Thermality of horizon through near horizon instability a path integral approach Gaurang Ramakant Kane1 2and Bibhas Ranjan Majhi1 1Department of Physics Indian Institute of Technology Guwahati Guwahati 781039 Assam India

2025-05-06 0 0 522.73KB 17 页 10玖币

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Thermality of horizon through near horizon instability: a path integral approach

Gaurang Ramakant Kane1, 2, ∗and Bibhas Ranjan Majhi1, †

1Department of Physics, Indian Institute of Technology Guwahati, Guwahati 781039, Assam, India

2Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU, United Kingdom

(Dated: October 17, 2023)

Recent investigations revealed that the near horizon Hamiltonian of a massless, chargeless outgoing

particle, for its particular motion in static as well as stationary black holes, is eﬀectively ∼xp kind.

This is unstable by nature and has the potential to explain a few interesting physical phenomena.

From the path integral kernel, we ﬁrst calculate the density of states. Also, following the idea of

[Phys. Rev. D 85, 025011 (2012)] here, in the vicinity of the horizon, we calculate the eﬀective path

corresponding to its Schrodinger version of Hamiltonian through the path integral approach. The

latter result appears to be complex in nature and carries the information of escaping the probability

of the particle through the horizon. In both ways, we identify the correct expression of Hawking

temperature. Moreover, here we successfully extend the complex path approach to a more general

black hole like Kerr spacetime. We feel that such a complex path is an outcome of the nature of near

horizon instability provided by the horizon and, therefore, once again bolstered the fact that the

thermalization mechanism of the horizon may be explained through the aforesaid local instability.

I. INTRODUCTION AND BASIC FORMALISM

A combination of Bekenstein’s proposal [1,2] and

Hawking’s explicit calculation [3,4] proposed black holes

as thermodynamical objects. The temperature of the

horizon is given by T= (ℏκ/2π) (in units of c=1=kB),

where κis known as surface gravity. Till now various

approaches (e.g. path integral of gravitational action

[5,6], tunneling formalism [7–10], gravitational anomaly

approach [11–14], extremization of action functional for

extremal black holes [15–17], Noether conserved charge

[18–22], etc.) have been put forward to get better under-

standing of the Hawking radiation phenomenon as well

as thermal behaviour of black holes. One of them was

to visualise the radiation of the particle as the escap-

ing of horizon barrier through a complex path – known

as quantum tunneling of particles [7–10]. In this ap-

proach, the wave function of the particle is chosen in

the semi-classical limit (the WKB approximation) as

Ψ∼exp[(i/ℏ)SHJ ], where SHJ is the Hamilton-Jacobi

action. It has been observed that SHJ picks up a complex

part across the horizon, providing a quantum mechanical

probability for the particle to escape from the black hole,

leading to a Boltzmann-like tunneling probability. An

analogy with the usual Boltzmann expression yielded an

exact expression for Hawking temperature. Interestingly,

the choice of the wave function under the WKB approxi-

mation has a close similarity with the path integral form

⟨x2, t2|x1, t1⟩=NRDx exp[(i/ℏ)S]∼ N′exp[(i/ℏ)S] in

the semi-classical limit along with saddle point approxi-

mation [23], wherein the left hand side the ket and bra

are basis vectors in Heisenberg picture. Here Sis the

classical action for the system. In the Schrodinger pic-

ture, this path integral can be interpreted as the wave

∗gsrkane@gmail.com

†bibhas.majhi@iitg.ac.in (Corresponding author)

function after time (t2−t1) in position representation

which was initially located at x1at time t1. Therefore

evaluation of it under the aforementioned approximations

must provide the wave function under WKB approxima-

tion, used in tunneling formalism. Hence obtention of the

imaginary part of the action along the horizon provides a

non-trivial signature about the black hole both in WKB

or path integration approaches.

The above discussion implies that the emission of the

particle can occur only when the action incorporates an

imaginary part in it. Hence in principle, the variational

principle of this action must provide a complex path at

the quantum level for the occurrence of such an event.

In original tunneling formalism, no one has addressed

the structure of the path. Interestingly, one can address

such questions in the path integral approach to quan-

tum mechanics. We already mentioned that at the semi-

classical limit, both WKB wave function and path inte-

gration carry similar information. Therefore one expects

that path integration must have a major role in provid-

ing a better understanding of this issue. The idea is as

follows. In the presence of source J(x, t), the vacuum to

vacuum transition amplitude (VVTA) can be expressed

in terms of path integration [23]:

0+0−J= lim

t1→−∞

t2→∞ NZDxe i

ℏS[x,J]= lim

t1→−∞

t2→∞

Z[J].

(1)

In this regard the eﬀective action W[J] is deﬁned by

W[J] = −iℏln Z[J]. Then VVTA can be expressed as

0+0−J= lim

t1→−∞

t2→∞

ℏW[J],(2)

and hence the transition probability is given by the imag-

inary part of the eﬀective action:

0+0−J

2= lim

t1→−∞

t2→∞

e−2WI[J]

ℏ.(3)

arXiv:2210.04056v2 [gr-qc] 14 Oct 2023

In the above, WIis the imaginary part of the eﬀective

action W. The above discussion implies that the prob-

ability other than unity occurs only when WIis non-

vanishing. In that situation, the particle jumps to other

states with ﬁnite (non-trivial) probability. Therefore the

transition of a quantum mechanical system happens due

to source Jonly if the eﬀective action Wis complex.

Applying this theory to a forced quantum oscillator by

a time-dependent source produces the correct transition

probability value (see a detailed discussion in [23,24]).

In fact, the energy transferred (E0) by the source is shown

to be related to the transition amplitude and hence can

be determined from the imaginary part of the eﬀective

action (E0=WI). This idea has also been introduced

in quantum ﬁeld theory, and it is found that the imagi-

nary part of the eﬀective action clearly explains the par-

ticle production (one of the well-known examples is the

Schwinger eﬀect [25]). Therefore a general consciousness

is – the appearance of complex eﬀective action is the sig-

nature of transition from the ground state to other states

in quantum mechanics and particle production in quan-

tum ﬁeld theory.

In this regard, the eﬀective path can be obtained by

extremising the aforesaid eﬀective action W. In fact, the

average path is determined by [26]

X(t) = NZDx x(t)ei

ℏS[x,J],(4)

and therefore alternatively we have

X(t) = −iℏ1

Z[J]

δZ[J]

δJ(t)J=0 =δW [J]

δJ(t)J=0

=⟨x2, t2|x(t)|x1, t1⟩

⟨x2, t2|x1, t1⟩.(5)

In the above, like earlier, Jacts as an external source

and therefore S[x, J] is expressed as S[x, J] = S[x] +

RdtJ(t)x(t). Here S[x] is the classical action for the sys-

tem under consideration. Hence (5) can be applied to

any quantum mechanical system. Hence the signature

of complex eﬀective action must be reﬂected in the path

X(t) as well, and so the complex nature of the path is

the conﬁrmation of the transition from the ground state

to other states. The conception of the path through the

above deﬁnition was initially put forward in [26] but did

not get attention until its application was made in [24].

The above form can be cast in terms of propagators as

well:

X(t) = R∞

−∞ dx x(t)K(x2, t2;x, t)K(x, t;x1, t1)

K(x2, t2;x1, t1).(6)

This has been extensively used in [24]. In fact in [24],

the authors showed for forced harmonic oscillator that

the imaginary part of Wis proportional to |X(t)|2in the

limit t→ ∞ and so |X(t)|2is directly connected to the

transition probability (see Eq. (3) for the general rela-

tion between transition probability and WI). Such an

example lets them conjecture that the probability transi-

tion or the particle production can be well studied through

the complex nature of the path X(t)as well. In fact, as

the energy transferred by the source is given by WI, they

drew an analogy through the results of the forced har-

monic oscillator that d|X(t)|2/dt provides information

about the rate of energy transferred.

As an application of the conjecture, the authors of [24]

calculated d|X(t)|2/dt for the eﬀective potential as seen

by the scalar modes in the near horizon limit. The eval-

uation of X(t) just around the horizon shows a complex

nature. More interestingly, d|X(t)|2/dt appears to be of

the form given by the energy spectrum of a quantum

harmonic oscillator ∼(N+ 1/2). For harmonic oscilla-

tor Ndenotes Nth quantum state and when it is in this

state the total energy is given by (N+ 1/2) times the

energy of a “photon”. Therefore Ncan be treated as the

number of emitted “photons” contained within emitted

energy when the excited harmonic oscillator decays to

ground state. With this analogy the value of N, calcu-

lated from complex path can be treated as the number

of the radiated particles due to the energy transferred by

the potential. Interestingly in this case Nis given by the

Planckian distribution. This led to the identiﬁcation of

the horizon temperature. Apart from the conjecture, a

few interesting observations came in the study [24], which

we want to mention below.

•The temperature was found to suﬀer from a factor

of 2 discrepancy. It has been argued in the pa-

per that such is due to the choice of Schwarzschild-

like coordinates. Moreover, the analysis was done

only for static spherically symmetric black holes

(SSSBH).

•An important comment has been made in the con-

clusion of [24] that a necessary (but not suﬃcient)

condition for obtaining a complex path in this

procedure is the Hamiltonian must be unbounded

and/or non-hermitian (actually, the Hamiltonian

needs to be non-self-adjoint operator, rather than

non-hermitian [27]).

In the present article, we want to follow up on the above

idea and see how far the idea can be extended. We aim

to extend the idea of the complex path beyond SSSBH

(e.g. Kerr black hole) and also see whether a correct

temperature value can be obtained.

Recently one of the authors of this paper has been try-

ing to understand the thermal nature of horizons using a

very novel idea based on a model consisting of a charge-

less massless particle moving very near the horizon. The

model shows that the near horizon Hamiltonian of the

particle is eﬀectively given by the following form

H=κxp , (7)

where xis the radial distance from the horizon and p

is its conjugate momentum. So x= 0 corresponds to

the location of the horizon. In the above, κis the sur-

face gravity. Such a Hamiltonian has been obtained in

Eddington- Finkelstein (EF) outgoing null coordinates

both for a general SSSBH as well as Kerr spacetime. In

the above, the classical path of the particle has been cho-

sen to be along the normal to null hypersurface, given by

Eddington null coordinate u= constant, where u=t−r∗

with r∗is the well known tortoise coordinate (see [28,29]

for details regarding the construction of (7)). The same

can also be obtained in Painleve coordinates for SSSBH

as well as Kerr BH, considering a particular trajectory

[30–32] (a generic null surface also provides such Hamilto-

nian [33]). We just mentioned that the form of the Hamil-

tonian (7) was obtained explicitly in two speciﬁc choices

of coordinates – one in EF and another in Painleve co-

ordinates. Therefore the time can be taken at this stage

as either EF time or Painleve time. Moreover, it must

be explicitly mentioned that such a Hamiltonian was ob-

tained for a speciﬁc choice of outgoing null path. For

instance in EF coordinates the tangent of the path is

normal to u= constant hypersurface, while in Painleve

we found this for radially outgoing null path. In this

sense the Hamiltonian is very speciﬁc to these choices of

null paths, and also it depends on a particular choice of

observer. We will see that the Hawking temperature can

be obtained by using the Hamiltonian. Since it is well

known that horizon thermodynamics is an observer de-

pendent concept, we can expect the underlying dynamics

may depend on choice of coordinates. However it would

be interesting to investigate the generality of such form of

Hamiltonian. Also note that here x= 0 is location of the

horizon. As the Hamiltonian has been obtained by ex-

panding the metric coeﬃcients around x= 0 and keeping

only the leading order term, the structure of the Hamilto-

nian retains for both x > 0 (just outside the horizon) and

x < 0 (just inside the horizon). Therefore the applicabil-

ity of the Hamiltonian is valid in the range −ϵ≤x≤ϵ

with ϵ > 0, where ϵis a very small quantity. We will use

this in our latter analysis.

Nonetheless, it must be noted that the above Hamil-

tonian, very near the horizon, is unstable. First discuss

for x > 0. The solutions of the equations of motions are

given by x∼eκt and p∼e−κt. Since the near horizon

limit x→0 is equivalent to t→ −∞, the momentum di-

verges there. Moreover for a ﬁxed energy (7) implies that

p→ ∞ as x→0. Both imply that the Hamiltonian has

radial instability in the near horizon regime. On the other

hand, (7) can be cast to that of an inverted harmonic os-

cillator (see [32]), which is unstable in nature. While for

x < 0, the momentum is given by p=−(H/κy), where

x=−ywith y > 0. This again diverges at the hori-

zon y= 0. In this case the classical solutions take the

forms as y∼e−κt and p∼eκt. Since here y→0 implies

t→ ∞, one observes that the classical value of palso

diverges at the horizon. Recently we explicitly showed

this “local instability” may cause the temperature to the

horizon at the semi-classical level [28,29,32,34].

Inspired by all these facts here, we like to investigate

whether the unstable Hamiltonian (7) can show a com-

plex path and thereby produce a correct form of Hawking

temperature. We will show by constructing an eﬀective

non-relativistic Hamiltonian in the Schrodinger descrip-

tion that this is indeed the case. Since this Hamilto-

nian is related to both Kerr and SSSBH spacetimes, our

model is capable of presenting a very general situation,

rather than restricted to SSSBH as was done originally

in [24]. Moreover, we will see that the identiﬁed temper-

ature does not suﬀer from the factor of two ambiguity.

This shows that our choice of coordinates in which (7) has

been obtained are suitable ones. The same is also being

conﬁrmed by obtaining the density of states directly from

original relativistic ∼xp Hamiltonian. Additionally, as

(7) is unstable (or unbounded) in nature, the outcome

from this again conﬁrms the robustness of the corollary,

suggested in [24] – the necessary (but maybe not suﬃ-

cient) condition that the Hamiltonian must be unbounded

in order to obtain a complex path.

Let us now move forward toward the calculation in

favour of our claim. In this analysis the Hamiltonian (7)

will be treated as that of a quantum mechanical system

and calculation will be done using the standard prescrip-

tions of path integral formalism. Before this, we will ﬁrst

calculate the density of states corresponding to (7) using

the path integral kernel and show that such is thermal

in nature. Later an eﬀective Hamiltonian will be con-

structed under the Schrodinger description in which case

the path will be calculated.

II. DENSITY OF STATES AND THE GROUND

STATE

The density of states (DOS) corresponding to our

present model can be evaluated from the kernel or prop-

agator K(x2, t2;x1, t1). In path integral approach DOS

is given by [35,36]

ρ(E) = −1

πImZG(X, E)dX,(8)

where Gis the Laplace-Fourier transformation of K:

G(X, E) = Z∞

dt e−iEt

ℏK(x2=X, t2=t;x1=X, t1= 0) .

(9)

The propagator for the Hamiltonian (7) is given by [34]

K(x2, t2=t;x1, t1= 0) = e−κt

2δx1−x2e−κt,(10)

with t > 0.

Then we ﬁnd

Z∞

−∞

dXG(X, E) = Z+∞

−∞

dX Z+∞

dte−iEt

ℏK(x2=X, t2=t;x1=X, t1= 0)

=Z+∞

−∞

dX Z+∞

dte−iEt

ℏe−κt

2δX−Xe−κt.(11)

The above integrant for t-integration has poles where the argument of the Dirac-Delta function vanishes. The t-

integration can be performed by the complex integration method. In order to do that, we ﬁrst replace t→t−iδ with

δ > 0, and after completing the integration, the limit δ→0 will be taken. Then the poles are given by eκ(t−iδ)= 1;

i.e. t=iδ ±(2πin)/κ with n= 0,1,2,3, . . . . Now for E > 0 in (11) only the poles on the lower imaginary axis will

contribute; i.e. we have now t=−(2πin)/κ with n= 1,2,3, . . . . In that case, the contributing part of the Dirac-Delta

function can be expressed as

δX−Xe−κt=X

n=1,2,3,...

δt+2πin

κ

dt (X−Xe−κt)t=−2πin

κX X

n=1,2,3,...

δt+2πin

κ.(12)

Substituting this in (11) and performing the integration one ﬁnds

Z∞

−∞

dXG(X, E) = Z∞

−∞

κX X

n=1,2,3,... −e−2πE

ℏκn=−1

e2πE

ℏκ+ 1 Z∞

−∞

=−iπ

e2πE

ℏκ+ 1 + Real part .(13)

In the step we have used the relation R∞

−∞

X=iπ + PR∞

−∞

X, where “P” stands for the principal value.

Therefore DOF states, by Eq. (8), is given by

ρ(E) = 1

e2πE

ℏκ+ 1 .(14)

The same can be obtained by transforming the Hamil-

tonian (7) in the form of that of an inverted harmonic

oscillator (see Appendix A). Note that it satisﬁes the

Kubo’s form of ﬂuctuation-dissipation relation [37]

ρ+(E) = cothβE

2ρ−(E),(15)

if one identiﬁes the inverse temperature as β= 2π/ℏκ=

1/T , where ρ±(E) = ρ(−E)±ρ(E). Note that βis the

inverse Hawking temperature.

It may be noted that (14) refers to the number of

modes at energy Ewhich are at temperature T. If

these are Hawking radiated ones, then the correspond-

ing entropy can be interpreted as that of the Hawking

radiation. In principle this can be calculated following

works of Page [38–40]. By integrating (14) over all pos-

sible energies, the energy emission rate dE/dt can be

obtained. Then introducing the law of thermodynamics

T dS/dt =dE/dt one ﬁnally calculates the rate of entropy

change of the radiation.

Let us now investigate another aspect of the Hamilto-

nian (7) through path integral or, in other words, through

the propagator. It is well formulated that in Euclidean

time t=−itEformalism, the ground state energy can be

determined from the propagator. It is given by

E0= lim

tE→∞ h−ℏ

ln ⟨tE,0|0,0⟩

|ϕ0(0)|2i.(16)

where ⟨tE,0|0,0⟩=GE(tE,0; 0,0). Here ϕ0(x) is the

ground state wave function, can be evaluated by the fol-

lowing relation:

ϕ0(x) = Nlim

tE→∞ KE(tE→ ∞,0; 0,x),(17)

with Nis the normalization factor. These deﬁnitions

lead to the required expressions when one retains only the

leading order terms in the limit tE→ ∞ (for a detailed

discussion, see section 1.2.3 of [25]). For the present case,

the propagator is given by (10). As it already has the

damping property with respect to t, we will use the above

Euclidean time formalism by taking tE=t. Under this

assumption setting x1=xand x2= 0 in (10) and using

(17) we ﬁnd

ϕ0(x) = Nlim

t→∞ e−κt

2δ(x).(18)

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Thermalityofhorizonthroughnearhorizoninstability:apathintegralapproachGaurangRamakantKane1,2,∗andBibhasRanjanMajhi1,†1DepartmentofPhysics,IndianInstituteofTechnologyGuwahati,Guwahati781039,Assam,India2RudolfPeierlsCentreforTheoreticalPhysics,UniversityofOxford,OxfordOX13PU,UnitedKingdom(Dated:Octobe...

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Thermality of horizon through near horizon instability a path integral approach Gaurang Ramakant Kane1 2and Bibhas Ranjan Majhi1 1Department of Physics Indian Institute of Technology Guwahati Guwahati 781039 Assam India.pdf

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Thermality of horizon through near horizon instability a path integral approach Gaurang Ramakant Kane1 2and Bibhas Ranjan Majhi1 1Department of Physics Indian Institute of Technology Guwahati Guwahati 781039 Assam India

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