# Comment on “Covariant quantum field theory of tachyons”

Krzysztof Jodłowski<EMAIL_ADDRESS>Particle Theory and Cosmology
Group, Center for Theoretical Physics of the Universe, Institute for Basic
Science (IBS), Daejeon, 34126, Korea

###### Abstract

Recently, Paczos et al. [1] proposed a covariant quantum field theory for free
and interacting tachyon fields. We show that the proposed Feynman propagator
is not Lorentz invariant, proper asymptotic (in/out) tachyon states do not
exist, and the proposed S-matrix describing interactions of tachyons and
subluminal matter is ill-defined. Since tachyons behave as bosons, interacting
tachyons may also self-interact, e.g., any interaction with ordinary matter
generates such terms. As a result, the physical vacuum, instead of being at
the origin of the potential, may correspond to the proper minimum of the
tachyon potential, or such state does not exist at all. Our analysis indicates
that quantum tachyon field does not describe a physical on-shell particle with
negative mass squared.

††preprint: CTPU-PTC-24-xx

##  I Introduction

Many works have investigated tachyons, superluminal particles characterized by
a negative mass squared parameter, due to their peculiar behavior, e.g.,
causality paradoxes or nondeterministic behavior. In fact, tachyons are
usually regarded as unwelcome artifacts, which signal inconsistency or at
least instability of the theory. Recently, however, tachyons have been studied
as a major part of the program to derive the fundamental postulates of quantum
mechanics (QM) by invoking the Galileo’s principle of relativity, provided
that the superluminal reference frames are also allowed [2]. While such
inertial observers are usually dismissed as unphysical, Ref. [2] has shown
that they can not only be mathematically viable, but also lead to
nondeterministic physics akin to QM, as they require a wave-like description
of Nature using complex numbers. Moreover, Ref. [3] has shown that the proper
formulation of superluminal reference frames must invoke field theory. In
fact, both classical [3] and quantum [1] field theory (QFT) of tachyons have
been developed. Since QFT is the most fundamental description of physical
interactions, the ultimate test of such a program is construction of a
consistent QFT of tachyons. If our results are correct, they indicate that
such a program cannot succeed.

The goal of this paper is to closely examine the expressions describing QFT of
tachyons given in the “Interacting theory” section of [1]. We aim to answer
the following questions: i) Is the proposed formalism consistent with the
robust standard QFT framework [4, 5, 7]?111Which is based on only a few
assumptions: relativistic covariance of fields, locality of interactions, and
Lorentz invariance, unitarity and cluster decomposition property of the S
matrix. ii) Is the reaction rate for the process of tachyon
emission/absorption shown in Fig. 1 of Ref. [1] well-defined?, and iii) Since
tachyons satisfy the Bose statistics222A scalar field is the only finite
irreducible unitary representation of the Poincare group for $m^{2}<0$. which
principle forbids its self-interactions, represented by cubic, quartic, etc.
terms in its potential? Due to quantum fluctuations, such terms could modify
the vacuum structure, i.e., what protects the origin of the tachyon potential,
corresponding to negative mass squared excitations, as the proper vacuum?
These questions are particularly relevant, since it have been suggested in [1,
6] that tachyonic fields might shed light on the electroweak symmetry breaking
(the Higgs mechanism). We show this cannot be the case since the proposed QFT
of interacting tachyons violates principle of locality of interactions in an
incurable way - asymptotic states of interacting tachyons do not exist.
Therefore, negative mass squared particles likely do not exist, because the QM
description of their interactions does not make sense.

##  II Tachyon interactions

We closely examine the expressions describing QFT of tachyons given in the
“Interacting theory” section of [1], and, whenever relevant, we compare them
to the formulas describing subluminal particles [4, 5, 7]. In the following,
we will use the elementary fact that the sign of the energy,
$\mathrm{sgn}(x^{0})=\theta(x^{0})-\theta(-x^{0})$, where $\theta$ is the
Heaviside function, is not Lorentz invariant (LI) for spacelike four-vector
$x$. In particular, the time-ordering operator is frame dependent, therefore
so is the Feynman propagator (FP) defined as the two-point time-ordered
correlation function (TOCF).

###  II.1 Feynman propagator

The following equation was given (Eq. 22) in [1] for the FP of a tachyon:

$\displaystyle\int_{\left|\vec{k}\right|>m}\frac{d^{4}k}{(2\pi)^{4}}\frac{i\,e^{-ik(x-y)}}{k^{2}+m^{2}+i\epsilon}=\int\frac{d^{4}k}{(2\pi)^{4}}\,\theta(|\vec{k}|-m)\frac{i\,e^{-ik(x-y)}}{k^{2}+m^{2}+i\epsilon},$
(1)

and it was stated that it is Lorentz invariant (LI). It is clear that it is
true if and only if the condition $\left|\vec{k}\right|>m$ is LI. Indeed, it
is the case for an on-shell tachyon moving with velocity $\vec{v}$, since we
can use Eqs. 1 and 22 from [3] to obtain the magnitude of the momentum
$\vec{k^{\prime}}$ after applying a subluminal boost with velocity $\vec{u}$:

$\displaystyle|\vec{k^{\prime}}|=\frac{|\vec{k}-E\vec{u}|}{\sqrt{1-u^{2}}}\geq\frac{|\vec{k}|-|E|u}{\sqrt{1-u^{2}}}=\frac{m(v-u)}{\sqrt{1-u^{2}}\sqrt{v^{2}-1}}\stackrel{{\scriptstyle?}}{{\geq}}m.$
(2)

Since $v=\left|\vec{v}\right|>1>u=\left|\vec{u}\right|$, we can square both
sides and show that our condition is equivalent to the true statement
$(1-uv)^{2}\geq 0$. We used the reverse triangle inequality for the length of
3-momentum, and the on-shell condition by using $\left|\vec{k}/E\right|=v$. In
particular, an off-shell tachyon with $E=|\vec{k}/u|$ has
$|\vec{k^{\prime}}|=0$, and therefore Eq. 1 is not LI for off-shell tachyons.

An important role of the FP of a scalar field is to obtain LI solution by
convolution of FP with the field describing the initial data. The ordinary FP
(corresponding to the Klein-Gordon (KG) equation with positive mass squared)
accomplishes this by propagating only the positive energy states, both
particles and antiparticles, towards the future, and negative energy states
towards the past. This is implemented by a specific choice of the contour,
which is equivalent to the definition of the two-point TOCF.

In fact, the TOCFs are the main object in QFT, since they allow for
perturbative computation of the S matrix elements due to the LSZ formula. They
also satisfy the Schwinger-Dyson equations, which are then used to derive the
Feynman rules for interactions in position space. In particular, the two-point
TOCFs should be a Green function for the KG operator, and not a solution,
which clearly shows that FP is an off-shell object. However, applying
$(\square-m^{2})$ to Eq. 1 results in

$\displaystyle-i\,\delta^{4}(x-y)-$
$\displaystyle\frac{i\,\delta(x^{0}-y^{0})}{2\pi^{2}}\times$ (3)
$\displaystyle\times\frac{\sin{(m|\vec{x}-\vec{y}|)}-m|\vec{x}-\vec{y}|\,\cos{(m|\vec{x}-\vec{y}|)}}{|\vec{x}-\vec{y}|^{3}}\,,$

instead. Therefore, Eq. 1 not only is not LI, but also does not satisfy the
equations of motion for a TOCF.

This is a serious flaw in the construction of Paczos et al. [1], which calls
into question the consistency of even non-interacting tachyon fields.
Moreover, free fields may be unphysical in the following sense: there is an
positive-definite energy-momentum tensor that can be contracted with the
metric tensor, giving gravitational interactions described at perturbative
level. For this reason, in the next section, we present an independent
argument for why the theory of interacting tachyons is ill-defined.

###  II.2 S matrix

We already noted that constructing the S matrix is problematic. In particular,
the usual definition using the Dyson series cannot be employed since it relies
on the time-ordering operator. Moreover, there is no guarantee that such an
expression is LI - since microcausality is violated, the interacting
Hamiltonian, which is expressed in terms of $\Phi$ and $\partial_{\mu}\Phi$,
has no reason to commute at spacelike separations either. On the other hand,
one can try not to construct the S matrix explicitly, but instead use the LSZ
formalism. We neglect a serious problem that scattering theory is defined only
for stable particles. Tachyons interacting with any particle make it unstable,
which begs the question, how is the S matrix of the known particles
constructed in such case. Since the LSZ formula allows perturbative
computation of the S matrix elements by means of TOCF, let us try to derive it
for the tachyon field, following the reasoning given for the ordinary KG field
[4, 5, 7]. In fact, assuming the existence of asymptotic (in/out) states, all
manipulations can be formally repeated, and it seems there is no change to the
results, provided we restrict the states to these satisfying
$\left|\vec{k}\right|>m$. Unfortunately, as we show below, the in/out states
for tachyons do not exist.

We follow the discussion of Lecture 14 from Ref. [5], in particular pages
160-162. In order to have a physical interpretation of scattering theory, we
need to consider interactions of particles which are far from each another in
space at initial and final times. Therefore, we represent the initial/final
states as wave packets normalized to a finite number because plane waves,
which are momentum eigenstates, cover the whole space (since they are
normalizable only in distributional sense) and thus cannot be used to model
scattering. If one demands that interactions have a finite range, the wave
packet associated with interacting field should have a compact support. This
is impossible for a tachyon field, since by the Paley-Wiener theorem the
Fourier transform of a compact function is analytical in the entire complex
plane. Thus, it cannot vanish on a compact set, which is violated due to the
$\left|\vec{k}\right|>m$ condition. For ordinary particles, nothing forbids
the wave packets to have compact support. This assumption guarantees that the
interacting field behaves as a free field in the $t\to\pm\infty$ limit.
Physically, compact wave packets are necessary since they are part of the
definition of microcausality of a quantum field. Indeed, requiring a set (of
positions) in Euclidean space to be compact is equivalent to making it
bounded.

Non-compact wave packets would be not only unphysical, but also would bring
additional mathematical problems. Strictly speaking, a quantum field is an
operator valued distribution, which results in a proper field (the value of a
field in any spacetime point is well-defined) only after smearing it with a
sufficiently regular test functions. For ordinary KG field, the test functions
are assumed to belong to the Schwartz space and have compact support. Then,
for the quantum fields which is a tempered distribution, i.e., the result of
acting on any test functions from Schwartz space is bounded, operations such
as integration by parts,333In particular, the equations of motion are derived
from the principle of least action under the assumption that fields decay at
the boundary (spatial infinity in field theory) fast enough. This is
guaranteed for ordinary fields because they have compact support. Giving up on
this property, one would need to construct the full formalism from scratch.
Fourier transform, etc., are well-defined. Introducing test functions without
compact support would not keep these desirable properties. Moreover, for
tachyons the LI measure contains non-analytical term $1/\omega_{\vec{k}}$,
which blows up for $|\vec{k}|\to m$, and hence it is not clear if $\Phi$ can
be a tempered distribution.

Let us show that the incompleteness of the tachyon mode expansion leads to a
phenomenon in which an interacting tachyon field does not behave as a free
field at asymptotic times. Therefore, one cannot construct local scattering
theory of tachyons, because these states are not localizable in space -
single-particle states and multiparticle states do not separate from each
other even after allowing for an infinite time of interactions. Let $f(x)$
represent the tachyon wave packet and let $\phi^{\prime}(x)$ be an interacting
state that satisfies $\langle 0|\phi^{\prime}(x)|0\rangle=0$ and $\langle
k|\phi^{\prime}(x)|0\rangle=e^{ikx}\,\theta(|\vec{k}|-m)$. We study matrix
elements of the operator $\phi^{{}^{\prime}f}(t)=i\int
d^{3}\vec{x}(\phi^{\prime}\partial_{0}f-f\partial_{0}\phi^{\prime})$, which
for $f(x)=e^{-ikx}$ and a free field $\phi^{\prime}(x)$ is proportional to the
creation operator $\hat{c}^{\dagger}_{\vec{k}}$. Direct computation shows that
at all times $\phi^{{}^{\prime}f}$ acts as a creation operator for the
momentum eigenstates used in the construction of the wave packet $f(x)$, and
that its overlap with the vacuum is $0$. We must also consider all
multiparticle states. The following overlap between a normalizable state
$\psi$, the operator $\phi^{{}^{\prime}f}(t)$, and the vacuum must vanish in
the infinite time limit, if the in/out states (the states appearing in the S
matrix elements, which are created from the vacuum by the operator
$\phi^{{}^{\prime}f}(t)$) exist:

$\displaystyle\lim_{t\to\pm\infty}\int
d^{3}\vec{x}\,\langle\psi|n\rangle\langle
n|\phi^{{}^{\prime}f}(t)|0\rangle=\lim_{t\to\pm\infty}\int
d^{3}\vec{x}f(x)g(x)e^{-iEt},$ (4)

where $E=\omega_{\vec{p}}-p_{n}^{0}\neq 0$ is a real number and $g(x)$ is a
regular function. The limit exists only if the function $h(x)=f(x)g(x)$ is
integrable. By regularity assumption, $g(x)$ is integrable. In particular, the
limit exists, and is equal to $0$, when $f(x)$ is a compact function. This
corresponds to the ordinary KG case, but is impossible for tachyons. For them,
we use the fact that the Fourier transform of an integrable function is
sufficiently regular (uniformly continuous), which is impossible to satisfy
because of the $\theta(|\vec{k}|-m)$ term. This shows that Eq. 4 does not
vanish for tachyons. Therefore, the multiparticle states produced by the
$\phi^{{}^{\prime}f}(t)$ operator do not get separated in space from the one-
particle wave packet in the infinite time limit. Such states would contribute
to any S matrix element, making it divergent (the limit does not exist).

Therefore, the nonlocality of tachyons has clear observable consequences - it
prohibits the existence of proper asymptotic states, if one uses the
restricted mode expansion. Obviously, one cannot use the full modes expansion
either, as it would lead to exponentially increasing terms in the wave
function normalization. In other words, complex energies would violate the
unitarity of the free theory.

###  II.3 Tachyon potential

Figure 1:  The tachyon potential terms generated by the exchange of virtual
ordinary particles denoted by solid lines. In the top row, we show one (left)
and two-point (right) TOCF, which correspond to tachyon vacuum expectation
value and its self-energy, respectively. The former can be set to $0$ by field
redefinition, while the latter renormalizes the mass squared parameter of the
tachyon, which we assume stays negative. The bottom row presents examples of
the diagrams generating tachyon self-interactions: cubic (left) and quartic
(right).

In this subsection, we study some consequences of introducing tachyon
interactions with ordinary matter, which are represented by fields $\phi$ and
$\psi$, respectively. We will assume the interaction Lagrangian is

$\displaystyle\mathcal{L}=\begin{cases}g\Lambda\,\phi\,|\psi|^{2}&\psi\text{\
is a boson, }\\\ g\,\phi\,\bar{\psi}\psi&\psi\text{\ is a
fermion},\end{cases}$ (5)

where $g$ is a dimensionless coupling, while $\Lambda$ is some energy scale.
The tachyon potential with radiatively generated terms is as follows:

$\displaystyle
V(\phi)=\frac{-m^{2}}{2}\phi^{2}+\frac{\lambda_{3}}{3!}\phi^{3}+\frac{\lambda_{4}}{4!}\phi^{4},$
(6)

where the cubic and quartic terms (which have a mass dimension $\leq 4$, so
perhaps they should be accounted for already at the tree level) not only
describe tachyon self-interactions, but also can change the vacuum structure
due to quantum fluctuations. We note that while a full 1-loop analysis would
require working with the Coleman-Weinberg potential, we expect it would lead
to similar conclusions.

The additional terms in the potential can change its character, from unbounded
to bounded from below, depending on the signs of $\lambda_{3}$, $\lambda_{4}$.
In fact, one studies such potential, e.g., in the Higgs mechanism and in
theories of inflation. Since these theories are not restricted to the
$\left|\vec{k}\right|>m$ modes, one shows that a positive $\lambda_{4}$ leads
to spontaneous electroweak symmetry breaking - the field settles in the proper
minimum of the potential which breaks/hides the gauge symmetry. On the other
hand, negative $\lambda_{4}$ causes the runaway ($|\phi|\to\infty$) in finite
time, which makes the whole theory unphysical. The tachyon case is more
involved, and detailed analysis is beyond the scope of this paper. However,
they at least present another challenge for constructing consistent QFT of
tachyons.

In Fig. 1, we present the relevant diagrams generated at 1-loop by interaction
given by Eq. 5. All the diagrams are UV divergent, and to renormalize the
theory, we use the $\overline{\mathrm{MS}}$ scheme with dimensional
regularization. We note that for the fermionic case, evaluation of such
diagrams is given as an exercise in Chapter 51 of [7]. We interpret the
interaction as an effective field theory valid below a high energy scale
$\Lambda^{\prime}\gg m_{\psi}$. Then, we can replace the divergence by a
finite value:
$1/\epsilon+2\log({\mu/m_{\psi}})\to\log({\Lambda^{\prime}/m_{\psi}})$. The
result is

$\displaystyle\lambda_{3}\simeq\begin{cases}\frac{g^{3}\Lambda^{3}}{4\pi^{2}}\log{(\frac{\Lambda^{\prime}}{m_{\psi}})}&\psi\text{\
is a boson, }\\\
\frac{-g^{3}}{4\pi^{2}}\log{(\frac{\Lambda^{\prime}}{m_{\psi}})}&\psi\text{\
is a fermion},\end{cases}$ (7)
$\displaystyle\lambda_{4}\simeq\begin{cases}\frac{g^{4}\Lambda^{4}}{4\pi^{2}}\log{(\frac{\Lambda^{\prime}}{m_{\psi}})}&\psi\text{\
is a boson, }\\\
\frac{-3g^{4}}{\pi^{2}}\log{(\frac{\Lambda^{\prime}}{m_{\psi}})}&\psi\text{\
is a fermion}.\end{cases}$

Note the opposite signs of the terms generated by bosons and fermions. This
difference stems from the fact that a closed fermion loop requires the
reordering of the anticommuting fermionic fields, resulting in a $(-1)$
factor. Our result is consistent with the fact that the top quark contribution
to the Coleman-Weinberg potential of the Higgs is negative. Eq. 7 implies that
the tachyon potential has a proper minimum if $\psi$ is a boson, while it
becomes even more unstable if $\psi$ is a fermion. The first case may result
in a Higgs-like mechanism transforming tachyon to an ordinary particle, while
the second one presents another problem to be solved if one desires to
describe stable particles with negative mass squared.

##  III Discussion and conclusions

The theory proposed in Ref. [1] is not LI, and the existence of S matrix with
physical properties has not been established. A serious obstacle is the lack
of LI of the sign of the energy (or time) component of a space-like four-
vector, which prevents the direct use of formulas used for subluminal
particles. While for subluminal particles, the mass shell, given by the set
$p=(E,\vec{p})$ satisfying $p^{2}=E^{2}-\left|\vec{p}\right|^{2}=m^{2}$, has
two sheets, each characterized in a LI way by the sign of the energy, tachyon
mass shell has only one sheet. Therefore, there is a geometrical obstacle to
the LI of time-ordering or LI definition of which states are initial/final, if
one desires that all tachyon/antitachyon states have positive energy. In
particular, the FP (and any TOCF), S matrix defined as the Dyson series, and
the phase space of tachyons are not well-defined.

In fact, Ref. [1] introduces the past-future symmetric two-state formalism for
this very reason. It is shown there that the amplitude describing the process
illustrated in Fig. 1 therein is LI. On the other hand, amplitudes are not
directly observable. To have a proper physical quantity, the amplitude needs
to be integrated over LI phase space. However, one cannot remove the negative
energy states in a LI way and preserve the notion which states are initial and
which are final - the Sudarshan’s reinterpretation principle alone does not
accomplish this, while the two-state formalism introduces different form of
nondeterminism than ordinary QM. Since superluminal observers were introduced
to deduce QM in the first place, it is puzzling (but not impossible) that a
covariant description of tachyons would require such a radical departure from
the known formalism.

For example, the aforementioned tachyon emission/absorption process is a $1\to
2$ decay in one frame and an inverse $2\to 1$ decay in another. The former is
described by a decay width, which is defined in the rest frame of the decaying
particle, while the latter - by a LI cross-section. These quantities cannot
describe the same process if there is a notion of a prepared initial state. In
ordinary QM, when the initial states are not fixed, a LI reaction rate can be
obtained by averaging over their phase space as well, e.g., one describes in
such a way a particle freeze-out from thermal plasma due to decays and inverse
decays. However, one can, at least in principle, prepare the initial states,
and particle being stable does not depend on the reference frame.

Therefore, to have a physical meaning, the formulas adapted from the “standard
toolkit of QFT” need to be adjusted, and there seems no simple way to do so,
which would not spoil other postulates of QFT. For example, abandoning
relativistic invariance and formulating the theory in a preferred reference
frame in which all negative mass squared states have positive energy could fix
most of these problems. However, the tachyon single-particle states remain
delocalized in space, as argued below.

Paraphrasing Sidney Coleman [5], one can say that, since $E=mc^{2}$, particle
number is not conserved in QFT. Therefore, one cannot consider quantum fields
acting only in a single-particle Hilbert space. Instead, an interacting
quantum field acts nontrivially also on a countable number of multi-particle
states. To interpret the state created from the vacuum by creation operator of
such field as a particle, it is mandatory that the overlap with any
multiparticle state and the vacuum must vanish in the infinite time limit. In
ordinary theory, it does due to Riemann-Lebesgue lemma, because the states can
be localized in space, which is impossible for tachyons due to the
incompleteness of the orthogonal basis. Therefore, interacting tachyon states
do not converge, in the weak sense, to the one-particle states. In particular,
the LSZ asymptotic condition cannot be satisfied, and the expressions used to
compute the S matrix elements are not well-defined. While the incompleteness
of such an expansion, which is directly responsible for the lack of
microcausality of the tachyon field, is well-known, our analysis of the in/out
states in the S matrix construction provides a different perspective on
whether tachyon non-locality has physical consequences.

Tachyons behave differently than ordinary fields also because they turn any
stable field they interact with into unstable one, or at least such
interactions cannot be predicted since the S matrix is ill-defined. Again,
this is different kind of nondeterminism from that exhibited by QM, since QM
allows for stable states, while any QM theory with tachyons does not.

In summary, tachyon non-locality has direct physical consequence, i.e., the S
matrix is ill-defined and it is not even formally LI. Moreover, a particle
interpretation of $\Phi$ is impossible if tachyons are propagating on-shell
states. While the tachyon fields are covariant in the twin space, the
postulate of locality of interactions is permanenty violated by considering a
truncated set of plane-waves as an (incomplete) orthogonal basis. In light of
the above arguments, formulation of a consistent relativistic theory of an
interacting tachyon field seems impossible.

###### Acknowledgements.

This work was supported by the Institute for Basic Science under the project
code, IBS-R018-D1. I thank Hanseul for motivation.

## References

  * [1] J. Paczos, K. Dębski, S. Cedrowski, S. Charzyński, K. Turzyński, A. Ekert, and A. Dragan, “Covariant quantum field theory of tachyons,” arXiv:2308.00450 [quant-ph].
  * [2] A. Dragan and A. Ekert, “Quantum principle of relativity,” New J. Phys. 22 no. 3, (2020) 033038, arXiv:1910.02780 [quant-ph]. [Addendum: New J.Phys. 25, 088002 (2023)].
  * [3] A. Dragan, K. Dębski, S. Charzyński, K. Turzyński, and A. Ekert, “Relativity of superluminal observers in 1 + 3 spacetime,” Class. Quant. Grav. 40 no. 2, (2023) 025013, arXiv:2209.01836 [physics.gen-ph].
  * [4] S. Weinberg, The Quantum Theory of Fields.  Cambridge University Press, 1995.
  * [5] S. Coleman, “Notes from Sidney Coleman’s Physics 253a: Quantum Field Theory,” arXiv:1110.5013 [physics.ed-ph].
  * [6] A. Dragan and A. Ekert, “Reply to the Comment on ‘Quantum principle of relativity’,” New J. Phys. 25 no. 12, (2023) 128002, arXiv:2309.00020 [quant-ph].
  * [7] M. Srednicki, Quantum field theory.  Cambridge University Press, 1, 2007.


# The cause of the difference in the propagation distances between compact and
transient jets in black-hole X-ray binaries

Andrzej A. Zdziarski Nicolaus Copernicus Astronomical Center, Polish Academy
of Sciences, Bartycka 18, PL-00-716 Warszawa, Poland<EMAIL_ADDRESS>Sebastian Heinz Department of Astronomy, University of Wisconsin-Madison, 475
N Charter St, Madison, WI 53726, USA

###### Abstract

Accreting black-hole binaries change their properties during evolution,
passing through two main luminous states, dominated by either hard or soft
X-rays. In the hard state, steady compact jets emitting multiwavelength
radiation are present. Those jets are usually observed in radio, and when
resolved, their extent is $\lesssim\\!10^{15}$ cm. Then, during hard-to-soft
transitions, powerful ejecta in the form of blobs appear. They are observed up
to distances of $\sim\\!10^{18}$ cm, which are $\gtrsim$1000 times larger than
the extent of hard-state jets. On the other hand, estimates of the accretion
rates during most luminous hard states and the hard-to-soft transitions are
very similar, implying that maximum achievable powers of both types of jets
are similar and cannot cause the huge difference in their propagation.
Instead, we explain the difference in the propagation length by postulating
that the ejecta consist of electron-ion plasmas, whereas the hard-state jets
consist mostly of electron-positron pairs. The inertia of the ejecta are then
much higher than those of compact jets, and the former are not readily stopped
by ambient media. A related result is that the accretion flow during the hard
state is of Standard and Normal Evolution (SANE), while it is a Magnetically
Arrested Disk (MAD) during transient ejections. The pairs in hard-state jets
can be produced by collisions of photons of the hard spectrum emitted by hot
accretion flows within the jet base. On the other hand, the X-ray spectra
during the state transitions are relatively soft and the same process produces
much fewer pairs.

revtex4-1.clsrevtex4-2.cls

##  1 Introduction

Accreting black-hole (BH) X-ray binaries (XRBs) have two main luminous X-ray
spectral states, hard and soft (see Done et al. 2007 for a review of their
X-ray properties). In radio, they show then two main types of activity:
compact steady jets in the hard state, and transient, discrete, jets appearing
during transitions from the hard intermediate state (i.e., the softest part of
the hard state, with the X-ray energy index $\alpha>1$, defined by
$F_{\nu}\propto\nu^{-\alpha}$) to the soft state, while there is usually no or
very weak radio emission during the soft state itself, see Fender et al.
(2004) for a review.

A striking difference between the two types of jets is in the distances to
which they are observed to propagate. Compact jets appear always relatively
small, in spite of the duration of their launching from the BH spanning from
weeks to even years. In most cases they remain unresolved in radio. Currently,
compact jets in only four BH XRBs have been resolved in radio. Three of them
are MAXI J1820+070 with the deprojected jet elongation of $\ell_{\rm
max}\approx\\!3\times 10^{13}$ cm at 15 GHz (Tetarenko et al., 2021), Cyg X-1
with $\ell_{\rm max}\approx\\!10^{15}$ cm at 8 GHz (Stirling et al., 2001),
and GRS 1915+105 (assuming the distance of $D\approx 8$ kpc; Reid et al. 2014)
with $\ell_{\rm max}\approx 3\times 10^{15}({\rm 1\,GHz}/\nu)$ (Dhawan et al.,
2000). Then, MAXI J1836–194 has a only rough (due to the distance and
inclination of this source being uncertain) estimate of the length of
$\ell_{\rm max}\sim\\!10^{16}$ cm at 2.3 GHz (Russell et al., 2015), which was
obtained in only one out of several VLBA observations after subtracting the
point source. Theoretically, we expect $\ell_{\rm max}\mathrel{\hbox{\raise
2.15277pt\hbox{$\propto$}\hbox to0.0pt{\hss\lower
2.15277pt\hbox{$\sim$}}}}\nu^{-1}$. Then, not a single discrete extension of a
hard-state jet was detected moving at large distances from the core. A single
case of an ejection was seen from Cyg X-1 at 5 GHz but only at a low distance
of $\approx 5\times 10^{15}$ cm from the core (Fender et al., 2006), as well a
it occurred in the soft state. Occasionally, we see Galactic analogs of radio
lobes at some parsecs from the core, mostly from persistent BH XRBs (e.g., Cyg
X-1; Gallo et al. 2005; Sell et al. 2015), but they are not moving.

On the other hand, transient jets are observed as discrete moving ejecta at
much larger distances, $\sim\\!10^{16}$–$10^{18}$ cm, in spite of the duration
of their launching being of the order of a day (e.g., Carotenuto et al. 2021,
2024). Often both the approaching and the receding components are seen (e.g.,
Mirabel & Rodríguez 1994; Fender et al. 1999; Bright et al. 2020), and they
are sometimes detected even at distances $\gtrsim\\!10^{18}$ cm (Carotenuto et
al., 2021, 2024). Both types of jets often appear during the same
observational campaigns, e.g., for MAXI J1820+070 and MAXI J1348–630. The
cause of the difference in the propagation distances has remained unexplained.

Another difference between the two types of jets is in their radio spectra.
The radio emission of the compact jets is hard, with $\alpha\sim 0$, which is
well explained by the partially synchrotron self-absorbed jet model of
Blandford & Königl (1979), see, e.g., Tetarenko et al. (2021), Zdziarski et
al. (2022b), while that of the transient jets is soft, $\alpha\sim 0.5$ (e.g.,
Rodriguez et al. 1995; Carotenuto et al. 2021), characteristic of optically
thin synchrotron emission by power-law electrons. This difference is well
explained by the observed relatively large lateral sizes of the ejecta (e.g.,
Mirabel & Rodríguez 1994; Rushton et al. 2017), causing the self-absorption
optical depth to be low, see, e.g., equations (15–16) in Zdziarski et al.
(2023).

On the other hand, the bulk Lorentz factors of both types of jets appear
relatively similar. These factors for compact jets remain difficult to
constrain, with only rough estimates of $\Gamma\gtrsim 1.5$–2 (Stirling et
al., 2001; Casella et al., 2010). Tetarenko et al. (2019) gave
$\Gamma=2.6^{+0.8}_{-0.6}$ for Cyg X-1 (note the caveats given in Zdziarski &
Egron 2022). In MAXI J1820+070, Zdziarski et al. (2022b) found $\Gamma\approx
1.5$–4. For transient jets, some jets are observed to decelerate, and we list
here the estimates for the initial Lorentz factor, $\Gamma_{0}$. Corbel et al.
(2005) and Steiner et al. (2012) gave $\Gamma_{0}\approx 1.4$–1.6 in
H1743–322, Steiner & McClintock (2012) found $\Gamma_{0}\gtrsim 1.6$ with no
upper limit for XTE J1550–564, Wang et al. (2003) obtained a good fit for the
same object with $\Gamma_{0}=3$, Wood et al. (2021) found $\Gamma_{0}>2.1$ and
$\Gamma_{0}\approx 1.05\pm 0.02$ for two ejecta of MAXI J1820+070, and
Zdziarski et al. (2023) obtained $\Gamma_{0}\approx 1.8\pm 0.1$ in MAXI
J1348–630. Thus, the transient jets appear to have the velocities similar to
those of the compact steady jets.

Also, existing estimates of the powers, $P_{\rm j}$, of the compact and
transient jets are relatively similar. A theoretical upper limit on $P_{\rm
j}$ is $\sim\dot{M}_{\rm accr}c^{2}$, where $\dot{M}_{\rm accr}$ is the
accretion rate (Davis & Tchekhovskoy 2020 and references therein). The
accretion luminosities, $L$, of the hard state before the state transition and
during it are quite similar, and the accretion efficiency, $\epsilon$, is
theoretically predicted to increase from the hard state to the soft soft one
(Yuan & Narayan, 2014). Then $\dot{M}_{\rm accr}=L/(\epsilon c^{2})$ is
unlikely to substantially increase at the hard-to-soft transition. For compact
jets, published estimates are mostly in the $P_{\rm j}\sim 10^{37-39}$ erg/s
range, e.g., Zdziarski et al. (2022b). In the case of transient jets, there
were some estimates of extreme powers, $P_{\rm j}\sim 10^{41}$ erg/s, in
particular those of Mirabel & Rodríguez (1994) and Carotenuto et al. (2022).
However, these values were revised and found much lower by Zdziarski (2014)
and Zdziarski et al. (2023). In the recent study of Carotenuto et al. (2024),
the kinetic energy of three transient jets were calculated, and they also
imply a $P_{\rm j}\sim 10^{37-39}$ erg/s range.

In a recent work, Sikora & Zdziarski (2023) proposed that the jets produced
during the hard-to-soft state transitions are transient because of an increase
of the accretion rate, $\dot{M}_{\rm accr}$, associated with the transitions.
If the jet before the transition was launched from a magnetically-arrested
accretion disk (MAD; McKinney et al. 2012) and the accumulated magnetic flux
remains constant during the transition (see fig. 1 in Sikora & Zdziarski
2023), the system can cease to be in the MAD state above certain $\dot{M}_{\rm
accr}$, which, in turn, can decollimate the outflow and stop the jet
production. The discrete ejection would correspond then to a part of the jet
produced just before the decollimation. This picture implies that the
transient jets are similar to the compact ones except for a shorter launching
time. This, however, does not explain the dramatic difference in the
propagation distances between the compact and transient jets, which thus is
still unknown. In this work, we attempt to explain this difference. We
concentrate our study on the case of MAXI J1820+070, whose detailed
observations allow relatively reliable estimates of the jet parameters for
both the compact jet and the transient ones.

##  2 The case of MAXI J1820+070

The deprojected elongation of the jet resolved at 15 GHz by VLBA on MJD 58193
(5 days after the source discovery in X-rays, Kawamuro et al. 2018) for the
distance of $D\approx 3.0\pm 0.3$ kpc (as measured by Atri et al. 2020) and
the inclination of $64\arcdeg$ (Wood et al., 2021) is $\ell_{\rm
max}\approx(2.6\pm 0.1)(D/3\,{\rm kpc})10^{13}$ cm (Tetarenko et al., 2021).
Extrapolating it to MJD 58220 based on the radio flux (with a model based on
Blandford & Königl 1979), the estimated length for that day was $\ell_{\rm
max}\approx 4\times 10^{13}$ cm (Zdziarski et al., 2022b). This also
approximately agrees with the time-lag estimates of Zdziarski et al. (2022b),
see their fig. 7, as well as with the estimates based on the breaks in the
power spectra (Tetarenko et al., 2021; Zdziarski et al., 2022b). Then, the
size estimates based on the method of core shift are still lower by a factor
of several (Prabu et al., 2023).

On the other hand, the main approaching ejection in that source, launched
during the hard-to-soft state transition (around MJD 58306; Wood et al. 2021)
was detected in X-rays at the distances from the compact object as large as
$\approx\\!6\times 10^{17}$ cm (Espinasse et al., 2020), which is $>$4 orders
of magnitude longer than the measured length of the hard-state jet. This is in
spite of the similar jet Lorentz factors, see below, and in spite both types
of jets apparently launched in the same directions.

The Lorentz factor of the compact jet was estimated as $\Gamma\approx$1.5–4
(Zdziarski et al., 2022b). Then, Carotenuto et al. (2024) estimated the
initial Lorentz factor as $\Gamma_{0}\approx 2.6^{+0.5}_{-0.4}$ for the main
transient ejection. Thus, there is no indication on which of the jets has the
higher $\Gamma$.

The main method for estimating the jet power in steady, compact, jets is to
use the observed synchrotron flux to calculate the flux of the emitting
electrons and positrons (e±). By assuming the ratio of the energy densities in
the e± to that in the magnetic field (the equipartition parameter), we can
calculate the power in the e± and the field, $P_{B{\rm e}}$. Another component
is the power in ions, $P_{\rm i}$, which depends on the fraction of e± pairs
and on the number of non-emitting electrons. In the case of MAXI J1820+070,
$P_{B{\rm e}}\sim 2\times 10^{37}$ erg s-1 while in the absence of both pairs
and non-emitting electrons, $P_{\rm i}\sim 1\times 10^{39}$ erg s-1, (see eqs.
45–46 of Zdziarski et al. 2022b). Those calculations were based on the fast-
timing and spectral results of Tetarenko et al. (2021), and the underlying
observations were done during the luminous hard state of that source (on MJD
58220, 2018-04-12).

Estimates of the jet power can be compared to the estimated $\dot{M}_{\rm
accr}c^{2}$, where $\dot{M}_{\rm accr}$ is the mass accretion rate. Assuming
isotropy, we have

$\dot{M}_{\rm accr}c^{2}=4\pi F_{\rm accr}D^{2}/\epsilon,$ (1)

where $F_{\rm accr}$ is the bolometric accretion flux, and $\epsilon$ is the
accretion efficiency. An upper limit on the power of both kinds of jets is
then given by

$P_{\rm j,MAD}\approx 1.3a_{*}^{2}\dot{M}_{\rm accr}c^{2},$ (2)

where $a_{*}$ is the BH spin parameter. This limit follows from numerical
simulations of jets launched from Magnetically Arrested Disks (MAD;
Bisnovatyi-Kogan & Ruzmaikin 1974; Narayan et al. 2003), which model gives the
highest jet power among existing ones (Davis & Tchekhovskoy 2020 and
references therein).

Shidatsu et al. (2019) found the unabsorbed 1–100 keV flux in the luminous
hard state to be between $\approx$0.5 and $\approx\\!1.4\times 10^{-7}$ erg
cm-2 s-1. The 0.1–200 keV flux as observed by XMM and INTEGRAL in the luminous
hard state on MJD 58220 (for which the hard-state jet power was estimated
above) was $\approx\\!1.1\times 10^{-7}$ erg cm-2 s-1 (J. Rodi, private
communication; Rodi et al. 2021). We find the estimated bolometric flux of
$\approx\\!1.2\times 10^{-7}$ erg cm-2 s-1, corresponding to $L_{\rm
bol}\approx 1.3(D/3\,{\rm kpc})^{2}10^{38}$ erg s-1. For the BH mass of
$M=6.8{\rm M}_{\sun}$ (Torres et al., 2020; Mikołajewska et al., 2022) and for
the H mass fraction of $X=0.7$, this $L_{\rm bol}$ represents 13% of the
Eddington luminosity. It implies $\dot{M}_{\rm accr}c^{2}\approx 1.3\times
10^{39}(D/3\,{\rm kpc})^{2}(\epsilon/0.1)^{-1}$ erg s-1. The initial estimates
of $a_{*}$ using the continuum method were $a_{*}\approx 0.2^{+0.2}_{-0.3}$
(Guan et al., 2021) and $a_{*}\approx 0.14\pm 0.09$ (Zhao et al., 2021), or
even $a_{*}<0$ (Fabian et al., 2020) at the binary inclination of
$\approx$66–$81\arcdeg$ (Torres et al., 2020). On the other hand, Bhargava et
al. (2021) obtained $a_{*}\approx 0.8$ from timing using the relativistic
precession model. Similarly, Banerjee et al. (2024) obtained $a_{*}=0.77\pm
0.21$. Scaling the power to $a_{*}=0.8$ gives $P_{\rm j,MAD}\approx
1.0(a_{*}/0.8)^{2}(\epsilon/0.1)^{-1}10^{39}$ erg s-1. Thus, the above
estimate for the jet compact power in ions, $P_{\rm i}$, in the absence of
pairs is equal to $P_{\rm j,MAD}$ at $a_{*}\approx 0.8$, and it is larger than
that for lower spins.

Table 1: Comparison of the source properties during the compact and transient jet launching in MAXI J1820+070 Jet Type | $L_{\rm bol}$ | $\Gamma$ | $P_{\rm j,MAD}$ | $\ell_{\rm max}$
---|---|---|---|---
| erg s-1 |  | erg s-1 | cm
Compact | $1.3\times 10^{38}$ | 1.5–4 | $1.0\left(\frac{a_{*}}{0.8}\right)^{2}\frac{0.1}{\epsilon}10^{39}$ | $4\times 10^{13}$
Transient | $1.8\times 10^{38}$ | $2.6^{+0.5}_{-0.4}$ | $1.5\left(\frac{a_{*}}{0.8}\right)^{2}\frac{0.1}{\epsilon}10^{39}$ | $6\times 10^{17}$

Just before and during the main transient ejection from MAXI J1820+070
(estimated to occur on MJD 58306, 2018-07-07; Wood et al. 2021), the source
was in the intermediate state (Shidatsu et al., 2019), which spectrum
approximately consists of a disk blackbody followed by a steep power law with
$\alpha>1$. The 0.1–78 keV flux on MJD 58306 was measured as
$\approx\\!1.62\times 10^{-7}$ erg cm-2 s-1 (Fabian et al., 2020), which,
given the shape of the spectrum is only slightly lower then the estimated
bolometric flux, $\approx 1.7\times 10^{-7}$ erg cm-2 s-1. The bolometric
luminosity is then $L_{\rm bol}\approx 1.8(D/3\,{\rm kpc})^{2}10^{38}$ erg s-1
(which is 18% of the Eddington luminosity), and $P_{\rm j,MAD}\approx
1.5(a_{*}/0.8)^{2}(\epsilon/0.1)^{-1}10^{39}$ erg s-1, similar to our
estimates for the hard state. The intermediate state has a stronger disk
component than the hard state, indicating the disc inner radius, $R_{\rm in}$,
is closer to the innermost stable circular orbit than in the hard state. Since
the efficiency of thin disks is $\approx R_{\rm g}/(2R_{\rm in})$ (Shakura &
Sunyaev, 1973) and the efficiency of hot flows is generally lower than that
(Yuan & Narayan, 2014), $\epsilon$ is expected to be higher in the
intermediate state than in the hard state. Then, the estimated $\dot{M}_{\rm
accr}c^{2}$ and $P_{\rm j,MAD}$ for the intermediate state are similar to
those for the hard state. Table 1 compares the accretion luminosity, the jet
properties and the maximum distance travelled for the two types of jet
launching.

The energy content of transient jets can be estimated from modelling their
motion in the surrounding medium (e.g., Wang et al. 2003; Steiner & McClintock
2012; Steiner et al. 2012; Zdziarski et al. 2023; Carotenuto et al. 2024),
which can be converted to the jet power if the duration of the ejection event
can be estimated. However, those estimates scale with the unknown density of
the surrounding medium, which appears much lower than the density of a warm
ISM, $n\ll 1$ cm-3, if the jet power is limited from above by $\dot{M}_{\rm
accr}c^{2}$ (Heinz, 2002). Since the densities of the surrounding media remain
unknown, we can only place upper limits on the jet power, equal to that of
Equation (2).

Taking $P_{\rm j}=P_{\rm j,MAD}\approx 1.5\times 10^{39}$ erg s-1 (as found
above), the kinetic energy of the ejection of MAXI J1820+070 estimated using
the ejection duration of 7 h (as in Carotenuto et al. 2024) becomes
$E_{0}\approx 4\times 10^{43}$ erg, which, given the Lorentz factor estimated
by Carotenuto et al. (2024), gives the mass of
$E_{0}/[(\Gamma_{0}-1)c^{2}]\approx 2.6\times 10^{22}$ g. If the ejection mass
were dominated by low-energy e±, there would $1.4\times 10^{49}$ pairs. For
the 7 h ejection duration, this requires the pair production rate of
$N_{+}\approx 6\times 10^{44}$ s-1. This is more than four orders of magnitude
larger than $\dot{N}_{+}\sim 2\times 10^{40}$ s-1 estimated for the hard state
of this source by considering pair production by accretion photons
($\gamma\gamma\rightarrow{\rm e}^{+}{\rm e}^{-}$) within the base of the
compact jet (Zdziarski et al., 2022b). Furthermore, the spectrum of that
intermediate state (Fabian et al., 2020) is much softer than that of the hard
state, and the number of photons available for pair production is
correspondingly much lower. Thus, we conclude that the ejection composition
was dominated by ions, similar to the case of MAXI J1348–630 (Zdziarski et
al., 2023).

Then, if its composition of compact jets were dominated by ions, its power
would be at the MAD limit, which would in turn leave the huge difference
between the propagation lengths of the compact and transient jets unexplained.
On the other hand, Zdziarski et al. (2022b) showed that the rate of pair
production in the hard state (see the paragraph above) approximately equals
the rate of the flow of the synchrotron-emitting leptons. Thus, the jet in
that state can easily be dominated by relativistic e± pairs, with only few
ions. The same conclusion was reached for Cyg X-1 (Zdziarski & Egron, 2022)
and the radio galaxy 3C 120 (Zdziarski et al., 2022a). Then, the hard-state
jet power is $P_{\rm j}\ll P_{\rm j,MAD}$, implying that the hard state
accretion flow contains a magnetic flux much lower than that of the MAD, and
thus is is of the Standard and Normal Evolution (SANE) type (Narayan et al.,
2012). The differences in both the jet composition and the power can then
explain the difference in the jet propagation length, see Section 3 below.
This conclusion also agrees with that of Fragile et al. (2023), based on
theoretical modelling of quasi-periodic oscillations, that the luminous hard
state cannot be MAD.

Summarizing the above, we find that the jet Lorentz factors, $\dot{M}_{\rm
accr}c^{2}$, and the maximum possible jet powers, $P_{\rm j,MAD}$, are similar
for both kinds of jets, see Table 1. Furthermore, if the compact jets were
dominated by ions, their powers would similar to $P_{\rm j,MAD}$, which is
unlikely. The power of transient jets cannot be uniquely determined, but given
their large propagation distances, it is much higher than that of the compact
jets.

Thus, considering possible differences between the two kinds of jet, we find
that the only possibility is a difference in the composition. Transient jets
are clearly dominated by normal plasma with at most few pairs. In contrast,
compact jets can be dominated by pairs, though this is not proven by their
modelling. We postulate here they are dominated by relativistic pairs and
magnetic field, and that this explains the very limited distances they
propagate, see Section 3. As a consequence of the pair dominance, their power
is dominated by e± and magnetic field, with only a minor possible contribution
from ions. In the case of MAXI J1820+070, that power is estimated as $P_{B{\rm
e}}\sim 2\times 10^{37}$ erg s-1, which is much below the MAD limit.

The power of the transient jet is, as discussed above, uncertain since it
scales with the unknown density of the surrounding medium. However, given
their very long propagation distances, it is likely that their power is at the
MAD limit, with $P_{\rm j,MAD}\approx 1.5(a_{*}/0.8)^{2}10^{39}$ erg s-1 in
the case of MAXI J1820+070. For $a_{*}=0.8$, is about two orders of magnitude
larger than the $P_{B{\rm e}}$.

##  3 Propagation of jets

The propagation of transient jets appears to be well described by the widely-
used formalism of Wang et al. (2003), which then requires the density of the
surrounding medium to be very low. Within that formalism, we have derived the
density of (similar to, but more accurate than eq. 5 of Heinz 2002)

$n\approx\frac{3E_{0}(1-k)\Gamma_{0}}{\pi m_{\rm
p}c^{2}l_{k}^{3}s(\Gamma_{0}-1)\phi^{2}(k^{2}\Gamma_{0}^{2}-1)},$ (3)

where $l_{k}$ is the distance at which the Lorentz factor of the ejection is
reduced to $k\Gamma_{0}$ ($1/\Gamma_{0}<k<1$), $\phi$ is the half-opening
angle, $s\approx 0.7$ (Wang et al., 2003), and $m_{\rm p}$ is the proton mass.
In the derivation, we made a simplifying assumption that the Lorentz factor of
the shock front equals that of the ejection. The stopping length is then

$l_{k}\approx\left[\frac{3E_{0}(1-k)\Gamma_{0}}{\pi nm_{\rm
p}c^{2}s(\Gamma_{0}-1)\phi^{2}(k^{2}\Gamma_{0}^{2}-1)}\right]^{1/3}.$ (4)

For $E_{0}=4\times 10^{43}$ erg s-1 (see Section 2), $k=0.5$, $\Gamma_{0}=2.6$
and $n=10^{-4}$ cm-3 (Carotenuto et al., 2024), we obtain $l_{1/2}\approx
1\times 10^{18}$ cm, which agrees quite well with the observations.

In the case of compact jets dominated by e±, the pairs in the jet may first
lose most of their energy, strongly reducing the mass. This may provide the
required reduction of the kinetic energy. Indeed, Zdziarski et al. (2022b)
found that the synchrotron power of MAXI J1820+070 is very similar to
$P_{B{\rm e}}$. If $P_{\rm j}\approx P_{B{\rm e}}$ (i.e., the bulk kinetic
power of ions can be neglected), the e± will indeed lose most of their
internal energy during the propagation. We will then have a dark jet, possibly
propagating to large distances. In the presence of both synchrotron and
adiabatic losses, the relativistic electrons lose all their energy. In the
absence of adiabatic losses, they reach a terminal Lorentz factor $>1$
(Kaiser, 2006).

Then (for the case of adiabatic losses) the jet power will be equal to that of
the energy flow of the cold pairs and the magnetic field. For MAXI J1820+070,
it can be estimated as $\approx 4\times 10^{34}$ erg s-1 (Zdziarski et al.,
2022b) provided the magnetic field is effectively dissipated, maintaining the
equipartition. In order to find the jet kinetic energy, we need to estimate
the characteristic time this power operates before the jet is slowed down. The
maximum estimate of this time is $l_{k}/c$, $k\sim 1/2$. Inserting
$E_{0}=P_{\rm j}l_{k}/c$ into Equation (4) gives

$l_{k}\approx\left[\frac{3P_{\rm j}(1-k)\Gamma_{0}}{\pi nm_{\rm
p}c^{3}s(\Gamma_{0}-1)\phi^{2}(k^{2}\Gamma_{0}^{2}-1)}\right]^{1/2}.$ (5)

For $P_{\rm j}=4\times 10^{34}$ erg s-1, $k=0.5$, $\Gamma_{0}=2.6$ and
$n=10^{-4}$ cm-3, we still find a very large $l_{1/2}\approx 2\times 10^{17}$
cm. Thus, the jet made of cold pairs would still be stopped at a large
distance from the BH, and we need some additional processes to efficiently
stop it. The main uncertainty in this estimate is in the assumption of the
time the jet kinetic energy effectively accumulates of $E_{0}=P_{\rm
j}l_{k}/c$. If the effective $E_{0}$ is much lower, the jet will stop at a
much closer distance. The jet stopping process will be considered in detail in
Heinz et al. (in preparation).

An additional effect increasing the efficiency of the jet stopping is its
lateral expansion, increasing $\phi$. We also speculate that the path along
the jet axis at distances $\ll\\!10^{17}$ cm contains a significant number of
remnants of previous ejections, effectively increasing $n$ in that part of the
jet path. These remnants can then stop the compact jets given their inferred
low power, while the transient jets with the power at the MAD limit can easily
overcome that.

##  4 Conclusions

We have shown that the accretion rate during launching of compact jets during
the luminous hard state is very similar to that in the case of transient jet
launching, which in turn, happens during transitions from the hard state to
the soft one. This implies that the maximum possible jet powers, achieved
during the MAD, are very similar. We used a specific example of the very well-
studied BH XRB, MAXI J1820+070. The estimated/measured bulk Lorentz factors of
both the compact and transient jets also appear similar. Furthermore, the
compact jet power estimated from its synchrotron emission assuming the absence
of e± pairs is at the MAD limit.

This would then leave unexplained the striking difference between the two
types of jets, with transient jets propagating to much larger distances than
the compact ones. We postulate here that compact jets consist mostly of e±
pairs, which strongly reduces both their powers and inertia. A sufficient
number of pairs can be produced by photon-photon collisions of hard
X-rays/soft $\gamma$-rays emitted by the accretion flow (as found in earlier
works). Consequently, the hard luminous accretion flow has the magnetic flux
much below the MAD limit and it is consequently of the SANE type. This
confirms the conclusion of Fragile et al. (2023) that the hard state is not
MAD.

Compact jets then propagate and lose their internal energy, further reducing
their power. This can then explain the compact jets propagating to much
shorter distances than the transient jets. Propagation of compact jets in BH
XRBs will be studied in detail in Heinz et al. (in preparation).

## Acknowledgements

We acknowledge M. Sikora for very important discussions. We also thank J.
Miller-Jones for extensive discussions, J. Rodi for providing us with the
broad-band X-ray luminosity of MAXI J1820+070 as observed by XMM and INTEGRAL,
and T. Russell for a discussion on MAXI J1836–194. We acknowledge support from
the Polish National Science Center under the grants 2019/35/B/ST9/03944,
2023/48/Q/ST9/00138, and from the Copernicus Academy under the grant
CBMK/01/24.

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# Causal simplicity and (maximal) null pseudoconvexity

J. Hedicke E-mail<EMAIL_ADDRESS>Fakultät für Mathematik,
Ruhr-Universität Bochum, Universitätstraße 150, D-44801 Bochum, Germany.  E.
Minguzzi E-mail<EMAIL_ADDRESS>Dipartimento di Matematica e
Informatica “U. Dini”, Università degli Studi di Firenze, Via S. Marta 3,
I-50139 Firenze, Italy.  B. Schinnerl E-mail<EMAIL_ADDRESS>Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090
Wien, Austria.  R. Steinbauer E-mail<EMAIL_ADDRESS>Fakultät
für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien,
Austria.  S. Suhr E-mail<EMAIL_ADDRESS>Fakultät für
Mathematik, Ruhr-Universität Bochum, Universitätstraße 150, D-44801 Bochum,
Germany.

###### Abstract

We consider pseudoconvexity properties in Lorentzian and Riemannian manifolds
and their relationship in static spacetimes. We provide an example of a
causally continuous and maximal null pseudoconvex spacetime that fails to be
causally simple. Its Riemannian factor provides an analogous example of a
manifold that is minimally pseudoconvex, but fails to be convex.

##  1 Introduction

A pseudo-Riemannian manifold is said to have a pseudoconvex class of
geodesics, if for each compact subset $K$ there is a larger compact subset
$K^{\prime}$, such that any geodesic segment of the class with endpoints in
$K$ lies entirely in $K^{\prime}$. Causal pseudoconvexity is a kind of an
“internal completeness” assumption for spacetimes akin to, but strictly weaker
than global hyperbolicity. Similar to the latter, it is a causality notion
with strong ties to the theory of PDE. Pseudoconvexity of bicharacteristics
characterizes the existence of a parametrix for pseudodifferential operators
of real principal type, cf. [12, Section 6], [6].

John Beem and his coauthors have used pseudoconvexity in several contexts in
causality theory. In [4] causal pseudoconvexity together with causal geodesic
non-imprisonment (inextendible causal geodesics leave every compact set)
appear as sufficient conditions for the stability of causal geodesic
completeness, cf. [5, Th. 7.35]. Geodesic non-imprisonment and pseudoconvexity
of all geodesics together with the absence of conjugate points implies
geodesic connectedness and serve as conditions for a pseudo-Riemannian version
of the Hadamard-Cartan theorem [7], [5, Ch. 11]. For a recent generalization
see [11].

Here we are especially interested in the relation between pseudoconvex and
causally simple spacetimes, i.e. causal spacetimes with closed causal
relation. Indeed, every causally simple spacetime is maximal null pseudoconvex
by [8, Th. 1], for which we provide a simplified proof in Theorem 2.7, below.
Concerning the reverse implication it has been conjectured in [3, Sec. 1] that
strongly causal (i.e. there are no “almost closed” causal curves) and null
pseudoconvex spacetimes are causally simple. Finally, in [20, Th. 2] it was
claimed that for strongly causal spacetimes maximal null pseudoconvexity and
causal simplicity are equivalent. Here we provide a counterexample to this
statement, i.e. we establish that

causal continuity and maximal null pseudoconvexity

do not imply causal simplicity.

Our counterexample is a static spacetime based on a corresponding Riemannian
counterexample, that enjoys a certain limiting property for minimizing
geodesics, but fails to be convex. This counterexample adds to the list of
recently found counterexamples involving the notion of causal simplicity [14,
10].

In Section 2 we provide some general results on pseudoconvexity and in Section
3 we specialize to static spacetimes. This will allow us to “lift” the
corresponding properties of the Riemannian counterexample to the spacetime
level in Section 4.

In the remainder of this introduction we fix some notations and conventions.
All manifolds are assumed to be smooth, connected, Hausdorff, second
countable, of arbitrary dimension $n\geq 2$, and without boundary. By a
minimizing geodesic in a Riemannian manifold $(\Sigma,h)$ we mean a geodesic
whose length equals the distance $d^{h}(x,y)$ between its endpoints $x$ and
$y$. We call $\Sigma$ convex if any pair of its points can be connected via a
minimizing geodesic.

A spacetime $(M,g)$ is a time oriented Lorentzian manifold, where we use the
signature $(-,+,\dots,+)$. A causal geodesic in $M$ is called maximizing if
its length equals the Lorentzian distance $d^{g}(p,q)$ between its endpoints
$p$ and $q$. We denote the chronological and the causal relation by $I$ and
$J$ respectively. A spacetime is called causal if there are no closed causal
curves. If in addition the causal relation $J$ is closed, it is called
causally simple. A spacetime is non-total imprisoning if no inextendible
causal curve is contained in a compact set. We shall only consider causal
spacetimes, in which case the non-total imprisonment property is equivalent
with the causal geodesic non-imprisonment property mentioned above [17, Prop.
4.41]. A spacetime is called strongly causal if for every point and for every
neighborhood of the point there is a smaller neighborhood such that no causal
curve intersects it more than once. A spacetime is called causally continuous
if it is strongly causal and reflective: $I^{+}(q)\subset
I^{+}(p)\Leftrightarrow I^{-}(p)\subset I^{-}(q)$. It is known that causal
simplicity $\Rightarrow$ causal continuity $\Rightarrow$ strong causality
$\Rightarrow$ non-total imprisonment $\Rightarrow$ causality. For other
results and conventions on causality not explicitly mentioned in this work, we
refer the reader to the review [17].

##  2 General results on pseudoconvexity

We start recalling some definitions, cf. [5, Chs. 7, 11].

######  Definition 2.1.

A spacetime $(M,g)$ is called (causal, null or maximally null) pseudoconvex,
if for any compact set $K$, there exists another compact set $K^{\prime}$,
such that each geodesic of the respective type with both endpoints in $K$ must
be entirely contained in $K^{\prime}$.

Clearly pseudoconvexity implies causal pseudoconvexity which implies null
pseudoconvexity which again is stronger than maximal null pseudoconvexity.
There is also a Riemannian version of the notion:

######  Definition 2.2.

A Riemannian manifold $(\Sigma,h)$ is called (minimally) pseudoconvex, if for
any compact set $C$, there exists another compact set $C^{\prime}$, such that
each (minimal) geodesic with endpoints in $C$ must be entirely contained in
$C^{\prime}$.

Clearly, pseudoconvexity implies minimal pseudoconvexity. Often it is useful
to relate pseudoconvexity to a certain limiting property of geodesic segments
which we define next.

######  Definition 2.3.

We say a pseudo-Riemannian manifold $M$ has the limit geodesic segment
property (LGS) if the following holds true: Given any pair of converging
sequences of points $p_{n}\to p$ and $q_{n}\to q\not=p$ and any sequence
$\sigma_{n}$ of geodesic segments connecting $p_{n}$ to $q_{n}$, there is a
subsequence of $\sigma_{n}$ (in a suitable affine reparametrization)
converging (locally uniformly) to a geodesic $\sigma$ from $p$ to $q$.

In the Riemannian case we will speak of the minimal LGS if all $\sigma_{n}$
(and hence $\sigma$) have the corresponding property. Similarly, in the
Lorentzian case, we will speak of the causal, null, maximal null LGS, if all
$\sigma_{n}$ (and hence $\sigma$) have the corresponding property.

Recall from [7, Def. 2] that a Riemannian manifold $(\Sigma,h)$ is
disprisoning if no forward inextensible geodesic
$\gamma\colon[0,\omega)\to\Sigma$ has compact closure.

######  Lemma 2.4.

A Riemannian manifold $(\Sigma,h)$ satisfies the LGS if and only if it is
disprisoning and pseudoconvex.

###### Proof.

First assume that LGS holds. Let $C\subset\Sigma$ compact be given. Choose a
compact exhaustion $\\{C_{n}\\}_{n\in\mathbb{N}}$ of $\Sigma$. If
pseudoconvexity fails for $C$ there exists a sequence
$\gamma_{n}\colon[0,1]\to\Sigma$ with endpoints in $C$ such that $\gamma_{n}$
leaves $C_{n}$. By the LGS $\\{\gamma_{n}\\}_{n\in\mathbb{N}}$ contains a
convergent subsequence. The limit geodesic $\gamma\colon[0,1]\to\Sigma$ is
contained in some $C_{N}$. This clearly contradicts the initial assumption.
Therefore $(\Sigma,h)$ is pseudoconvex.

Next assume that LGS holds and there exists a inextensible geodesic ray
$\gamma\colon[0,\omega)\to\Sigma$ with compact closure. Then we have
$\omega=\infty$ and $\gamma$ has infinite length. In that case the sequence
$\gamma_{n}\colon[0,1]\to\Sigma$ where $\gamma_{n}(t):=\gamma(nt)$ has no
convergent subsequence.

Finally, assume that $(\Sigma,h)$ is pseudoconvex and disprisoning. Let
$p_{n},q_{n}\in\Sigma$ be sequences converging to $p$ and $q\neq p$
respectively and $\gamma_{n}\colon[0,1]\to\Sigma$ be a sequence of geodesics
from $p_{n}$ to $q_{n}$. Let $C$ be a compact set given by the union of a
compact neighborhood of $p$ with a compact neighborhood of $q$. Without loss
of generality we can assume $p_{n},q_{n}\in C$ for each $n$. There exists a
compact set $C^{\prime}\subset\Sigma$ such that $\gamma_{n}\subset C^{\prime}$
for all $n\in\mathbb{N}$ by pseudoconvexity. Since the manifold is
disprisoning the length $L^{h}(\gamma_{n})$ is uniformly bounded from above.
Then by a standard argument for geodesics it follows that there exists a
convergent subsequence. ∎

By the previous result every compact (closed) Riemannian manifold is trivially
pseudoconvex but fails to satisfy the LGS as, certainly, it is not
disprisoning. For instance, $T^{2}$ is pseudoconvex but does not satisfy the
LGS. Choosing a geodesic $\sigma$ with irrational slope, which hence is dense
in $T^{2}$, we may choose a point $x\not\in\sigma$ and $t_{n}$ such that
$\sigma(t_{n})\to x$. But the sequence of geodesic segments
$\sigma|_{[0,t_{n}]}$ has no subsequence converging to a geodesic between
$\sigma(0)$ and $x$.

In this connection we mention a Theorem of Serre, see [19], implying that any
pair of points in a noncontractible complete Riemannian manifold are connected
by a sequence of geodesics whose lengths are diverging.

Completeness implies convexity by the Hopf-Rinow theorem [15], but
completeness (hence convexity) does not imply pseudoconvexity (e.g. a complete
surface with infinitely many holes), cf. [18]. For another example of convex
but not pseudoconvex space see Theorem 2.7 in [14].

The next result is analogous to the previous one, but for the “minimal” case.

######  Lemma 2.5.

A Riemannian manifold $(\Sigma,h)$ satisfies the minimal LGS if and only if it
is minimally pseudoconvex.

###### Proof.

We already know that if LGS holds then pseudoconvexity holds, which implies
minimal pseudoconvexity.

For the converse, let $\Sigma$ be minimally pseudoconvex. Further let
$\sigma_{n}:[0,b_{n}]\to\Sigma$ be minimizing geodesics from $x_{n}$ to
$y_{n}$, parametrized by arc length, and let $x_{n}\to x$, $y_{n}\to y\not=x$.
Let $C$ be a compact set given by the union of a compact neighborhood of $x$
with a compact neighborhood of $y$. Without loss of generality we can assume
$x_{n},y_{n}\in C$ for each $n$. By compactness $\sigma_{n}^{\prime}(0)$
converges up to a subsequence to a unit vector $v\in T_{x}\Sigma$. Let
$\sigma:[0,\beta)\to\Sigma$ be the maximally extended geodesic with initial
vector $v$. Then $\sigma_{n}$ converges to $\sigma$ locally uniformly on
$[0,\beta)$. By minimal pseudoconvexity all $\sigma_{n}$ are contained in some
compact $C^{\prime}$ and so
$L^{h}(\sigma_{n})\leq\text{diam}^{h}(C^{\prime})$. Hence, again up to a
subsequence, $b_{n}$ converge to some $b<\beta$. Hence
$\sigma_{n}(b_{n})\to\sigma(b)=q$ and we are done. ∎

The next result is a Lorentzian analog to 2.4. It slightly improves [5, Lem.
11.20].

######  Lemma 2.6.

$(M,g)$ satisfies the (maximal) null LGS iff it is non-total imprisoning and
(maximal) null pseudoconvex.

A similar results with “causal” replacing the two instances of “null” holds.

###### Proof.

The proof of “(maximal) null LGS $\Rightarrow$ (maximal) null pseudoconvex”,
can be done in the same way as the proof of Lemma 2.4 (first paragraph). If
non-total imprisonment were violated we could find a future inextendible
lightlike geodesic $\gamma:[0,a)\to M$, achronal hence maximizing, contained
in a compact set $C$ cf. [17, Thm. 2.77]. Let $a_{n}\to a$, and pass to a
subsequence (denoted in the same way) so that $\gamma(a_{n})\to q$. But then
the maximizing lightlike geodesics $\gamma_{n}=\gamma|_{[0,a_{n}]}$ with
$a_{n}\to a$, do not converge to a geodesic $\eta$ connecting $p:=\gamma(0)$
to $q$, for if that were the case, as each $\gamma_{n}$ coincides with a
segment of $\gamma$, $\eta$ would coincide with a segment of $\gamma$, which
is impossible since certainly $\gamma_{n}$ converges on a domain larger than
any subinterval $[0,b]\subset[0,a)$.

For the converse, assume non-total imprisonment and (maximal) null
pseudoconvexity. Let $\sigma_{n}$ be (maximal) null geodesics with endpoints
$p_{n}$, $q_{n}$ converging to $p$ and $q$, respectively. Let $C$ be a compact
set given by the union of a compact neighborhood of $p$ with a compact
neighborhood of $q$. Without loss of generality we can assume $p_{n},q_{n}\in
C$ for each $n$. By (maximal) null pseudoconvexity there is a compact set
$C^{\prime}$ containing all the curves $\sigma_{n}$. By the limit curve
theorem (two endpoints case [17]), the sequence must admit a converging
subsequence, otherwise we could find a future inextendible causal curve
$\sigma^{p}\subset C^{\prime}$ starting from $p$ to which some, suitably
parametrized, subsequence of $\sigma_{n}$ converges uniformly on compact
subset. But then $\sigma^{p}$ would contradict non-total imprisonment. ∎

In a globally hyperbolic spacetime for every compact subset $K$, $J^{+}(K)\cap
J^{-}(K)$ is compact. As a consequence, global hyperbolicity trivially implies
causal pseudoconvexity (and hence maximal causal pseudoconvexity). However,
causal pseudoconvexity does not imply global hyperbolicity, e.g. a strip
$|x|<1$ in Minkowski 1+1 spacetime [5].

We now give a simplified proof of the next Lorentzian result originally proved
in [8, Th. 1]. In the last section we shall show that the reverse implication
does not hold.

######  Theorem 2.7.

If $(M,g)$ is causally simple, then it is maximally null pseudoconvex
(equivalently, it satisfies the maximal null LGS).

The equivalence in the different formulations of the conclusion is given by
Lemma 2.6, since causal simplicity implies non-total imprisonment.

###### Proof.

Suppose that the claim is false, then we can find a compact set $K$, and
maximal null geodesic segments $\gamma_{n}$ with endpoints $p_{n},q_{n}\in K$,
such that for some $r_{n}\in\gamma_{n}$, $r_{n}$ escapes every compact set.
Further we can assume $p_{n}\to p\in K$, and $q_{n}\to q\in K$. As
$(p_{n},q_{n})\in J$ we have by causal simplicity $(p,q)\in J$. Moreover, it
cannot be $(p,q)\in I$ otherwise for sufficiently large $n$, $(p_{n},q_{n})\in
I$ which contradicts the maximality of $\gamma_{n}$. Thus $(p,q)\in
E=J\setminus I$.

By the limit curve theorem [17, Th. 2.53(ii)] there are lightlike rays
$\sigma^{p}$ starting from $p$ and $\sigma^{q}$ ending at $q$ such that for
every $p^{\prime}\in\sigma^{p}$ and $q^{\prime}\in\sigma^{q}$, we have
$(p^{\prime},q^{\prime})\in\bar{J}=J$ where we used causal simplicity. For
every $p^{\prime}\in\sigma^{p}\backslash\\{p\\}$ we have $(p^{\prime},q)\in J$
but the causal curve connecting $p^{\prime}$ to $q$ must be the prolongation
of the lightlike ray $\sigma^{p}$ otherwise $(p,q)\in I$, a contradiction.
Thus $q\in\sigma^{p}$ and repeating the argument by taking $p^{\prime}$ along
$\sigma^{p}$ after $q$ one gets that $\sigma^{p}$ passes through $q$ several
times, which violates causality. ∎

Next we establish that for Riemannian manifolds convexity is stronger than the
minimal LGS. Again, in the last section we shall show that the reverse
implication does not hold.

######  Theorem 2.8.

If $\Sigma$ is convex, then it satisfies the minimal LGS (equivalently, it is
minimal pseudoconvex).

The equivalence in the different formulations of the conclusion is given by
Lemma 2.5.

###### Proof.

Let $\sigma_{n}:[0,a_{n}]\to\Sigma$ be minimizing unit speed geodesics such
that $p_{n}:=\sigma_{n}(0)\to p$ and $q_{n}:=\sigma_{n}(a_{n})\to q\neq p$,
$p,q\in\Sigma$. Without loss of generality we can assume that
$\dot{\sigma}_{n}(0)\to u\in T_{p}\Sigma$ and that $a_{n}\to a>0$. Let
$\sigma:I\to\Sigma$ be the unit speed geodesic that starts from $p$ with
tangent $\dot{\sigma}(0)=u$. For every $t\in I$ by the continuity of the
exponential map

$\sigma_{n}(t)=\exp_{p_{n}}(\dot{\sigma}_{n}(0)t)\to\exp_{p}(ut)=\sigma(t).$

Moreover for every $c\in I$, $\sigma_{n}|_{[0,c]}$ is minimizing and so
$\sigma|_{[0,c]}$ is minimizing. If there is $t\in I$ such that $\sigma(t)=q$,
then it must be $t\geq a$ otherwise, as $t$ is the length of a curve (the
curve $\sigma$) connecting $p$ to $q$, no $\sigma_{n}$ could be minimizing for
sufficiently large $n$. However, $t>a$ cannot happen since otherwise it would
be shorter to go from $p$ to $q$, passing from some $\sigma_{n}$, which would
contradict that $\sigma|_{[0,t]}$ is minimizing. We conclude that if there is
$t$ such that $\sigma(t)=q$ then $t=a$.

We can now assume that $q\notin\sigma$ otherwise we have finished, hence we
can assume $I=[0,b)$. Observe that $d(p_{n},q_{n})=a_{n}$ and by the
continuity of distance $d(p,q)=a$. Now, for $0<\epsilon<b$, we have
$d(\sigma_{n}(\epsilon),q_{n})=a_{n}-\epsilon$ as $\sigma_{n}$ is minimizing,
and hence
$d(\sigma(\epsilon),q)=\lim_{n}d(\sigma_{n}(\epsilon),q_{n})=a-\epsilon$ as
$d$ is continuous. Thus any chosen minimizing curve $\gamma$ connecting
$\sigma(\epsilon)$ with $q$ has length $a-\epsilon$. But
$d(p,q)<\ell(\sigma|_{[0,\epsilon]})+\ell(\gamma)\leq\epsilon+a-\epsilon=a$
where the first inequality is strict because there must be a corner at
$\sigma(\epsilon)$ between $\sigma$ and $\gamma$, otherwise it would be
$q\in\sigma$. The contradiction proves that the connecting case is the only
option. ∎

We do not known if pseudoconvexity implies convexity. The next result
represents an attempt in this direction. It is not used in what follows.

######  Proposition 2.9.

Let $(\Sigma,h)$ be a Riemannian manifold which admits an equidimensional
embedding into a complete manifold $(\tilde{\Sigma},\tilde{h})$. Then
$(\Sigma,h)$ is minimally pseudoconvex if and only if it is convex.

######  Remark 2.10.

The proof shows that a positive lower bound on the convexity radius on bounded
sets of $(\Sigma,h)$ suffices at least for the conclusion that minimal
pseudoconvexity implies convexity.

###### Proof.

(i) By Theorem 2.8 convexity implies minimal pseudoconvexity without further
assumptions.

(ii) Assume that $(\Sigma,h)$ is minimally pseudoconvex. We will show that the
closure $\overline{\Sigma}$ of $\Sigma$ in $\tilde{\Sigma}$ is locally convex,
i.e. for any $x\in\overline{\Sigma}$ there exists $\epsilon>0$ such that
$\overline{\Sigma}\cap B^{\tilde{h}}_{\epsilon}(x)$ is convex in
$\tilde{\Sigma}$ (i.e. any two points in this set are connected by a geodesic
segment which is minimizing among all the connecting curves contained in the
same set).

We define the convexity radius for $(\tilde{\Sigma},\tilde{h})$ as in [16,
Corollary 1.9.11]. All we need to know is that this function
$r\colon\tilde{\Sigma}\to(0,\infty]$ is continuous and that in
$B^{\tilde{h}}_{r(\tilde{x})}(\tilde{x})$ any two points are connected by one
and only one geodesic contained in the ball, and this geodesic is minimizing
among all the connecting curves in $\tilde{\Sigma}$.

We claim that for all $\tilde{x}\in\tilde{\Sigma}$ and all $y,z$ in the same
connected component of $B^{\tilde{h}}_{r(\tilde{x})}(\tilde{x})\cap\Sigma$ the
unique minimal $\tilde{h}$-geodesic between $y$ and $z$ lies in $\Sigma$. This
can be seen as follows. Let $\eta\colon[0,1]\to
B^{\tilde{h}}_{r(\tilde{x})}(\tilde{x})\cap\Sigma$ be a curve from $y$ to $z$.
The set of parameters $t\in[0,1]$ for which the unique minimal
$\tilde{h}$-geodesic between $y$ and $\eta(t)$ lies in $\Sigma$ is nonempty.
It is also open, since $\Sigma\subset\tilde{\Sigma}$ is open and
$\exp^{\tilde{h}}_{y}$ is a diffeomorphism defined on
$\exp^{\tilde{h}}_{y}{}^{-1}[B^{\tilde{h}}_{r(\tilde{x})}(\tilde{x})]$.
Finally it is closed by minimal pseudoconvexity. Therefore the unique minimal
$\tilde{h}$-geodesic between $y$ and $\eta(1)=z$ lies in $\Sigma$. This
geodesic is also a minimal geodesic for $(\Sigma,h)$ as
$\text{dist}^{\tilde{h}}\leq\text{dist}^{{h}}$, and hence the unique minimal
geodesic for $(\Sigma,h)$ connecting $y$ and $z$. By considering limits of
geodesics we infer that, for every $x,y\in
B^{\tilde{h}}_{r(\tilde{x})}(\tilde{x})$ belonging to the closure of the same
connected component $C$ of $B^{\tilde{h}}_{r(\tilde{x})}(\tilde{x})\cap\Sigma$
there is a $\tilde{h}$-minimizing geodesic entirely contained in
$\overline{C}\subset\overline{\Sigma}$ connecting them.

Observe that for every point $\overline{x}\in\overline{\Sigma}$ we have that
every connected component of
$B^{\tilde{h}}_{r(\overline{x})}(\overline{x})\cap\Sigma$ is convex, and that
the established properties provide a “local convexity” property for
$\overline{\Sigma}$.

Now let $x,y\in\Sigma$ be given. We will show that there exists a minimal
geodesic in $\Sigma$ connecting the two points.

The following reasoning is similar to [1, Satz 2.8(i)]. Let
$\gamma_{n}\colon[0,1]\to\Sigma$ be a minimizing sequence of curves connecting
$x$ with $y$. According to the argument above we can assume that the curves
are geodesic polygons and that the length of the individual arcs is bounded
from below by some positive constant. By the completeness of
$(\tilde{\Sigma},\tilde{h})$ a subsequence converges to a geodesic polygon
$\gamma\colon[0,1]\to\overline{\Sigma}$. Using a standard argument involving
the triangle inequality we see that $\gamma$ is a $\tilde{h}$-geodesic.

We claim that $\gamma$ does not intersect the boundary
$\partial\Sigma:=\overline{\Sigma}\setminus\Sigma$. Assume otherwise, i.e.
there exists $t\in(0,1)$ with $\gamma(t)\in\partial\Sigma$. W.l.o.g. we can
assume that $t$ is minimal in $(0,1)$. Denote with $\Sigma_{t}$ the connected
component of $B^{\tilde{h}}_{r(\gamma(t))/4}(\gamma(t))\cap\Sigma$ which
contains a segment $\gamma|_{[s,t)}$ for some $s<t$. Note that $\Sigma_{t}$ is
convex with $\gamma(t)\in\partial\Sigma_{t}$. The following argument is an
adaptation of [9, Lemma 8.6] to the present situation. Choose an arbitrary
small smooth hypersurface $W\subset\Sigma_{t}$ transversal to $\gamma$ and
containing $\gamma(s)$. Choose $u>t$ such that $\gamma|_{[t,u]}\subset
B^{\tilde{h}}_{r(\gamma(t))/4}(\gamma(t))$ and consider the set

$V:=\\{\exp_{r}(\lambda w)|\;\tilde{h}(w,w)<r(\gamma(t))^{2},\;\exp_{r}(w)\in
W,\;\lambda\in(0,1)\\}.$

Here $r\in\Sigma$ is a point sufficiently close to
$\gamma(u)\in\overline{\Sigma}$ which stays in the connected component
$\Sigma_{t}$ (it is possible to show that it exists perturbing
$\gamma_{[s,u]}$). The set $V$ is open and contained in ${\Sigma_{t}}$ by
convexity. Further $V$ is an open neighborhood of $\gamma(t)$, (remember that
$\gamma$ is a $\tilde{h}$-geodesic hence $C^{1}$) a contradiction to the
assumption that $\gamma(t)\in\partial\Sigma_{t}$. Therefore $\gamma$ is
contained in $\Sigma$. The lower semicontinuity of the length functional
implies that the curve is minimal in $\Sigma$. Note that $\gamma$ is i.g. not
minimal in $\tilde{\Sigma}$. ∎

##  3 Pseudoconvexity in static spacetimes

Let $(\Sigma,h)$ be a Riemannian manifold. In this section we investigate the
relation of convexity and pseudoconvexity properties of $(\Sigma,h)$ to
causality and pseudoconvexity properties of the static spacetime $(M,g)$ with

$M:=\mathbb{R}\times\Sigma,\quad\text{and}\quad g=-dt^{2}+h.$ (1)

The projection onto the first factor $t:M\to\mathbb{R}$ is a temporal
function.

Observe that in semi-Riemannian product manifolds a path is, up to
parametrisation, a geodesic if and only if the projections onto the factors
are geodesics. Hence in our case, geodesics are always of the form
$\gamma=(\alpha,\sigma)$ where $\alpha$ is some linear function in
$\mathbb{R}$ and $\beta$ is a geodesic in $\Sigma$. Moreover, clearly $\gamma$
is causal if and only if $|\alpha^{\prime}|\geq\|\sigma^{\prime}\|_{h}$. We
observe the following relation of minimizing and maximizing geodesics.

######  Lemma 3.1.

A causal geodesic $\gamma=(\alpha,\sigma)$ in $M$ is maximizing if and only if
$\sigma$ is minimizing in $\Sigma$.

###### Proof.

It suffices to consider geodesics $\gamma=(\alpha,\sigma)$ of the form
$\alpha:[0,1]\to{\mathbb{R}}$, $\alpha(s)=bt$, and $\sigma:[0,1]\to\Sigma$,
i.e., $\|\sigma^{\prime}(s)\|_{h}=L^{h}(\sigma)$, connecting $p=(0,x)$ and
$q=(b,y)$ with $b>0$. First observe that $\gamma$ is causal iff $b\geq
L^{h}(\sigma)$ and so

$L^{g}(\gamma)=\sqrt{b^{2}-L^{h}(\sigma)^{2}}.$ (2)

Now suppose $\sigma$ is not minimizing, then there is
$\tilde{\sigma}:[0,1]\to\Sigma$ connecting $x$ to $y$ with
$L^{h}(\tilde{\sigma})<L^{h}(\sigma)$. But then for
$\tilde{\gamma}=(bt,\tilde{\sigma})$ we find
$L^{g}(\tilde{\gamma})=\sqrt{b^{2}-L^{h}(\tilde{\sigma})^{2}}>\sqrt{b^{2}-L^{h}(\sigma)^{2}}=L^{g}(\gamma)$
and so $\gamma$ is not maximizing.

Conversely, suppose that $\gamma$ is not maximizing, then there is a future
pointing timelike curve $\tilde{\gamma}:[0,1]\to M$,
$\tilde{\gamma}=(bt,\tilde{\sigma})$ with
$L^{g}(\tilde{\gamma})>L^{g}(\gamma)$. Since $\tilde{\gamma}$ need not be a
geodesic we can in general not parametrize it both linear in the first factor
and constant speed in the second. However, we have

$L^{g}(\tilde{\gamma})=\int_{0}^{1}\sqrt{b^{2}-\|\tilde{\sigma}^{\prime}\|_{h}^{2}}dt>L^{g}(\gamma)=\sqrt{b^{2}-(L^{h}(\sigma))^{2}}.$

By applying the Cauchy-Schwarz inequality to
$\int_{0}^{1}\sqrt{b^{2}-\|\tilde{\sigma}^{\prime}\|_{h}^{2}}dt$ and by using
the previous inequality we get the second inequality in the next expression

$L^{h}(\tilde{\sigma})^{2}\leq\int_{0}^{1}\|\tilde{\sigma}^{\prime}\|_{h}^{2}dt<L^{h}(\sigma)^{2}.$

The first inequality is obtained through another standard application of the
Cauchy-Schwarz inequality. In conclusion, $\sigma$ is not minimizing between
$x$ and $y$. ∎

######  Lemma 3.2.

If $M$ is maximally null pseudoconvex, then $\Sigma$ is minimal pseudoconvex.
Conversely, if $\Sigma$ is minimal pseudoconvex, then $M$ is maximally causal
(hence null) pseudoconvex. $M$ is pseudoconvex if and only if $\Sigma$ is
pseudoconvex.

###### Proof.

Let $\Sigma$ be pseudoconvex and take any $K\subseteq M$. Then the projections
of $K$ onto $\mathbb{R}$ and $\Sigma$ are compact, i.e. we have
$K\subseteq[c,d]\times C$ for $c,d\in\mathbb{R}$ and some compact
$C\subseteq\Sigma$. By pseudoconvexity of $\Sigma$ there exists a compact set
$C^{\prime}$ such that any geodesic starting and ending in $C$ is contained in
$C^{\prime}$. Now let $\gamma$ be a geodesic in $M$, starting and ending in
$K$. We can write $\gamma(t)=(a+bt,\sigma(t))$, hence $c\leq a,a+b\leq d$ with
$\sigma$ a geodesic starting and ending in $C$. This however implies that
$\gamma\subseteq K^{\prime}:=[c,d]\times C^{\prime}$.

The proof that “$\Sigma$ is minimally pseudoconvex” implies “$M$ is maximally
causal pseudoconvex” follows from a very similar argument which makes use of
Lemma 3.1.

For the converse direction, we first look at the maximal-minimal case: Let $M$
be maximally null pseudoconvex and let $C\subseteq\Sigma$ be compact. Any
minimizing unit speed geodesic $\sigma$ starting and ending in $C$ fulfills
$L^{h}(\sigma)\leq c:=\text{diam}^{h}(C)$. For the compact set $K:=[0,c]\times
C$ by maximal null pseudoconvexity there exists some compact set $K^{\prime}$
containing all maximal null geodesics starting and ending in $K$. Again by
construction we must have $K^{\prime}\subseteq[0,c]\times C^{\prime}$ for some
compact set $C^{\prime}$ in $\Sigma$. By Lemma 3.1 the null geodesic
$\gamma(t):=(t,\sigma(t))$ is maximizing and hence contained in $K^{\prime}$,
but then also $\sigma\subseteq C^{\prime}$.

If $M$ is pseudoconvex then $\Sigma$ is too by noticing that for any compact
set $C\subseteq\Sigma$ the set $\\{0\\}\times C$ is compact in $M$. ∎

######  Remark 3.3.

While pseudoconvexity of $\Sigma$ clearly implies causal and null
pseudoconvexity of $M$ the converse does not hold. This is due to the fact
that any causal geodesic starting and ending in a compact set in $M$ has a
projection with a length bounded by the difference of the $t$ components of
the causal curve endpoints. So one cannot expect to gain control over
sequences in $\Sigma$ with unbounded lengths.

######  Lemma 3.4.

$\Sigma$ is convex if and only if $M$ is causally simple.

###### Proof.

Assume $\Sigma$ is convex. Observe that $(M,g)$ is causal as $t$ is a time
function. By translational invariance over the time fiber and by time
reflection symmetry, we need only to show that $J^{+}(p)$ is closed for any
point of the form $p=(0,x)$.

Let $q_{n}=(b_{n},y_{n})\in J^{+}(p)$ with $q_{n}=(b_{n},y_{n})\to(b,y)=q$.
There exist causal paths $\sigma_{n}$ from $p$ to $q_{n}$ of the form
$(\frac{b_{n}}{d^{h}(x,y_{n})}\,t,\sigma_{n}(t))$ with $\sigma_{n}$ from $x$
to $y_{n}$. The causality condition reads $\|\sigma_{n}^{\prime}\|_{h}\leq
b_{n}/d^{h}(x,y_{n})$. Hence $d^{h}(x,y_{n})\leq
L^{h}(\sigma_{n})=\int_{0}^{d^{h}(x,y_{n})}\|\sigma_{n}^{\prime}\|_{h}\leq
b_{n}$, which by continuity gives $d^{h}(x,y)\leq b$.

By convexity there exists a unit speed geodesic
$\sigma:[0,d^{h}(x,y)]\to\Sigma$ connecting $x$ to $y$. Now
$\gamma:[0,d^{h}(x,y)]\to M,\gamma(t):=(\frac{b}{d^{h}(x,y)}\,t,\sigma(t))$ is
a path from $p$ to $q$ that is causal iff, $d^{h}(x,y)\leq b$, which is the
case as we just proved. Thus $q\in J^{+}(p)$ and, by the arbitrariness of $q$,
$J^{+}(p)$ is closed.

Conversely, let $M$ be causally simple and let $x,y\in\Sigma$. There exist
curves $\sigma_{n}$ from $x$ to $y$ with $L^{h}(\sigma_{n})=:l_{n}\searrow
l:=d^{h}(x,y)$ with $\|\sigma_{n}^{\prime}\|_{h}=1$. The curves
$\gamma_{n}:[0,l_{n}]\to M,\gamma_{n}(t):=(t,\sigma_{n}(t))$ from $(0,x)$ to
$(l_{n},y)$ are null and hence $(l,y)\in J^{+}((0,x))$ by causal simplicity.
So there exists some causal curve $\alpha:[0,l]\to M,\alpha(t)=(t,\beta(t))$
with $\beta$ a path in $\Sigma$ from $x$ to $y$. Moreover $0\geq
g(\alpha^{\prime},\alpha^{\prime})=-1+\|\beta^{\prime}\|_{h}^{2}$ and so
$\|\beta^{\prime}\|_{h}^{2}\leq 1$, which leads to $L^{h}(\beta)\leq l$ and
hence $\beta$ must be a minimizing geodesic from $x$ to $y$. ∎

##  4 The counterexamples

With Thm. 2.8 we learned that on a Riemannian manifold convexity implies the
minimal LGS. In this section we show that the converse does not hold.

Then we shall consider the Lorentzian Thm. 2.7, analogous to Thm. 2.8,
according to which causal simplicity implies the maximal null LGS. By using
the results of the previous section on static spacetimes, we shall show, once
again, that the converse implication does not hold.

Let us give an example of Riemannian space $(\Sigma,h)$ which satisfies the
minimal LGS but is not convex. A very similar Riemannian space was constructed
in [2, Sec. 2.1] as a counterexample to another statement also related to the
notions of convexity and connectedness in Riemannian spaces.

Figure 1: The counterexample consists of two cylinders $H_{0}$ and $H_{1}$,
closed above with cups and connected by a sequence of immersed tubes $T_{k}$,
$k\geq\bar{k}$. This figure is inspired by that in [2, Sec. 2.1].

######  Example 4.1 (A non-convex Riemannian manifold possessing the minimal
LGS).

The space $\Sigma$ consists of two cylinders $H_{0}$ and $H_{1}$ closed above
with cups and then connected in the flat region by a sequence of immersed
tubes $T_{k}$, $k\geq\bar{k}>0$. The Cauchy boundary of the space can be
identified with the union of two circles, which are the lower boundaries of
the cylinders used in the construction. The mouths of the tubes converge to
points $e_{0}$ and $e_{1}$ on the Cauchy boundary of the space.

Let the geodesic distance between the center of the mouths of $T_{k}$ and
$T_{k+1}$ in $H_{0}$ before the discs (tube mouths) are excised, be
$\frac{1}{2^{k+2}}$, let the diameter of the mouths (discs) in the same
geometry be $\frac{1}{4^{k}}$ (basically is goes so fast to zero that in our
arguments this distance becomes negligible), and let the length of the tube
$T_{k}$ from mouth to mouth in $(\Sigma,h)$ be $\frac{1}{2^{k-1}}$. Let
analogous conditions hold in $H_{1}$. If $\gamma_{1}$ is a curve passing
through $T_{k}$ from mouth to mouth (here $\gamma_{1}$ might have endpoints
belonging to some tubes that might coincide with $T_{k}$ or not) then we can
find a curve $\gamma_{2}$ connecting the same endpoints of $\gamma_{1}$
passing through $T_{k+1}$ from mouth to mouth and such that

$\ell(\gamma_{2})\leq\ell(\gamma_{1})+2\frac{1}{2^{k+2}}-\frac{1}{2^{k-1}}+\frac{1}{2^{k}}+O(\frac{1}{4^{k}})\leq\ell(\gamma_{1})-\frac{1}{2^{k+1}}+O(\frac{1}{4^{k}}).$

This implies that, provided the space is defined with $\bar{k}$ sufficiently
large, $\ell(\gamma_{2})<\ell(\gamma_{1})$, hence a minimizing geodesic cannot
pass through a tube from mouth to mouth and so that there are certainly pairs
of points not connected by a minimizing geodesic, for instance any pair
$(x_{0},x_{1})$ where $x_{0}$ belongs to the first cylinder and $x_{1}$
belongs to the second cylinder. As a consequence, convexity does not hold
(this fact was already pointed out in [2]).

We want to show that the minimal LGS holds. Let
$\sigma_{n}\colon[0,a_{n}]\to\Sigma$ be minimizing unit speed geodesics such
that $p_{n}:=\sigma_{n}(0)\to p$ and $q_{n}:=\sigma_{n}(a_{n})\to q$,
$p,q\in\Sigma$. Without loss of generality we can assume that the tangents
$\dot{\sigma}_{n}$ converge to some unit vector $u\in T_{p}\Sigma$, and since
the diameter of $(\Sigma,h)$ is bounded we can also assume that $a_{n}\to a$.
Let $\sigma\colon I\to\Sigma$ be the geodesic that starts from $p$ with
tangent $u$. If its domain interval includes $[0,a]$, then by the continuity
of the exponential map we can conclude that

$q_{n}=\exp_{p_{n}}(\dot{\sigma}_{n}(0)a_{n})\to\exp_{p}(ua)=\sigma(a)$

that is $q=\sigma(a)$. Observe that $a_{n}$ is the length of $\sigma_{n}$ and
it converges to the length $a$ of $\sigma$. Thus $\sigma$ must be minimizing
otherwise for sufficiently large $n$, $\sigma_{n}$ would not be minimizing.
This would give the desired result so we have only to show that the domain of
$\sigma$ includes $[0,a]$.

Suppose not. The maximal domain of $\sigma$ is $[0,b)$ for some $b\leq a$, and
by standard ODE theory [13] $\sigma(t)$ escapes every compact set as $t\to a$
(though possibly returning indefinitely to it).

Let $T^{p}=T_{s}$ be the tube whose mouth is closest to $p$ (which could be
the tube to which $p$ belongs), and let $T^{q}=T_{t}$ be the tube whose mouth
is closest to $q$ (if there are more choices for the closest tube we choose
that with larger $s$, resp. $t$). For sufficiently large $k$ namely for
$k>K>\max(s,t)$, with suitable $K>0$ the minimizing geodesics $\sigma_{n}$
cannot transverse $T_{k}$ from mouth to mouth (as they are minimizing, cf. the
above argument).

Let us section the figure by cutting with a horizontal plane at a height $y>0$
selected so that above it we have the points $p$, $q$, and the tubes $T_{k}$,
$k\leq K$, thus including $T^{p}$ and $T^{q}$. Then the geodesics $\sigma_{n}$
stay entirely above the horizontal section (because if they cross the plane
then they can be replaced by a shorter curve running on the plane for a
geodesic segment, a fact which would contradict the minimizing property). As a
consequence $\sigma$ being a limit of $\sigma_{n}$ is also entirely above the
plane and so it cannot escape every compact set of $\Sigma$. The contradiction
proves that the domain of $\sigma$ is $[0,a]$ and so that the minimal LGS
holds.

We are ready to give the Lorentzian example.

######  Example 4.2 (A maximally causal pseudoconvex and causally continuous
but non-causally simple spacetime).

We consider the static spacetime $(M,g)$ with $M={\mathbb{R}}\times\Sigma$ and
$g=-dt^{2}+h$, where $(\Sigma,h)$ is as in Example 4.1. By construction, $M$
is strongly causal (actually stably causal, because of the time function $t$).
Due to the timelike Killing vector $\partial_{t}$ it is reflective and hence
causally continuous [17].

Now, since $\Sigma$ possesses the minimal LGS, it is also minimal pseudoconvex
by Lemma 2.5 and hence $M$ is maximal causal (hence null) pseudoconvex by
Lemma 3.2.

On the other hand $\Sigma$ is not convex and so, by Lemma 3.4, $M$ is not
causally simple.

## Acknowledgements

B.S. is supported by the Uni:Docs program of the University of Vienna. R.S. is
supported by FWF-grants P28770 and P33594. E.M. is partially supported by GNFM
of INDAM. J.H. and S.S. are partially supported by the SFB/TRR 191 “Symplectic
Structures in Geometry, Algebra and Dynamics”, funded by the Deutsche
Forschungsgemeinschaft. Part of this work has been developed while S.S. was a
visiting professor at Politecnico di Bari; he thanks Politecnico di Bari and
the Department of Mechanics, Mathematics and Management for the support.

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