Review on the matching conditions for the tidal problem: towards the application to more general contexts

Gene al Rela i i y and G a i a ion (2024) 56:50
h ps://doi.o g/10.1007/s10714-024-03237-5
REVIEW
Re iew on he ma ching condi ions o he idal p oblem:
owa ds he applica ion o mo e gene al con ex s
Eneko A angu en1·Raül Ve a1
Recei ed: 31 Janua y 2024 / Accep ed: 29 Ma ch 2024 / Published online: 22 Ap il 2024
© The Au ho (s) 2024
Abs ac
The idal p oblem is used o ob ain he idal de o mabili y (o Lo e numbe ) o s a s.
The semi-analy ical s udy is usually ea ed in pe u ba ion heo y as a i s o de pe -
u ba ion p oblem o e a sphe ically symme ic backg ound con igu a ion consis ing
o a s ella in e io egion ma ched ac oss a bounda y o a acuum ex e io egion
ha models he idal ield. The ield equa ions o he me ic and ma e pe u ba ions
a he in e io and ex e io egions a e complemen ed wi h co esponding bounda y
condi ions. The da a o he wo p oblems a he common bounda y a e ela ed by he
so called ma ching condi ions. These condi ions o he idal p oblem a e known in
he con ex s o pe ec luid s a s and supe luid s a s modelled by a wo- luid. He e
we e iew he ob aining o he ma ching condi ions o he idal p oblem s a ing om
a pu ely geome ical se ing, and p esen hem so ha hey can be eadily applied o
mo e gene al con ex s, such as o he ypes o ma e ields, di e en mul iple laye s
o phase ansi ions. As a guide on how o use he ma ching condi ions, we eco e
he known esul s o pe ec luid and supe luid neu on s a s.
Keywo ds Gene al ela i i y ·Pe u ba ion heo y ·Tidal p oblem ·Ma ching
condi ions
Con en s
1 In oduc ion ............................................. 2
1.1 Ma ching in pe u ba ion heo y ................................ 4
2 Geome ical pe u bed ma ching condi ions ............................. 5
2.1 Backg ound ........................................... 5
BEneko A angu en
[email p o ec ed]
BRaül Ve a
[email p o ec ed]
1Fisica Saila, Uni e si y o he Basque Coun y UPV/EHU, Ba io Sa iena s/n, 48940 Leioa, Biscay,
Spain
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50 Page 2 o 16 E. A angu en, R. Ve a
2.2 Fi s o de ............................................ 6
2.3 Ma ching condi ions in e ms o he Eins ein enso ...................... 6
3 Applica ion o gene al ela i i y ................................... 7
3.1 S ep (i): geome ical pe u bed ma ching condi ions using he backg ound ield equa ions .. 8
3.2 S ep (ii): adding he ield equa ions a i s o de ........................ 10
4 Tidal p oblem and Lo e numbe ................................... 12
5 Conclusions ............................................. 14
Re e ences ................................................ 15
1 In oduc ion
The pu pose o his pape is o b ie ly e iew he ma ching condi ions in pe u ba ion
heo y o i s o de in ol ed in he e en-pa i y idal p oblem, and p esen hem in a
way ha a e eadily applicable o gene aliza ions o he han in he known pe ec luid
[1–3] o wo- luid ( o model supe luid) s a s [4] con ex s.
The aim o s udying he idal p oblem is o ob ain he Lo e numbe , o , equi alen ly,
he idal de o mabili y, o s a s. The global p oblem consis s o a egion o model he
s ella in e io which is ma ched o an ex e io acuum egion ha models a idal
ield, p oduced e.g. by a companion s a . Using pe u ba ion heo y he model is buil
as a i s o de pe u ba ion on op o a s a ic and sphe ically symme ic backg ound
con igu a ion consis ing o an in e io egion ball o adius R∗wi h a Schwa zschild
ex e io . The Lo e numbe is hen de e mined om he alue ha ce ain unc ion o
he i s o de me ic pe u ba ion, ha depends only on he adial coo dina e, y−( ),
akes a he ou e su ace ( he su ace as seen om he ex e io ) o he s a , speci ically
y−(R∗). Tha alue is ound by in eg a ing an analogous unc ion y+( )in he in e io
egion om he o igin ou wa ds o ob ain y+(R∗), and hen use wha e e in o ma ion
we ha e on he jump [y]:=y+(R∗)−y−(R∗) o de e mine y−(R∗)(see [5]).
In he con ex o pe ec luid models, i was a gued in [1,2] ha he jump is gi en
by (see Eq. (15) in [2])
[y]=κ
R3
∗
2ME(R∗), (1)
whe e Eis he ene gy densi y (o he backg ound con igu a ion), Mis he mass o
he s a , and we use κ o he g a i a ional coupling cons an (κ=8πin na u al
uni s G=c=1). The a gumen uses he ac ha he unc ion y( )is de ined
as he quo ien H( )/H( ), whe e His a unc ion ha desc ibes pa o he i s
o de pe u ba ion ( ha is he ha monic =2 pa o he H0 unc ion o he Regge–
Wheele decomposi ion [6]). The p oblem is se on he whole domain ∈(0,∞),
and His implici ly assumed o be con inuous a =R∗, jus as well as he es o he
pe u ba ion me ic unc ions. The key idea is ha he Eins ein ield equa ions (EFE)
a i s o de can be de ined (in s ic e ms a leas in a dis ibu ional sense), and H( )
hus sa is ies a second o de ODE ha con ains a e m wi h dE/dP, whe e Pis he
p essu e o he backg ound con igu a ion. The e o e, since P( )mus be con inuous
a he su ace, i E( )has a jump a =R∗, hen dE/dP p esen s a Di ac del a e m
he e p opo ional o E(R∗). As a esul , he solu ion o H( )yields a con inuous
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Re iew on he ma ching condi ions o he idal p oblem… Page 3 o 16 50
unc ion bu discon inuous H( )a =R∗,wi hajump[H]p opo ional o E(R∗).
Comple ing he chain [y]=[ H/H]=R∗[H]/H(R∗)wi h he explici exp ession
o [H]leads o (1).
Al hough he esul is co ec , he p ocedu e has wo d awbacks. The i s comes
om a p ac ical poin o iew. The p ocedu e elies comple ely on he use o he EFEs
o a pe ec luid a bo h sides ( acuum aken as a i ial pa icula case). I one needs
o conside o he ma e ields a he in e io , o e en phase ansi ions in di e en
egions, he de i a ion o [y]mus be ca ied ou o each di e en case. Le us s ess
ha a e [1,2] we e published, se e al wo ks on o he ypes o s ella in e io s miss
he de i a ion o [y] o p ope ly jus i y ha yis aken o be “con inuous".
The second d awback is concep ual, since he p ocedu e also elies on he se ing
o he in e io and ex e io p oblems as one single p oblem on a common domain,
∈(0,∞), and hen implici ly assumes ha he pe u bed me ic unc ions ha e he
con inui y p ope ies needed o de ise he EFEs in a dis ibu ional sense. This se ing
has i s basis in he o iginal Ha le-Tho ne pe u ba i e amewo k [7,8], whe e all
he pe u bed me ic unc ions we e assumed o be con inuous in ha sense. Howe e ,
he s udy o ma chings in pe u ba ion heo y has shown ha ha cons uc ion is no
necessa y, and no desi able in some cases o in e es . In pe u bed ma ching heo y,
he in e io and ex e io i s o de p oblems (deno ed wi h a +and − espec i ely)
a e mo e con enien ly ea ed as wo sepa a e (gauge ield) p oblems wi h ela ed
bounda y da a a he common bounda y += −=: R∗, wi h basis on a pu ely
geome ical cons uc ion. In ac , some o he unc ions may indeed p esen jumps, ha
can be made o anish by pa ially ixing he gauges, whe eas some o he s necessa ily
p esen jumps in gene al, as o he wise he se ing becomes inconsis en . In pa icula ,
he amendmen needed o he o iginal Ha le-Tho ne model has consequences in he
compu a ion o he s ella mass o second o de [9].
Al hough pe u bed ma ching heo y may seem o be a me e ma hema ical a i ac ,
apa om p o iding i m and igo ous g ounds o he esul s on pe u bed ma chings
based on he Ha le-Tho ne amewo k (a e he needed amendmen s), i p o ides,
on he one hand, ull con ol o e he gauges a ei he side independen ly (see [10]).
This ac is key in he p oo s o uniqueness and exis ence o compac o a ing con-
igu a ions in GR in pe u ba ion heo y o second o de [11,12]. On he o he hand,
he geome ical basis p o ides a di ec way o gene alize he de i a ion, in pa icula ,
o [y] o any ma e ield con en , e en including di e en kind o laye s and phase
ansi ions.
In his pape we e iew he ma ching condi ions o i s o de in pe u ba ion heo y
(based on he wo ks [9,12,13]) aimed a he idal p oblem, s a ing om a pu e
geome ical se ing, and hus eady o be used o ob ain he ma ching condi ions o
gene al s ella ma e ield con en s. We ake he oppo uni y o include a o mal way
o dealing wi h mul iple concen ically dis ibu ed egions. To show how o use he
condi ions, we e iew he ob aining o he pe u bed ma ching condi ions o i s o de
o he idal p oblem o pe ec luid s a s [3], hus eco e ing (and pu ing on i m
g ounds) he condi ion (1), and wo- luid supe luid s a s [4]. Fo comple eness, we
also p o ide in Sec .4 he usual p ocedu e o compu e he Lo e numbe and how he
jump [y]en e s he calcula ion.
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50 Page 4 o 16 E. A angu en, R. Ve a
We include in he ollowing subsec ion a b ie accoun on he ma ching in pe -
u ba ion heo y o se he g ounds, p o ide some ele an e e ences, and ix some
no a ion.
1.1 Ma ching in pe u ba ion heo y
In essence, he p oblem o ma ching in pe u ba ion heo y o i s o de in Gen-
e al Rela i i y s a s wi h a se o wo space imes wi h bounda y (M+,g+,+)and
(M−,g−,−) ha ha e been ma ched ac oss +=−=:  o c ea e a single
space ime (M,g)wi h wo egions, whe e M=M+∪M−and gequals g+o g−
on each co esponding egion. Each egion is now endowed wi h a symme ic enso ,
K+
1and K−
1, which desc ibe he me ic pe u ba ion on he co esponding egion. The
pe u ba ion pa ame e εis chosen so ha he amilies o me ics g±
ε:= g±+εK±
1
desc ibe he me ic o i s o de a each egion. By aking he me ic gεon Mde ined
o be g+
εon M+and g−
εon M−, one can p oceed o cons uc i s Riemann enso
Riem(gε). We say ha he wo egions o he gi en backg ound (M,g)pe u bed
wi h K±
1ma ch ( o i s o de ) when Riem(gε) o i s o de in εcan be cons uc ed
as a dis ibu ion and does no ha e a Di ac del a e m. The gene al ea men o he
ma ching o i s o de , explici ly applicable in e ms o any gauge, was done in [14]
and [15] (see e.g. [16] and [10] ega ding ea men s in e ms o gauge in a ian s).
I was shown ha he wo pe u bed egions ma ch o i s o de i and only i he e
exis a couple ec o s Z±
1de ined a poin s on so ha a ce ain se o condi ions on
a e sa is ied, which depend on {g±,K±
1,Z±
1}. We e e o hose as he i s o de
pe u bed ma ching condi ions, and he no mal pa o Z±
1 o , ha we shall deno e
by ±
1, desc ibe he de o ma ion o ( o i s o de ) as seen om each side [13]in
e ms o he gauges (o class o gauges) chosen.
The second o de p oblem is cons uc ed analogously in e ms o an ex a symme ic
enso o each egion K±
2 o desc ibe he second o de pe u ba ions, and he co e-
sponding second o de ma ching condi ions ( ound in [13]) demand he exis ence o
wo ex a ec o s Z±
2such ha a ce ain se o condi ions o {g±,K±
1,K±
2,Z±
1,Z±
2}
on a e sa is ied.
The pa icula iza ion o second o de s a iona y and axially symme ic pe u ba ions
a ound s a ic and sphe ically symme ic backg ounds was pe o med in [9](seealso
[12]). I mus be s essed ha he pe u ba ion enso ha sui s he e en-pa i y idal
p oblem scena io en e s hose cons uc ions a second o de . The i s o de ma ching
condi ions sui able o he idal p oblem co espond o he condi ions a second o de
wi h a anishing i s o de p oblem in he ea men in [9,12](see[3,4]).
The local na u e o he ma ching p ocedu e allows us o i ially de ise he (pe -
u bed) ma ching p ocedu e needed o a s ella in e io made up o laye s wi h di e en
ma e con en s. Fo he sake o o mali y we conside a cons uc ion based on a se o
N+1 egions {M(i)}wi h bounda y whe e i=1, ..., N+1, so ha M(1)con ains
he o igin and a e o de ed om inne o ou e , ma ched oge he ac oss he ma ching
hype su aces (i)wi h i=1,...,N o o m a global s a ic and sphe ically symme -
ic backg ound space ime as depic ed in Fig. 1. No e ha one may include (N+1) o
deno e “in ini y”. A any (i)sepa a ing wo egions, say M+=M(i)(inne ) and
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Re iew on he ma ching condi ions o he idal p oblem… Page 5 o 16 50
Fig. 1 Schema ic ep esen a ion o he ma ching be ween di e en mani olds. The i- h mani old M(i)is
ma ched o M(i+1) h ough he hype su ace (i).A (i)M(i)plays he ole o he inne pa o he
ma ching, M+=M(i),andM(i+1) he ou e pa , so ha M−=M(i+1). The inne mos and ou e mos
egions, M(1)and M(N+1)a e no linked o any o he space ime
M−=M(i+1)(ou e ), o any pai o unc ions +and −de ined on he co e-
sponding egion, we will use [ ]i:= +|(i)− −|(i) o deno e he “jumps” o
he e.
2 Geome ical pe u bed ma ching condi ions
In his sec ion we se o h he pe u bed ma ching condi ions o i s o de o s a ic
and axially symme ic e en pa i y pe u ba ions a ound a s a ic and sphe ically sym-
me ic backg ound a any hype su ace (i)in pu ely geome ical e ms. Wi h he idal
p oblem in mind we es ic he analysis, as usual, o he ≥2 sec o . We e e o [4],
based in u n on [12] and [9], o he p o ing de ails.
In p inciple, each M(i)is endowed wi h he usual sphe ical coo dina es
{ i, i,θ
i,φ
i}, and we may indica e by i ha a unc ion is de ined on M(i). Howe e ,
o a oid looding all equa ions wi h iindices, in wha ollows we only show explici ly
he i-dependence a he jumps o unc ions.
2.1 Backg ound
The me ic o he backg ound con igu a ion in sphe ical coo dina es is gi en by
g=−eν( )d 2+eλ( )d 2+ 2(dθ2+sin2θdφ2), (2)
(i-subindexes ommi ed in he unc ions and coo dina es) on each M(i).Gi en he
sphe ical symme y o he whole con igu a ion, he ma ching hype su aces (i)inhe i
he symme ies and hey can be pa ame ized by common alues o { ,θ,φ}a each
side (see e.g. [17]). Then, he ma ching condi ions o he backg ound con igu a ion
a e [ ]i=0, ha es ablishes ha (i)is de ined by he same alue, say Ri,o he
espec i e adial coo dina es , and
[λ]i=0,[ν]i=0,(3)
[ν]i=0.(4)
Obse e ha he condi ion [ν]i=0 is, in ac , a consequence o he choice o he
coo dina e a each side, which ha e been aken so ha hei alues coincide on (i).
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50 Page 6 o 16 E. A angu en, R. Ve a
I is in ha sense ha we say ha he ma ching condi ion jus se es o accommoda e
he choice o “gauges” on each side.
2.2 Fi s o de
We ake he i s o de pe u ba ion enso K1desc ibing he s a ic, axially symme ic
and e en pa i y idal ield as gi en in he Regge-Wheele gauge, and decomposed in
Legend e polynomials,
K1=
eν( )H0( )d 2+eλ( )H2( )d 2+ 2K( )(dθ2+sin2θdφ2)P(cos θ),
(5)
on each egion. The ma ching condi ions o ≥2 on each (i) ead [see Eqs. (37)-(39)
in [4]]
[K]i=0,[H0]i=0,(6)
[H2]i−Ri[K
]i=e−λ(Ri)/2i[λ]i,(7)
[H
0]i+Ri
2ν(Ri)[K
]i=−e−λ(Ri)/2i[ν]i,(8)
o some unc ions ia each ≥2. As men ioned in he In oduc ion, he unc ions
iwill desc ibe he de o ma ion o he hype su ace (i)in he class o gauges
compa ible wi h he p oblem (and i is equal a bo h sides) [9,12]. The i s o de
p oblems o ≥2 will hen ma ch a (i)i and only i he e exis s a quan i y i
such ha Eqs. (6)-(8) a e sa is ied.
Al hough he ma ching condi ions o =0,1 a e i ele an as a as he idal
p oblem is conce ned, hey can be ob ained om [9]o [12], and ind, in pa icula ,
ha he jumps become a bi a y enough as o accommoda e all he gauge eedom le
a his poin .
2.3 Ma ching condi ions in e ms o he Eins ein enso
Ou goal now is o exp ess he ma ching condi ions in e ms o he Eins ein enso o
he backg ound geome y Gαβ:= Ein(g)αβ. The only independen and nonze o com-
ponen s sa is y he ela ions (we d op he -dependence he e, p ime deno es de i a i e
wi h espec o )
λ=+1−eλ
− eλG ,(9)
ν=−1−eλ
+ eλG ,(10)
ν =1
2 (λ−ν)(2+ ν)+2eλGθθ,(11)
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Re iew on he ma ching condi ions o he idal p oblem… Page 7 o 16 50
on any egion M(i). Compu ing he jumps o he abo e ela ions on any (i)yields
[λ]i=−Rieλ(Ri)[G ]i,(12)
[ν]i=+Rieλ(Ri)[G ]i,(13)
[ν]i=1+Riν(Ri)
2[λ]i
Ri+2eλ(Ri)[Gθθ]i.(14)
As a esul , using also ha he mass unc ion is gi en by
M( )=
21−e−λ( ),(15)
he ma ching condi ions (3)-(4) can be ew i en as
[M]i=0,[ν]i=0,(16)
[G ]i=0,(17)
while (6)-(8) ead
[K]i=0,[H0]i=0,(18)
[H2]i−Ri[K
]i=−Rieλ(Ri)/2i[G ]i,(19)
[H
0]i−[K
]i=−2eλ(Ri)/2i[Gθθ]i−1+Riν(Ri)
2[H2]i
Ri
.(20)
3 Applica ion o gene al ela i i y
In his sec ion we e iew how he abo e cons uc ion o he pe u bed ma ching o
i s o de applies o p oblems in pe u ba ion heo y o i s o de in Gene al Rela i -
i y (wi hou cosmological cons an ) in he p esen con ex o he idal p oblem, i.e.
pe u ba ions o he o m (5) a ound s a ic and sphe ically symme ic backg ounds.
Fo any gene al case, anyway, he p ocedu e consis s in he h ee ollowing s eps.
(i) As a i s s ep, since only he Eins ein enso o he backg ound geome y en e s
he pe u bed ma ching o i s o de , i su ices o in oduce he Eins ein ield
equa ions (EFE) a he backg ound le el (we omi he iindex o each egion)
Gαβ=κTαβ,(21)
in he equa ions (16)-(20) o ob ain he geome ical pe u bed ma ching condi ions
in e ms o quan i ies ela ed o he desi ed ma e ield con en s o he model. We
a e using G=Ein(g),T o he ene gy-momen um enso a he backg ound le el,
and κ o he g a i a ional coupling cons an .
(ii) The second s ep consis s in using he EFEs a i s o de , oge he wi h he pe u bed
ma ching condi ions om he i s s ep. This may p o ide addi ional condi ions
on he jumps o he pe u ba ion me ic unc ions {H0,H2,K}.
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50 Page 8 o 16 E. A angu en, R. Ve a
(iii) The inal s ep co esponds o he addi ion o he ma e ield ma ching condi-
ions go e ning he beha iou o he ma e ields ac oss laye s, such as su aces
sepa a ing di e en media, phase ansi ions, cha ged su aces, e c...
Be o e we s a wi h he i s s ep, obse e ha he Eins ein enso o he backg ound
geome y gi en by (2) only has (a mos ) h ee non- anishing componen s G ,G ,
and Gθθ=Gφφ(in all egions). The o m o he ene gy-momen um enso compa ible
wi h ha , c. . (21), is many imes e e ed o as an “aniso opic luid” wi h “ adial
p essu e” T =κ−1G . Equa ion (17) hus s a es ha he adial p essu e canno
ha e a jump on any ma ching hype su ace (i). I he ou e mos egion is he ex e io
acuum o a compac objec , his condi ion es ablishes he alue RNo he adial
coo dina e whe e he bounda y common o he wo p oblems is loca ed. On he o he
hand, no ice, howe e , ha nei he [Gθθ]ino [G ]ia e necessa ily ze o in gene al,
and he e o e nei he a e [Tθθ]ino [T ]i. I he e a e no equa ions o he ma e
ields, no addi ional ield ma ching condi ions o hem, he necessa y and su icien
condi ions o “aniso opic luids” o ma ch a i s o de a e he backg ound ma ching
equa ions (16)-(17) plus he wo i s o de condi ions in (18). The e o e, in ha case,
he jumps o H2and he de i a i es o all he pe u ba ion me ic unc ions a e no
cons ained, a p io i.
In he ollowing we conside pe ec luid and wo- luid supe luid egions, using
s eps (i) and (ii) o eco e he known ma ching condi ions in he idal p oblem o
neu on and qua k s a s (see e.g. [1–3]) and supe luid neu on s a s (see e.g. [18]).
3.1 S ep (i): geome ical pe u bed ma ching condi ions using he backg ound
ield equa ions
Le us s a by pe o ming he i s s ep (i) o pe ec luid and wo- luid egions.
Pe ec luid: To ix some no a ion, we w i e he ene gy-momen um enso o a pe ec
luid as
Tαβ=(E+P)UαUβ+Pδα
β,(22)
o some uni luid low U, whe e Eand Pa e he co esponding ene gy densi y and
p essu e, espec i ely. Gi en he o m o he Eins ein enso o (2), as men ioned
abo e, he EFEs on he backg ound imply ha U=e−ν/2∂ , and equi e only he
equa ion
G =Gθθ.(23)
The ene gy densi y and p essu e a e hen gi en by he ela ions
G =−κE,G (=Gθθ=Gφφ)=κP.(24)
The ma ching condi ions (16)-(20) applied o a pe ec luid hus ead
[M]i=0,[ν]i=0,(25)
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Re iew on he ma ching condi ions o he idal p oblem… Page 9 o 16 50
[P]i=0,(26)
[K]i=0,[H0]i=0,(27)
[H2]i−Ri[K
]i=κRieλ(Ri)/2i[E]i,(28)
[H
0]i−[K
]i=−1+Riν(Ri)
2[H2]i
Ri
.(29)
Obse e ha because o he ield equa ions a he backg ound le el, c. . (23), we now
ha e [Gθθ]i=0 and he e o e (20), ha becomes (29), in ol es pe u ba ion me ic
unc ions only.
I he ou e egion is acuum, he jump o he ene gy is simply [E]i=N=E+(RN),
whe e RNsa is ies P+(RN)=0. In gene al, he alues o he ene gy densi y a
he ma ching hype su aces (i), and hus he jumps [E]i, will be de e mined by he
ba o opic EOS E=E(P) ha go e n he s ella con igu a ions on he di e en
egions. In o he cases wi h no ba o opic EOS, such as homogeneous s a s [19] and
some Sky me s a s [20], hose alues will be de e mined by wha e e ela ions used
o close he sys em o equa ions.
No e ha a his poin he unc ion H2may p esen jumps a he ma ching hype -
su aces (see (29)). La e we a e going o see how he pe ec luid equa ions a i s
o de (in combina ion wi h he es o condi ions) imply he anishing o hose jumps.
Supe luid: Fo comple eness, le us in oduce e y b ie ly he wo- luid supe luid
o malism, as desc ibed in [21](seealso[22]). Fo b e i y, we e e o his o malism
simply as supe luid. The low o neu ons and p o ons is gi en by nα=nuαand pα=
p α, whe e nand pa e he numbe densi ies o neu ons and p o ons, espec i ely,
and uαand αa e uni imelike ec o s. All he model is de e mined by he mas e
unc ion
=(n2,p2,x2),
whe e n2:= nαnα,p2:= pαpα, and x2:= −pαnαis he in e ac ion e m. Wi h he
help o he de ini ions
μα:= Bnα+Apα,χ
α:= Cpα+Anα,
wi h
A:= −∂(n2,p2,x2)
∂x2,B:= −2∂(n2,p2,x2)
∂n2,C:= −2∂(n2,p2,x2)
∂p2,
and
:= −nαμα−pαχα,(30)
he ene gy-momen um enso eads
Tαβ=δα
β+pαχβ+nαμβ.(31)
123
50 Page 16 o 16 E. A angu en, R. Ve a
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(2018). h ps://doi.o g/10.1103/PhysRe D.98.084010.a Xi :1806.10986
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105008
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Publishe ’s No e Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in published maps
and ins i u ional a ilia ions.
123