Eu . Phys. J. C (2022) 82:299
h ps://doi.o g/10.1140/epjc/s10052-022-10275-3
Regula A icle - Theo e ical Physics
Ch onology p o ec ion implemen a ion in analogue g a i y
Ca los Ba celó1,a, Jokin Eguia Sánchez2, Ge a do Ga cía-Mo eno1,b, Gil Jannes3,c
1Ins i u o de As o ísica de Andalucía (IAA-CSIC), Glo ie a de la As onomía, 18008 G anada, Spain
2Depa men o Cell Biology and His ology, Facul y o Medicine and Nu sing, Uni e si y o he Basque Coun y (UPV/EHU),
Ba io Sa iena S/N, 48940 Leioa, Spain
3Depa men o Financial and Ac ua ial Economics and S a is ics, Uni e sidad Complu ense de Mad id, Campus Somosaguas s/n,
28223 Pozuelo de Ala cón, Mad id, Spain
Recei ed: 31 Janua y 2022 / Accep ed: 27 Ma ch 2022
© The Au ho (s) 2022
Abs ac Analogue g a i y sys ems o e many insigh s
in o g a i a ional phenomena, bo h a he classical and a
he semiclassical le el. The exis ence o an unde lying
Minkowskian s uc u e (o Galilean in he non- ela i is ic
limi ) in he labo a o y has been a gued o di ec ly o bid
he simula ion o geome ies wi h Closed Timelike Cu es
(CTCs) wi hin analogue sys ems. We will show ha his is no
s ic ly he case. In p inciple, i is possible o simula e space-
imes wi h CTCs whene e his does no en ail he p esence o
a ch onological ho izon sepa a ing egions wi h CTCs om
egions ha do no ha e CTCs. We ind an Analogue-g a i y
Ch onology p o ec ion mechanism e y simila in spi i o
Hawking’s Ch onology P o ec ion hypo hesis. We iden i y
he uni e sal beha iou o analogue sys ems nea he o ma-
ion o such ho izons and discuss he u he implica ions ha
his analysis has om an eme gen g a i y pe spec i e. Fu -
he mo e, we build explici geome ies con aining CTCs, o
ins ance space imes cons uc ed om wo wa p-d i e con ig-
u a ions, ha migh be use ul o u u e analysis, bo h om
a heo e ical and an expe imen al poin o iew.
Con en s
1 In oduc ion ......................
2 A emp s o simula e Gödel space ime .........
3 A su ey o some space imes displaying CTCs
amenable o analogue g a i y simula ion .......
3.1 CTCs enginee ing h ough wa p-d i e bubbles ..
3.2 Gene alized wa p-d i e egions ..........
3.3 Misne and Misne -like space imes ........
4 Analogue g a i y simula ions: a emp s o simula e CTCs
ae-mail: [email p o ec ed]
be-mail: [email p o ec ed] (co esponding au ho )
ce-mail: [email p o ec ed]
5 Ch onology p o ec ion: lessons om analogue g a i y
6 Summa y and conclusions ...............
Re e ences .........................
1 In oduc ion
I is by now well known ha many sys ems akin o condensed
ma e sys ems, in he sense o being composed by a la ge
amoun o elemen a y building blocks (a oms o abs ac pa -
icles), exhibi a beha iou in ce ain egimes which can be
cha ac e ized by he p esence o some e ec i e ields, classi-
cal o quan um, mo ing in an e ec i ely cu ed Lo en zian
geome y. These beha iou s a e collec i ely called “analogue
g a i y” [1,2]. In i s b oades desc ip ion, he analogue g a -
i y p og am in ends o ob ain new insigh s in o g a i a ional
beha iou s by analyzing hei equi alen coun e pa s wi hin
hese analogue amewo ks. The e e se di ec ion: acqui ing
new ideas abou labo a o y sys ems by impo ing g a i a-
ional no ions and echniques, is also pa o he analogue
g a i y ealm.
The mos pa adigma ic analysis wi hin his p og am has
been he heo e ical and expe imen al e i ica ion o Hawk-
ing adia ion wi hin black hole con igu a ions e en when
hese ake place in he con ex o an e ec i e and collec-
i e phenomenon. The appea ance o spon aneous Hawking
adia ion in Bose-Eins ein condensa es has been obse ed
in [3] as o iginally sugges ed in [4]. Apa om black holes,
since he la e 90s, many o he ypes o geome ies ha e also
been p oposed in di e en labo a o y se ings [1], such as
o a ing geome ies [5–7], cosmological solu ions [8] wi h a
e y ecen expe imen al ealiza ion wi h luids o ligh [9],
an i-de Si e space ime [10], and e en wa p-d i e geome-
ies [11,12]. Fo a mo e exhaus i e lis , see [1] and e e -
ences he ein.
0123456789().: V,- ol 123
299 Page 2 o 17 Eu . Phys. J. C (2022) 82:299
In his pape we a e in e es ed p ecisely in some puzzles
ha appea when playing wi h hese wa p d i e geome ies.
I is well known ha one can in p inciple use wa p d i es o
build ime machines [13]. This aises he ques ion whe he
one could simula e a geome y wi h Closed Timelike Cu es
(CTCs) wi hin an analogue sys em inspi ed by wa p d i es.
The labo a o y sys ems used o build analogue g a i y con-
igu a ions a e always embedded in ou locally Minkowskian
wo ld. Howe e , he Minkowski s uc u e o ou cu en un-
damen al desc ip ion o Na u e ypically does no e en play
a ole in he analogue geome ies: hese geome ies can be
ob ained di ec ly wi hin a Galilean desc ip ion o he lab-
o a o y. In bo h cases, i seems ha he causal s uc u e o
he backg ound (Minkowski o Galilean) p ohibi s he sim-
ula ion o causally pa hological space imes in he embedded
analogue g a i y sys em [14]. Howe e , by analyzing di e -
en si ua ions, we will show ha his is in ac no s ic ly
he case. The simula ion o space imes wi h CTCs pe se
does no p esen insu moun able obs acles. The eal p oblem
appea s when he ele an space imes posses a ch onological
ho izon [15], ha is, a su ace sepa a ing a egion wi h CTCs
om ano he wi h a s anda d causali y. Fo ex e nal o labo-
a o y obse e s, he inabili y o gene a e a egion wi h CTCs
mani es s i sel in he o m o di e gences in ce ain p op-
e ies o he local physics ha hey expe ience. On he o he
hand, o an in e nal obse e wi hou di ec access o he
unde lying causal s uc u e, he inabili y o p oduce CTCs
mani es s i sel h ough some e ec i e p o ec ion mecha-
nism. As we will discuss, hese mechanisms a e eminiscen
o Hawking’s Ch onology P o ec ion Conjec u e.
Th oughou ou discussion, we will e ise se e al sim-
ple con igu a ions o space imes wi h CTCs, such as wa p
d i e space imes, Gödel space ime and Misne space ime.
We p esen e sions o hese geome ies ha a e amenable o
u he analysis bo h in he con ex o analogue g a i y bu
also om a pu ely geome ical pe spec i e. In o de o main-
ain an explici connec ion wi h po en ial labo a o y ealiza-
ions o such CTC space imes, we will ocus on a conc e e
subs a um, namely a gene alized Bose-Eins ein condensa e
wi h aniso opic masses [1,16]. The in e se acous ic me ic
o linea pe u ba ions on his quan um luid can be w i en
gμν =μ
ρ0c⎛
⎝
−1− i
− jc2hij − i j⎞
⎠.(1)
He e ρ0is he backg ound luid densi y, c he local speed
o sound and i(x) he eloci y o he luid, which is simply
he g adien o he phase o he mac oscopic BEC wa e unc-
ion. The ma ix hij (o i s in e se hij) akes in o accoun any
aniso opy acqui ed by he e ec i e masses o he bosons in
he condensa e: mij =μhij, whe e μis an a bi a y con o -
mal cons an . In simple (iso opic) BECs, hij is jus a mul iple
o he iden i y ma ix. Fo a weakly in e ac ing BEC he e is
also he ollowing ela ion be ween c,μ and he e ec i e
coupling cons an o he condensa e λ:c2=λρ/μ.
Theme ic(1) au oma ically inhe i s he s able causali y
p ope y [15,17] om he backg ound s uc u e [14]. In pa -
icula , i con ains a globally de ined ime unc ion since
gμν∂μ ∂ν =−μ
ρ0c≤0.(2)
This appea s o au oma ically ule ou he possibili y o
simula ing me ics con aining CTCs. As s a ed abo e, we
will see ha wi hou u he quali ica ion his is in ac no
s ic ly ue. Mo eo e , in he cases in which we eally ind
an obs uc ion, i is in e es ing o analyze when and how
hese e ec i e-me ic desc ip ions b eak down, and how
hese b eakdowns a e ela ed o mechanisms o Ch onology
p o ec ion.
A b ie ou line o he emainde o his wo k is he ollow-
ing. We begin in Sec . 2wi h he wa m-up exe cise o ying
o simula e a Gödel space ime and mild de o ma ions he eo
in he sys em desc ibed abo e. We will ind ha , al hough i
is possible o simula e CTCs, hey a e i ial in a sense ha
we will speci y conc e ely. Fu he mo e, we will ind ha i
is impossible o simula e a modi ica ion o Gödel’s space-
ime such ha a ch onologically well-beha ed egion wi h
no CTCs e ol es in o a egion wi h CTCs, due o he di e -
gence o he speeds o he luid equi ed. Mo i a ed by his
exe cise, we y o analyze whe he his is a gene ic ea u e o
space imes con aining CTCs. Fo ha pu pose, we in oduce
in Sec . 3a ca alogue o geome ies amenable o simula ion
in analogue g a i y. Some o hem do no ha e a Gene al
Rela i is ic coun e pa . In Sec . 3.1 we desc ibe space imes
con aining CTCs enginee ed om wo wa p-d i e bubbles.
We discuss he impossibili y o doing i in 1 +1 space ime
dimensions, wi h special emphasis on he poin ha CTCs
in such dimensionali y equi e non- i ial opologies. Based
on hese wa p d i e ube geome ies, we in oduce a am-
ily o simple geome ies which a e quali a i ely simila o
hem bu much easie o handle in Sec . 3.2. We conclude
Sec . 3wi h a discussion o 1 +1-dimensional space imes in
Sec . 3.3. We in oduce he a che ypal example o a space ime
con aining a ch onological ho izon, Misne ’s space ime, and
hen discuss how an e e nal cylinde wi h a la me ic can be
unde s ood as ha ing “ i ial” CTCs by a simple in e change
o he ime and space coo dina es. A eade in e es ed jus
in he Ch onology P o ec ion mechanism in Analogue g a -
i y can sa ely skip hese i s sec ions and jump di ec ly o
Sec . 4, which con ains a de ailed desc ip ion o he possibil-
i y o simula ing i ial CTCs in ou analogue model, and he
impossibili y o simula ing ch onological ho izons. Fu he -
mo e, we iden i y he insu moun able di icul y ha e e y
123
Eu . Phys. J. C (2022) 82:299 Page 3 o 17 299
s anda d analogue g a i y model would ace when ying o
simula e a ch onological ho izons. In Sec . 5we discuss he
in e play o ou analysis and Hawking’s Ch onology P o-
ec ion conjec u e, i s implica ions o he eme gen g a i y
p og am and we also connec wi h ecen ela ed discussions
in he li e a u e. Finally, we inish in Sec . 6by summa izing
he con en o he a icle and desc ibing po en ial di ec ions
o u u e wo k.
No a ion and con en ions. We will use he signa u e
(−,+,...,+) o he space ime me ic and ollow he
Misne –Tho ne–Wheele con en ions o he cu a u e en-
so s [18]. G eek indices (μ,ν,...)will un om 0 o D, ep-
esen ing space ime indices, whe eas La in indices (i,j...)
will un om 1 o Dand ep esen spa ial indices. Eins ein’s
summa ion con en ion is used h oughou he wo k unless
o he wise s a ed.
2 A emp s o simula e Gödel space ime
As a wa m-up exe cise we will desc ibe Gödel’s me ic
as he a che ypal example o a geome y which con ains
CTCs [15,19]. The pu pose o his sec ion is wo old. Fi s ,
we will desc ibe he geome ic p ope ies o Gödel’s space-
ime. Many o hese p ope ies will be sha ed by any space-
ime con aining CTCs, hus allowing us o ocus he dis-
cussion on he essen ial ea u es o any success ul simu-
la ion o CTCs in an analogue model. Second, we will dig
in o he p oblems ha appea when one a emp s o sim-
ula e such ch onologically pa hological space imes. A mo e
gene al discussion conce ning gene ic space imes displaying
CTCs will be p o ided la e . The s a ing poin o his sec ion
has a subs an ial o e lap wi h he unpublished wo k [20]. An
analysis simila o he one p esen ed he e o he simula ion
o Gödel geome y in an op ic sys em was p esen ed in [21].
The iden i ica ion o he me ic componen s wi h he physical
pa ame e s o he analogue sys em do no seem o be done in
he co ec way, and hence he di e gences ha we obse e
in he ho izons he e a e absen .
Gödel’s space ime is a solu ion o he Eins ein equa ions
wi h sui able sou ces, namely a nega i e cosmological con-
s an and he ene gy momen um enso o a p essu eless
pe ec luid wi h a densi y: ρ∝−. In app op ia e coo di-
na es ( , ,φ,z), i can be w i en as ollows [22]:
ds2=−d 2+d 2
1+ 2
4ω2+ 21− 2
4ω2dφ2
+dz2−√2
ω 2d dφ(3)
whe e ωis a pa ame e cha ac e izing he solu ion and ela ed
o he densi y and hence also i ially ela ed o he cosmo-
logical cons an as ω2=−. Tha his geome y con ains
CTCs can be seen as ollows: o ≥ C=2ω, he (Killing)
ec o ield ∂φbecomes imelike. Since such a ec o needs
o be pe iodically iden i ied o a oid a conical singula i y a
=0, we ha e ha φ∼φ+2π. Hence, his ec o ield
has closed o bi s. Since i becomes imelike a ≥ C,i is
i ial o conclude ha he o bi s o φ o > Ca e CTCs.
I seems ha no CTCs pass h ough he egion < C.
Howe e , we mus ake in o accoun ha his geome y is
comple ely homogeneous, in ac i con ains a g oup o i e
Killing ec o ields ac ing ansi i ely on he mani old [15].
This means ha e e y poin o he mani old can be mapped
by a symme y ans o ma ion o any o he poin on he man-
i old. Hence, CTCs pass h ough e e y single poin in his
space ime. Howe e , he e a e no CTCs con ined o he egion
< C, in ac e e y CTC passing h ough he egion < C
c osses he cylinde = Can e en numbe o imes.
F om he poin o iew o an acous ic me ic, we can eal-
ize ha his p ecise sys em o coo dina es allows o a di ec
ealiza ion o Gödel’s me ic wi h sui able luid pa ame e s.
The non- anishing componen s o he in e se Gödel me ic
in hese coo dina es a e
g =−F( ),
g φ=− 1
√2ω
1
1− 2
4ω2F( ),
gφφ =1
2
1
1− 2
4ω2F( ),
g =1+ 2
4ω22
1− 2
4ω2F( ),
gzz =1+ 2
4ω2
1− 2
4ω2F( ). (4)
wi h F( )de ined as
F=1− 2
4ω2
1+ 2
4ω2.(5)
Compa ing Eq. (4) wi h Eq. (1) we ealize ha we simply
need a mo ion o he luid in he φ-di ec ion i= φδi
φ. Tak-
ing in o accoun he change o coo dina es o a cylind ical-
like coo dina e sys em we ha e ha he azimu hal eloci y
mus be
φ=1
√2ω
1− 2
4ω2
.(6)
123
299 Page 4 o 17 Eu . Phys. J. C (2022) 82:299
On he o he hand he h ee p incipal di ec ions o he
aniso opy ma ix hij need o obey
c2h =1+ 2
4ω22
1− 2
4ω2,
c2hφφ =1+ 2
4ω2
1− 2
4ω22,
c2hzz =1+ 2
4ω2
1− 2
4ω2.(7)
These quan i ies can be in e p e ed as h ee aniso opic sound
speeds c2
,c2
φand c2
z, espec i ely. In addi ion, we no ice ha
o he weakly in e ac ing BEC we ha e λ/c3=F, o equi -
alen ly
c2=λ2/31+ 2
4ω22/3
1− 2
4ω22/3,(8)
we inally ob ain
h =λ−2/31+ 2
4ω24/3
1− 2
4ω21/3,
hφφ =λ−2/31+ 2
4ω21/3
1− 2
4ω24/3,
hzz =λ−2/31+ 2
4ω21/3
1− 2
4ω21/3.(9)
Le us analyze he p ope ies o his luid equi ed o simu-
la e he geome y. The i s hing we no ice is ha he eloci y
o he luid becomes in ini e a = C, whe e i also changes
i s sign. Hence he luid sys em is singula ; he < Cand
> Cpa s o he sys em a e disconnec ed. Howe e , he
CTCs a e li ing en i ely in he ex e io egion o he me ic
so i may s ill appea ha he analogue sys em can locally
simula e CTCs. Howe e , looking a he speeds o sound we
iden i y an addi ional issue. The speeds o sound also di e ge
a = C, bu mo eo e c2
=c2h and c2
z=c2hzz become
nega i e o > C,soc and czbecome pu ely imagina y.
This means ha we no longe ha e wa e-like beha iou s, o
in o he wo ds causal signalling, in hose di ec ions wi hin
he analogue sys ems. The whole acous ic pic u e appea s o
b eak down. In ha sense, he = Ccylinde can be unde -
s ood as a so o “domain wall”: i sepa a es he in e io
egion in which we ha e causal signalling om he egion in
which we ha e abno mal (exponen ially ampli ied o a en-
ua ed) beha iou o he pu a i e sound-like exci a ions.
To design a ealis ic si ua ion, imagine ha we s a wi h
a luid a es and inci e a o a ion a ound he z-axis wi h he
in en ion o e ol ing owa ds he Gödel geome y. In o de
o do so, we need o ob ain a con igu a ion in which he e
is a sepa a ion be ween a clockwise and an an iclockwise
o a ing pa o he luid, sepa a ed by a su ace a = C
whe e he eloci y needs o app oach in ini y and mo eo e
he speed o sound also blows up. These equi emen s essen-
ially imply ha he hyd odynamic desc ip ion o he BEC
b eaks down. Fu he mo e, because o he in ini e luid eloc-
i y a he = Csu ace, no signal could c oss his su -
ace. Howe e , he physical pa ame e s o he BEC a e well
de ined o > C, whe e one inds CTCs o he BEC
exci a ions. On he one hand, one would need an imagina y
sound speed in ha egion. This can be a ained in BECs
wi h a ac i e in e ac ions [23,24]. On he o he hand, wo
o he e ec i e aniso opic masses o he BEC mus be nega-
i e. The peculia i y o a pa icle wi h a nega i e mass is ha
i accele a es backwa ds when pushed o wa d.1Howe e ,
i has been shown expe imen ally ha i is indeed possible
o achie e such s ange beha iou and c ea e pa icles wi h
nega i e e ec i e masses [25].
The e o e, we ha e a su p ising si ua ion. The exci a ions
o a qui e s ange BEC, wi h a ac i e in e ac ions (which in
p inciple would appea o o bid wa e phenomena) combined
wi h some nega i e aniso opic masses, end up beha ing as
i hese exci a ions li e in a pe ec ly Lo en zian wo ld dis-
playing CTCs. The si ua ion would be equi alen in any o he
aniso opic luid, no necessa ily quan um.2One would jus
need ha c2
and c2
zbecome nega i e in some egion while
c2
φs ays posi i e. Roughly speaking, his ensu es ha he
and zcoo dina es acqui e he same signa u e as he coo -
dina e, lea ing he angula coo dina e φas he coo dina e o
di e en signa u e, i.e. he ime coo dina e, and CTCs will
de elop. F om he pe spec i e o he in e nal obse e s inside
he luid, “ ime” would be wha o an ex e nal (labo a o y)
obse e is simply he angula coo dina e.
Going back o Gödel’s me ic, one could be emp ed o
modi y he pa ame e s o he analogue model and egula ize
he di e gences. A simple example would be he ollowing
p o iles whe e a sui able egula ing pa ame e 1isin o-
duced ( o simplici y we es ic ou luid o be e ec i ely
wo-dimensional):
1No ice ha such nega i e masses a e no undamen al, and hus need
no esul in achyonic ins abili ies.
2I is ue, howe e , ha enginee ing a classical luid o display “nega-
i e mass” exci a ions migh be much mo e complica ed, pe haps e en
impossible in p ac ice.
123
Eu . Phys. J. C (2022) 82:299 Page 5 o 17 299
Fig. 1 The le panel ep esen s he aniso opic speeds o sound and
he azimu hal eloci y o he luid equi ed o pe o m an analogue
simula ion o Gödel’s geome y o ω=1. All o hem display a e -
ical asymp o e a = C. The igh panel displays he co esponding
aniso opic speeds o sound and he azimu hal eloci y o he luid o
he egula ized Gödel geome y in oduced in he ex , also o ω=1
and o he egula iza ion pa ame e =0.01. Whe eas he g aphics
on he le panels all blow up a = C, he ones in he igh panels a e
smoo h e e ywhe e. All unc ions a e no malized o a maximum alue
o 1 in he plo
φ=
√2ω
1− 2
4ω2
1− 2
4ω22+2
,(10)
c2
=1+ 2
4ω221− 2
4ω2
1− 2
4ω22+2
,(11)
c2
φ=1+ 2
4ω2
1− 2
4ω22+2
.(12)
Howe e , i is s aigh o wa d o see ha he associa ed
acous ic me ic (a cousin o Gödel’s me ic) is no a egu-
la Lo en zian me ic now, i is degene a e a = C. S ill,
s ange as i may seem, his acous ic sys em does app oach
Gödel me ic o C. These speeds o sound and eloci y
o he luid as well as he co esponding ones o pu e Gödel
a e plo ed in Fig. 1.
A no e o cau ion migh be in o de a his poin . Al hough
his analysis sugges s ha i is possible o simula e CTCs in
an analogue sys em (a luid in his case), hese CTCs a e,
in a sense, i ial. F om he poin o iew o he ex e nal
obse e , he c ea ion o hese CTCs co esponds simply o
decla ing ha he in e nal obse e is using in he in e nal
sys em an angula coo dina e as ime coo dina e. F om now
on, we will e e o his kind o CTCs as i ial, o dis inguish
hem om CTCs ha appea in he causal u u e o a causally
well-beha ed egion, which a e he mos in e es ing ones
om a physical poin o iew. Hawking cha ac e ized his
ype o space imes geome ically as hose wi h a compac ly
gene a ed Cauchy ho izon [26].3
Space imes wi h non- i ial CTCs can hus be unde s ood
as hose wi h a smoo h ansi ion om a egion wi hou CTCs
o one wi h CTCs. F om an analogue poin o iew, i seems
ha he simula ion o such non- i ial cases is no possible.
Indeed, hey would equi e ei he a non- egula Lo en zian
me ic, which could be ep oduced wi hin an analogue model,
o a well-de ined Lo en zian me ic bu equi ing some di e -
gences in he analogue model. The o me case would no
eally cons i u e a clea p oo o p inciple o he possibili y
o simula ing space imes wi h CTCs, since he CTCs could
be unde s ood o be an a i ac o he non-smoo hness o he
3A compac ly gene a ed Cauchy ho izon is a Cauchy ho izon such ha
he pas ex ension o i s gene a o s en e s and emains wi hin a compac
subse o he mani old.
123
299 Page 6 o 17 Eu . Phys. J. C (2022) 82:299
me ic and hus, in a sense, spu ious o a leas no di ec ly
ela ed o a (non-analogue) ela i is ic equi alen . On he
o he hand, he la e case has al eady been discussed and,
based on he example o Gödel’s me ic, seems o co e-
spond o i ial CTCs a bes , since he wo egions need o
be causally disconnec ed.
To inish his sec ion le us conside he possibili y o sim-
ula ing a geome y which app oaches Gödel’s me ic only in
a ini e ange o he labo a o y ime . To ou knowledge, his
geome y does no ha e a Gene al Rela i is ic coun e pa ,
in he sense ha i is no a solu ion o he Eins ein equa ions
wi h he ene gy-momen um enso o a known ma e con-
en . Such beha iou could be achie ed wi h he help o a
modula ing unc ion ( ), such ha we can w i e a me ic
ds2=−d 2+d 2
1+ ( ) 2
4ω2+ 21− ( ) 2
4ω2dφ2
+dz2− ( )√2
ω 2d dφ, (13)
whe e we can choose ( ) o be a unc ion wi h compac
suppo , o ins ance
( )=exp −σ
( B− )( − A),(14)
which is non- anishing o ∈( A, B). Hence, he me ic
ep esen s a la space ime ou side his in e al, and de el-
ops CTCs wi hin he egion ( A, B). In o de o con ine he
CTCs o a compac egion o space also, one could o ce he
me ic componen gφφ o ake nega i e alues only wi hin a
ini e in e al o he -componen , o example h ough he
ollowing eplacemen
gφφ = 21− ( ) 2
4ω2−→ 21− ( )e− 2
σ2 2
4ω2,
(15)
whe e σmus o be su icien ly la ge in o de o he unc ion
1−e− 2
σ2 2
4ω2 o display wo ze os. This new geome y exhibi s
CTCs ha a e con ined wi hin a ini e egion o space ime.
Howe e , o he same a gumen s explained abo e, i is no
possible o simula e hem as acous ic me ics since he luid
would be equi ed o de elop a singula eloci y. This again
illus a es ou mo e gene al poin ha i seems impossible o
gene a e me ics wi h non- i ial CTCs h ough an analogue
me ic.
3 A su ey o some space imes displaying CTCs
amenable o analogue g a i y simula ion
In his sec ion we a e going o p esen some geome ies con-
aining CTCs which we hink a e concep ually simple han
Gödel space ime. Mos o hese geome ies can be ound
somehow in he li e a u e. Howe e , we hink ha i is wo -
hy o e ise hem and p esen hem in a uni ied way, so
ha hey a e amenable o be analyzed om he analogue
g a i y pe spec i e. Fi s , we will s a desc ibing he geom-
e y ha esul s om combining wo wa p d i es and dis-
plays CTCs. Mo i a ed by he p ope ies o his geome y, we
will in oduce a amily o space imes which a e simple bu
encapsula e hei main geome ic ea u es. Finally, we will
discuss p obably he mos pa adigma ic example o space-
ime con aining CTCs: he so-called Misne space ime. This
space ime is used as a p oxy o mo e con olu ed analysis,
since i s ch onological ho izon is usually conside ed o ha e
he gene al p ope ies a ch onological ho izon has. Al hough
o mos p ac ical pu poses his is ue, we will pu special
emphasis he e on he ac ha Misne space ime is opologi-
cally non- i ial as a mani old (o he wise i could no con ain
CTCs as we will explain). In highe dimensions, (like 3 +1
space ime dimensions) i is possible o ha e space imes wi h
CTCs exhibi ing a i ial opology. F om an analogue g a i y
pe spec i e, his makes li e much easie o hei simula ion.
3.1 CTCs enginee ing h ough wa p-d i e bubbles
Wa p d i es we e o iginally in oduced by Alcubie e [27].
They a e based on dis u bing a gi en space ime wi hin a
compac egion in such a way ha o obse e s ou side ha
egion, obse e s inside o i mo e wi h supe luminal speeds.
They can be hough as “ achyonic” bubbles ha p opaga e
as e han ligh o ex e nal obse e s. The simples me ics
ep esen ing wa p d i es can be w i en using he Ná a io’s
line elemen :
ds2=−d 2+δijc−2dxi− id dxj− jd .(16)
The e a e some wa p d i es wi h non-ze o lapse unc ion o
wi h a non-Euclidean me ic, al hough we will no ocus on
hem.
In his way, he me ic o a wa p d i e is nicely adap ed o
be simula ed wi h he acous ic me ic o a BEC, as desc ibed
abo e, o acous ic me ics showing up in o he luids. This
idea has al eady been sugges ed in he li e a u e, see o
ins ance [11,12]. Essen ially, we need o iden i y he eloci y
o he luid wi h he shi unc ions en e ing he wa p d i e
elemen . Fo conc e eness, we can hink o a wa p-d i e bub-
123
Eu . Phys. J. C (2022) 82:299 Page 7 o 17 299
ble whose p o ile acqui es he ollowing o m
i( ,xi)=δi
xu( ) (x−x( ))2+y2+z2(17)
whe e (x)is a compac suppo unc ion, desc ibing he
p o ile o he bubble which is peaked a ound he poin s o
he ajec o y
x( )=x(0)+
0
d u( ), y=z=0.(18)
As we jus say, he simula ion o a single wa p d i e in a
acous ic analogue sys em is di ec [1]. One jus need o gen-
e a e a space ime egion a which he eloci y o he low
exceeds he speed o sound. F om he in e nal pe spec i e
his allows o a el om one poin o ano he a eloci ies
highe han ha o sound (which emembe akes he ole o
he speed o ligh ). I is con enien o ew i e he wa p d i e
me ic in Eq. (16) as a pe u ba ion o he la space ime me -
ic ημν as
gμν =ημν +bμν,(19)
wi h bμν ha ing he ollowing non- anishing componen s:
b00 =u2( ) 2(x−x( ))2+y2+z2,
b01 =b10 =−u( ) (x−x( ))2+y2+z2.(20)
Now, al hough a single bubble wa p d i e by i sel does no
esul in any ch onology o causali y iola ions, as al eady
no iced in [13] i is ela i ely easy o enginee a space ime
ha con ains CTCs by aking ad an age o ha ing wo wa p-
d i e bubbles in dimensions highe han 1 +1. The idea is
simila o he way in which one can send in o ma ion o he
pas wi h a pai o achyon pa icles in la space ime [28].
Wha is i hen he clash, i any, be ween wa p d i e me ics
and hei analogue simula ions?
We wan o cons uc egula space ime geome ies based
on a combina ion o wo wa p d i es, in such a way ha
hey con ain CTCs. A e inding such geome ies we will
analyze whe he i is possible o ep oduce hem wi hin an
analogue model in Sec . 4. The simples such cons uc ion
ha one can hink o a p io i in ol es wo wa p-d i e bub-
bles in a 1 +1 dimensional se ing wi h i ial R2 opology.
In he emain o his subsec ion, we a e going o discuss
o a momen his 1 +1 po en ial cons uc ion. Fi s , we will
nai ely p esen i . Then, we will illus a e he obs uc ion ha
one inds when one ies o o malize his cons uc ion. A e
ha , we will show ha his p oblem canno be ci cum en ed
by p esen ing some heo ems showing ha his cons uc ion
Fig. 2 The igu e ep esen s a wa p d i e ube ha s a s in a loca ion
A and ends in loca ion B. The unde lying space ime can be conside ed
D+1 dimensional wi h he wa p-d i e bubble mo ing in a s aigh
line (along he x-coo dina e, in he pic u e). In 1 +1 dimensions he
causal cone would become jus wo c ossing lines bu we keep he cone
symbol o cla i y. This is he simples cons uc ion o a wa p d i e and
as desc ibed in he ex , i can be simula ed in an analogue g a i y model
wi hou u he p oblems
is ac ually no possible. Finally, we will conclude his sub-
sec ion by explaining how his cons uc ion can be ex ended
o D+1 space ime dimensions wi h i ial opology wi hou
p oblems.
Le us begin wi h he mos nai e way in which one migh
y o make his con igu a ion. Le us conside a 1+1 dimen-
sional Minkowski backg ound. Le us choose an ine ial e -
e ence ame Swi h Ca esian coo dina es ( ,x). Fu he -
mo e, le us choose wo e en s Aand Bsuch ha hey a e
spacelike sepa a ed, wi h coo dina es ( A,xA)and ( B,xB),
espec i ely. Wi hou loss o gene ali y, le us assume ha
B> A. This se up is ep esen ed pic o ially in Fig. 2. We can
enginee a wa p d i e ube connec ing he wo e en s. No ice
ha he ligh cones inside he ube a e modi ied wi h espec
o he Minkowskian e e ence. Fu he mo e, we emphasize
ha he ube has some hick walls a which he ligh cones
expe ience a il ing e ec . I is p ecisely on hose walls whe e
he s ess ene gy enso suppo ing hese con igu a ions nec-
essa ily de elops some ene gy condi ions iola ions [27]. We
emphasize ha he ajec o y as seen om ou side he ube
appea s o be spacelike. O cou se, obse e s going om A
o Bwi hin he bubble would be ollowing s ic ly imelike
ajec o ies.
I one can cons uc his wa p d i e, om a pu ely gen-
e al ela i is ic pe spec i e i is also possible o cons uc
an equi alen wa p d i e con igu a ion in which he coo di-
na e ime goes o he pas ins ead o he u u e [13]. Le us
w i e down such a me ic explici ly. Le us s a cons uc -
ing a wa p d i e me ic as he one jus desc ibed bu using
123
299 Page 8 o 17 Eu . Phys. J. C (2022) 82:299
ano he ine ial e e ence ame Smo ing wi h eloci y
in he xdi ec ion. Le us deno e wi h a p ime he ca esian
coo dina es o he e e ence ame S, i.e. ( ,x). In hese
coo dina es, he me ic o he bubble akes he simple o m
g
μν o Eq. (16). To ind he me ic in he coo dina es ( ,x)
adap ed o he ine ial ame S, we simply need o pe o m a
boos o eloci y − in he x-axis, wi h he ela i e eloci y
be ween Sand S. In his way, he esul ing me ic eads
gμν =ημν +cμν,(21)
whe e cμν is ans o med om he p ime coo dina es o he
unp imed ones by an o dina y Lo en z ans o ma ion. We
highligh ha i s unc ional o m di e s om he bμν enso
in oduced in Eq. (19). Ac ually, we can ind i s unc ional
o m by pe o ming a Lo en z boos in he x-di ec ion o
eloci y − , whe e is he ela i e eloci y be ween bo h
ames Sand S. Explici ly, i is desc ibed by he ollowing
linea ans o ma ion in:
()μ
ν=cosh φsinh φ
sinh φcosh φ,wi h anh φ= , (22)
W i ing down he ans o ma ion we ob ain he ollowing cμν
enso
c00 =u2( ( ,x), x( ,x), y−y0,z)cosh2φ
−2u( ( ,x), x( ,x), y−y0,z)cosh φsinh φ, (23)
c01 =c10 =u2( ( ,x), x( ,x), y−y0,z)cosh φsinh φ
−u( ( ,x), x( ,x), y−y0,z)cosh2φ
−u( ( ,x), x( ,x), y−y0,z)sinh2φ, (24)
c11 =u2( ( ,x), x( ,x), y−y0,z)sinh2φ
−2u2( ( ,x), x( ,x), y−y0,z)sinh φcosh φ. (25)
No ice ha he unc ions and xdepend on he coo dina es
,xin a non i ial manne , and we need o ew i e hem in
e ms o such coo dina es. In ac , he coo dina es {xμ}a e
ela ed o he coo dina es {xμ} h ough a Lo en z ans o -
ma ion om S o Sin which he Lo en z ma ix is p ecisely
μ
ν. Explici ly, he change o coo dina es eads
= cosh φ−xsinh φ, (26)
x= sinh φ+xcosh φ. (27)
W i ing e e y hing explici ly wi hou ixing a pa icula a-
jec o y and shape o he bubbles would no be e y illus-
a i e, hence we simply keep e e y hing indica ed as done
abo e. We emphasize ha one jus needs o choose a p o ile
o he bubble and he eloci ies in o de o be able o w i e
down explici ly he me ic in global coo dina es by subs i-
u ing in he exp essions abo e. In gene ic e ms we ha e
Fig. 3 The igu e ep esen s a wa p d i e ube ha s a s in a loca ion
Aand ends in loca ion B. The unde lying space ime can be conside ed
D+1 dimensional wi h he wa p-d i e bubble mo ing in a s aigh line
(along he x-coo dina e, in he pic u e). In 1 +1 dimensions he causal
cone would become jus wo c ossing lines bu we keep he cone symbol
o cla i y
build a wa p d i e a elling backwa ds in coo dina e ime ,
pic o ially ep esen ed in Fig. 3. No e howe e ha one can
easily check ha he Lo en z ans o ma ion we ha e applied
make he new wa p d i e me ic o ake a di e en o m om
Na a io’s line elemen . Fo he a gumen s ha ollow he p e-
cise o m o he ubes will no be ele an .
Now se ing up a combina ion o a “ o wa d” and a “back-
wa d” wa p-d i e bubbles one can a emp o build a ime
machine. Le us explici ly illus a e his. We can i s se up a
“ o wa d” wa p d i e allowing a as e - han-ligh a el om
A o B. Once he a elle has exi he bubble a B he could
immedia ely en e in a new wa p d i e, now o a “backwa d”
ype, and a el om A o B. Using ano he spacelike a-
jec o y, as seen om he ex e nal Minkowski space ime, his
second wa p d i e can ake he ime- a elle o an e en B
in pass o he ini ial e en A. In his way a CTC is comple ed.
This se up is pic o ially ep esen ed in Fig. 4.
Howe e , he e is a p oblem conce ning his cons uc ion.
The e is a egion, he c ossing egion C, a which one would
need o ha e wo di e en me ics. Ac ually, his ansla es
in o a singula poin whe e he me ic is no de ined. I is
na u al hen o pose he ollowing ques ion: is i possible o
disen angle he c ossing poin mo ing a ound he s a ing and
ending poin s o he bubbles o /and playing wi h hei shapes
and he speci ic o ms o hei eloci ies 1( ), 2( ), in such
a way ha we ind a comple ely egula Lo en zian me ic
con aining CTCs? The answe o his ques ion is nega i e.
To unde s and why, i is use ul o conside a oy geome y
which nicely illus a es he obs uc ion. Imagine ha we w i e
down a geome y which is ha o la space ime a e e y
123
Eu . Phys. J. C (2022) 82:299 Page 9 o 17 299
Fig. 4 We ep esen he e wo wa p d i es in 1 +1 dimensions and
in such a con igu a ion ha hey appea o allow o he o ma ion o
CTCs. The pu ple cu e ep esen s a gene ic CTC on his backg ound.
The p oblem wi h his 1+1 con igu a ion is ha he me ic in he egion
whe e he wo wa p-d i e bubbles c oss is ill-de ined. As desc ibed in
he ex he simples geome y wi h CTCs is ei he one wi h a S1×R
opology o wi h opology RD+1in D+1 dimensions, wi h D>1. In
his la e case, we jus need o enginee he wo wa p d i es bubbles in
di e en pa allel planes o a oid he c ossing
poin excep o a ci cle o adius one a ound he o igin in a
gi en se o Ca esian coo dina es ( ,x). A such ci cle, we
make he ligh cones o make an angle o 45 deg ees wi h he
ci cle a e e y poin . Clea ly such geome y is singula since
he e is a jump in he me ic. E en i we y o smoo h such
geome y by gi ing he ci cle a ini e size and con e ing i
in o a disk, we can ind a cu e along which he ligh cones
make a o a ion o 360 deg ees and, hence, i is impossible
o ha e a smoo h me ic in he egion enclosed by such cu e
o egula ize i in some way. This is depic ed schema ically
in Fig. 5.
Now we a e in posi ion o s a ing he ollowing heo em:
Theo em Le (M,g)be a wo dimensional simply con-
nec ed space ime (being M he smoo h mani old and g i s
me ic). Then, he causali y condi ion au oma ically holds.
Recall [15] ha a space ime is said o sa is y he ch ono-
logical condi ion i i does no con ain any closed imelike
cu es, and i is said o obey he causali y condi ion i he e
a e no closed non-spacelike cu es. The idea o he p oo
is al eady con ained in he obse a ion ha we ha e made
abo e: ha ing he s uc u e o ligh cones enclosing a com-
pac egion, i is impossible o push hem inwa ds o ou wa ds
ha egion wi hou making hem ze o o singula . The o -
maliza ion o his s a emen can be ound in [29].
I is possible o e en p o e a s onge esul . In Lo en zian
geome y he e exis a hie a chy o causali y condi ions
Fig. 5 We ep esen he e he se up desc ibed in he ex ha al eady
shows he di icul y p esen when ying o build CTCs in a space ime
wi h a i ial opology. The shaded egion ep esen s he egion o abno -
mal beha iou o he ligh cones. I seems impossible o egula ize he
ligh cones wi hou emo ing poin s om he space ime, o he wise he
me ic would need o anish a some poin and hence i would no be a
egula space ime
whe e each o hem is s onge han he p e ious ones.
Al hough he ch onology condi ion is he weakes o such
condi ions, ollowed by he causali y condi ion, and hey a e
enough o ule ou closed non-spacelike cu es, one can s ill
hink o space imes ha a e “a bi a ily” close o con aining
closed causal cu es. Hence, hese se o s onge condi ions
a emp s o o malize hese no ion o “almos ha ing closed
cu es” [15,17].
The s ong causali y condi ion [17], which is obeyed
by a space ime i o e e y poin pand e e y neighbou -
hood No p, he e exis s a neighbou hood Ocon ained
in Nsuch ha no causal cu e in e sec s Omo e han
once. By essen ially he same a gumen s exhibi ed in ou
p oo , one can s eng hen he esul and p o e ha e e y wo
dimensional ime-o ien able simply connec ed space ime is
s ongly causal (see Lemma 14.34 o [29]).
Ac ually, i is e en possible o s eng hen his esul unde
he same hypo hesis. A space ime is said o be s ably causal
i he e exis s a imelike ec o ield μsuch ha he Lo en z
me ic de ined as ˜gμν =gμν − μ ν(which has la ge ligh -
cones han gμν a e e y poin ) con ains no closed causal
cu es. I can be p o ed ha a space ime is s ably causal
i and only i i admi s a globally de ined ime- unc ion,
i.e. a unc ion ha is s ic ly inc easing along each u u e
di ec ed causal cu e [15]. I is also possible o p o e ha
s able causali y implies s ong causali y, i.e. i is a s onge
condi ion. Hence, s able causali y is a s onge condi ion and
one migh wonde whe he i is possible o p o e ha e e y
wo dimensional simply connec ed space ime obeys i wi h-
ou u he assump ions. In [30] he a i ma i e answe is
p o ided in he o m o Theo em 3.43, whe e i is shown ha
e e y simply connec ed wo dimensional space ime (M,g)
123
299 Page 16 o 17 Eu . Phys. J. C (2022) 82:299
o he in e p e a ion o he physical sys em as a g a i a ional
analogue in he i s place. We ha e ocused on an acous-
ic sys em in which i is essen ially he eloci y o he luid
which mus di e ge in o de o c ea e he equi ed il ing o
he sound cones. We ha e shown ha i is pe ec ly possible o
simula e geome ies allowing supe luminal beha iou s such
as wa p d i es. Howe e , his does no imply di ec ly ha one
can build an analogue ime machine, as we ha e discussed in
de ail. I is in ac he o ma ion o a ch onological ho izon
which is o bidden in he analogue implemen a ion, since i
is no possible o c ea e a wa p d i e a elling backwa d in
labo a o y ime.
The obs uc ions ound by explo ing he analogue g a i y
implemen a ion o CTCs esona e wi h Hawking’s idea o a
Ch onology P o ec ion mechanism in semiclassical Gene al
Rela i i y. F om he poin o iew p esen ed he e, such p o-
ec ion mechanisms a ise na u ally in amewo ks o Eme -
gen G a i y. In hese eme gen amewo ks, he e exis s a
backg ound s uc u e wi h a mo e undamen al unde lying
causali y, which na u ally p e en s he ype o manipula ions
equi ed o c ea e ch onological pa hologies.
Finally, we sum up h ee impo an lessons om he
p esen wo k. (i) Supe luminali y i sel does no imply he
possibili y o abno mal causal beha iou such as ime a el;
(ii) P oblems in he analogue implemen a ion o ch onolog-
ical ho izons appea due o he ela i e il ing o he causal
cones; (iii) The cu en obse a ional absence o ch onolog-
ical pa hologies in ou uni e se is na u ally explained in
amewo ks in which he e is a ixed unde lying causali y
beyond he local Gene al Rela i is ic modi ica ions explo ed
so a .
Acknowledgemen s The au ho s hank G isha Volo ik o use ul con-
e sa ions and o sha ing unpublished no es on he simula ion o
Gödel space ime. The au ho s hank Ca los Sabín and F anco Fio ini
o help ul co espondence. GGM and CB hank Luis Ga ay, Miguel
Sánchez and Valen ín Boyano o e y use ul con e sa ions. GGM
hanks Robe o Empa án o an enligh ening con e sa ion. Financial
suppo was p o ided by he Spanish Go e nmen h ough he p ojec s
PID2020-118159GB-C43 and PID2020-118159GB-C44, and by he
Jun a de Andalucía h ough he p ojec FQM219. C.B. and G.G.M.
acknowledge inancial suppo om he S a e Agency o Resea ch o
he Spanish MCIU h ough he “Cen e o Excellence Se e o Ochoa”
awa d o he Ins i u o de As o ísica de Andalucía (SEV-2017-0709).
GGM is unded by he Spanish Go e nmen ellowship FPU20/01684.
Da a A ailabili y S a emen This manusc ip has no associa ed da a
o he da a will no be deposi ed. [Au ho s’ commen : The e is no da a
in ou wo k o be deposi ed.]
Open Access This a icle is licensed unde a C ea i e Commons A i-
bu ion 4.0 In e na ional License, which pe mi s use, sha ing, adap a ion,
dis ibu ion and ep oduc ion in any medium o o ma , as long as you
gi e app op ia e c edi o he o iginal au ho (s) and he sou ce, p o-
ide a link o he C ea i e Commons licence, and indica e i changes
we e made. The images o o he hi d pa y ma e ial in his a icle
a e included in he a icle’s C ea i e Commons licence, unless indi-
ca ed o he wise in a c edi line o he ma e ial. I ma e ial is no
included in he a icle’s C ea i e Commons licence and you in ended
use is no pe mi ed by s a u o y egula ion o exceeds he pe mi -
ed use, you will need o ob ain pe mission di ec ly om he copy-
igh holde . To iew a copy o his licence, isi h p://c ea i ecomm
ons.o g/licenses/by/4.0/.
Funded by SCOAP3.
Re e ences
1. C. Ba celó, S. Libe a i, M. Visse , Li ing Re . Rela i . 14, 159
(2011)
2. G. Volo ik, The Uni e se in a Helium D ople In e na ional Se ies
o Monog aphs on Physics. (OUP, Ox o d, 2009)
3. V.I. Kolobo , K. Golubko , J.R. Muñoz de No a, J. S einhaue ,
Na . Phys. 17, 362 (2021)
4. L.J. Ga ay, J.R. Anglin, J.I. Ci ac, P. Zolle , Phys. Re . Le . 85,
4643 (2000). a Xi :g -qc/0002015
5. G. Jannes, G.E. Volo ik, JETP Le . 102, 73 (2015).
a Xi :1506.00882 [g -qc]
6. T. To es, S. Pa ick, A. Cou an , M. Richa z, E.W. Ted o d, S.
Wein u ne , Na . Phys. 13, 833 (2017). a Xi :1612.06180 [g -qc]
7. M.C. B aido i, R. P izia, C. Mai land, F. Ma ino, A. P ain, I.
S a shyno , N. Wes e be g, E.M. W igh , D. Faccio, Measu e-
men o Pen ose supe adiance in a pho on supe luid (2021).
a Xi :2109.02307 [physics.op ics]
8. C. Ba celo, S. Libe a i, M. Visse , In . J. Mod. Phys. D 12, 1641
(2003). a Xi :g -qc/0305061
9. J. S einhaue , M. Abuza li, T. Aladjidi, T. Bienaimé, C. Pieka ski,
W. Liu, E. Giacobino, A. B ama i, Q. Glo ieux, Analogue cos-
mological pa icle c ea ion in an ul acold quan um luid o ligh
(2021). a Xi :2102.08279 [cond-ma .quan -gas]
10. C. Ba celo, A. Campos, Phys. Le . B 563, 217 (2003).
a Xi :hep- h/0206217
11. U.R. Fische , M. Visse , Eu ophys. Le . 62, 1 (2003).
a Xi :g -qc/0211029
12. S. Finazzi, Analogue g a i a ional phenomena in Bose–Eins ein
condensa es. Ph.D. hesis, SISSA, T ies e (2011). a Xi :1208.4729
[g -qc]
13. A.E. E e e , Phys. Re . D 53, 7365 (1996)
14. M. Visse , Class. Quan um G a i y 15, 1767 (1998).
a Xi :g -qc/9712010
15. S.W. Hawking, G.F.R. Ellis, The La ge Scale S uc u e o Space-
Time Camb idge Monog aphs on Ma hema ical Physics. (Cam-
b idge Uni e si y P ess, Camb idge, 2011)
16. C. Ba celo, S. Libe a i, M. Visse , Class. Quan um G a i y 18, 1137
(2001). a Xi :g -qc/0011026
17. R.M. Wald, Gene al Rela i i y (Chicago Uni e si y P ess, Chicago,
1984)
18. C.W. Misne , K.S. Tho ne, J.A. Wheele , G a i a ion (W.H.F ee-
man, San F ancisco, 1973)
19. K. Godel, Re . Mod. Phys. 21, 447 (1949)
20. G. Jannes, G.E. Volo ik, Gödel’s uni e se and ime-like cu es in
condensed ma e . Unpublished
21. F. Fio ini, S.M. He nandez, E.L. Losada, Phys. Re . D 104, 124009
(2021). a Xi :2108.12049 [g -qc]
22. E. Kaja i, R. Walse , W.P. Schleich, A. Delgado, Gen. Rela i .
G a i . 36, 2289 (2004). a Xi :g -qc/0404032
23. E.A. Donley, N.R. Claussen, S.L. Co nish, J.L. Robe s,
E.A. Co nell, C.E. Wieman, Na u e 412, 295–299 (2001).
a Xi :cond-ma /0105019
24. R.A. Duine, H.T.C. S oo , Phys. Re . Le . 86, 2204 (2001).
a Xi :cond-ma /0007320
25. M.A. Khamehchi, K. Hossain, M.E. Mossman, Y. Zhang, T. Busch,
M.M. Fo bes, P. Engels, Phys. Re . Le . 118, 155301 (2017)
123
Eu . Phys. J. C (2022) 82:299 Page 17 o 17 299
26. S.W. Hawking, Phys. Re . D 46, 603 (1992)
27. M. Alcubie e, Class. Quan um G a i y 11, L73 (1994).
a Xi :g -qc/0009013
28. W.B. Rolnick, Phys. Re . 183, 1105 (1969)
29. B. O’Neill, Semi-Riemannian Geome y wi h Applica ions o Rel-
a i i y, Pu e and Applied Ma hema ics, ol. 103. (Academic P ess,
Camb idge, 1983)
30. J. Beem, P. Eh lich, K. Easley, Global Lo en zian Geome y, Chap-
man & Hall/CRC Pu e and Applied Ma hema ics, 2nd edn. (Taylo
& F ancis, Mil on Pa k, 1996)
31. M. Sánchez, Some coun e examples abou globally hype bolic
space imes (2021). a Xi :2110.13672 [g -qc]
32. C.W. Misne , Taub–NUT space as a coun e example o almos any-
hing, in Rela i i y Theo y and As ophysics. Vol. 1: Rela i i y and
Cosmology, ol. 8, ed. by J. Ehle s (Ame ican Ma hema ical Soci-
e y, Rhode Island, 1967), p. 160
33. I.D. No iko , Phys. Re . D 45, 1989 (1992)
34. F. Eche e ia, G. Klinkhamme , K.S. Tho ne, Phys. Re . D 44,
1077 (1991)
35. J. F iedman, M.S. Mo is, I.D. No iko , F. Eche e ia, G.
Klinkhamme , K.S. Tho ne, U. Yu se e , Phys. Re . D 42, 1915
(1990)
36. G.E. Volo ik, Pisma Zh. Eksp. Teo . Fiz. 104, 660 (2016).
a Xi :1610.00521 [cond-ma .o he ]
37. J. Nissinen, G.E. Volo ik, JETP Le . 105, 447 (2017).
a Xi :1702.04624 [cond-ma .s -el]
38. C. Sabín, Phys. Re . D 94, 081501 (2016). a Xi :1603.00639
[quan -ph]
39. G. Ma ín-Vázquez, C. Sabín, Class. Quan um G a i y 37, 045013
(2020). a Xi :1810.05124 [quan -ph]
40. R. Pen ose, R.D. So kin, E. Woolga , A posi i e mass heo em
based on he ocusing and e a da ion o null geodesics (1993).
a Xi :g -qc/9301015
41. K.D. Olum, Phys. Re . Le . 81, 3567 (1998). a Xi :g -qc/9805003
42. M. Visse , B. Basse , S. Libe a i, Nucl. Phys. B P oc. Suppl. 88,
267 (2000). a Xi :g -qc/9810026
43. M. Visse , B. Basse , S. Libe a i, AIP Con . P oc. 493, 301 (1999).
a Xi :g -qc/9908023
44. S. Gao, R.M. Wald, Class. Quan um G a i y 17, 4999 (2000).
a Xi :g -qc/0007021
45. S.W. Kim, K.S. Tho ne, Phys. Re . D 43, 3929 (1991)
46. M. Visse , in Wo kshop on Con e ence on he Fu u e o Theo e -
ical Physics and Cosmology in Hono o S e en Hawking’s 60 h
Bi hday (2002). a Xi :g -qc/0204022
47. C. Ba celó, R. Ca ballo-Rubio, L.J. Ga ay, G. Ga cía-Mo eno,
Appl. Sci. 11, 8763 (2021). a Xi :2108.06582 [g -qc]
48. N. Rosen, Phys. Re . 57, 147 (1940)
49. N. Rosen, Phys. Re . 57, 150 (1940)
50. E. Al a ez, D. Blas, J. Ga iga, E. Ve dague , Nucl. Phys. B 756,
148 (2006). a Xi :hep- h/0606019
51. R. Empa an, M. Tomaše i´c, Holog aphy o ime machines (2021).
a Xi :2107.14200 [hep- h]
52. S. Ca lip, S ud. His . Philos. Sci. B 46, 200 (2014).
a Xi :1207.2504 [g -qc]
53. A.D. Dolgo , I.D. No iko , Phys. Le . B 442, 82 (1998).
a Xi :g -qc/9807067
54. A. Adams, N. A kani-Hamed, S. Dubo sky, A. Nicolis, R. Ra azzi,
JHEP 10, 014 (2006). a Xi :hep- h/0602178
55. X.O. Camanho, J.D. Edels ein, J. Maldacena, A. Zhiboedo , JHEP
02, 020 (2016). a Xi :1407.5597 [hep- h]
56. S. Libe a i, S. Sonego, M. Visse , Ann. Phys. 298, 167 (2002).
a Xi :g -qc/0107091
57. E. Babiche , V. Mukhano , A. Vikman, JHEP 02, 101 (2008).
a Xi :0708.0561 [hep- h]
58. R. Ge och, AMS/IP S ud. Ad . Ma h. 49, 59 (2011).
a Xi :1005.1614 [g -qc]
123
