Lorentzian Aubry Set in Compact Spacetimes

• The Lorentzian Aubry set is defined as the subset of future-pointing unit tangent vectors whose calibrated pregeodesics maximize proper time in a class A spacetime.
• It replaces action minimization with proper time maximization and uses Lorentzian calibrations in place of weak KAM solutions to characterize invariant maximizing measures.
• The structure exhibits graph properties with Lipschitz regularity, linking the supports of invariant measures to calibrated orbits and paralleling classical Aubry–Mather theory.

Searching arXiv for the specified paper and closely related Lorentzian Aubry–Mather work. First, I’ll confirm the main source paper by arXiv id, then check for directly related follow-up work by Suhr and others. Using the arXiv search tool now. The Lorentzian Aubry set is the Lorentzian analogue of the Aubry set from classical Aubry–Mather theory, formulated for the length functional of causal curves on compact Lorentzian manifolds. In the setting of a class A spacetime and a cohomology class $\alpha \in (\mathfrak T^*)^\circ$, it is naturally realized as the subset $V(\tau)\subset T^1M$ consisting of future-pointing tangent vectors whose pregeodesics are calibrated by a calibration $\tau$ representing $\alpha$; the paper does not single out a separate symbol named “Aubry set,” but identifies $V(\tau)$ as the structure playing precisely that role (Suhr, 2011). The construction replaces action minimization by proper-time maximization, stable norms by stable time separation, and weak KAM solutions by Lorentzian calibrations.

1. Geometric setting and dynamical framework

Suhr works on smooth time-oriented Lorentzian manifolds $(M,g)$ in a restricted causal class designed to parallel the compact Tonelli setting. A class A spacetime is a closed spacetime $(M,g)$ such that $M$ is time-orientable, $M$ is vicious, and the Abelian cover

$\bar M:=\tilde M/[\pi_1(M),\pi_1(M)]$

is globally hyperbolic (Suhr, 2011). Here vicious means that every point lies on a timelike loop, or equivalently that $V(\tau)\subset T^1M$0 for some, hence every, $V(\tau)\subset T^1M$1.

The variational object is the Lorentzian length of a future-pointing piecewise $V(\tau)\subset T^1M$2 causal curve,

$V(\tau)\subset T^1M$3

together with the time separation

$V(\tau)\subset T^1M$4

with $V(\tau)\subset T^1M$5 when no causal curve joins $V(\tau)\subset T^1M$6 to $V(\tau)\subset T^1M$7. A future-pointing causal curve is a maximizer if it attains this supremum between its endpoints. Maximizers are geodesics up to reparametrization.

To avoid incompleteness issues, the paper does not use the ordinary geodesic flow. Instead it introduces the pregeodesic flow $V(\tau)\subset T^1M$8 by reparametrizing all geodesics with respect to a fixed complete background Riemannian metric $V(\tau)\subset T^1M$9. For each geodesic $\tau$0 with initial velocity $\tau$1, one lets $\tau$2 be its reparametrization by constant $\tau$3-speed and defines

$\tau$4

This is a complete flow on $\tau$5, and the discussion is restricted to the unit tangent bundle $\tau$6 of $\tau$7. The future-directed phase space is the set of future-pointing causal, or timelike, unit vectors.

2. Stable time separation and asymptotic homology

The stable time geometry is encoded by the stable time cone $\tau$8, defined as the closure of the cone generated by homology classes of future-pointing loops. It is a closed convex cone. On the Abelian cover $\tau$9, the homology difference $\alpha$0 is defined via integration of a basis of closed $\alpha$1-forms.

For class A spacetimes there exists a unique function

$\alpha$2

with four structural properties: approximation of time separation on the cone interior away from the boundary, positive homogeneity, concavity, and upper semicontinuity at the boundary (Suhr, 2011). Explicitly, for every $\alpha$3 there exists $\alpha$4 such that

$\alpha$5

where

$\alpha$6

The homogeneity and concavity are

$\alpha$7

and

$\alpha$8

This function is the Lorentzian analogue of Mather’s $\alpha$9-function. Its concavity reflects the fact that the theory maximizes length rather than minimizing action.

For a future-pointing curve $V(\tau)$0, the rotation vector is

$V(\tau)$1

If $V(\tau)$2 is an admissible sequence of maximizers with $V(\tau)$3, then

$V(\tau)$4

Thus $V(\tau)$5 admits a dynamical characterization as the asymptotic maximal proper time per unit parameter along maximizers with prescribed homological direction.

3. Invariant measures, rotation vectors, and Lorentzian Mather sets

Let $V(\tau)$6 denote the finite $V(\tau)$7-invariant Borel measures supported in future-pointing $V(\tau)$8-unit vectors. For $V(\tau)$9, the rotation vector $(M,g)$0 is defined by

$(M,g)$1

for every closed $(M,g)$2-form $(M,g)$3 representing $(M,g)$4. The average length is

$(M,g)$5

A central structural statement is that

$(M,g)$6

and for each $(M,g)$7,

$(M,g)$8

Hence $(M,g)$9 is the maximal average length among invariant measures with prescribed homology, and the measures attaining this supremum are the Lorentzian Mather measures (Suhr, 2011). The paper also proves that if $(M,g)$0, then the pregeodesic flow admits at least $(M,g)$1 distinct maximal ergodic measures.

For each $(M,g)$2, the set of $(M,g)$3-maximizing invariant measures is

$(M,g)$4

The corresponding Mather set is the union of supports

$(M,g)$5

This is the Lorentzian analogue of the classical Mather set: the union of supports of invariant maximizing measures for a given cohomology class.

4. Duality, calibrations, and calibrated pregeodesics

The dual stable time cone is

$(M,g)$6

Since $(M,g)$7 is positively homogeneous and concave, the paper defines its dual on $(M,g)$8 by

$(M,g)$9

A second dual quantity is built from $M$0-forms. For a covector $M$1,

$M$2

and for a $M$3-form $M$4,

$M$5

Then

$M$6

A class $M$7 satisfies $M$8 if and only if it contains a smooth representative $M$9 such that $M$0 is everywhere future-directed timelike; equivalently, $M$1 carries a temporal function on $M$2. Precisely the interior of the dual cone has this property:

$M$3

On $M$4 one has the identification

$M$5

For $M$6, a function $M$7 is $M$8-equivariant if

$M$9

An $\bar M:=\tilde M/[\pi_1(M),\pi_1(M)]$0-pseudo-time function is one which, near each point, satisfies

$\bar M:=\tilde M/[\pi_1(M),\pi_1(M)]$1

in some convex normal neighborhood $\bar M:=\tilde M/[\pi_1(M),\pi_1(M)]$2. A calibration representing $\bar M:=\tilde M/[\pi_1(M),\pi_1(M)]$3 is then an $\bar M:=\tilde M/[\pi_1(M),\pi_1(M)]$4-equivariant Lipschitz function

$\bar M:=\tilde M/[\pi_1(M),\pi_1(M)]$5

which is a $\bar M:=\tilde M/[\pi_1(M),\pi_1(M)]$6-pseudo-time function. Existence is proved by a Busemann-type formula: if $\bar M:=\tilde M/[\pi_1(M),\pi_1(M)]$7 is a smooth closed $\bar M:=\tilde M/[\pi_1(M),\pi_1(M)]$8-form representing $\bar M:=\tilde M/[\pi_1(M),\pi_1(M)]$9 and $V(\tau)\subset T^1M$00 is a primitive of $V(\tau)\subset T^1M$01, then

$V(\tau)\subset T^1M$02

is a calibration representing $V(\tau)\subset T^1M$03 (Suhr, 2011).

A pregeodesic $V(\tau)\subset T^1M$04 is calibrated by $V(\tau)\subset T^1M$05 if for some, hence any, lift $V(\tau)\subset T^1M$06,

$V(\tau)\subset T^1M$07

The set

$V(\tau)\subset T^1M$08

collects the tangent vectors to calibrated pregeodesics. Every calibrated pregeodesic is a maximizer.

5. The Lorentzian Aubry set

For $V(\tau)\subset T^1M$09, the set $V(\tau)\subset T^1M$10 is the natural Lorentzian analogue of the Aubry set. The paper states that the terminology “Aubry set” is not introduced as a separate symbol, but that the structure given by $V(\tau)\subset T^1M$11 together with the graph theorems is precisely the Lorentzian analogue. In that sense, the Lorentzian Aubry set associated with $V(\tau)\subset T^1M$12 is the subset of the unit future-directed timelike tangent bundle consisting of vectors tangent to pregeodesics calibrated by a calibration representing $V(\tau)\subset T^1M$13 (Suhr, 2011).

The relation to maximizing measures is explicit. If $V(\tau)\subset T^1M$14 is a calibration representing $V(\tau)\subset T^1M$15, then

$V(\tau)\subset T^1M$16

Thus all pregeodesics in the support of an $V(\tau)\subset T^1M$17-maximizing measure are calibrated by $V(\tau)\subset T^1M$18, and in particular $V(\tau)\subset T^1M$19. Conversely, if $V(\tau)\subset T^1M$20 is a future-pointing maximizer calibrated by $V(\tau)\subset T^1M$21, then all limit measures of $V(\tau)\subset T^1M$22 lie in $V(\tau)\subset T^1M$23. Moreover, the tangent curve remains uniformly separated from the light cone:

$V(\tau)\subset T^1M$24

Therefore calibrated maximizers are strictly timelike, and their asymptotic behavior is described by $V(\tau)\subset T^1M$25-maximizing invariant measures.

These properties reproduce the characteristic features of the classical Aubry set. The Lorentzian Aubry set is the union of calibrated orbits, contains the supports of maximizing measures, and is organized by the dual functional $V(\tau)\subset T^1M$26. A plausible implication is that $V(\tau)\subset T^1M$27 functions simultaneously as a dynamical, geometric, and measure-theoretic core of the maximizing dynamics for the cohomology class $V(\tau)\subset T^1M$28.

6. Graph theorems, comparison with classical theory, and optimality

The graph property is one of the main structural outcomes. For each $V(\tau)\subset T^1M$29, the projection $V(\tau)\subset T^1M$30 is injective on $V(\tau)\subset T^1M$31, and the inverse on its image is $V(\tau)\subset T^1M$32-Hölder continuous:

$V(\tau)\subset T^1M$33

On the uniformly timelike part

$V(\tau)\subset T^1M$34

the inverse is Lipschitz. For $V(\tau)\subset T^1M$35 and any calibration $V(\tau)\subset T^1M$36 representing $V(\tau)\subset T^1M$37, the restriction

$V(\tau)\subset T^1M$38

is injective, and there exists $V(\tau)\subset T^1M$39 such that

$V(\tau)\subset T^1M$40

Thus the Lorentzian Aubry set $V(\tau)\subset T^1M$41 is a Lipschitz graph over its projection.

The comparison with classical Aubry–Mather theory is systematic. Classical theory studies a Tonelli Lagrangian and minimizes action; the Lorentzian theory studies the length functional on causal curves and maximizes proper time. Classical $V(\tau)\subset T^1M$42 is convex; $V(\tau)\subset T^1M$43 is concave. Weak KAM solutions are replaced by $V(\tau)\subset T^1M$44-equivariant pseudo-time functions, namely calibrations, and minimizing semistatics are replaced by maximizing timelike geodesics. The stable time cone $V(\tau)\subset T^1M$45 replaces the relevant positive cone of homological directions, while the dual stable time separation $V(\tau)\subset T^1M$46 plays the role analogous to the critical slope governing calibration identities.

The Lorentzian Hedlund examples on the $V(\tau)\subset T^1M$47-torus provide the optimality statements. In these examples, the quotient $V(\tau)\subset T^1M$48 with the induced Lorentzian metric is class A, and the stable time separation is linear on

$V(\tau)\subset T^1M$49

There are three families of parallel timelike lines, each family consisting of globally maximizing geodesics, and outside small tubular neighborhoods causal curves are forced to stay close to these tubes in order to be maximizing. These examples show that the lower bound $V(\tau)\subset T^1M$50 for the number of maximal ergodic measures is sharp, and that the graph regularity cannot in general be improved beyond Lipschitz (Suhr, 2011). This suggests that the Lorentzian Aubry set is not only the formal analogue of the classical Aubry set, but also inherits the same kind of sharp regularity and multiplicity limitations.