Displacement Convexity of Boltzmann's Entropy ... › mccann › Talk2.pdf · between x and y (as in the twin paradox). Apart from sign restrictions, this is quite analogous to the

Displacement Convexity of Boltzmann’s EntropyCharacterizes Positive Energy in General Relativity

Robert J McCann

University of Toronto

on arXiv and at

www.math.toronto.edu/mccann/

Toronto 8 Dec 2019

Robert J McCann (Toronto) Entropic displ. convexity ⇔ strong energy Toronto 8 Dec 2019 1 / 28

Two famous sign laws in physics

• gravity is always attractive, never repulsive

• entropy always goes up, never down

Might these be related?

• Bekenstein ’73: 2nd law of black hole dynamicsarea of horizons can only increase

• Jacobson ’95: Einstein’s equation follows from Entropy := Horizon Area

• E Verlinde ’11: Gravity as an emergent entropic (i.e. statistical) force

My purpose today is to describe anew connection between gravity and entropy using optimal transportleading toward a nonsmooth theory of gravity


Two famous sign laws in physics

• gravity is always attractive, never repulsive

• entropy always goes up, never down

Might these be related?

• Bekenstein ’73: 2nd law of black hole dynamicsarea of horizons can only increase

• Jacobson ’95: Einstein’s equation follows from Entropy := Horizon Area

• E Verlinde ’11: Gravity as an emergent entropic (i.e. statistical) force

My purpose today is to describe anew connection between gravity and entropy using optimal transportleading toward a nonsmooth theory of gravity






Terminology and conventions

0 6= v ∈ TxM is

(a) timelike if g(v , v) > 0(b) lightlike (or null) if ” = 0(c) spacelike if ” < 0(d) causal if (a) or (b) hold, in which case(e) future-directed if it lies in the green cone(f) past-directed if it lies in the red cone

A C 1 curve σ : (a, b)→ M is said to have the property (a-f) if each of itstangent vectors does.

Particles with mass follow timelike future-directed curves on M.


Positive Energy Conditions of Hawking and Penrose ’70

Weak energy condition: Gijviv j ≥ 0 for all timelike (v , x)∈ TM

(believed to be satisfied in all physical geometries)

Strong energy condition: Rijviv j ≥ 0 for all timelike (v , x) ∈ TM, where

Gij = Rij −1

n − 2Rgij

here Rij is the Ricci curvature tensor and R = g ijRij is its scalar trace.

- less universally satisfied- does not imply weak energy condition- implies gravity is attractive- was used by Hawking and Penrose to show “trapped” spacelike surfaces(whose areas decrease instantaneously in all possible futures) implysingularities


We’ll assume global hyperbolicity of (M, g), meaning• (M, g) is smooth, connected, Hausdorff, timelike-orientable• has no closed future-directed curves (i.e. no ‘back to the future’)• J+(x) ∩ J−(y) is compact for all x , y ∈ M, whereJ+(x) is the set of points reached from x along future-directed curvesJ−(y) is the set of points reached from y along past-directed curves

For q ∈ (0, 1] introduce the convex(!) Lagrangian on TM

L(v , x ; q) :=

{− 1

qg(v , v)q/2 if v is future-directed

+∞ else.

and its convex dual Hamiltonian H = L∗ given by 1q + 1

q′ = 1 (if q 6= 1)and

H(p, x ; q) :=

{− 1

q′ g(p, p)q′/2 if p ∈ T ∗xM is past-directed

+∞ else.

- notice q′ < 0


We’ll assume global hyperbolicity of (M, g), meaning• (M, g) is smooth, connected, Hausdorff, timelike-orientable• has no closed future-directed curves (i.e. no ‘back to the future’)• J+(x) ∩ J−(y) is compact for all x , y ∈ M, whereJ+(x) is the set of points reached from x along future-directed curvesJ−(y) is the set of points reached from y along past-directed curves

For q ∈ (0, 1] introduce the convex(!) Lagrangian on TM

L(v , x ; q) :=

{− 1

qg(v , v)q/2 if v is future-directed

+∞ else.

and its convex dual Hamiltonian H = L∗ given by 1q + 1

q′ = 1 (if q 6= 1)and

H(p, x ; q) :=

{− 1

q′ g(p, p)q′/2 if p ∈ T ∗xM is past-directed

+∞ else.

- notice q′ < 0



associated convex action on C 1 curves σ : [0, 1] −→ M,

A[σ; q] :=

∫ 1

0L(σ′(s), σ(s); q)ds.

The (q-dependent) Lorentz distance is defined by least action

`(x , y ; q) := − infσ(0)=xσ(1)=y

A[σ; q]

=1

q`(x , y ; 1)q (Jensen′s 6=)

∈ {−∞} ∪ [0,∞)

where `(x , y) := `(x , y ; 1) is also called the time-separation function, anddenotes the maximum a particle can age between x and y . Throughout weadopt the convention

(−∞)q := −∞ =: (−∞)1/q


associated convex action on C 1 curves σ : [0, 1] −→ M,

A[σ; q] :=

∫ 1

0L(σ′(s), σ(s); q)ds.

The (q-dependent) Lorentz distance is defined by least action

`(x , y ; q) := − infσ(0)=xσ(1)=y

A[σ; q]

=1

q`(x , y ; 1)q (Jensen′s 6=)

∈ {−∞} ∪ [0,∞)

where `(x , y) := `(x , y ; 1) is also called the time-separation function, anddenotes the maximum a particle can age between x and y . Throughout weadopt the convention

(−∞)q := −∞ =: (−∞)1/q


The time-separation function satisfies a backwards triangle inequality:

`(x , z) + `(z , y) ≤ `(x , y) ∀x , y , z ∈ M.

Global hyperbolicity ensures the infimum defining `(x , y ; q) > 0 isattained; for q < 1, the curve σ which attains it is a geodesic segment, i.e.

`(σ(s), σ(t)) = |t − s|`(σ(0), σ(1)) ∀ 0 ≤ s < t ≤ 1

The resulting curve σ is the proper-time- (i.e. age-) maximizing trajectorybetween x and y (as in the twin paradox).

Apart from sign restrictions, this is quite analogous to the construction ofshortest length curves achieving the Riemannian distance d(x , y) in thecase of a positive definite metric g on M, except that in that case theconvex Lagrangian would be defined using an exponent q ≥ 1 (e.g. q = 2)instead of q ∈ (0, 1].


The time-separation function satisfies a backwards triangle inequality:

`(x , z) + `(z , y) ≤ `(x , y) ∀x , y , z ∈ M.

Global hyperbolicity ensures the infimum defining `(x , y ; q) > 0 isattained; for q < 1, the curve σ which attains it is a geodesic segment, i.e.

`(σ(s), σ(t)) = |t − s|`(σ(0), σ(1)) ∀ 0 ≤ s < t ≤ 1

The resulting curve σ is the proper-time- (i.e. age-) maximizing trajectorybetween x and y (as in the twin paradox).

Apart from sign restrictions, this is quite analogous to the construction ofshortest length curves achieving the Riemannian distance d(x , y) in thecase of a positive definite metric g on M, except that in that case theconvex Lagrangian would be defined using an exponent q ≥ 1 (e.g. q = 2)instead of q ∈ (0, 1].


In the Riemannian case, given arclength-parameterized geodesics σ(s) andτ(t) through a common point σ(0) = τ(0), a local Taylor expansion yields

d2(σ(s), τ(t)) = s2 + t2 − 2stg(σ(0), τ(0))

−s2t2

6Rijkl σ

i τ j σk(0)τ l(0) + O(|s|5 + |t|5)

where Rijkl is the Riemannian curvature tensor. It measures the leadingcorrection to Pythagoras’ law, and also

the failure of covariant derivativeswrt g ’s Levi-Civita connection (∇i gjk = 0) to commute:

Rijklvk = −[∇i , ∇j ]v

l

Its trace Rik := g jl Rijkl gives the Ricci tensor associated to gij .


In the Riemannian case, given arclength-parameterized geodesics σ(s) andτ(t) through a common point σ(0) = τ(0), a local Taylor expansion yields

d2(σ(s), τ(t)) = s2 + t2 − 2stg(σ(0), τ(0))

−s2t2

6Rijkl σ

i τ j σk(0)τ l(0) + O(|s|5 + |t|5)

where Rijkl is the Riemannian curvature tensor. It measures the leadingcorrection to Pythagoras’ law, and also the failure of covariant derivativeswrt g ’s Levi-Civita connection (∇i gjk = 0) to commute:

Rijklvk = −[∇i , ∇j ]v

l

Its trace Rik := g jl Rijkl gives the Ricci tensor associated to gij .


The Riemann and Ricci tensors Rijkl and Rik associated to a Lorentzianmetric gij can be defined analogously.Gravitational attractivity stems from positivity of Rij in timelike directions.But what has this to do with entropy or the second law?

In the Riemannian setting, a line of developments starting fromM. ’94Otto & Villani ’00Cordero-Erausquin, M., & Schmuckenschlager ’01 ledvon Renesse & Sturm ’04 to characterize Rij ≥ 0 via the convexity ofBoltzmann’s entropy along L2-Kantorovich-Rubinstein-Wassersteingeodesics given by optimal transportation of probability measures.

This inspired Sturm ’06, Lott and Villani ’09 to adopt such convexity asthe definition of lower Ricci bounds in a (non-smooth) metric-measuresetting, leading to the blossoming study of curvature-dimension spacesCD(K ,N) developed by Ambrosio, Gigli, Savare, Erbar, Kuwada, Sturm, ...










CD(K,N) and RCD(K,N) spaces


Can something similar be done in the Lorentzian setting?

Use Lorentz distance `(x , y) to lift the geometry from M to the set Pc(M)of (compactly supported for simplicity) Borel probability measures on M:Given 0 < q ≤ 1 and µ0, µ1 ∈ Pc(M), define

`q(µ0, µ1) :=

(sup

γ∈Γ(µ0,µ1)

∫M×M

`(x , y)qdγ(x , y)

)1/q

,

where the supremum is over joint measures γ ≥ 0 on M ×M having µ0

and µ1 as left and right marginals

- this is a (Kantorovich ’42) optimal transport problem

with lowersemicontinuous cost −`q whose gradient diverges as the boundary of thecausal set J+ = `−1([0,∞)) is approached, and which jumps to +∞outside J+.

- these singularities reflects the degeneration of both strict convexity andof smoothness for the Lagrangian L(v , x ; q) at the light cone

- still the supremum is attained by some γ which will be called `q-optimal(unless `q(µ0, µ1) = −∞).


Can something similar be done in the Lorentzian setting?

Use Lorentz distance `(x , y) to lift the geometry from M to the set Pc(M)of (compactly supported for simplicity) Borel probability measures on M:Given 0 < q ≤ 1 and µ0, µ1 ∈ Pc(M), define

`q(µ0, µ1) :=

(sup

γ∈Γ(µ0,µ1)

∫M×M

`(x , y)qdγ(x , y)

)1/q

,

where the supremum is over joint measures γ ≥ 0 on M ×M having µ0

and µ1 as left and right marginals

- this is a (Kantorovich ’42) optimal transport problem with lowersemicontinuous cost −`q whose gradient diverges as the boundary of thecausal set J+ = `−1([0,∞)) is approached, and which jumps to +∞outside J+.

- these singularities reflects the degeneration of both strict convexity andof smoothness for the Lagrangian L(v , x ; q) at the light cone

- still the supremum is attained by some γ which will be called `q-optimal(unless `q(µ0, µ1) = −∞).


A close variant of `q(µ0, µ1) was defined in Eckstein & Miller ’17, whoshow `q inherits the reverse triangle inequality from `(x , y):

`q(µ0, µ1) ≥ `q(µ0, ν) + `q(ν, µ1).

DEFN: We say (µs)s∈[0,1] is a q-geodesic in Pc(M) iff

`q(µs , µt) = |t − s|`q(µ0, µ1)> 0 ∀ 0 ≤ s < t ≤ 1.

- q-geodesics exist, if ` > 0 a.e. wrt a Kantorovich maximizer γ ∈ Γ(µ0, µ1)

- When ` > 0 on spt[µ0 × µ1] (:= smallest closed set of full mass),so that µ1 lies entirely in the timelike future of µ0, we’ll characterize theq-geodesic joining them uniquely provided µ0 ∈ Pac

c (M), meaning µ0 isabsolutely continuous wrt the Lorentzian volume

dvolg (x) ( := | det gij(x)|1/2dnx in coordinates).


A close variant of `q(µ0, µ1) was defined in Eckstein & Miller ’17, whoshow `q inherits the reverse triangle inequality from `(x , y):

`q(µ0, µ1) ≥ `q(µ0, ν) + `q(ν, µ1).

DEFN: We say (µs)s∈[0,1] is a q-geodesic in Pc(M) iff

`q(µs , µt) = |t − s|`q(µ0, µ1)> 0 ∀ 0 ≤ s < t ≤ 1.

- q-geodesics exist, if ` > 0 a.e. wrt a Kantorovich maximizer γ ∈ Γ(µ0, µ1)

- When ` > 0 on spt[µ0 × µ1] (:= smallest closed set of full mass),so that µ1 lies entirely in the timelike future of µ0, we’ll characterize theq-geodesic joining them uniquely provided µ0 ∈ Pac

c (M), meaning µ0 isabsolutely continuous wrt the Lorentzian volume

dvolg (x) ( := | det gij(x)|1/2dnx in coordinates).


DEFN Setting dm(x) := e−V dvolg (x) where V ∈ C 2(M) we define therelative entropy by

EV (µ) :=

{ ∫M ρ log ρdm if µ ∈ Pac

c (M) and ρ := dµdm ,

+∞ if µ ∈ Pc(M) \ Pac(M).

- When V = 0 the Boltzmann-Shannon entropy E0(µ) results.

- our sign convention is opposite to that of the physicists’ entropy


THM 1: (Positive energy = entropic displacement convexity)Fix 0 < q < 1 and a globally hyperbolic spacetime (Mn, g).

(a) If Rijviv j < K ∈ R for SOME unit timelike vector (v , x) ∈ TM, there

is a q-geodesic (µs)s∈[0,1] supported arbitrarily close to x and with ` > 0 onspt[µ0 × µ1] along which e(s) := E0(µs) ∈ C 2 and e ′′(0) < K`q(µ0, µ1)2.

(b) Conversely, if Rijviv j ≥ Kg(v , v) ≥ 0 for EVERY timelike

(v , x) ∈ TM, then

e ′′(s) ≥ 1

ne ′(s)2 + K`q(µ0, µ1)2 ≥ 0

holds (distributionally) along ALL q-geodesics (µs)s∈[0,1] having finiteentropy endpoints and ` > 0 on spt[µ0 × µ1].


THM 1: (Positive energy = entropic displacement convexity)Fix 0 < q < 1 and a globally hyperbolic spacetime (Mn, g).

(a) If Rijviv j < K ∈ R for SOME unit timelike vector (v , x) ∈ TM, there

is a q-geodesic (µs)s∈[0,1] supported arbitrarily close to x and with ` > 0 onspt[µ0 × µ1] along which e(s) := E0(µs) ∈ C 2 and e ′′(0) < K`q(µ0, µ1)2.

(b) Conversely, if Rijviv j ≥ Kg(v , v) ≥ 0 for EVERY timelike


e ′′(s) ≥ 1

ne ′(s)2 + K`q(µ0, µ1)2 ≥ 0

holds (distributionally) along ALL q-geodesics (µs)s∈[0,1] having finiteentropy endpoints and ` > 0 on spt[µ0 × µ1].




DEFN (N-Bakry-Emery modified Ricci tensor; cf. Erbar-Kuwada-Sturm’15)Given N ∈ (n,∞] and V ∈ C 2(M) define

R(N,V )ij := Rij +∇i∇jV −

1

N − n(∇iV )(∇jV )

THM 1’ Fix 0 < q < 1 and a globally hyperbolic spacetime (Mn, g).

(a) If R(N,V )ij v iv j < K ∈ R for some unit timelike vector (v , x) ∈ TM,

there is a q-geodesic (µs)s∈[0,1] supported arbitrarily close to x and with` > 0 on spt[µ0 × µ1] along which e(s) := EV (µs) ∈ C 2 ande ′′(0) < K`q(µ0, µ1)2.

(b) Conversely, if R(N,V )ij v iv j ≥ Kg(v , v) ≥ 0 for every timelike


e ′′(s) ≥ 1

Ne ′(s)2 + K`q(µ0, µ1)2(≥ 0)

holds (distributionally) along all q-geodesics (µs)s∈[0,1] having finiteentropy q-separated endpoints


DEFN (N-Bakry-Emery modified Ricci tensor; cf. Erbar-Kuwada-Sturm’15)Given N ∈ (n,∞] and V ∈ C 2(M) define

R(N,V )ij := Rij +∇i∇jV −

1

N − n(∇iV )(∇jV )

THM 1’ Fix 0 < q < 1 and a globally hyperbolic spacetime (Mn, g).

(a) If R(N,V )ij v iv j < K ∈ R for some unit timelike vector (v , x) ∈ TM,

there is a q-geodesic (µs)s∈[0,1] supported arbitrarily close to x and with` > 0 on spt[µ0 × µ1] along which e(s) := EV (µs) ∈ C 2 ande ′′(0) < K`q(µ0, µ1)2.

(b) Conversely, if R(N,V )ij v iv j ≥ Kg(v , v) ≥ 0 for every timelike


e ′′(s) ≥ 1

Ne ′(s)2 + K`q(µ0, µ1)2(≥ 0)

holds (distributionally) along all q-geodesics (µs)s∈[0,1] having finiteentropy q-separated endpoints


Remarks and related developments

1. Although the Ricci tensor is only defined in the smooth setting, as inthe Riemannian case the entropic characterization can be used to definethe strong energy condition in a metric measure setting (future work).

2. The smooth manifolds which satisfy it do not depend on q ∈ (0, 1];whether or not the nonsmooth spaces which satisfy it depend on q isunclear, even in the analogous Riemannian problem (c.f. Kell ’17; w.i.p)

3. A few months later, a related but independent local construction wasannounced by Mondino and Suhr ’18+ which also gives the complementarylower bound — thus a weak formulation to the full Einstein equation!

4. Kunzinger & Samann ’17+ give a synthetic approach to two-sidedsectional curvature bounds via triangle comparison.

5. Loeper ’06 first observed the enhancement of displacement convexity byNewtonian gravity in the Euclidean setting.

6. Gomes & Senici ’18+ give a heuristic proof that displacement convexityextends to the planning problem from mean fields games with localcongestion interactions.


To prove this, used linear programming duality to analyze the optimaltransportation problem defining

1

q`q(µ, ν)q = inf

u⊕v≥ 1q`q

∫Mudµ+

∫Mvdν,

Unfortunately, the singularities of ` may prevent attainment of thisKantorovich dual infimum by (lsc) potentials (u, v) satisfying

(∗) u(x) + v(y) ≥ 1

q`(x , y)q ∀ (x , y) ∈ spt[µ× ν]

DEFN: Fix q ∈ (0, 1]. We say (µ, ν) ∈ Pc(M) is q-separated byγ ∈ Γ(µ, ν) and (u, v) if (∗) is satisfied and spt γ ⊂ S ⊂ {` > 0} whereS is the set of (x , y) ∈ spt[µ× ν] producing equality in (∗)

LEMMA: (a) q-separation of (µ0, ν0) implies optimality of γ and (u, v).(b) ` > 0 on spt[µ0 × µ1] is sufficient (but not necessary) for q-separation.(c) (µs , µt) inherit q-separation along the q-geodesic from µ0 to µ1.



1

q`q(µ, ν)q = inf

u⊕v≥ 1q`q

∫Mudµ+

∫Mvdν,


(∗) u(x) + v(y) ≥ 1

q`(x , y)q ∀ (x , y) ∈ spt[µ× ν]





1

q`q(µ, ν)q = inf

u⊕v≥ 1q`q

∫Mudµ+

∫Mvdν,


(∗) u(x) + v(y) ≥ 1

q`(x , y)q ∀ (x , y) ∈ spt[µ× ν]




THM (Lagrangian characterization of q-geodesics)Fix 0 < q < 1. If (µ0, µ1) ∈ P2

c (M) is q-separated by (γ, u, v) andµ0 << volg then the map Fs(x) := expx sDH(Du(x); q) induces theunique q-geodesic s ∈ [0, 1] 7→ µs in Pc(M) linking µ0 to µ1.Moreover, µs << volg if s < 1 (by using uniform convexity of L away fromlight cone to adapt Monge-Mather ‘shortening’ estimate).

Here µs := (Fs)#µ0 is defined by

µs [V ] := µ0[F−1s (V )] ∀V ⊂ M,

Fs is an optimal (i.e. Monge) map between µ0 and µs , andγ = (id × F1)#µ0 uniquely maximizes the Kantorovich problem defining`q(µ0, µ1).

Setting ρs := dµsdvolg

yields the Monge-Ampere type equation

ρ0(x) = ρs(Fs(x))|JFs(x)| ρ0 − a.e.,

where JFs(x) = detDFs(x) is the (approximate) Jacobian of Fs and

∂

∂s

∣∣∣s=0

(DFs) = D2H|DuD2u.


THM (Lagrangian characterization of q-geodesics)Fix 0 < q < 1. If (µ0, µ1) ∈ P2

c (M) is q-separated by (γ, u, v) andµ0 << volg then the map Fs(x) := expx sDH(Du(x); q) induces theunique q-geodesic s ∈ [0, 1] 7→ µs in Pc(M) linking µ0 to µ1.Moreover, µs << volg if s < 1 (by using uniform convexity of L away fromlight cone to adapt Monge-Mather ‘shortening’ estimate).

Here µs := (Fs)#µ0 is defined by

µs [V ] := µ0[F−1s (V )] ∀V ⊂ M,

Fs is an optimal (i.e. Monge) map between µ0 and µs , andγ = (id × F1)#µ0 uniquely maximizes the Kantorovich problem defining`q(µ0, µ1).

Setting ρs := dµsdvolg

yields the Monge-Ampere type equation

ρ0(x) = ρs(Fs(x))|JFs(x)| ρ0 − a.e.,

where JFs(x) = detDFs(x) is the (approximate) Jacobian of Fs and

∂

∂s

∣∣∣s=0

(DFs) = D2H|DuD2u.



Remarks and related developments

1. A posteriori, it is possible to relax the q-separation hypothesis a la Gigli’12 to extend these results provided γ ∈ Γ(µ0, µ1) exists with ` > 0holding γ-a.e.

2. An independent alternative approach to solving the dual problem incase q = 1 was announced a few months later by Kell and Suhr ’18+,following Suhr ’16+’s earlier study of that case (which was in turn inspiredby work on the relativistic heat equation by Brenier ’03, M. & Puel ’09,and Bertrand & Puel ’13 with Pratelli ’18)

3. Optimal transport with respect to smooth, strictly convex Hamiltonianswas studied in varying degrees of generality Bernard & Buffoni ’06, Villani’09, Fathi & Figalli ’10 and Agueh ’02-, Ohta ’09, Lee ’13, Kell ’17,Schachter ’17.

4. Not much beyond #2 above on singular Lagrangians, apart from thesubRiemannian case investigated by Ambrosio & Rigot ’04, Agrachev &Lee ’09, Figalli & Rifford ’10 and Lee, Li & Zelenko ’16, Balogh, Kristaly& Sipos ’18.


Conclusions: optimal transport relates gravity to entropy

1. Fractional powers 0 < q < 1 of the time-separation `(x , y) come from aLagrangian L, smooth and strictly(!) convex away from the light cone.

2. Optimal transport with respect to this cost lifts the geometry fromspacetime events M to probability measures on M.

3. q-separation of the target and source makes this transportation problemand its dual analytically tractable.

4. Convexity properties of Boltzmann’s entropy along timelike geodesics ofprobability measures provide a robust formulation of the strong energycondition of Hawking and Penrose ’70 — and via Mondino & Suhr 18+’sparallel work, of Einstein’s field equations.

5. This provides a new approach to gravity without smoothness — muchdesired in view of the singularity theorems from general relativity.

6. Whereas the second law of thermodynamics is encoded in the firsttime-derivative of entropy, the Einstein equations of gravity are encoded inits second time-derivative along q-geodesics.

THANK YOU!


Conclusions: optimal transport relates gravity to entropy

1. Fractional powers 0 < q < 1 of the time-separation `(x , y) come from aLagrangian L, smooth and strictly(!) convex away from the light cone.

2. Optimal transport with respect to this cost lifts the geometry fromspacetime events M to probability measures on M.

3. q-separation of the target and source makes this transportation problemand its dual analytically tractable.

4. Convexity properties of Boltzmann’s entropy along timelike geodesics ofprobability measures provide a robust formulation of the strong energycondition of Hawking and Penrose ’70 — and via Mondino & Suhr 18+’sparallel work, of Einstein’s field equations.

5. This provides a new approach to gravity without smoothness — muchdesired in view of the singularity theorems from general relativity.

6. Whereas the second law of thermodynamics is encoded in the firsttime-derivative of entropy, the Einstein equations of gravity are encoded inits second time-derivative along q-geodesics.

THANK YOU!Robert J McCann (Toronto) Entropic displ. convexity ⇔ strong energy Toronto 8 Dec 2019 28 / 28

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Displacement Convexity of Boltzmann's Entropy ... › mccann › Talk2.pdf · between x and y (as in the twin paradox). Apart from sign restrictions, this is quite analogous to the