THE CAUCHY PROBLEM FOR LORENTZIAN DIRAC OPERATORS UNDER NON-LOCAL BOUNDARY CONDITIONS CHRISTIAN BÄR AND PENELOPE GEHRING

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THE CAUCHY PROBLEM FOR LORENTZIAN DIRAC OPERATORS

UNDER NON-LOCAL BOUNDARY CONDITIONS

CHRISTIAN BÄR AND PENELOPE GEHRING

ABSTRACT. Non-local boundary conditions, such as the Atiyah-Patodi-Singer

(APS) conditions, for Dirac operators on Riemannian manifolds are well under-

stood while not much is known for such operators on spacetimes with timelike

boundary. We deﬁne a class of Lorentzian boundary conditions that are local in

time and non-local in the spatial directions and show that they lead to a well-

posed Cauchy problem for the Dirac operator. This applies in particular to the

APS conditions imposed on each level set of a given Cauchy temporal function.

INTRODUCTION

In General Relativity spacetime is mathematically modeled by a, generally curved,

Lorentzian manifold. To describe wave propagation we must be able to solve initial

value problems for wave equations. If the spacetime has no boundary, this is always

possible if the underlying manifold is globally hyperbolic. This means that there

exist Cauchy hypersurfaces on which initial values can be imposed. Well-posedness

of the Cauchy problem for linear second order equations can e.g. be found in [11].

It is then not hard to also deduce it for Dirac equations, which are of ﬁrst order.

Assuming for a moment that the spacetime is spatially compact, i.e. the Cauchy hy-

persurfaces are compact manifolds without boundary, we can consider the space-

time region between two (smooth spacelike) Cauchy hypersurfaces, one lying in

the future of the other. The Cauchy hypersurfaces being closed Riemannian mani-

folds, their Dirac operators are elliptic and, in the selfadjoint case, spectral projec-

tors make sense. We can therefore impose the famous Atiyah-Patodi-Singer (APS)

boundary conditions on both Cauchy hypersurfaces. Bär and Strohmaier showed

that these spatial boundary conditions turn the Dirac operator into a Fredholm op-

erator and gave a geometric index formula ([14,15]). As an application, they com-

puted the chiral anomaly in algebraic quantum ﬁeld theory on curved spacetimes in

[13]. Bär and Hannes [12] investigated to what extend these boundary conditions

can be replaced by more general ones and how the index then changes. An anal-

ogous result to [14] was obtained by Shen and Wrochna ([36]) for asymptotically

static spacetimes with only one spacelike Cauchy boundary hypersurface.

The Anti-de Sitter (AdS) and the asymptotically AdS spacetimes became increas-

ingly important in recent years - especially in the context of studying the properties

of Green-hyperbolic operators like the wave, the Klein-Gordon, or the Dirac oper-

ator; see for example [4,30,40,41]. Green hyperbolic here means that advanced

and retarded Green’s operators exist; this can be deduced from well-posedness of

arXiv:2210.15052v1 [math.DG] 26 Oct 2022

2 CHRISTIAN BÄR AND PENELOPE GEHRING

the Cauchy problem. The manifolds considered in these results are spacetimes with

timelike boundary, i.e. the Cauchy hypersurfaces have a boundary themselves. Fur-

ther studies of Green-hyperbolic operators on such spacetimes were accomplished,

for example, in [20–22,25,27]. These results are concerned with local boundary

conditions. Since analytically reasonable local boundary conditions do not always

exist for ﬁrst-order operators, it is necessary to study global boundary conditions

such as APS boundary conditions.

APS boundary conditions cannot be imposed on the boundary of the spacetime

itself because this boundary is Lorentzian rather than Riemannian. To overcome

this diﬃculty, Drago, Große and Murro suggested in [23] to choose a time function

and to impose APS on each level set. They stated well-posedness of the Cauchy

problem for the spinorial Dirac operator under slicewise APS conditions assuming

a few further conditions on the geometry of the spacetime. However, it seems that

there is a gap in their existence proof.

We also take this local-in-time, global-in-space approach and ﬁx a time function.

The boundary conditions which we impose on the level sets need not be APS but

APS is included in our class of admissible boundary conditions. Another prominent

example consists of the transmission conditions - this may potentially allow for cut-

and-paste arguments.

In Section 1, we recall the most important facts on elliptic boundary problems for

Dirac-type operators on compact Riemannian manifolds with boundary. This the-

ory was initiated by Atiyah, Patodi, and Singer in their famous paper [3]. Here they

introduced the APS boundary conditions. There is a huge literature on this topic.

We follow the approach laid out by Bär and Ballmann in [7,8] because it describes

all possible boundary conditions systematically.

This describes what can be done on a ﬁxed spacelike Cauchy hypersurface. We

then discuss families of boundary value problems in Section 2. We need to study

such families as the time function sweeps out our spacetime.

Next,we recall the most important facts of spin geometry and the spinorial Dirac

operator on spacetimes with timelike boundary in Section 3. We deﬁne what we

mean by an admissible boundary conditions.

Section 4is the core of the present paper. Here we prove well-posedness of the

Cauchy problem for the spinorial Dirac operator under admissible boundary condi-

tions, see Theorem 4.1.

We apply these results in Section 5and study the support of the solutions to the

Cauchy problem. The result essentially says that a wave propagates with the speed

of light at most. There is an interesting violation of this principle, however. As

soon as the wave hits the boundary somewhere, the whole boundary radiates oﬀ

instantenously (w.r.t. the given time function). This is due to the global nature

of the boundary conditions and violates the causal principle that no signal should

propagate fast than with the speed of light. We show by example that this eﬀect

really occurs.

Then we use well-posedness together with ﬁnite propagation speed to construct

advanced and retarded Green’s operators.

CAUCHY PROBLEM UNDER NON-LOCAL BOUNDARY CONDITIONS 3

We conclude the paper by discussing some examples in Section 6. In particular, we

introduce Grassmannian boundary conditions which are special admissible bound-

ary conditions, but the assumptions are easier to check in practice, see Theorem 6.1.

APS boundary conditions fall into this class.

Acknowledgments. We would like to thank Lashi Bandara, Nicoló Drago, Nadine

Große, Sebastian Hannes, Rubens Longhi, Jan Metzger, Simone Murro, Miguel

Sánchez, and Mehran Seyedhosseini for helpful discussions.

This research is supported by the International Max Planck Research School for

Mathematical and Physical Aspects of Gravitation, Cosmology and Quantum Field

Theory and by the focus program on Geometry at Inﬁnity (SPP 2026 funded by

Deutsche Forschungsgemeinschaft).

1. BOUNDARY CONDITIONS FOR DIRAC-TYPE OPERATORS ON RIEMANNIAN

MANIFOLDS

In this preliminary section we collect the relevant facts about boundary conditions

for Dirac-type operators. Most of the material is well known which is why we just

display the results without proof. We mostly follow [7,8]. In [9] the theory is

developed for general elliptic ﬁrst-order operators, but we will not need this level

of generality.

Throughout this section let (Σ, 𝑔)be a compact Riemannian manifold with smooth

boundary 𝜕Σ. We denote its unit conormal ﬁeld by ŋ. Furthermore, let (𝐸, ℎ𝐸)→Σ

be a Hermitian vector bundle. Recall that a diﬀerential operator 𝐷∶𝐶∞(Σ, 𝐸)→

𝐶∞(Σ, 𝐸)of order one is said to be of Dirac type if its principal symbol 𝜎𝐷satisﬁes

the Cliﬀord relations

𝜎𝐷(𝜁)𝜎𝐷(𝜉) + 𝜎𝐷(𝜉)𝜎𝐷(𝜁) = −2𝑔(𝜁, 𝜉)⋅id𝐸𝑥,

for all 𝑥∈ Σ and 𝜁, 𝜉 ∈𝑇∗

𝑥Σ.

The Riemannian spin Dirac operator acting on spinor ﬁelds is an important example

of a Dirac-type operator. More generally, Dirac operators in the sense of Gromov

and Lawson as introduced in [26,31] are of Dirac-type.

Dirac-type operators are elliptic, i.e. 𝜎𝐷(𝜁)is invertible for all 𝑥∈ Σ and all 𝜁∈

𝑇∗

𝑥Σ⧵{0}, since by the Cliﬀord relations the inverse is explicitly given as

𝜎𝐷(𝜁)−1 = − 𝜁−2

𝑔𝜎𝐷(𝜁).

Before discussing boundary conditions for Dirac-type operators, let us spend some

time on introducing the maximal and minimal domain of an operator and the range

of the restriction map to the boundary mapping from these domains.

The maximal extension 𝐷max is deﬁned to be the distributional extension of 𝐷,

restricted to the maximal domain

dom(𝐷max) ∶= {𝜓∈𝐿2(Σ, 𝐸); 𝐷𝜓 ∈𝐿2(Σ, 𝐸)}.

The maximal domain together with the graph norm deﬁned by

⋅2

𝐷∶= 𝐷⋅2

𝐿2+⋅2

𝐿2

4 CHRISTIAN BÄR AND PENELOPE GEHRING

forms a Banach space.

The minimal extension 𝐷min is the closure of 𝐷when 𝐷is given the domain 𝐶∞

cc (Σ, 𝐸).

Here 𝐶∞

cc (Σ, 𝐸)is the space of smooth sections with compact support contained in

the interior of Σ, i.e. supp 𝜓∩𝜕Σ=∅for 𝜓∈𝐶∞

cc (Σ, 𝐸). Hence the domain of

𝐷min, called the minimal domain, is the closure of 𝐶∞

cc (Σ, 𝐸)with respect to the

graph norm of 𝐷. The minimal domain dom(𝐷min)equipped with the graph norm

⋅𝐷is a Banach space as well.

Deﬁnition 1.1. A ﬁrst-order diﬀerential operator 𝐴∶𝐶∞(𝜕Σ, 𝐸𝜕Σ)→𝐶∞(𝜕Σ, 𝐸𝜕Σ)

is called a boundary operator for 𝐷if its principal symbol is given by

𝜎𝐴(𝑥, 𝜁) = 𝜎𝐷(𝑥, ŋ(𝑥))−1◦𝜎𝐷(𝑥, 𝜁)

for all 𝑥∈𝜕Σand 𝜁∈𝑇∗

𝑥𝜕Σ.

Remark 1.2. (a) The boundary operator is unique up to zero-order terms. It can be

chosen such that it is selfadjoint and that it anticommutes with 𝜎𝐷(ŋ), see Lemma 2.2

in [8]. For Dirac operators in the sense of Gromov and Lawson there is a natural

choice of a selfadjoint boundary operator 𝐴which anticommutes with 𝜎𝐷(ŋ). This

operator has a lower order term involving the mean curvature of the boundary, see

for example [8]. For the spin Dirac operator this can also be seen directly, see (15).

(b) Since 𝜕Σis compact and without boundary, the boundary operator 𝐴, if chosen

selfadjoint, has discrete real spectrum. The case of noncompact 𝜕Σhas been studied

in [28] under suitable geometric conditions.

Assumption 1.3. For the rest of this section, we assume that 𝐷is a selfadjoint

Dirac-type operator and that 𝐴is a selfadjoint boundary operator for 𝐷which an-

ticommutes with 𝜎𝐷(ŋ).

Let 𝜒+(𝐴)∶ 𝐿2(𝜕Σ, 𝐸𝜕Σ)→𝐿2(𝜕Σ, 𝐸𝜕Σ)and 𝜒−(𝐴) ∶ 𝐿2(𝜕Σ, 𝐸𝜕Σ)→𝐿2(𝜕Σ, 𝐸𝜕Σ)

be the spectral projections onto the spectral subspaces corresponding to the positive

and the nonpositive eigenvalues of 𝐴, respectively. These projections are pseudo-

diﬀerential operators of order zero and therefore

𝜒±(𝐴)𝐻𝑠(𝜕Σ, 𝐸𝜕Σ)

are closed subspaces of the Sobolev spaces 𝐻𝑠(𝜕Σ, 𝐸𝜕Σ)for all 𝑠∈ℝ.

Deﬁnition 1.4. We deﬁne the check space corresponding to the boundary operator

𝐴as

𝐻(𝐴) ∶= 𝜒−(𝐴)𝐻1

2(𝜕Σ, 𝐸𝜕Σ)⊕ 𝜒+(𝐴)𝐻−1

2(𝜕Σ, 𝐸𝜕Σ),

with norm

𝜓2

𝐻(𝐴)∶= 𝜒−(𝐴)𝜓2

𝐻1

+

𝜒+(𝐴)𝜓

2

𝐻−1

The check space arises naturally as the image of the extension of the trace map

𝑅∶𝐶∞(Σ, 𝐸)→𝐶∞(𝜕Σ, 𝐸𝜕Σ), 𝜓 ↦𝜓𝜕Σto the maximal domain of 𝐴. More

precisely:

Theorem 1.5 (Theorem 6.7 in [7]).The following holds:

(1) 𝐶∞(Σ, 𝐸)is dense in dom(𝐷max)with respect to ⋅𝐷.

CAUCHY PROBLEM UNDER NON-LOCAL BOUNDARY CONDITIONS 5

(2) The trace map extends uniquely to a continuous surjection

𝑅∶ dom(𝐷max)→̌

𝐻(𝐴)

with kernel ker 𝑅= dom(𝐷min). In particular, 𝑅induces an isomorphism

𝐻(𝐴) ≅ dom(𝐷max)

dom(𝐷min).

(3) For all 𝜙, 𝜓 ∈ dom(𝐷max)

∫Σ

ℎ𝐸(𝐷max𝜙, 𝜓) − ℎ𝐸(𝜙, 𝐷max𝜓) d𝜇Σ= − ∫𝜕Σ

ℎ𝐸(𝜎𝐷(ŋ)𝑅𝜙, 𝑅𝜓) d𝜇𝜕Σ.(1)

(4) 𝐻1(Σ, 𝐸) ∩ dom(𝐷max)={𝜓∈ dom(𝐷max); 𝑅𝜓 ∈𝐻1

2(𝜕Σ, 𝐸𝜕Σ)}.

Note that the RHS of (1) is well deﬁned because 𝜎𝐷(ŋ)maps ̌

𝐻(𝐴)to ̌

𝐻(−𝐴),

since it anticommutes with 𝐴. Theorem 1.5 gives us the necessary tools to deﬁne

boundary conditions.

Deﬁnition 1.6. Aboundary condition is a closed linear subspace 𝐵 ⊆ ̌

𝐻(𝐴). The

domains of the associated operators are

dom(𝐷max,𝐵) ∶= {𝜓∈ dom(𝐷max); 𝑅𝜓 ∈𝐵},and

dom(𝐷𝐵) ∶= {𝜓∈ dom(𝐷max) ∩ 𝐻1(Σ, 𝐸); 𝑅𝜓 ∈𝐵}.

We have a 1-1 relation between boundary conditions and closed extensions of 𝐷

between the minimal and the maximal extension. Moreover, (dom(𝐷max,𝐵),⋅𝐷)

is a Banach space for any boundary condition 𝐵. A boundary condition 𝐵satisﬁes

𝐵 ⊆ 𝐻 1

2(𝜕Σ, 𝐸𝜕Σ)if and only if 𝐷𝐵=𝐷max,𝐵. Motivated by (1), the boundary

condition adjoint to 𝐵is deﬁned as

𝐵∗∶= 𝜙∈̌

𝐻(𝐴); ∫𝜕Σ

ℎ𝐸(𝜎𝐷(ŋ)𝜓, 𝜙) d𝜇𝜕Σ= 0 ∀ 𝜓∈𝐵.

We call a boundary condition 𝐵selfadjoint if 𝐵∗=𝐵. By Subsection 7.2 in [7],

the domain of the adjoint of 𝐷max,𝐵 is given by

dom((𝐷max,𝐵)∗)={𝜓∈ dom(𝐷max); 𝜓𝜕Σ∈𝐵∗} = dom(𝐷max,𝐵∗).

In particular, if 𝐵is selfadjoint boundary condition then 𝐷max,𝐵 is a selfadjoint

operator.

Deﬁnition 1.7. Let 𝐵 ⊆ 𝐻 1

2(𝜕Σ, 𝐸𝜕Σ)be a linear subspace such that 𝐵 ⊆ ̌

𝐻(𝐴)

is closed and 𝐵∗⊆ 𝐻 1

2(𝜕Σ, 𝐸𝜕Σ), then 𝐵is called elliptic.

Theorem 1.8 (Theorem 7.11 [7]).Let 𝐵 ⊆ 𝐻 1

2(𝜕Σ, 𝐸𝜕Σ)be a linear subspace.

Then the following are equivalent:

(1) dom(𝐷max,𝐵)⊆ 𝐻1(Σ, 𝐸)and dom(𝐷max,𝐵∗)⊆ 𝐻1(Σ, 𝐸);

(2) 𝐵is elliptic.

Moreover, for any elliptic boundary condition 𝐵the adjoint boundary condition 𝐵∗

is elliptic as well.

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THECAUCHYPROBLEMFORLORENTZIANDIRACOPERATORSUNDERNON-LOCALBOUNDARYCONDITIONSCHRISTIANBÄRANDPENELOPEGEHRINGABSTRACT.Non-localboundaryconditions,suchastheAtiyah-Patodi-Singer(APS)conditions,forDiracoperatorsonRiemannianmanifoldsarewellunder-stoodwhilenotmuchisknownforsuchoperatorsonspacetimeswithtimeli...

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THE CAUCHY PROBLEM FOR LORENTZIAN DIRAC OPERATORS UNDER NON-LOCAL BOUNDARY CONDITIONS CHRISTIAN BÄR AND PENELOPE GEHRING

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