Construction of gauge-invariant variables Construction of gauge-invariant variables for linear-order metric perturbationfor linear-order metric perturbation on general background spacetime on general background spacetime

Kouji NakamuraKouji Nakamura (NAOJ) (NAOJ)

References : K.N. Prog. Theor. Phys., 110 (2003), 723. (arXiv:gr-qc/0303039). K.N. Prog. Theor. Phys., 113 (2005), 413. (arXiv:gr-qc/0410024). K.N. Phys. Rev. D 74 (2006), 101301R. (arXiv:gr-qc/0605107). K.N. Prog. Theor. Phys., 117 (2007), 17. (arXiv:gr-qc/0605108). K.N. Phys. Rev. D 80 (2009), 124021. (arXiv:0804.3840[gr-qc]). K.N. Prog. Theor. Phys. 121 (2009), 1321. (arXiv:0812.4865[gr-qc]). K.N. Adv. in Astron. 2010 (2010), 576273. (arXiv:1001.2621[gr-qc]). K.N. preprint (arXiv:1011.5272[gr-qc]). K.N. preprint (arXiv:1101.1147[gr-qc]).K.N. preprint. (arXiv:1103.3092[gr-qc]).

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The higher order perturbation theory in general relativity has very wide physical motivation.

– Cosmological perturbation theory• Expansion law of inhomogeneous universe (CDM v.s. inhomogeneous cosmology)• Non-Gaussianity in CMB (beyond WMAP)

– Black hole perturbations• Radiation reaction effects due to the gravitational wave emission.• Close limit approximation of black hole - black hole collision

(Gleiser, et al. (1996))

– Perturbation of a star (Neutron star)• Rotation – pulsation coupling (Kojima 1997)

There are many physical situations to which higher order perturbation theory should be applied.

I. IntroductionI. Introduction

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The first order approximation of our universe The first order approximation of our universe from a homogeneous isotropic one is revealed by from a homogeneous isotropic one is revealed by the recent observations of the CMB.the recent observations of the CMB.

• It is suggested that the fluctuations are adiabatic and Gaussian at least in the first order approximation.

• One of the next research is to clarify the accuracy of this result.– Non-Gaussianity, non-adiabaticity …

and so on.(Bennett et al., (2003).)

• To carry out this, it is necessary to discuss perturbation theories beyond linear order.

• The second-order perturbation theory is one of such perturbation theories.

c.f. Non-Gaussianity is a topical subject also in Observations.– E. Komatsu, et al., APJ Supp. 180 (2009), 330;

arXiv:1001.4538[astro-ph.CO].– The first full-sky map of Planck was press-released on 5 July 2010!!! ----->

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The higher order perturbation theory in general relativity has very wide physical motivation.

– Cosmological perturbation theory• Expansion law of inhomogeneous universe (CDM v.s. inhomogeneous cosmology)• Non-Gaussianity in CMB (beyond WMAP)

– Black hole perturbations• Radiation reaction effects due to the gravitational wave emission.• Close limit approximation of black hole - black hole collision

(Gleiser, et al. (1996))

– Perturbation of a star (Neutron star)• Rotation – pulsation coupling (Kojima 1997)

There are many physical situations to which higher order perturbation theory should be applied.

I. IntroductionI. Introduction

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However, general relativistic perturbation theory requires very delicate treatments of “gauges”.

It is worthwhile to formulate the higher-order gauge-invariant perturbation theory from general point of view.

• According to this motivation, we have been formulating the general relativistic second-order perturbation theory in a gauge-invariant manner. – General framework:

• Framework of higher-order gauge-invariant perturbations:• K.N. PTP110 (2003), 723; ibid. 113 (2005), 413.

• Construction of gauge-invariant variables for the linear order metric perturbation:• K.N. arXiv:1011.5272[gr-qc]; 1101.1147[gr-qc]; 1103.3092[gr-qc].

– Application to cosmological perturbation theory :• Einstein equations :

• K.N. PRD74 (2006), 101301R; PTP117 (2007), 17.• Equations of motion for matter fields:

• K.N. PRD80 (2009), 124021.• Consistency of the 2nd order Einstein equations :

• K.N. PTP121 (2009), 1321.• Summary of current status of this formulation:

• K.N. Adv. in Astron. 2010 (2010), 576273.• Comparison with a different formulation:

• A.J. Christopherson, K. Malik, D.R. Matravers, and K.N. arXiv:1101.3525 [astro-ph.CO]

Our general framework of the second-order gauge invariant perturbation theory is based on a single assumption.

linear order (assumption, decomposition hypothesis) :

metric expansion :

, metric perturbation : metric on PS : metric on BGS :

Suppose that the linear order perturbation is decomposed as

so that the variable and are the gauge invariant and the gauge variant parts of , respectively. These variables are transformed as

under the gauge transformation .

In cosmological perturbations, this is almost correct and we may choose as (longitudinal gauge, J. Bardeen (1980))

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Problems in decomposition hypothesisProblems in decomposition hypothesis• In cosmological perturbations, .....

– Background metric :

– Zero-mode problem :

• This decomposition is based on the existence of Green functions , , .• In our formulation, we ignored the modes (zero modes) which belong to

the kernel of the operators , , .• How to include these zero modes into our consideration?

• On general background spacetime, … --->Generality problem :– Is the decomposition hypothesis also correct in general

background spacetime?

: metric on maximally symmetric 3-space

: curvature constant associated with

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• In this talk, .....– We partly resolve this generality problem using ADM

decomposition.

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• In this talk, .....– We partly resolve this generality problem using ADM decomposition.

• In our proof, we assume the existence of Green functions of two derivative operator in the simple case :

----> the zero-mode problem remains and it should be examined carefully.• Although our main explanation is for the case , a similar argument is applicable to the general case in which (The Green function of is necessary.) ---> the zero-mode problem is more delicate in general case.

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V. V. SummarySummary

We may say that the decomposition hypothesis

is almost correct for the linear metric perturbation on general background spacetime.

• The first kind “gauge” is a coordinate system on a single spacetime manifold.

• The second kind “gauge” appears in the perturbation theory.

This is a point identification between the physical spacetime and the background spacetime.– To explain this second kind “gauge”, we have to remind what

we are doing in perturbation theory.

II. “Gauge” in general relativity II. “Gauge” in general relativity

• There are two kinds of “gauge” in general relativity.– The concepts of these two “gauge” are closely related to the

general covariance.– “General covariance” : There is no preferred coordinate system in nature.

(R.K. Sachs (1964).)

III. The second kind gauge in GR.III. The second kind gauge in GR.

“Gauge degree of freedom” in general relativistic perturbationsarises due to general covariance.

In any perturbation theories, we always treat two spacetimes :

– Physical Spacetime (PS);– Background Spacetime (BGS).

Physical spacetime (PS)

Background spacetime (BGS)

(Stewart and Walker, PRSL A341 (1974), 49.)

In perturbation theories, we always write equations like

Through this equation, we always identify the points on these two spacetimes and this identification is called “gauge choice” in perturbation theory.

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• Expansion of gauge choices :

We assume that each gauge choice is an exponential map.

------->　　　• Expansion of the variable :• Order by order gauge transformation rules :

Gauge transformation rules of each orderGauge transformation rules of each order

(Sonego and Bruni, CMP, 193 (1998), 209.)

Through these understanding of gauges and the gauge-transformation rules,

we developed second-order gauge-invariant perturbation theory.

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III. Construction of gauge invariant variables in III. Construction of gauge invariant variables in higher order perturbationshigher order perturbations

Our general framework of the second-order gauge invariant perturbation theory WAS based on a single assumption.

linear order (decomposition hypothesis) :

metric expansion :

, metric perturbation : metric on PS : metric on BGS :

Suppose that the linear order perturbation is decomposed as

so that the variable and are the gauge invariant and the gauge variant parts of , respectively. These variables are transformed as

under the gauge transformation .

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is gauge invariant part and is gauge variant part.

where

Under the gauge transformation , the vector field is transformed as

Second order :order :

Once we accept the above assumption for the linear order metric perturbation , we can always decompose the second order metric perturbations as follows :

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Perturbations of an arbitrary matter field Q :

Using gauge variant part of the metric perturbation of each order, gauge invariant variables for an arbitrary fields Q other than metric are defined by

First order perturbation of Q :

Second order perturbation of Q :

These implies that each order perturbation of an arbitrary field is always decomposed as

: gauge invariant part

: gauge variant part15

Comments : gauge-variant parts v.s. gauge-invariant parts

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As a corollary of these decomposition formulae, any order-by-order perturbative equation is automatically given in gauge-invariant form. (Gauge-variant parts are unphysical.)

The decomposition of the metric perturbation into gauge-invariant and gauge-variant parts is not unique. (This corresponds to the fact that there are infinitely many gauge

fixing procedure. Christopherson, et al., arXiv:1101.3525 [astro-ph.CO])

Gauge-variant parts of metric perturbations also play an important role in the systematic construction of gauge-invariant variables for any perturbations. (In this sense, gauge-variant parts are also necessary.)

III. Construction of gauge invariant variables III. Construction of gauge invariant variables in higher-order perturbationsin higher-order perturbations

Our general framework of the second-order gauge invariant perturbation theory WAS based on a single assumption.

linear order (decomposition hypothesis) :

metric expansion :

, metric perturbation : metric on PS : metric on BGS :

Suppose that the linear order perturbation is decomposed as

so that the variable and are the gauge invariant and the gauge variant parts of , respectively. These variables are transformed as

under the gauge transformation .

This decomposition hypothesis is an important premise of our general framework of higher-order gauge-invariant perturbation theory. 17

IV. Construction of gauge-invariant variables for linear metric perturbations on general background spacetime

metric expansion :

, metric perturbation : metric on PS : metric on BGS :

Gauge-transformation for the linear metric perturbation

For simplicity, we first consider the case

(extrinsic curvature),

ADM decomposition of BGS : , .

, .

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(K.N. arXiv:1011.5272[gr-qc])

Decomposition of the linear metric perturbation [cf. J. W. York Jr., JMP14 (1973), 456; AIHP21 (1974), 319.]

We assume the existence of these variables at the starting point, but this existence is confirmed later soon.

Here, and are the variables whose gauge transformation rules are given by

To accomplish this decomposition, we have to assume the existence of the Green functions of the derivative operators

: Ricci curvature on

. ,

This situation is equivalent to the case of cosmological perturbations.-----> zero-mode problem. 19

Gauge-variant parts of

Gauge transformation rule :

We have confirmed the existence of these variables,

which was assumed at the starting point.

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The above construction of gauge-variant parts implies that we may start from the decomposition of the components of :

Even if we start from this decomposition of , we reach to the same conclusion.

Gauge-invariant parts of

• tensor mode :

• vector mode :

• scalar mode :

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Expression of the components of the original in terms of gauge-invariant and gauge-variant variables.

Defining the gauge-invariant and gauge-variant part as

The above components are summarized as the covariant form :

General case (1/4)

• Gauge-transformation rules:

(K.N. arXiv:1101.1147[gr-qc], K.N. arXiv:1103.3092[gr-qc])

Inspecting these gauge-transformation rules, we construct gauge-variant, and gauge-invariant variables as in the previous simple case

General case (2/4) Through the similar logic to the simple case , we can derive the following decomposition of hti and hij :

We may start from this decomposition to derive the gauge-transformation rules for perturbative variables.

General case (3/4) • We can derive gauge-transformation rules for variables as

To reach to these gauge-transformation rules, the Green function ofthe elliptic derivative operator is necessary.

General case (4/4) • Gauge-variant variables :

• Gauge-invariant variables :

---->

• Definitions of gauge-invariant and gauge-variant parts :

• In this talk, .....– We partly resolve this generality problem using ADM decomposition.

• In our proof, we assume the existence of Green functions of two derivative operator in the simple case :

----> the zero-mode problem remains and it should be examined carefully.• Although our main explanation is for the case , a similar argument is applicable to the general case in which (The Green function of is necessary.) ---> the zero-mode problem is more delicate in general case.

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V. V. SummarySummary

We may say that the decomposition hypothesis

is almost correct for the linear metric perturbation on general background spacetime.