Correspondences of matter fluctuations in semiclassical and classical gravity for cosmological spacetime

TL;DR

This paper establishes a fundamental correspondence between quantum matter fluctuations in semiclassical gravity and classical fluid fluctuations in cosmological spacetime, enhancing understanding of mesoscopic phenomena in gravitating systems.

Contribution

It introduces a novel link between quantum stress tensor fluctuations and classical matter fluctuations, expanding the theoretical framework of stochastic gravity in cosmology.

Findings

Quantum fluctuations correspond to covariances of pressure and density.

The correspondence aids in statistical analysis of curved spacetime systems.

It provides insights into mesoscopic phenomena in gravitating systems.

Abstract

A correspondence between fluctuations of conformally invariant quantum fields and that of classical fields finally reducing to perfect fluid matter content is shown to exist. Previously a similar correspondence between the stress tensors was known and well established in the 70's. Using recent results obtained in semiclassical stochastic gravity regarding exact definition and significance given to quantum fluctuations of the stress tensor, we obtain this correspondence, which is fundamental to statistical analysis of systems in curved spacetime. This is of immense importance, in that the fluctuations of stress tensor play a similar central role in stochastic gravity, as that of the stress tensor in classical and semiclassical gravity. A relation between the semiclassical and classical fluctuations therefore, gives insight to the mesoscopic phenomena for gravitating systems and would further enhance the perturbative analysis for cosmological spacetime and astrophysical objects, which is an expansive area of research. Interestingly we see that the quantum fluctuations have a correspondence with covaiances of pressure and density of the gravitating system in the stochastic analysis.

Equations

$$T_{ab}(x)={\varphi }_{;a}{\varphi }_{;b}-\frac{1}{2}{g}_{ab}{\varphi }^{;c}{\varphi }_{;c}+1/6(g_{ab}\square -{\nabla }_a{\nabla }_b+G_{ab}){\varphi }^2$$

$$d{s}^2=-d{t}^2+a^2(t)d{x}_1^2+b^2(t)d{x}_2^2+c^2(t)d{x}_3^2$$

$$G_{ab}=<{\hat{T}}_{ab}>$$

$$8{\hat{N}}_{ab{c}^{\prime }{d}^{\prime }}(x,y)$$

$${\hat{N}}_{ab{c}^{\prime }{d}^{\prime }}=Re\{{\bar{K}}_{ab{c}^{\prime }{d}^{\prime }}+g_{ab}{\bar{K}}_{c^{\prime }{d}^{\prime }}+g_{c^{\prime }{d}^{\prime }}{\bar{K}}_{ab}^{\prime }+g_{ab}{g}_{c^{\prime }{d}^{\prime }}\bar{K}\}$$

$$9{\bar{K}}_{ab{c}^{\prime }{d}^{\prime }}=4({\hat{G}}_{;c^{\prime }b}{\hat{G}}_{;d^{\prime }a}+{\hat{G}}_{;c^{\prime }a}{\hat{G}}_{d^{\prime }b})+{\hat{G}}_{;c^{\prime }{d}^{\prime }}{\hat{G}}_{;ab}$$

$$+\hat{G}{\hat{G}}_{;ab{c}^{\prime }{d}^{\prime }}-2({\hat{G}}_{;b}{\hat{G}}_{;c^{\prime }a{d}^{\prime }}+{\hat{G}}_{;a}{\hat{G}}_{;c^{\prime }b{d}^{\prime }}+{\hat{G}}_{;d^{\prime }}$$

$${\hat{G}}_{;ab{c}^{\prime }}+{\hat{G}}_{;c^{\prime }}{\hat{G}}_{;ab{d}^{\prime }})+2({\hat{G}}_{;a}{\hat{G}}_{;b}{R}_{c^{\prime }{d}^{\prime }}+{\hat{G}}_{;c^{\prime }}{\hat{G}}_{;d^{\prime }}{R}_{ab})$$

$$-({\hat{G}}_{;ab}{R}_{c^{\prime }{d}^{\prime }}+{\hat{G}}_{c^{\prime }{d}^{\prime }}{R}_{ab})\hat{G}+\frac{1}{2}{R}_{c^{\prime }{d}^{\prime }}{R}_{ab}{\hat{G}}^2$$

$$36{\bar{K}}_{ab}^{\prime }=8(-{\hat{G}}_{;p^{\prime }b}\text{tensor}\hat{G}{}_{;}{}_{p^{\prime }}{}_a^{p^{\prime }}+{\hat{G}}_{;a}\text{tensor}\hat{G}{}_{;}{}_{p^{\prime }}{}_b^{p^{\prime }})+4({\hat{G}}_{;}^{p^{\prime }}{\hat{G}}_{;ab{p}^{\prime }}$$

$$-{\hat{G}}_{;p^{\prime }}^{p^{\prime }}{\hat{G}}_{;ab}-{\hat{G}}_{ab{p}^{\prime }}^{p^{\prime }})-2R^{\prime }(2{\hat{G}}_{;a}{\hat{G}}_{;b}-\hat{G}{\hat{G}}_{;ab})-$$

$$2({\hat{G}}_{;p^{\prime }}{\hat{G}}_{;}^{p^{\prime }}-2\hat{G}{\hat{G}}_{;p^{\prime }}^{p^{\prime }})R_{ab}-R^{\prime }{R}_{ab}{\hat{G}}^2$$

$$36\bar{K}=2{\hat{G}}_{;p^{\prime }q}{\hat{G}}_{;}^{p^{\prime }q}+4({\hat{G}}_{;p^{\prime }}^{p^{\prime }}{\hat{G}}_{;q}^q+\hat{G}\text{tensor}\hat{G}{}_{;}^p{}_{q^{\prime }}^{q^{\prime }})-$$

$$4({\hat{G}}_{;p}\text{tensor}\hat{G}{}_{;}{}_{q^{\prime }}^p{}^{q^{\prime }}+{\hat{G}}_{;}^{p^{\prime }}\text{tensor}\hat{G}{}_{;}{}_q^q{}_{p^{\prime }})R{\hat{G}}_{;p^{\prime }}{\hat{G}}_{;}^{p^{\prime }}+R^{\prime }{\hat{G}}_{;p}{\hat{G}}^{;p}$$

$$-2(R{\hat{G}}_{;p^{\prime }}^{p^{\prime }}+R^{\prime }{\hat{G}}_{;p}^p)\hat{G}+\frac{1}{2}R{R}^{\prime }{\hat{G}}^2.$$

$$\hat{G}(x,y)=<\{\hat{\varphi }(x),\hat{\varphi }(y)\}>.$$

$$\hat{G}(x,y)\to G(x,y)=<\varphi (x)\varphi (y)>,$$

$$<T_{ab}(x)T_{cd}(y)>-<T_{ab}(x)><T_{cd}(y)>$$

$$4\text{tensor}N{}_b^a{}_d^c(x,y)=<T_b^a(x)T_{d^{\prime }}^{c^{\prime }}(x)>-<T_b^a(x)><T_b^a(y)>$$

$$4\text{tensor}N{}_0^0{}_0^0(t,t^{\prime })$$

$$4\text{tensor}N{}_0^0{}_i^i(t,t^{\prime })$$

$$4\text{tensor}N{}_i^i{}_0^0(t,t^{\prime })$$

$$4\text{tensor}N{}_i^i{}_j^j(t,t^{\prime })$$

Full text

In this letter, inspired by the correspondence between classical and quantum stress tensors which was established in 70’s and is well known in literature dodleson ; zeldo2 , we attempt to establish a similar correspondence between the fluctuations of the two. This is an enhanced result that we obtain by using the exact form of noise kernel phillipnk defining fluctuations of a quantum field, recently obtained in semiclassical stochastic gravity bei1 , and relate it those of a perfect fluid stress tensor in the classical limit. Thus we show an equivalence of the two noise kernels in classicalized limit, which opens up possibliltiy for simplifying mathematically more intricate quantum stress tensor issues and replacing them by simpler fluctuations of classical variables which are equivalent for statistical analysis. This is the first attempt to show any such relation between the quantum and classical fluctuations thus providing a link between semiclassical and classical gravity to be treated equivalently in the stochastic domain. Of interest towards possible applications of such a correspondence would be the cosmological and astrophysical spacetimes and their analysis in both the equilibrium and non-equilibrium limits.

Stress tensors describe the matter part of the Einstein’s equations in classical and semiclassical gravity. The issues of renormalization and regularization are important to get physically meaningful results for the quantum stress tensors birrel . However the effect of such a renormalized quantum stress tensor on the spacetime geometry is realized only through its average or expectation value which acts as a classical entity.

On the other hand classical matter described by scalar fields and perfect fluids etc. are commonly used to model matter content of the astrophysical objects or the universe dodleson ; friedman ; madsen . The correspondences between stress tensors of scalar fields, perfect fluid and expectation of quantum stress tensor is a well established for relevant cases. Specially for a cosmological spacetime, the relation between expectation of quantum stress tensor with conformally invariant scalar fields and a perfect fluid model is well known and widely used dodleson ; zeldo2 ; lukash .

While the importance of the stress tensors, whether quantum or classical, for General Relativity cannot be underestimated, fluctuations of the stress tensor are finding increasing applications in astrophysics and cosmology bei1 ; sukbh ; sukratu ; seema1 . In this letter we are concerned about statistical fluctuations on the background spacetime and do not touch upon perturbations of the matter fields, as in a perturbative analysis. Such fluctuations of the quantum stress tensor have been defined and given explicit form and meaning in semiclassical stochastic gravity phillipnk . On the other hand, recently proposed classical stochastic gravity shows fluctuations of the classical stress tensor useful for analysing statistical properties of astrophysical objects seema2 . We further attempt to obtain a correspondence between the two .

The procedure that we follow to show this correspondence, takes the quantum fluctuations to classicalized form via transition from quantum scalar fields to classical scalar fields (the reverse of quantizing the classical fields), then using the point splitting method as described later, and finally using the relation between classical scalar fields and pressure and energy density variables as in the perfect fluid desciption of matter. The key relation that enables us to obtain this very correspondence is the noise kernel for quantum fields.

A quantum stress tensor is obtained by quantizing a scalar field ( here we consider a conformally invariant scalar field)

[TABLE]

where $\phi$ is the classical scalar field which has to be quantized, to get a quantum stress tensor $\hat{T}_{ab}$.

Here we review the case studied in zeldo2 and work on Kasner metric, which decribes the anisoptropic but homogenous cosmological spacetime, while our results also extend to the FRW metric for the isotropic and homogenoeus case. Thus we use the general form

[TABLE]

where $a,b$ and $c$ are certain non negative functions of time ( including the case where $a=b=c$, for FRW ). We are interested in the epoch where, in the early universe, $t_{0}>>t_{Pl}$, such that $t<t_{0}$, is the time when there is no particle production thus for $t=t_{0}$ one can define a vacuum state correctly and particle production is switched on. Then for $t>t_{0}$ one considers the expectation value of the quantum stress tensor to have non-zero values. The epoch when the influence of the quantum particles on the metric becomes important, one can neglect quantum effects and consider only the classicalized effects given by the expectation of stress tensor, thus falling in the regime of

[TABLE]

the semiclassical gravity.

We are interested thus in the case, where $t>t_{0}$ hence the classicalized system, where the formulation of quantum problem reduces to considering classical wave equation for the fields $\phi$. Then, only the following average values of the quantum energy momentum tensor are different from zero and are given by $<0|\hat{T}^{0}_{0}|0>=-\epsilon,<0|\hat{T}^{1}_{1}|0>=<0|\hat{T}^{2}_{2}|0>=<0|\hat{T}^{3}_{3}|0>=p$. The vacuum expectation values given above diverge due to zero-point oscillations of the vacuum and are hence renormalized, giving renormalized values of the energy density and pressure zeldo2 . From now on we would address only the renormalized energy density and pressure. Thus one can imagine a perfect fluid given by $T^{a}_{b}=\mbox{ diag}\{-\epsilon,p,p,p\}$ to represent the matter part of the Einstein equations for this model.

We begin with the two point noise kernel as has been obtained in phillipnk and attempt to show that it can be directly related to that of the perfect fluid case for a Kasner and an FRW metric.

The semiclassical two point noise kernel is defined by

[TABLE]

In phillipnk point splitting method in the above equation was used to obtain the following expression for the noise kernel.

[TABLE]

where,

[TABLE]

where the Wightmann functions denoted by $\hat{G}$ are defined as

[TABLE]

Point splitting is usually used for dealing with ill defined quantum operators, which are in the form of $\hat{\phi}^{2}$ birrel ; bei1 ; phillipnk . The prescription is to take $\hat{\phi}(x)\rightarrow\hat{\phi}(x)\hat{\phi}(x^{\prime})$, and use this to evaluate the final expressions, which at the end are consistently put back in a much desired form by taking back $x^{\prime}\rightarrow x$ for the functions and operators. However the quantum nature of the operators here has nothing to do with the mathematical artefact of using point splitting. Thus, we employ the same for classical case as well and get consistent results. This is discussed further.

The Wightmann function given above by (7), in the classical limit reduces to

[TABLE]

with $\phi$ classical. Replacing $\hat{G}$ by $G$ in equation (Correspondences of matter fluctuations in semiclassical and classical gravity for cosmological spacetime), gives us corresponding noise kernel in the classicalized system. The expression of such a noise kernel (classicalized) thus obtained can be seen to match, with

[TABLE]

It can be (conversely) formally shown, that if one employs the point splitting procedure also for the classical fields given by the stress tensor (1) in the above expression we obtain a similar expression as that in (5) and (Correspondences of matter fluctuations in semiclassical and classical gravity for cosmological spacetime), with $\hat{G}$ replaced by $G$. Hence one obtains the correspondence: $\hat{N}_{abcd}(x,y)\rightarrow N_{abcd}(x,y)$. We now work on the latter form. It can be directly verified from (4) that in the classical limit,

[TABLE]

Here the averages are over the classical field distribution.

For the metric given by (2) , $a(t),b(t),c(t)$ the scalar field $\phi(t)$ and the fluid variables $\epsilon(t),p(t)$ are homogeneous. For such a case, only the diagonal elements of the averaged quantum stress tensor are important and give non-zero contribution as mentioned earlier. Further, using the relation between scalar fields and pressure/density, one can easily see that the two corresponding forms of stress tensors are interchangeable and thus one can get the noise kernel in terms of perfect fluid variables here. This can be modeled by a stress tensor with diagonal non-zero elements $T^{0}_{0}=-\epsilon(t),T^{1}_{1}=T^{2}_{2}=T^{3}_{3}=p(t)$ of the classical stress tensor for homogeneous case. The procedure we have followed here, takes $\hat{T}_{ab}$ (quantum)$\rightarrow T_{ab}$ (classical $\phi$ ) $\rightarrow T_{ab}$ (perfect fluid).

The perfect fluid, has collisons and microscopic effects which give rise to non-zero pressures. This enables one to consider the stress tensor as a random variable seema2 ; friedman , and introduce a statistical treatment for the same, such that averages of fluid variables are well defined in the distributional sense.

Putting this in (9) we get the following non-zero components of the noise kernel.

[TABLE]

Thus the covarainces of pressure and density capture the microscopic nature of the fluid in a statistical description which are thus related to quantum fluctuation inherently.

It is known that the noise kernel is of central importance to stochastic gravity, which is a step up from semiclassical gravity in the perturbative approach. In addition to being used in the perturbative theory as noise, these fluctuations characterize the microscopic effects in homogeneous cosmological spacetime itself. Thus our results indicate that, fluctuations of quantum fields induce mesoscopic classicalized effects in the fluid description of the same model, given by covariances of pressures and energy density in the background spacetime.

Here we have extended the correspondence between quantum and classical stress tensors to the fluctuations (two point) of the same. We emphasise, that this holds in the classical limit, where the purely quantum effects can be ignored, and thus only the classicalized effect of quantum stress tensor is important. This is by no means a trivial case, since for semiclassical gravity, one always uses the expectation of the quantum stress tensor in the Einstein’s equation.

The noise kernel $\hat{N}_{abcd}$ due to quantum fluctuations, is classical and stochastic in nature. One can thus raise a question about the correspondence worked out here. The importance of what we have shown here, lies in realizing that the quantum sourced noise due to quantum fluctuations can also be captured partially via the classical noise driven by perfect fluid model of matter for the homogenoeus Kasner and FRW metric. Hence a basic link between quantum and classical describtion of the fluctuations, on the lines of corresponding stress tensors, has been established in the sense of treating matter fields and their properties in the two domains.