Transcript Higher (second) order gauge invariant perturbation theory

Construction of gauge-invariant variables
for linear-order metric perturbation
on general background spacetime
Kouji Nakamura (NAOJ)
References :
K.N. Prog. Theor. Phys., 110 (2003), 723.
K.N. Prog. Theor. Phys., 113 (2005), 413.
K.N. Phys. Rev. D 74 (2006), 101301R.
K.N. Prog. Theor. Phys., 117 (2007), 17.
K.N. Phys. Rev. D 80 (2009), 124021.
K.N. Prog. Theor. Phys. 121 (2009), 1321.
K.N. Adv. in Astron. 2010 (2010), 576273.
K.N. preprint
K.N. preprint
K.N. preprint.
(arXiv:gr-qc/0303039).
(arXiv:gr-qc/0410024).
(arXiv:gr-qc/0605107).
(arXiv:gr-qc/0605108).
(arXiv:0804.3840[gr-qc]).
(arXiv:0812.4865[gr-qc]).
(arXiv:1001.2621[gr-qc]).
(arXiv:1011.5272[gr-qc]).
(arXiv:1101.1147[gr-qc]).
(arXiv:1103.3092[gr-qc]).
1
I. Introduction
The higher order perturbation theory in general relativity
has very wide physical motivation.
– Cosmological perturbation theory
• Expansion law of inhomogeneous universe
(LCDM 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.
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The first order approximation of our universe
from a homogeneous isotropic one is revealed by
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!!!
----->
3
3
I. Introduction
The higher order perturbation theory in general relativity
has very wide physical motivation.
– Cosmological perturbation theory
• Expansion law of inhomogeneous universe
(LCDM 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.
4
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]5
Our general framework of the second-order gauge
invariant perturbation theory is based on a single
assumption.
metric perturbation : metric on PS :
, metric on BGS :
metric expansion :
linear order (assumption, decomposition hypothesis) :
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))
6
Problems in decomposition hypothesis
• In cosmological perturbations, .....
– Background metric :
: metric on maximally symmetric 3-space
– Zero-mode problem :
• This decomposition is based on the existence of Green functions
,
,
.
: curvature constant associated with
• 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?
7
• In this talk, .....
– We partly resolve this generality problem using ADM
decomposition.
8
V. Summary
• 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.
We may say that the decomposition hypothesis
is almost correct for the linear metric perturbation
on general background spacetime.
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II. “Gauge” in general relativity
(R.K. Sachs (1964).)
• 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.
• 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.
III. The second kind gauge in GR.
(Stewart and Walker, PRSL A341 (1974), 49.)
Physical spacetime (PS)
“Gauge degree of freedom” in
general relativistic perturbations
arises due to general covariance.
In any perturbation theories, we
always treat two spacetimes :
– Physical Spacetime (PS);
– Background Spacetime (BGS).
Background spacetime (BGS)
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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•
Gauge transformation rules of each order
Expansion of gauge choices :
We assume that each gauge choice is an exponential map.
(Sonego and Bruni, CMP, 193 (1998), 209.)
------->
• Expansion of the variable :
• Order by order gauge transformation rules :
Through these understanding of gauges
and the gauge-transformation rules,
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we developed second-order gauge-invariant perturbation theory.
III. Construction of gauge invariant variables in
higher order perturbations
metric perturbation : metric on PS :
, metric on BGS :
metric expansion :
Our general framework of the second-order gauge invariant
perturbation theory WAS based on a single assumption.
linear order (decomposition hypothesis) :
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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Second order :
Once we accept the above assumption for the linear order metric
perturbation
, we can always decompose the second order
metric perturbations
as follows :
where
is gauge invariant part and
Under the gauge transformation
is transformed as
is gauge variant part.
, the vector field
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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 part
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Comments : gauge-variant parts v.s. gauge-invariant parts
 As a corollary of these decomposition formulae, any orderby-order perturbative equation is automatically given in
gauge-invariant form.
(Gauge-variant parts are unphysical.)
 The decomposition of the metric perturbation into gaugeinvariant 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 gaugeinvariant variables for any perturbations.
(In this sense, gauge-variant parts are also necessary.)
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III. Construction of gauge invariant variables
in higher-order perturbations
metric perturbation : metric on PS :
, metric on BGS :
metric expansion :
Our general framework of the second-order gauge invariant
perturbation theory WAS based on a single assumption.
linear order (decomposition hypothesis) :
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
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perturbation theory.
IV. Construction of gauge-invariant variables for
linear metric perturbations on general background
spacetime
(K.N. arXiv:1011.5272[gr-qc])
metric perturbation : metric on PS :
, metric on BGS :
metric expansion :
ADM decomposition of BGS :
,
.
,
Gauge-transformation for the linear metric perturbation
.
For simplicity, we first consider the case
,
(extrinsic curvature)
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Decomposition of the linear metric perturbation
[cf. J. W. York Jr., JMP14 (1973), 456; AIHP21 (1974), 319.]
Here,
given by
and
are the variables whose gauge transformation rules are
We assume the existence of these variables at the starting point,
but this existence is confirmed later soon.
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.
.
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Gauge-variant parts of
Gauge transformation rule :
We have confirmed the existence of these variables,
which was assumed at the starting point.
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.
,
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Gauge-invariant parts of
•
scalar mode :
•
vector mode :
•
tensor mode :
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 :
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General case
(1/4)
(K.N. arXiv:1101.1147[gr-qc], K.N. arXiv:1103.3092[gr-qc])
• Gauge-transformation rules:
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
derive the following decomposition of hti and hij :
, we can
We may start from this decomposition to derive the gaugetransformation 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 of
the elliptic derivative operator
is necessary.
General case
•
•
Gauge-variant variables :
•
Gauge-invariant variables :
(4/4)
Definitions of gauge-invariant and gauge-variant parts :
---->
V. Summary
• 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.
We may say that the decomposition hypothesis
is almost correct for the linear metric perturbation
on general background spacetime.
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