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    • Abstract: Manifestly gauge-invariant framework for General Relativity. Application to Cosmology (FRW and perturbation ... Problem of Time in General Relativity. Observables in General Relativity. Observables are by definition gauge invariant quantities ...

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Observables in General Relativity
Application to Cosmology
Quantisation
Conclusions & Outlook
Manifestly Gauge – Invariant Relativistic
Perturbation Theory
Kristina Giesel
Albert – Einstein – Institute
ILQGS
25.03.2008
References:
K.G., S. Hofmann, T. Thiemann, O.Winkler, arXiv:0711.0115, arXiv:0711.0117
K.G., T. Thiemann, arXiv:0711.0119
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General Relativity
Application to Cosmology
Quantisation
Conclusions & Outlook
Plan of the Talk
Content
Application of Relational framework to General Relativity
Special Case of Deparametrisation: Two examples
Manifestly gauge-invariant framework for General Relativity
Application to Cosmology (FRW and perturbation around FRW)
Quantisation: Reduced Phase Space Quantisation
Conclusions & Outlook
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Problem of Time in General Relativity
Observables in General Relativity
Observables are by definition gauge invariant quantities
The gauge group of GR is Diff(M)
Canonical picture:
Constraints c, c generate spatial and ’time’ gauge transformations
O gauge invariant ⇔ {c, O} = {c, O} = 0
’Hamiltonian’ hcan for Einstein Equations is linear combination of
constraints and thus constrained to vanish
Consequently: O gauge invariant ⇔ {hcan , O} = 0
Frozen picture, contradicts experiments, problem of time in GR
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Relational Formalism
Basic Idea [Bergmann ’60, Rovelli ’90]
Einstein Equations are no physical evolution equations
Rather describe flow of unphysical quantities under gauge transf.
Relational formalism:
Take two gauge variant f , g and choose T := g as a clock
Define gauge invariant extension of f denoted by Ff ,T in relation to
values T takes
Ff ,T : Values of f when clock T = g takes values 5, 17, 23, 42, ...
Solve αt (T ) = τ for t, then use solution tT (τ ) for Ff ,T which
becomes a function of τ
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Relational Formalism: Idea
f , g move along gauge orbit
PSfrag replacements f (t3 ) g (t4 )
g (t2 )
f (t1 )
gauge orbit f gauge orbit g
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Relational Formalism
Basic Idea [Bergmann ’60, Rovelli ’90]
Einstein Equations are no physical evolution equations
Rather describe flow of unphysical quantities under gauge transf.
Relational formalism:
Take two gauge variant f , g and choose T := g as a clock
Define gauge invariant extension of f denoted by Ff ,T in relation to
values T takes
Ff ,T : Values of f when clock T = g takes values 5, 17, 23, 42, ...
Solve αt (T ) = τ for t, then use solution tT (τ ) for Ff ,T which
becomes a function of τ
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Relational Formalism
Explicit Form for Ff ,T [Dittrich ’04]
Take as many clocks TI as they are CI then Ff ,T (τ ) can be
expressed as powers series in T I with coefficients involving multiple
Poisson brackets of CI and f .
Explicit form in general quite complicated
But: One has explicit strategy how to construct observables
Analysed in several examples, application to cosmology and
cosmological perturbations [Dittrich, Dittrich & Tambornino]
Automorphism property
{Ff ,T (τ ), Ff ,T (τ )} = F{f ,f },T (τ ),
If f (q, p) then Ff ,T = f (Fq,T , Fp,T )
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Strategy of the Formalism
Steps to obtain EOM for observables
Consider a physical System for instance gravity & some standard
matter
We would like to derive EOM for the observables associated to
(qa , p a ) of gravity & matter
Add additional action to the system which become clocks T
We have
c tot = c geo + c matter + c clock =: c + c clock = 0
tot geo matter clock clock
ca = c a + c a + ca =: ca + ca =0
Construct observables wrt to these constraints: Fqa ,T (τ ) & Fpa ,T (τ )
Construct so called physical Hamiltonian Hphys which generates
true evolution of Fqa ,T (τ ), Fpa ,T (τ )
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Special Case of Deparametrisation
Steps technically simplify
Deparametrisation: c tot and ca can be solved for p clock
tot
Expressions for Fqa /pa ,T (τ ) and Hphys simplify
Note: Hphys is in general different for each chosen clock system
Evolution of observables is generated by Hphys
EOM for observables are clock – dependent
Consider two examples for clarification:
scalar field without potential
k-essence
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Scalar field as a Clock (LQC-Model)
Deparametrisation for scalar field φ
Constraints:
1 π2
c tot = c(qa , p a ) + 2λ ( √q + q ab φ,a φ,b )
ca = ca (qa , p a ) + πφ,a
tot
Using ca = 0 we get q ab φ,a φ,b = 1/π 2 q ab ca cb (more details later)
tot
Using c tot = 0 we get
√ √
π = | − qλc − q λ2 c 2 − q ab ca cb | =: hφ (qa , p a )
Equivalent Hamiltonian constraint: c tot = π − hφ (qa , p a )
˜
Construct observables Qa (τφ ) := Fqa ,φ (τ ) and P a (τφ ) := Fpa ,φ (τ )
Evolution:
˙ ˙
Qa (τφ ) = {Hphys , Qa (τφ )} and P a (τφ ) = {Hphys , P a (τφ )}
√ √
Hφ := d 3 σ | − QλC − Q λ2 C 2 − Q ab Ca Cb |
phys
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
K-essence (Thiemann ’06)
Deparametrisation for k-essence field ϕ: Case I
Constraints:

c tot = c(qa , p a ) − [1 + q ab ϕ,a ϕ,b ][π 2 + α2 q], α > 0
tot a
ca = ca (qa , p ) + πϕ,a
Using ca = 0 we get again q ab ϕ,a ϕ,b = 1/π 2 q ab ca cb
tot
Using c tot = 0 we get π = −hϕ (qa , p a )
hϕ (qa , p a ) :=
1 2 1 2
2 (c − q ab ca cb − α2 q) + 4 (c − q ab ca cb − α2 q)2 − α2 q ab ca cb q
Equivalent Hamiltonian constraint: c tot = π + hϕ (qa , p a )
˜
Construct observables Qa (τϕ ) := Fqa ,ϕ (τ ) and P a (τϕ ) := Fpa ,ϕ (τ )
˙ ˙
Qa (τφ ) = {Hphys , Qa (τφ )} and P a (τϕ ) = {Hphys , P a (τφ )}
ϕ 3 a
Hphys := d σhϕ (Qa , P )
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
K-essence (Thiemann ’06)
Deparametrisation for k-essence field ϕ: Case II
Constraints:

c tot = c (qa , p a ) − [1 + q ab ϕ,a ϕ,b ][π 2 + α2 q], α > 0
tot a
ca = ca (qa , p ) + πϕ,a
Using ca = 0 we get again q ab ϕ,a ϕ,b = 1/π 2 q ab ca cb
tot
Using c tot = 0 we get π = −hϕ (qa , p a )
h (qa , p a ) :=
1 1
2 ((c )2 − q ab ca cb − α2 q) + 4 ((c )2 − q ab ca cb − α2 q)2 − α2 q ab ca cb q
Equivalent Hamiltonian constraint: c tot = π + hϕ (qa , p a )
˜
Construct observables Qa (τϕ ) := Fqa ,ϕ (τ ) and P a (τϕ ) := Fpa ,ϕ (τ )
˙ ˙
Qa (τφ ) = {Hphys , Qa (τφ )} and P a (τϕ ) = {Hphys , P a (τφ )}
Hϕ := d 3 σhϕ (Qa , P a )
phys
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Comparison of both physical Hamiltonian
Comparison of Hφ and Hϕ
phys phys
Physical Hamiltonians: (D 2 := Q ab Ca Cb ), (D )2 := Q ab Ca Cb )
√ √ √
Hφ =
phys d 3σ |− QλC − Q λ2 C 2 − D 2 |
Hϕ =
phys
1 1
d 3σ 2 ((C )2 − (D )2 − α2 Q) + 4 ((C )2 − (D )2 − α2 Q)2 − α(D )2 Q
Specialise both Hphys to cosmology (FRW – symmetry)
Then D 2 = (D )2 = 0 and

Hφ =
phys d 3σ | − 2λ QCFRW | and Hϕ =
phys d 3 σCFRW
tot
Note that CFRW = 0 here only CFRW = 0
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Clocks for General Relativity
Choose Clock and Ruler for GR
Choose clock and ruler to give time & space physical meaning
We need 1 × ∞ clocks and 3 × ∞ rulers: 4 scalar fields
Chosen clocks & rulers such that good for cosmology:
Free falling observer
Standard cosmology CFRW as true Hamiltonian
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Brown-Kuchaˇ-Mechanism
r
Dust Lagrangian
Add dust Lagrangian to Gravity & Standard Model
1
Sdust = − d 4X | det(g )|ρ(g µν Uµ Uν + 1)
2 M
j
where Uµ = −T,µ + Wj S,µ , ρ energy density
U µ = g µν Uν is a geodesic, fields Wj , S j are constant along
geodesics, T defines proper time along each geodesic
Tµν of a pressureless perfect fluid
αt (T ) = τ becomes clock, αx (S j ) = σ j becomes ruler
Dust serves as a physical reference system
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
A few Words on Notation
Canonical (3+1) split of Gravity + Standard Model + Dust
Dust variables time αt (T ) = τ and space αx (S j ) = σ j : Conjugate
momenta P and Pj , j = 1, 2, 3
Remaining gravity & matter degrees of freedom qab , p ab and φ, π
are denoted by q a , pa
Gauge variant quantities: Lower case letters q a , pa
Gauge invariant quantities: Capital letters Q a , Pa
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Brown-Kuchaˇ-Mechanism
r
Deparametrisation of the Constraints in GR
Canonical 3+1 split: (P,T ),(Pj ,S j ) & remaining non dust (pa , q a )
Detailed constraints analysis, then 1st class constraints
c tot = c + c dust
j j
with c dust = − P 2 + q ab (PT,b + Pj S,b )(PT,b + Pj S,b )
tot dust dust j
ca = c a + c a with ca = PT,a + Pj S,a
Brown-Kuchaˇ-Mechanism:
r
c dust = − dust dust
P 2 + q ab ca cb
tot dust
Use ca = 0 and replace ca by −ca in c dust
Then solve c tot for P and ca for Pj
tot
Need to assume S,a is invertible with inverse Sja
j
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Deparametrisation of the Constraints in GR
(Partial) Deparametrisation of the Constraints in GR
Constraints in (partial) deparametrised form
c tot = P + h with h(pa , q a ) := c 2 − q ab ca cb
˜
ca = Pj + hj with hj (T , S j , pa , q a ) = Sja (ca − hT,a )
˜tot
c tot , ca mutually commute
˜ ˜tot
Here Ff ,T simplifies a lot
Construction of Ff ,T in two steps
˜tot
1.) Reduction wrt to ca : qab (x, t) −→ qij (σ, t)
˜
2.) Reduction wrt to c tot : qij (σ, t) −→ Qij (σ, τ )
˜ ˜
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Observables with respect to Dust Clock & Rulers
Space time points are labled by τ and σ j
τ proper time on each geodesic
x x
PSfrag replacements
(σ 1 =1,σ 2 =4,σ 3 =35)
(σ 1 =8,σ 2 =0.3,σ 3 =44)
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Construction of Observables
Explicit Form of Observables
1.) Spatial diff’-invariant quantities
qij (σ, t) =
˜ d 3 x| det(∂S(x) ∂x)|δ(S(x), σ)qab (x)Sia (x)Sjb (x)
local in σ but ultra – non – local in x
2.) Full Observables

1 ˜
Qij (σ, τ ) = n! {h(τ ), qij (σ)}(n)
˜
n=0
where {f , g }(0) = g , {f , g }(n) := {f , {f , g }(n−1) }}
˜ ˜
and h(τ ) := d 3 σ(τ − T (σ))h(σ) ˜
S
S=range(σ) so called ’dust space’
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Physical Hamiltonian for GR
Physical Hamiltonian Hphys
We have a strategy to construct gauge invariant extension for all
pa , q a and get Pi , Q i
Due to automorphism property of Ff ,T , we can extend this to
functions of pa , q a which just become functions of Pi , Q i
However, we would like to have so called physical Hamiltonian
Hphys for GR that generates evolution of observables
Recall: We cannot use canonical Hamiltonian hcan from Einstein
equations because {hcan, P i } = {hcan, Q i } = 0
Hphys should itself be gauge invariant
Hphys can be derived from deparametrised constraints
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Reduced Phase Space & Physical Hamiltonian
Physical Hamiltonian
We have c tot = P + h(pa , q a ) with h =
˜ c 2 − q ab ca cb
H(σ, τ ) := Fh,T = C 2 (τ, σ) − Q ij (τ, σ)Ci (σ)Cj (σ)
Physical Hamiltonian is given by Hphys = S
d 3 σH(σ, τ )
(S dust space)
dFf ,T (σ,τ )
Physical Physical time evolution: dτ = {Hphys , Ff ,T (σ, τ )}
Symmetries of Hphys : {Hphys , Cj (σ)} = 0, {Hphys , H(σ)} = 0
Hphys no τ dependence: conservative system
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Reduced Phase space of Gravity + Scalar field and Dust
Comparison with Unreduced Phase Space
Standard unreduced framework: Gravity & scalar field
Einstein Equations: EOM for qab , p ab and matter dof
’Hamiltonian’ hcan = Σ d 3 x (n(x)c(x) + na (x)ca (x))
Constraints c := c geo + c matter = 0 and ca := ca + ca
geo matter
=0
Reduced framework with additional Dust: [K.G., Hofmann, Thiemann, Winkler]
Manifestly gauge invariant EOM for Qij , P ij and matter dof
Physical Hamiltonian Hphys = S d 3σ C 2 − Q ij Ci Cj (σ)
Energy & momentum conservation H = − , Cj = − j
j ij
Lapse & Shift dynamical: N = C /H, N = −Q Ci /H
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Equation of Motion for Unreduced Case
Second Order Time Derivative Equation of Motion for q ab
¨
n ( det(q))˙
˙ n det(q)
¨
qab = − + Ln qab − Ln q
˙ ab
n det(q) det(q) n
+q cd qac − Ln q
˙ ac
qbd − Ln q
˙ bd
2
n κ κ κ
+qab − C + n2 2Λ + v (ξ) + n2 ξ,a ξ,b − 2Rab
2 det(q) 2λ λ
+2n Da Db n + 2 Ln q
˙ ab
+ Ln q
˙ ab
− Ln Ln q ab
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Reduced Phase space of Gravity + Scalar field and Dust
Comparison with Unreduced Phase Space
Standard unreduced framework: Gravity & scalar field
Einstein Equations: EOM for qab , p ab and matter dof
’Hamiltonian’ hcan = Σ d 3 x (n(x)c(x) + na (x)ca (x))
Constraints c := c geo + c matter = 0 and ca := ca + ca
geo matter
=0
Reduced framework with additional Dust: [K.G., Hofmann, Thiemann, Winkler]
Manifestly gauge invariant EOM for Qij , P ij and matter dof
Physical Hamiltonian Hphys = S d 3σ C 2 − Q ij Ci Cj (σ)
Energy & momentum conservation H = − , Cj = − j
j ij
Lapse & Shift dynamical: N = C /H, N = −Q Ci /H
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Equation of Motion for Reduced Case
¨
Second Order Time Derivative Equation of Motion for Qjk
√ √
˙
N ( det Q)˙ N det Q
¨
Qjk = − √ +√ LN ˙
Qjk − LN Q
N N jk
det Q det Q
˙
+Q mn Qmj − L Q ˙
Qnk − L Q
N mj N nk
N 2κ κ κ
+Qjk − √ C + N 2 2Λ + v (Ξ) + N 2 Ξ,j Ξ,k − 2Rjk
2 det Q 2λ λ
˙
+2N Dj Dk N + 2 LN Q jk + L ˙ Q jk − LN LN Q jk
N
NH
−√ Gjkmn N m N n
det Q
¨
Qjk refers to derivative with respect to dust time τ here
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Reduced Phase space of Gravity + Scalar field and Dust
Comparison with Unreduced Phase Space
Standard unreduced framework: Gravity & scalar field
Einstein Equations: EOM for qab , p ab and matter dof
’Hamiltonian’ hcan = Σ d 3 x (n(x)c(x) + na (x)ca (x))
Constraints c := c geo + c matter = 0 and ca := ca + ca
geo matter
=0
Reduced framework with additional Dust: [K.G., Hofmann, Thiemann, Winkler]
Manifestly gauge invariant EOM for Qij , P ij and matter dof
Physical Hamiltonian Hphys = S d 3σ C 2 − Q ij Ci Cj (σ)
Energy & momentum conservation H = − , Cj = − j
j ij
Lapse & Shift dynamical: N = C /H, N = −Q Ci /H
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Problem of Time in GR
Observables in General Relativity Relational formalism
Application to Cosmology Clocks for GR
Quantisation Brown-Kuchaˇ-Mechanism
r
Conclusions & Outlook Physical Hamiltonian
Summary: Classical Setup
Equation of Motion for Reduced Case
¨
Second Order Time Derivative Equation of Motion for Qjk
√ √
˙
N ( det Q)˙ N det Q
¨
Qjk = − √ +√ LN ˙
Qjk − LN Q
N N jk
det Q det Q
mn ˙ ˙ nk − L Q
+Q Qmj − L Q QN mj N nk
2
N κ κ κ
+Qjk − √ C + N 2 2Λ + v (Ξ) + N 2 Ξ,j Ξ,k − 2Rjk
2 det Q 2λ λ
˙
+2N Dj Dk N + 2 LN Q jk + L ˙ Q jk − LN LN Q jk
N
NH
−√ Gjkmn N m N n
det Q
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General Relativity
Application to Cosmology Specialisation to FRW spacetimes
Quantisation Linear Cosmological Perturbation Theory
Conclusions & Outlook
Application to Cosmology
Apply Manifestly Gauge Invariant Framework to FRW
¨
1.) Specialise Qij equations to FRW spacetime
2.) Consider linear perturbations around FRW spacetime
3.) Compare with standard results and check that dust clocks do not
contradict current experiments
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General Relativity
Application to Cosmology Specialisation to FRW spacetimes
Quantisation Linear Cosmological Perturbation Theory
Conclusions & Outlook
Check Manifestly Gauge Invariant Equations for FRW Case
Standard Framework: FRW Spacetime
ds 2 = −dt 2 + a(t)2 δab dx a dx b = a(x 0 )2 ηµν dx µ dx ν
Metric qab = a2 (t)δab , Momenta p ab = −2aδ ab , ca = 0,
˙
FRW eqn from qab = {hcan , qab } and p ab = {hcan , p ab }, c(q, p) = 0


Use: 0.2879