,�,≤) is a set X endowed with a preorder ≤ and atransitive relation � contained in ≤.We write x < y if x ≤ y and x 6= y. When x� y (resp. x ≤ y), we say thatx and y are time-like (resp. causally) related.


We define the chronological (resp. causal) future of a subset E ⊂ X as

I+(E) := {y ∈ X : ∃x ∈ E, x� y}J+(E) := {y ∈ X : ∃x ∈ E, x ≤ y}

respectively. Analogously, we define I−(E) (resp. J−(E)) the chronological(resp. causal) past of E. In case E = {x} is a singleton, with a slight abuseof notation, we will write I±(x) (resp. J±(x)) instead of I±({x}) (resp.J±({x})).

Recall that a metric space (X, d) is said to be proper if closed and boundedsubsets are compact.

Definition B.1 (Lorentzian pre-length space (X, d,�,≤, τ)). A Lorentzianpre-length space (X, d,�,≤, τ) is a casual space (X,�,≤) additionallyequipped with a proper metric d and a lower semicontinuous function τ :X ×X → [0,∞], called time-separation function, satisfying

τ(x, y) + τ(y, z) ≤ τ(x, z) ∀x ≤ y ≤ z reverse triangle inequality

τ(x, y) = 0, if x 6≤ y, τ(x, y) > 0⇔ x� y.(B.1)

We will consider X endowed with the metric topology induced by d.We say that X is (resp. locally) causally closed if {x ≤ y} ⊂ X × X is

a closed subset (resp. if every point x ∈ X has neighbourhood U such that{x ≤ y} ∩ U × U is closed in U × U).

Throughout this appendix, I ⊂ R will denote an arbitrary interval. Anon-constant curve γ : I → X is called (future-directed) time-like (resp.causal) if γ is locally Lipschitz continuous (with respect to d) and if for allt1, t2 ∈ I, with t1 < t2, it holds γt1 � γt2 (resp. γt1 ≤ γt2).

The length of a causal curve is defined via the time separation function,in analogy to the theory of length metric spaces: for γ : [a, b] → X future-directed causal we set

Lτ (γ) := inf

{N−1∑i=0

τ(γti , γti+1) : a = t0 < t1 < . . . < tN = b, N ∈ N

}.

A future-directed causal curve γ : [a, b] → X is maximal (also calledgeodesic) if it realises the time separation, i.e. if Lτ (γ) = τ(γa, γb).A Lorentzian pre-length space (X, d,�,≤, τ) is called

• non-totally imprisoning if for every compact set K b X there is con-stant C > 0 such that the d-arc-length of all causal curves containedin K is bounded by C;• globally hyperbolic if it is non-totally imprisoning and for every x, y ∈X the set J+(x) ∩ J−(y) is compact in X;• K-globally hyperbolic if it is non-totally imprisoning and for everyK1,K2 b X compact subsets, the set J+(K1) ∩ J−(K2) is compactin X;• geodesic if for all x, y ∈ X with x ≤ y there is a future-directed

causal curve γ from x to y with τ(x, y) = Lτ (γ), i.e. a (maximizing)geodesic from x to y.


For a globally hyperbolic Lorentzian geodesic (actually length would suffice)space (X, d,�,≤, τ), the time-separation function τ is finite and continu-ous, see [54, Theorem 3.28]. Moreover, any globally hyperbolic Lorentzianlength space (for the definition of Lorenzian length space see [54, Definition3.22], we omit it for brevity since we will not use it) is geodesic [54, Theorem3.30].A measured Lorentzian pre-length space (X, d,m,�,≤, τ) is a Lorentzianpre-length space endowed with a Radon non-negative measure m with suppm =X.

Examples entering the class of Lorentzian synthetic spaces. In thisshort section we briefly recall some notable examples entering the aforemen-tioned framework of Lorentzian synthetic spaces.

Spacetimes with a continuous Lorentzian metric. Let M be a smooth man-ifold endowed with a continuous Lorentzian metric g. Assume that (M, g)is time-oriented, i.e. there is a continuous time-like vector field. Observethat, for C0-metrics, the natural class of differentiability of the underlyingmanifolds is C1; now, C1 manifolds always admit a C∞ subatlas, and onecan pick such sub-atlas whenever convenient.

A causal (respectively time-like) curve in M is by definition a locallyLipschitz curve whose tangent vector is causal (resp. time-like) almost ev-erywhere. One could also start from absolutely continuous (AC for short)curves, but since causal AC curves always admit a Lipschitz re-parametrisation[63, Sec. 2.1, Rem. 2.3], we do not loose in generality with the above con-vention. Set Lg(γ) to be the g-length of a causal curve γ : I →M , i.e.

Lg(γ) :=

ˆI

√−g(γ, γ) dt.

The time separation function τ : M ×M → [0,∞] is then defined in theclassical way, i.e.

τ(x, y) := sup{Lg(γ) : γ is future directed causal from x to y}, if x ≤ y,

and τ(x, y) = 0 otherwise. The reverse triangle inequality (B.1) followsdirectly from the definition. Moreover, every Lg-maximal curve γ is also Lτ -maximal, and Lg(γ) = Lτ (γ) (see for instance [54, Remark 5.1]). In orderto have an underlying metric structure, we also fix a complete Riemannianmetric h on M and denote by dh the associated distance function.

For a spacetime with a Lorentzian C0-metric:

• Global hyperbolicity implies causal closedness and K-global hyper-bolicity [72, Proposition 3.3 and Corollary 3.4].• Recall that a Cauchy hypersurface is a subset which is met exactly

once by every inextendible causal curve. Every Cauchy hypersur-face is a closed acasual topological hypersurface [72, Proposition5.2]. Global hyperbolicity is equivalent to the existence of a Cauchyhypersurface [72, Theorem 5.7, Theorem 5.9].• By [54, Proposition 5.8], if g is a causally plain (or, more strongly,

locally Lipschitz) Lorentzian C0-metric on M then the associatedsynthetic structure is a pre-length Lorentzian space. More strongly,


from [54, Theorem 3.30 and Theorem 5.12], if g is a globally hyper-bolic and causally plain Lorentzian C0-metric on M then the asso-ciated synthetic structure is a causally closed, K-globally hyperbolicLorentzian geodesic space.

Closed cone structures. Closed cone structures provide a rich source of ex-amples of Lorentzian pre-length and length spaces, which can be seen as thesynthetic-Lorentzian analogue of Finsler manifolds (see [54, Section 5.2] formore details). See Minguzzi’s review paper [63] for a comprehensive analy-sis of causality theory in the framework of closed cone structures, includingembedding and singularity theorems.

Some examples towards quantum gravity. The framework of Lorentzian pre-length spaces allows to handle situations where one may not have the struc-ture of a manifold or a Lorentz(-Finsler) metric. A remarkable example ofsuch a situation is given by certain approaches to quantum gravity, see forinstance [60] where it is shown that it is possible to reconstruct a globallyhyperbolic spacetime and the causality relation from a countable dense setof events, in a purely order theoretic manner.Two approaches to quantum gravity, particularly linked to Lorentzian pre-length spaces, are the theory of causal Fermion systems [36, 37] and thetheory of causal sets [13]. The basic idea in both cases is that the structureof spacetime needs to be adjusted on a microscopic scale to include quantumeffects. This leads to non-smoothness of the underlying geometry, and theclassical structure of Lorentzian manifold emerges only in the macroscopicregime. For the connection to the theory of Lorentzian (pre-)length spaceswe refer to [54, Section 5.3], [36, Section 5.1].

B.2. The p-Lorentz-Wasserstein distance on a Lorentzian pre-lengthspace. We denote with P(X) (resp. Pc(X), or Pac(X)) the collection ofall Borel probability measures (resp. with compact support, or absolutelycontinuous with respect to m).Given a probability measure µ ∈ P(X) we define its relative entropy by

Ent(µ|m) =

ˆXρ log(ρ) dm,

if µ = ρm ∈ Pac(X) and (ρ log(ρ))+ is m-integrable. Otherwise we setEnt(µ|m) = +∞. We set Dom(Ent(·|m)) the finiteness domain of Ent(·|m).

We next recall some basics of optimal transport theory in Lorentzian pre-length spaces, following the approach and the notation in [25].Given µ, ν ∈ P(X), denote

Π(µ, ν) := {π ∈ P(X ×X) : (P1)]π = µ, (P2)]π = ν},Π≤(µ, ν) := {π ∈ Π(µ, ν) : π(X2

≤) = 1},Π�(µ, ν) := {π ∈ Π(µ, ν) : π(X2

�) = 1}

where X2≤ := {(x, y) ∈ X2 : x ≤ y} and X2

� := {(x, y) ∈ X2 : x� y}.


Definition B.2 (The p-Lorentz-Wasserstein distance). Let (X, d,�,≤, τ)be a Lorentzian pre-length space and let p ∈ (0, 1]. Given µ, ν ∈ P(X), thep-Lorentz-Wasserstein distance is defined by

(B.2) `p(µ, ν) := supπ∈Π≤(µ,ν)

(ˆX×X

τ(x, y)p dπ(x, y)

)1/p

.

When Π≤(µ, ν) = ∅ we set `p(µ, ν) := −∞.

Note that Definition B.2 extends to Lorentzian pre-length spaces the cor-responding notion given in the smooth Lorentzian setting in [29] (see also[62], and [78] for p = 1); when Π≤(µ, ν) = ∅ we adopt the convention ofMcCann [62] (note that [29] set `p(µ, ν) = 0 in this case).

A key property of `p is the reverse triangle inequality. This was proved inthe smooth Lorentzian setting by Eckstein-Miller [29, Theorem 13] and inthe present synthetic setting in [25, Proposition 2.5]. Such a property is thenatural Lorentzian analogue of the fact that the Kantorovich-Rubinstein-Wasserstein distances Wp, p ≥ 1, in the metric space setting satisfy theusual triangle inequality (see for instance [81, Section 6]).

Proposition B.3 (`p satisfies the reverse triangle inequality). Let (X, d,�,≤, τ) be a Lorentzian pre-length space and let p ∈ (0, 1]. Then `p satisfiesthe reverse triangle inequality:

(B.3) `p(µ0, µ1) + `p(µ1, µ2) ≤ `p(µ0, µ2), ∀µ0, µ1, µ2 ∈ P(X),

where we adopt the convention that ∞−∞ = −∞ to interpret the left handside of (B.3).

A coupling π ∈ Π≤(µ, ν) maximising in (B.2) is called `p-optimal. The

set of `p-optimal couplings from µ to ν is denoted by Πp-opt≤ (µ, ν). We also

set Πp-opt� (µ, ν) := Π�(µ, ν) ∩Πp-opt

≤ (µ, ν).

We say that (µs)s∈[0,1] ⊂ P(X) is an `p-geodesic if and only if

(B.4) `p(µs, µt) = (t− s) `p(µ0, µ1) ∈ (0,∞), for all 0 ≤ s < t ≤ 1.

Note that, with the convention above, `p-geodesics are implicitly future-directed and time-like.

The next lemma follows by the classical theory of optimal transport, sincethe assumptions allow to localise the argument on suppµ0 × suppµ1 ⊂ X2

≤where the optimal transport problem becomes standard.

Lemma B.4. Assume suppµ× supp ν ⊂ X2≤. Then

(1) Every `p-optimal coupling π ∈ Πp-opt≤ (µ, ν) is also τp-cyclically mono-

tone;(2) If π ∈ Π≤(µ, ν) is τp-cyclically monotone then π ∈ Πp-opt

≤ (µ, ν) is`p-optimal.

B.3. Synthetic time-like Ricci upper bounds. Inspired by Sturm’s ap-proach to Riemannian/metric Ricci curvature upper bounds [76] and byTheorem 4.7 (see also Remark 4.8), we propose the following synthetic no-tion of time-like Ricci curvature bounded above. We denote the metric ballin (X, d) with center x and radius r by Bd

r (x).


Definition B.5 (Synthetic time-like Ricci curvature bounded above). Fixp ∈ (0, 1), K ∈ R. We say that a measured Lorentzian pre-length space(X, d,m,�,≤, τ) has time-like Ricci curvature bounded above by K in asynthetic sense if there exists r0 > 0 and a function ω : [0, r0)→ [0,∞) withlimr↓0 ω(r) = 0 such that for every r ∈ [0, r0) the following holds.

• For every x, y ∈ X with d(x, y) = r > 0 and such that Bdr4(x) ×

Bdr2(y) ⊂ X2

�,

• for every µ0 ∈ Dom(Ent(·|m) with suppµ0 ⊂ Bdr4(x),

there exists an `p-geodesic (µt)t∈[−1,1] satisfying

• suppµ1 ⊂ Bdr2(y),

• suppµ−1 × suppµ1 ⊂ X2≤,

•⋃t∈[−1,1] suppµt ⊂ Bd

10r0(x),

• Ent(µ−1|m)− 2Ent(µ0|m) + Ent(µ1|m) ≤ (K + ω(r)) r2.

A key property of the above notion of time-like Ricci bounded aboveis the stability under weak convergence of measured Lorentzian pre-lengthspaces. The stability of Riemannian/metric Ricci upper bounds via optimaltransport was proved by Sturm [76]. The notion of convergence we use belowis a slight reinforcement (we ask that the immersion maps are isometries withrespect to the metric structures instead of merely topological embeddingmaps) of the weak convergence used in [25] to show stability of synthetictime-like Ricci lower bounds; in any case it is a natural adaptation to theLorentzian setting of the mGH convergence used for metric measure spaces(see for instance [40, Section 3] for an overview of equivalent formulationsof convergence for metric measure spaces).

Theorem B.6 (Stability of time-like Ricci curvature upper bounds). Let{(Xj , dj ,mj ,�j ,≤j , τj)}j∈N∪{∞} be a sequence of measured Lorentzian geo-desic spaces satisfying the following properties :

(1) There exists a locally causally closed, globally hyperbolic Lorentziangeodesic space (X, d,�,≤, τ) such that each (Xj , dj ,mj ,�j ,≤j , τj),j ∈ N ∪ {∞}, is isomorphically embedded in it, i.e. there exist in-clusion maps ιj : Xj ↪→ X such that for every x1

j , x2j ∈ Xj, for every

j ∈ N ∪ {∞}, the following holds:• d(ιj(x

1j ), ιj(x

2j )) = dj(x

1j , x

2j );

• x1j ≤j x2

j if and only if ιj(x1j ) ≤ ιj(x2

j );

• τ(ιj(x1j ), ιj(x

2j )) = τj(x

1j , x

2j ).

(2) The measures (ιj)]mj converge to (ι∞)]m∞ weakly in duality withCc(X) in X, i.e.

(B.5)

ˆϕ d(ιj)]mj →

ˆϕ d(ι∞)]m∞ ∀ϕ ∈ Cc(X),

where Cc(X) denotes the set of continuous functions with compactsupport.

(3) Volume non-collapsing: there exists a function v : (0,∞) → (0,∞)

such that for every xj ∈ Xj it holds mj(Bdjr (xj)) ≥ v(r) > 0.

(4) There exists a function ω : [0,∞) → [0,∞) with limr↓0 ω(r) = 0and there exist p ∈ (0, 1),K ∈ R such that (Xj , dj ,mj ,�j ,≤j , τj)


has time-like Ricci curvature bounded above by K with respect top ∈ (0, 1), r0 > 0 and with remainder function ω in the syntheticsense of Definition B.5.

Then also the limit space (X∞, d∞,m∞,�∞,≤∞, τ∞) has time-like Riccicurvature bounded above by K with respect to p ∈ (0, 1), r0 + 1 and withremainder function ω in the synthetic sense of Definition B.5.

Proof. For simplicity of notation, we will identify Xj with its isomorphicimage ιj(Xj) ⊂ X and the measure mj with (ιj)]mj , for each j ∈ N ∪ {∞}.

We will write Bdjr (x) for the ball in (Xj , dj) centred at x ∈ Xj . Notice

that Bdjr (x) = Bd

r (x) ∩Xj .

Fix arbitrary x∞, y∞ ∈ X∞ with d∞(x∞, y∞) = r > 0 and Bd∞r4 (x∞) ×

Bd∞r2 (y∞) ⊂ (X2

∞)�. Since (X2∞)� ⊂ X∞ × X∞ is an open subset, there

exist ε > 0 such that

(B.6) Bd∞(r+ε)4(x∞)×Bd∞

(r+ε)2(y∞) ⊂ (X2∞)� .

Fix µ∞0 ∈ Dom(Ent(·|m∞)) with suppµ∞0 ⊂ Bd∞r4 (x∞).

Step 1. We claim that, for j ∈ N sufficiently large, there exist xj , yj ∈ Xj

such that

(1) xj → x∞ and yj → y∞ in (X, d);(2) dj(xj , yj) =: rj → r;

(3) Bdjr4j(xj)×B

djr2j(yj) ⊂ (X2

j )� .

Since by assumption mj ⇀ m∞ weakly as measures in (X, d), by the lower-semicontinuity over open subset we have that for every r > 0 it holds

0 < m∞(Bd∞r (x∞)) = m∞(Bd

r (x∞)) ≤ lim infj→∞

mj(Bdr (x∞)).

In particular, for every r > 0 there exists J(r) > 0 such thatBdr (x∞)∩Xj 6= ∅

for all j ≥ J(r). By a diagonal argument we obtain that there exist xj ∈ Xj

with xj → x∞ in (X, d). The analogous argument gives a sequence yj ∈ Xj

with yj → y∞.Using that the inclusion maps ιj are isometric, we also get

rj := dj(xj , yj) = d(xj , yj)→ d(x∞, y∞) = r.

We are left to show the third claim. For every j ∈ N, let x′j ∈ Bdjr4j(xj) and

y′j ∈ Bdjr2j(yj) be arbitrary.

Combining the volume non-collapsing assumption with the weak conver-gence mj ⇀ m∞ and the properness of the metrics, we obtain that there

exist x′∞ ∈ Bd∞(r+ε)4(x∞) and y′∞ ∈ Bd∞

(r+ε)2(y∞) such that x′j → x′∞ and

y′j → y′∞ up to a subsequence.

Recalling (B.6) and that ι∞ is an isomorphic embedding, we infer that(x′∞, y

′∞) ∈ X2

�. Since X2� ⊂ X2 is an open subset, we conclude that

for j sufficiently large it holds

(x′j , y′j) ⊂ X2

� ∩X2j = (X2

j )�


as desired.

Step 2. We claim that, for every j ∈ N, there exists µj0 ∈ Dom(Ent( · |mj))

with suppµj0 ⊂ Bdjr4j(xj) such that µj0 → µ∞0 narrowly and

(B.7) Ent(µ∞0 |m∞) ≥ lim supj→∞

Ent(µj0|mj).

Since mj ⇀ m∞ weakly, there exists R ∈ (2r4, 3r4) such that

(B.8) z−1j := mj(B

dR(x∞))→ m∞(Bd

R(x∞)) =: z−1∞ .

For j ∈ N ∪ {∞}, denote mj := zj mjxBdR(x∞) and observe that mj ⇀ m∞

weakly and thus (since now the supports are uniformly bounded in X) in

W(X,d)q for some (or equivalently every) q ∈ [1,∞). In particular mj → m∞

in W(X,d)2 . Let

(B.9) ηj ∈ Π(m∞, mj) be a W(X,d)2 -optimal coupling.

Let us first consider the case µ∞0 = ρ m∞ with ρ ∈ L∞(m∞).

Define η′j ∈ P(X) as dη′j(x, y) := ρ(x) dηj(x, y) and µj0 := (p2)]η′j ∈ P(Xj).

By construction, η′j � ηj , hence µj0 � (p2)]ηj = mj . Let µj0 = ρj0mj . It is

readily checked from the definition that it holds ρj0(y) =´ρ(x) d(ηj)y(x),

where {(ηj)y} is the disintegration of ηj w.r.t. the projection on the secondmarginal. By Jensen’s inequality applied to the convex function u(s) :=s log(s), we have:

Ent(µj0|mj) =

û(ρj0) dmj =

û

(ˆρ(x) d(ηj)y(x)

)dmj(y)

≤ˆ ˆ

u(ρ(x)) d(ηj)y(x) dmj(y) =

ˆ û(ρ(x)) dηj(x, y)

=

û(ρ) d(p1)]ηj =

û(ρ) dm∞ = Ent(µ∞0 |m∞).

(B.10)

Since by construction we have γ′j ∈ Π(µ∞0 , µj0), it holds

W 22 (µ∞0 , µ

j0) ≤

ˆd2(x, y) dη′j(x, y) =

ˆρ(x) d2(x, y) dηj(x, y)

≤ ‖ρ‖L∞(m∞)W22 (m∞, mj),

and therefore W2(µ∞0 , µj0)→ 0. In particular, µj0 → µ∞0 narrowly in (X, d).

Using that xj → x∞, rj → r and suppµ∞0 ⊂ Bd∞r4 (x∞), it follows that c−1

j :=

µj0(B

dj(r′j)

4(xj))→ 1, where we set r′j := rj− 1

j . Define µj0 := cj µj0xB

dj(r′j)

4(xj).

Clearly, µj0 ∈ Dom(Ent( · |mj)), suppµj0 ⊂ Bdjr4j(xj), W2(µ∞0 , µ

j0)→ 0 and

(B.11) limj→∞

∣∣Ent(µj0|mj)− Ent(µj0|mj)∣∣ = 0.

The combination of (B.8), (B.10) and (B.11) gives the desired (B.7).


If ρ is not bounded, for k ∈ N define ρk := Ck min{ρ, k}, Ck being suchthat µk∞ := ρk m∞ ∈ P(X∞). Clearly, it holds

lim supk→∞

Ent(µk∞|m∞) ≤ Ent(µ∞|m∞), limk→∞

W2(µk∞, µ∞) = 0.

Then apply the previous procedure to µk∞ and conclude with a diagonalargument.

Step 3. Conclusion.By assumption, for every j ∈ N the space (Xj , dj ,mj ,�j ,≤j , τj) has time-like Ricci curvature bounded above by K with respect to p ∈ (0, 1), r0 > 0

and with remainder function ω. Thus there exists an `p-geodesic (µjt )t∈[−1,1]

with suppµj1 ⊂ Bdjr2j(yj),

⋃t∈[−1,1] suppµjt ⊂ B

dj10r0

(xj) such that

(B.12) Ent(µ−1|mj)− 2Ent(µ0|mj) + Ent(µ1|mj) ≤ (K + ω(r)) r2.

Since by construction xj → x∞, there exists J > 0 such that

(B.13)⋃j≥J

⋃t∈[−1,1]

suppµjt ⊂ Bd11r0(x∞).

Let ηj ∈ P(C[−1, 1], (X, d)) be the dynamical plan associated to the geodesic

(µjt )t∈[−1,1], i.e. such that (et)]ηj = µjt for every t ∈ [0, 1], j ∈ N. From thenon-total imprisoning property of X, we infer that there exists C > 0 suchthat

sup{Ld(γ) : γ ∈ supp ηj , j ∈ N} ≤ C <∞.From [56, Theorem 4], we infer that (µjt )t∈[−1,1] is an absolutely continuouscurve in the W1-Kantorovich Wasserstein space (P(X),W1) w.r.t. d, withbounded length:

LW1((µjt )t∈[−1,1]) ≤ˆ

Ld(γ) dηj(γ) ≤ C <∞, ∀j ∈ N.

By the metric Arzela-Ascoli Theorem we deduce that there exists a limit

continuous curve (µ∞t )t∈[−1,1] ⊂ P(X∞) ∩ P(X,W(X,d)1 ) such that (up to a

sub-sequence) W(X,d)1

(µjt , µ

∞t

)→ 0 and thus µjt → µ∞t narrowly in X, as

j →∞ for every t ∈ [0, 1]. From (B.13) we have⋃t∈[−1,1]

suppµ∞t ⊂ Bd11r0(x∞) ⊂ Bd

10(r0+1)(x∞).

By assumption, we also know that for every j ∈ N it holds

(B.14) suppµj−1 × suppµj1 ⊂ (X2j )≤ = X2

≤ ∩X2j , ∀j ∈ N.

Since by assumption X is locally causally closed, we infer that

(B.15) suppµ∞−1 × suppµ∞1 ⊂ (X2∞)≤ = X2

≤ ∩X2∞.

For every j ∈ N, let πj ∈ Πp-opt≤ (µj−1, µ

j1). Thanks to Lemma B.4 we know

that πj is τp-cyclically monotone.

From (B.13), we have that suppπj ⊂ Bd11r0

(x∞)2 for j ≥ J and thus (byProkhorov Theorem) there exists π∞ ∈ P(X × X) such that πj → π∞narrowly. Using the continuity of the projection maps, it is readily seen


that π∞ ∈ Π(µ∞−1, µ∞1 ) and thus π∞ ∈ Π≤(µ∞−1, µ

∞1 ), thanks to (B.15).

Since by global hyperbolicity τp is continuous, the τp-cyclical monotonicityis preserved under narrow convergence. We infer that π∞ is τp-cyclicallymonotone and thus `p-optimal thanks to (B.15) and Lemma B.4. Therefore:

`p(µj−1, µ

j1) =

(ˆτ(x, y)p dπj(x, y)

)1/p

−→(ˆ

τ(x, y)p dπ∞(x, y)

)1/p

= `p(µ∞−1, µ

∞1 ).(B.16)

For intermediate t ∈ (−1, 1), using that (µjt )t∈[−1,1] is an `p-geodesic, wehave(B.17)

`p(µ∞−1, µ

∞t ) ≥ lim

j→∞`p(µ

j−1, µ

jt ) = (t+1) lim

j→∞`p(µ

j−1, µ

j1) = (t+1) `p(µ

∞−1, µ

∞1 ).

By reverse triangle inequality (B.3), we get that the curve (µ∞t )t∈[−1,1] is an`p-geodesic from µ∞−1 to µ∞1 .The joint lower semicontinuity of Ent(·|·) under narrow convergence of prob-ability measures (this is a well-known fact, for a proof see for instance [3,Lemma 9.4.3]) yields:

Ent(µ∞t |m∞) ≤ lim infj∈N

Ent(µjt |mj), ∀t ∈ [0, 1],

and thus, using (B.8):

(B.18) Ent(µ∞t |m∞) ≤ lim infj∈N

Ent(µjt |mj), ∀t ∈ [0, 1].

The combination of (B.7), (B.12) and (B.18) gives that

Ent(µ∞−1|m∞)− 2Ent(µ∞0 |m∞) + Ent(µ∞1 |m∞) ≤ (K + ω(r)) r2.

as desired.�

B.4. Synthetic time-like Ricci lower bounds. In this subsection webriefly recall some basics of the synthetic theory of time-like Ricci lowerbounds developed in [25].

Definition B.7 (Time-like p-dualisable measures). Let (X, d,�,≤, τ) be aLorentzian pre-length space and fix p ∈ (0, 1]. We say that (µ, ν) ∈ P(X)2

is time-like p-dualisable (by π ∈ Π�(µ, ν)) if

(1) `p(µ, ν) ∈ (0,∞);

(2) π ∈ Πp-opt≤ (µ, ν) and π(X2

�) = 1;

(3) there exist measurable functions a, b : X → R, with a⊕b ∈ L1(µ⊗ν)such that `p ≤ a⊕ b on suppµ× supp ν.

The pair (µ, ν) ∈ (P(X))2 is said to be strongly time-like p-dualisable if inaddition there exists a measurable `p-cyclically monotone set Γ ⊂ X2

� ∩(suppµ × supp ν) such that a coupling π ∈ Π≤(µ, ν) is `p-optimal if andonly if π is concentrated on Γ, i.e. π(Γ) = 1.


The motivation for considering (strongly) time-like p-dualisable pairs ofmeasures is twofold: firstly the p-optimal coupling dπ(x, y) matches eventsdescribed by dµ(x) with events described by dν(y) so that x� y; secondlyKantorovich duality holds (see [78, Proposition 2.7] in smooth Lorentziansetting and in case p = 1 and [25, Section 2.4] for the non-smooth settingand general p ∈ (0, 1]).

Motivated by the work of McCann [62] and by Theorem 4.3, in [25] thefollowing synthetic notion of time-like Ricci curvature bounded below byK ∈ R and dimension bounded above by N ∈ (0,∞] was proposed:

Definition B.8 (TCDep(K,N) and wTCDep(K,N) conditions). Fix p ∈ (0, 1),K ∈ R, N ∈ (0,∞]. We say that measured Lorentzian pre-length space(X, d,m,�,≤, τ) satisfies TCDep(K,N) (resp. wTCDep(K,N)) if the follow-

ing holds. For any couple (µ0, µ1) ∈ (Dom(Ent(·|m)))2 which is time-likep-dualisable (resp. (µ0, µ1) ∈ [Dom(Ent(·|m)) ∩ Pc(X)]2 which is strongly

time-like p-dualisable) by some π ∈ Πp-opt� (µ0, µ1), there exists an `p-geodesic

(µt)t∈[0,1] such that the function [0, 1] 3 t 7→ e(t) := Ent(µt|volg) is semi-convex (and thus in particular it is locally Lipschitz in (0, 1)) and it satisfies

(B.19) e′′(t)− 1

Ne′(t)2 ≥ K

ˆX×X

τ(x, y)2 dπ(x, y),

in the distributional sense on [0, 1], where we adopt the convention that thesecond adding term in the left hand side is zero if N =∞.

The following stability result for synthetic time-like Ricci lower boundswas proved in [25, Theorem 3.14].

Theorem B.9 (Weak stability of TCDep(K,N)). Let {(Xj , dj ,mj ,�j ,≤j, τj)}j∈N∪{∞} be a sequence of measured Lorentzian geodesic spaces satisfyingthe following properties:

(1) There exists a locally causally closed, K-globally hyperbolic Lorentziangeodesic space (X, d,�,≤, τ) such that each (Xj , dj ,mj ,�j ,≤j , τj),j ∈ N∪{∞}, is isomorphically embedded in it (as in (1) of TheoremB.6).

(2) The measures (ιj)]mj converge to (ι∞)]m∞ weakly in duality withCc(X) in X, i.e. (B.5) holds.

(3) There exist p ∈ (0, 1),K ∈ R, N ∈ (0,∞] such that (Xj , dj ,mj ,�j

,≤j , τj) satisfies TCDep(K,N), for each j ∈ N.

Then the limit space (X∞, d∞,m∞,�∞,≤∞, τ∞) satisfies the wTCDep(K,N)condition.

B.5. Synthetic vacuum Einstein’s equations, Combining the syntheticupper and lower bounds on the time-like Ricci curvature, i.e. DefinitionsB.5 and B.8, it is natural to propose the following synthetic version for thevacuum Einstein’s equations (with possibly non-zero cosmological constant):

Definition B.10 (Synthetic vacuum Einstein’s equations). Fix p ∈ (0, 1),Λ ∈ R, N ∈ (0,∞]. We say that the measured Lorentzian pre-length space(X, d,m,�,≤, τ) satisfies the (resp. weak) synthetic formulation of the vac-uum Einstein equations Ric ≡ Λ with cosmological constant Λ ∈ R and hassynthetic dimension ≤ N if


• (X, d,m,�,≤, τ) satisfies the TCDep(Λ, N) (resp. wTCDep(Λ, N))condition;• There exists r0 > 0 and a function ω : [0, r0)→ [0,∞) with limr↓0 ω(r) =

0 such that (X, d,m,�,≤, τ) has time-like Ricci curvature boundedabove by Λ, with respect to p ∈ (0, 1), r0 and ω.

The combination of the stability of time-like Ricci lower and upper bounds(i.e. Theorem B.6 and Theorem B.9) gives the stability of the syntheticvacuum Einstein’s equations under the aforementioned natural Lorentzianvariant of measured Gromov-Hausdorff convergence.

Theorem B.11 (Weak stability of synthetic vacuum Einstein’s equations).Let {(Xj , dj ,mj ,�j ,≤j , τj)}j∈N∪{∞} be a sequence of measured Lorentziangeodesic spaces satisfying the following properties:

(1) There exists a locally causally closed, K-globally hyperbolic Lorentziangeodesic space (X, d,�,≤, τ) such that each (Xj , dj ,mj ,�j ,≤j , τj),j ∈ N∪{∞}, is isomorphically embedded in it (as in (1) of TheoremB.6).

(2) The measures (ιj)]mj converge to (ι∞)]m∞ weakly in duality withCc(X) in X, i.e. (B.5) holds.

(3) Volume non-collapsing: there exists a function v : (0,∞) → (0,∞)

such that for every xj ∈ Xj it holds mj(Bdjr (xj)) ≥ v(r) > 0.

(4) There exist p ∈ (0, 1),Λ ∈ R, N ∈ (0,∞], r0 > 0 and ω : [0, r0) →[0,∞) with limr↓0 ω(r) = 0 such that (Xj , dj ,mj ,�j ,≤j , τj) satisfiesthe synthetic formulation of the vacuum Einstein equations Ric ≡ Λwith cosmological constant Λ ∈ R, with synthetic dimension ≤ Nwith respect to p ∈ (0, 1), r0 and ω as in Definition B.10.

Then the limit space (X∞, d∞,m∞,�∞,≤∞, τ∞) satisfies the weak syntheticformulation of the vacuum Einstein equations Ric ≡ Λ with cosmologicalconstant Λ ∈ R, with synthetic dimension ≤ N with respect to p ∈ (0, 1), r0+1 and ω as in Definition B.10.

Remark B.12. By [25, Theorem 3.1] after [62] (see also Corollary 4.4)and by Theorem 4.7 (see also Remark 4.8), if (X, d,m,�,≤, τ) is a (forsimplicity say a compact subset in a) smooth Lorentzian manifold, then(X, d,m,�,≤, τ) satisfies the Einstein’s equations Ric ≡ Λ in the smoothclassical sense if and only if (X, d,m,�,≤, τ) satisfies the Einstein’s equa-tions in the synthetic sense of Definition B.10. Therefore, Theorem B.11gives that the corresponding limits of smooth solutions to Einstein’s equa-tion Ric ≡ Λ satisfy the weak synthetic Einstein’s equations Ric ≡ Λ in thesense of Definition B.10. In other terms, the vacuum Einstein’s equationsare stable under the conditions (and with respect to the notion of conver-gence) of Theorem B.11.

Let us mention that the stability of the Einstein’s equations under variousnotions of (weak) convergence is a topic of high interest in General Relativity.Classically, the problem is phrased in terms of convergence of a sequence ofLorentzian metrics gj converging to a limit Lorentzian metric g∞, on a fixedunderlying manifold.It is well known that, if gj are solutions of the vacuum Einstein equations,


gj → g∞ in C0loc and the derivatives of gj converge in L2

loc, then the limitg∞ satisfies the vacuum Einstein equations as well.However, if the gj → g∞ in C0

loc and the derivatives of gj converge weaklyin L2

loc, explicit examples are known (see [20, 42] for examples in symmetryclasses) where the limit g∞ may satisfy the Einstein equations with a non-vanishing stress energy momentum tensor. Burnett [20] conjectured that, ifthere exist C > 0 and λj → 0 such that

|gj − g∞| ≤ λj , |∂gj | ≤ C, |∂2gj | ≤ Cλ−1j ,

then g∞ is isometric to a solution to the Einstein-massless Vlasov systemfor some appropriate choice of Vlasov field. Such a conjecture remains open,although there has been recent progress [47, 48] when gj are assumed to beU(1)-symmetric. We also mention the recent work [58] where concentrations(at the level of ∂gj) are allowed in addition to oscillations.

Theorem B.11 gives a new point of view on the stability of the vacuumEinstein’s equations. Indeed, while in the aforementioned results the metricsgj are converging on a fixed underlying manifold, in Theorem B.11 also theunderlying space X may vary (along the sequence and at the limit), allowingchange in topology in the limit, as one may expect in case of formation ofsingularities. Moreover, the notion of convergence is quite different in spirit:while in the aforementioned results gj → g∞ in a suitable functional analyticsense, in Theorem B.11 the spaces are converging in a more geometric sense(inspired by the measured Gromov-Haudorff convergence).

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