Relativistic eﬀects on the Schr odinger-Newton equation David Brizuela1and Albert Duran-Cabac es2 Department of Physics and EHU Quantum Center University of the Basque Country UPVEHU

2025-04-26 0 0 996.51KB 18 页 10玖币

侵权投诉

Relativistic eﬀects on the Schr¨odinger-Newton equation

David Brizuela 1and Albert Duran-Cabac´es 2

Department of Physics and EHU Quantum Center, University of the Basque Country UPV/EHU,

Barrio Sarriena s/n, 48940 Leioa, Spain

Abstract

The Schr¨odinger-Newton model describes self-gravitating quantum particles, and it is often

cited to explain the gravitational collapse of the wave function and the localization of macroscopic

objects. However, this model is completely nonrelativistic. Thus, in order to study whether

the relativistic eﬀects may spoil the properties of this system, we derive a modiﬁcation of the

Schr¨odinger-Newton equation by considering certain relativistic corrections up to the ﬁrst post-

Newtonian order. The construction of the model begins by considering the Hamiltonian of a

relativistic particle propagating on a curved background. For simplicity, the background metric is

assumed to be spherically symmetric and it is then expanded up to the ﬁrst post-Newtonian order.

After performing the canonical quantization of the system, and following the usual interpretation,

the square of the module of the wave function deﬁnes a mass distribution, which in turn is the

source of the Poisson equation for the gravitational potential. As in the nonrelativistic case,

this construction couples the Poisson and the Schr¨odinger equations and leads to a complicated

nonlinear system. Hence, the dynamics of an initial Gaussian wave packet is then numerically

analyzed. We observe that the natural dispersion of the wave function is slower than in the

nonrelativistic case. Furthermore, for those cases that reach a ﬁnal localized stationary state, the

peak of the wave function happens to be located at a smaller radius. Therefore, the relativistic

corrections eﬀectively contribute to increase the self-gravitation of the particle and strengthen the

validity of this model as an explanation for the gravitational localization of the wave function.

1 Introduction

The Schr¨odinger-Newton (SN) equation [1] describes nonrelativistic quantum objects under self-gravitation.

In this model, the Schr¨odinger equation is coupled to a Newtonian gravitational potential term, which, in

turn, is sourced by the mass density given by the square of the module of the wave function. Therefore,

contrary to the usual Schr¨odinger equation, this system is nonlinear. The SN equation was ﬁrst proposed

to study self-gravitating bosonic stars [2]. However, interesting applications of this equation to describe

the gravitational collapse of the wave function and the decoherence of macroscopic objects have also been

proposed later [3,4].

Despite its nonlinearity, the SN system has been shown to have an inﬁnite family of deﬁned stationary

states with negative energies [5]. These eigenstates have been explicitly found both numerically [5–8] and with

approximate analytical methods [9]. The studies of the dynamical evolution of a general wave packet given by

the SN equation have been mostly based on numerical methods due to the complexity of the nonlinear system

(see, e.g., Refs. [8,10–15]). In general, in this dynamical scenario, two regimes can be distinguished depending

on the features of the initial state. On the one hand, there is the weak self-gravitational regime, where

the natural quantum spreading of the wave function dominates. In this regime, the wave function evolves

similarly to, but somehow slower than, a free particle. On the other hand, there is the case dominated by

self-gravitation, or, as we will refer to it, the strong self-gravitational regime. In this latter regime, the wave

function tends to decay into its corresponding ground state through the so-called gravitational cooling [10,11].

Furthermore, certain modiﬁcations of the SN equation have also been studied in the literature. For

instance, a dark-energy term was coupled to this equation in Refs. [9, 16]. The addition of dark energy

changes the eigenstates of the Hamiltonian, as well as the dynamical evolution of the wave function, but the

main features of the model are kept intact.

1E-mail address: david.brizuela@ehu.eus

2E-mail address: albertdurancabaces@gmail.com

arXiv:2210.06195v2 [gr-qc] 3 Jan 2023

All in all, the SN equation is a promising eﬀective model to describe macroscopic quantum objects, which

are expected to be well localized. Nonetheless, due to the Newtonian treatment of the gravitational inter-

action, relativistic eﬀects, which could strengthen or undermine the validity of this model, are completely

missing from this picture. In fact, the Newtonian limit of fully relativistic self-gravitating models, like the

Einstein-Klein-Gordon system, has been analyzed [10], and certain variations of the Klein-Gordon equa-

tion with a Newtonian gravitational potential have been proposed [17]. In addition, there are studies in

which relativistic and gravitational corrections from external ﬁelds are considered in the usual Schr¨odinger

equation [18, 19], though discarding the self-gravitational interaction. In Ref. [20], the SN equation was

derived in the context of the gravitoelectromagnetism approximation to general relativity. While this ap-

proximation includes the relativistic corrections produced by the gravitomagnetic vector potential up to ﬁrst

post-Newtonian (1PN) order (that is, up to order c−2, with cbeing the speed of light), other terms of the

same order are excluded. Therefore, the main goal of the present paper is to construct a model that describes

the SN system with the 1PN relativistic corrections given by nonlinear terms of the Newtonian potential and

kinetic energies, which have not been previously considered, and to determine the physical consequences of

such corrections on the dynamics of the wave packets.

For such a purpose, we will ﬁrst derive the Hamiltonian of a relativistic particle propagating on a curved

background. After assuming spherical symmetry, and performing a post-Newtonian expansion of the metric in

inverse powers of the speed of light up to the order c−2, the basic variables will be promoted to operators, and

the canonical quantization of the Hamiltonian will be performed. In this way, we will obtain the Schr¨odinger

equation with relativistic corrections, which will depend on a certain Newtonian gravitational potential. This

potential will then be assumed to obey the Poisson equation with a mass density given by the probability

distribution of the particle. This procedure will lead to a closed nonlinear system of equations. We will then

analyze the dynamics of an initial Gaussian state given by these equations of motion, and obtain the physical

eﬀects produced by the relativistic correction terms by comparing this evolution with the one given by the

usual nonrelativistic SN system.

The paper is organized as follows. In Sec. 2, the SN equation with relativistic corrections up to the ﬁrst

post-Newtonian order is derived. Section 3 presents the analysis of the dynamics of an initial Gaussian wave

packet. In Sec. 4 we summarize and discuss the main physical results of the model. Finally, in Appendix A

we include a description and some technical aspects of the numerical methods used to solve the equations,

while in Appendix B we present the results found with a modiﬁed version of the Poisson equation.

2 Relativistic corrections on the Schr¨odinger-Newton equation

This section is divided into three subsections. In Sec. 2.1 the Hamiltonian of a relativistic particle

propagating on a certain curved background is obtained. In Sec. 2.2 a post-Newtonian expansion of the

Hamiltonian is considered up to 1PN order. Finally, in Sec. 2.3, the canonical quantization of the system is

performed in order to obtain the Schr¨odinger-Newton equation with relativistic corrections.

2.1 Hamiltonian of a relativistic particle on curved backgrounds

The action of a particle with mass mpropagating on a spacetime described by the metric tensor gµν is

given by

S=Zds L(xµ,˙xµ),(1)

with the Lagrangian

L(xµ,˙xµ) = −mc(−gµν ˙xµ˙xν)1/2,(2)

where xµ=xµ(s) is the trajectory of the particle, and the dot stands for a derivative with respect to the

parameter s. In this setup, all four coordinates xµ= (x0, x1, x2, x3) are dynamical variables. For convenience,

we will also use the notation x0:= ct for the time coordinate. In order to obtain the Hamiltonian, one ﬁrst

needs to deﬁne the conjugate momenta,

pµ:= ∂L

∂˙xµ=mc

(−gαβ ˙xα˙xβ)1/2gµν ˙xν.(3)

It is easy to check that these momenta are not independent, since they obey the constraint

pµpµ+m2c2= 0.(4)

This constraint is ﬁrst class, and it is related to the reparametrization invariance of the system, that is, the

freedom to choose the parameter s. By performing a Legendre transformation and following the usual Dirac

procedure for constrained systems, which states that one should add the diﬀerent constraints multiplied by

certain Lagrange multiplier to the Hamiltonian, one then ﬁnds the generalized Hamiltonian,

C=α

2(pµpµ+m2c2),(5)

with the Lagrange multiplier α. The equations of motion can be readily obtained by computing the Poisson

brackets between the diﬀerent variables and the Hamiltonian,

˙xµ={xµ,C} =αgµν pν,(6)

˙pµ={pµ,C} =−α

2pαpβ

∂gαβ

∂xµ.(7)

Note that the Hamiltonian (5) is vanishing on shell and its canonical quantization would lead to the Klein-

Gordon equation. Instead, in order to construct our model, we will ﬁx the gauge before quantization by

imposing the condition x0=cs, or equivalently t=s, on the time coordinate. The conservation of this

condition all along the evolution, ˙x0=c, establishes the value α=c/(g0µpµ) for the Lagrange multiplier.

The gauge-ﬁxing procedure is then completed by solving the constraint (4) to write the conjugate momentum

p0in terms of the other variables. This leads to the Hamiltonian

H:= −cp0=c

p−g00 m2c2+gij −1

g00 g0ig0jpipj1/2

+cg0i

g00 pi,(8)

where we have chosen the positive sign in front of the square root in order to get the correct sign in the

nonrelativistic limit and latin letters stand for spatial indices running from 1 to 3. Now, the dynamical

variables are the three spatial coordinates xi, for which we will also use the notation x:= (x1, x2, x3), and

the equations of motion take the form

˙xi=c(g00H−cg0kpk)−1(g0iH−cgij pj),(9)

˙pi=1

2g00H−cg0kpk−1H2∂g00

∂xi−2cHpj

∂g0j

∂xi+c2pjpl

∂gjl

∂xi.(10)

2.2 Parametrized post-Newtonian formalism

In the nonrelativistic limit, every metric theory of gravity can be written in terms of small deviations from

the Newtonian gravitational equations. The post-Newtonian framework encompasses the mathematical tools

used for this purpose. An especially useful setup is the so-called parametrized post-Newtonian formalism [21],

which encodes these deviations using explicit parameters. These parameters have speciﬁc physical meaning,

as they describe diﬀerent properties of the spacetime, and their numerical values are ﬁxed by the particular

theory of gravity under consideration.

However, there is nothing like a parametrized post-Newtonian formalism for a generic spacetime. In order

to construct such a framework, one needs to assume certain particular physical scenario with corresponding

symmetries and, more importantly, a speciﬁc matter content. The usual parametrized post-Newtonian for-

malism is written for a perfect ﬂuid, which is a suitable assumption for most applications, and it is described

in terms of ten real-valued parameters along with the same number of metric potentials that obey Poisson-like

equations. This formalism includes all possible 1PN-order terms in the metric. However, the resolution of

such a system is very complicated due to the large number of equations involved. In addition, since our

goal is to describe the evolution of a mass density given by the norm of the wave function, in order to apply

the complete formalism, one would need to deﬁne diﬀerent kinematic and thermodynamic quantities (like

the velocity, the pressure, the internal energy,...) in terms of the wave function, as well as the equation of

state of such a ﬂuid, and such deﬁnitions might not be obvious. Therefore, we leave the construction and

analysis of the complete framework for future work. Here, in order to construct the simplest possible model

that encodes the 1PN corrections that only involve the Newtonian potential, we will assume the same metric

as considered by Eddington, Robertson, and Schiﬀ [21], which describes a spherically symmetric vacuum

spacetime. By doing so, we are neglecting all other post-Newtonian potentials. Some of these potentials are

related to the velocity of the ﬂuid, while others are sourced by thermodynamic magnitudes as the pressure or

the internal energy. Therefore, our formalism will be valid near the equilibrium and as long as such termody-

namic quantities do not have a large impact on the dynamics. Finally, let us comment that, instead of using

the parametrized post-Newtonian framework to study the relativistic eﬀects, another alternative would be to

consider the metric obtained from the expansion of the Einstein-Klein-Gordon system presented in Ref. [22],

which leads to the SN system.

In the coordinates centered at the gravitational source, the components of the Eddington-Robertson-Schiﬀ

metric are

g00 =−1+2Φ

c2+ 2βΦ2

c4+O(c−5), g0i=O(c−4), gij =1−2γΦ

c2qij +O(c−3),(11)

where qij is the three-dimensional Euclidean metric, Φ is the Newtonian gravitational potential, and the

truncation order is chosen so that all the components of the line element gµν dxµdxνare accurate up to an

order c−2. In particular, this diﬀers from the gravitoelectromagnetism approximation used in Ref. [20], which

does not include the quadratic term Φ2in the expansion of the g00 component of the metric. Note that this

metric depends on the two parameters βand γ. The parameter βdescribes how nonlinear the superposition

law for gravitational ﬁelds is, whereas γis related to the spatial curvature. Both of them have ﬁxed values

for any given theory of gravity and, in the particular case of general relativity, β=γ= 1. However, in order

to track their contribution along the diﬀerent equations, we will keep them explicitly and only replace them

by their value in general relativity β=γ= 1 for the numerical implementation that will be performed in the

next section.

It is now straightforward to obtain the components of the inverse metric,

g00 =−1−2Φ

c2−(2β−4)Φ2

c4+O(c−5), g0i=O(c−4), gij =1+2γΦ

c2qij +O(c−3),(12)

with qij being the inverse Euclidean metric, that is, qij qjk =δik. Replacing the elements (12) into the

expression (8), and performing an expansion in inverse powers of c, the Hamiltonian up to 1PN order is

found to be

H=mc2+p2

2m+mΦ + m

2c2(2β−1)Φ2+(2γ+ 1)

2mc2Φp2−p4

8m3c2+O(c−3),(13)

with p2:= pipjqij . The ﬁrst term is the rest energy of the particle and will be absorbed simply by redeﬁning

the origin of the energy. The second and third terms are, respectively, the Newtonian kinetic and potential

energies. The last three terms are thus the relativistic corrections, which are essentially the three possible

quadratic combinations of the Newtonian kinetic and potential energies. That is, the term that depends on

βis quadratic on the potential energy and, as commented above, represents how nonlinear the superposition

of potential energies is, which is exactly linear in the Newtonian regime. The term parametrized by γshows

a coupling between the potential and kinetic energies, while the last term is quadratic in the kinetic energy

and encodes the usual relativistic correction to the kinetic energy. These relativistic terms are not present

in the Hamiltonian obtained in the context of the gravitoelectromagnetism approach (see equation (30) of

Ref. [20]), while in contrast, the geometromagnetic potential vector is absent from our Hamiltonian since it

corresponds to one of the neglected post-Newtonian potentials.

2.3 Canonical quantization

In order to perform the canonical quantization of the system, the classical position xiand momentum pi

variables are promoted to quantum operators xi→ˆxiand pi→ˆpi. These operators do not commute,

[ˆxi,ˆpj] = i~δij,(14)

and they act on states of a given Hilbert space H. For any two vectors ϕ, φ ∈ H, the inner product of this

space will be deﬁned as

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

RelativisticeectsontheSchrodinger-NewtonequationDavidBrizuela1andAlbertDuran-Cabaces2DepartmentofPhysicsandEHUQuantumCenter,UniversityoftheBasqueCountryUPV/EHU,BarrioSarrienas/n,48940Leioa,SpainAbstractTheSchrodinger-Newtonmodeldescribesself-gravitatingquantumparticles,anditisoftencitedtoexplain...

展开>> 收起<<

Relativistic eﬀects on the Schr odinger-Newton equation David Brizuela1and Albert Duran-Cabac es2 Department of Physics and EHU Quantum Center University of the Basque Country UPVEHU.pdf

共18页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Relativistic eﬀects on the Schr odinger-Newton equation David Brizuela1and Albert Duran-Cabac es2 Department of Physics and EHU Quantum Center University of the Basque Country UPVEHU

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: