partially massless (pm) cédric deffayet graviton on curved ... · kiroro, feb the 18th 2018. from...
TRANSCRIPT
![Page 1: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/1.jpg)
PACIFIC 2018
Kiroro, Feb the 18th 2018.
From a Massive to a
Partially Massless (PM)
Graviton
on Curved Space-times
1. Massive gravity 101:
Massive graviton on
Einstein space-times
2. Massive graviton on arbitrary
spacetimes.
3. PM graviton on non Einstein
spacetimes.
Cédric Deffayet
(IAP and IHÉS, CNRS Paris)
FP7/2007-2013
« NIRG » project no. 307934
L. Bernard, C.D., K. Hinterbichler and M. von Strauss
arXiv:1703.02538 (PRD)
L. Bernard, C.D., M. von Strauss + A. Schmidt-May
(2015-2016, PRD, JCAP)
![Page 2: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/2.jpg)
Consider an Einstein space-time obeying
Fierz-Pauli theory (1939)
is the (only correct) theory of a
massive graviton which
propagates on this space-time
1. Massive gravity 101:
Fierz-Pauli linear theory for an Einstein space-time
![Page 3: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/3.jpg)
1. Massive gravity 101:
Fierz-Pauli linear theory for an Einstein space-time
Consider an Einstein space-time obeying
Fierz-Pauli theory is defined by
Fierz-Pauli (1939), Deser Nepomechie (1984), Higuchi (1987),
Bengtsson (1995), Porrati (2001)
Field equations
with on shell
Kinetic
operator
Mass
term
Cosmological
constant
![Page 4: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/4.jpg)
Kinetic
operator
Mass
term
Cosmological
constant
Comes from expanding the
Einstein-Hilbert action
![Page 5: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/5.jpg)
Kinetic
operator
Mass
term
Cosmological
constant
Comes from expanding the
Einstein-Hilbert action
on shell
ana
Analogous to Proca equations
for a massive photon :
![Page 6: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/6.jpg)
Kinetic
operator
Mass
term
Cosmological
constant
Comes from expanding the
Einstein-Hilbert action
Interest: Cosmology ?
The mass term leads to a large distance
modification of gravity (C.D., Dvali, Gabadadze 2001)
![Page 7: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/7.jpg)
DOF (= « polarizations » ) counting
The Fierz Pauli theory for a massive graviton
of mass m propagates
• 2 DOF if m = 0
Massless graviton (of GR)
![Page 8: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/8.jpg)
The Fierz Pauli theory for a massive graviton
of mass m propagates
• 2 DOF if m = 0
• 5 DOF if m 0 and m2 2 ¤ /3
Generic massive graviton
Massless graviton (of GR)
DOF (= « polarizations » ) counting
![Page 9: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/9.jpg)
The Fierz Pauli theory for a massive graviton
of mass m propagates
• 2 DOF if m = 0
• 5 DOF if m 0 and m2 2 ¤ /3
Generic massive graviton
Massless graviton (of GR)
DOF (= « polarizations » ) counting
ana the Proca « photon »
has 3 polarizations
![Page 10: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/10.jpg)
The Fierz Pauli theory for a massive graviton
of mass m propagates
• 2 DOF if m = 0
• 5 DOF if m 0 and m2 2 ¤ /3
Generic massive graviton
Massless graviton (of GR)
DOF (= « polarizations » ) counting
![Page 11: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/11.jpg)
The Fierz Pauli theory for a massive graviton
of mass m propagates
• 2 DOF if m = 0
• 5 DOF if m 0 and m2 2 ¤ /3
• 4 DOF if m2 = 2 ¤ /3
Generic massive graviton
Partially Massless
graviton
Massless graviton (of GR)
DOF (= « polarizations » ) counting
![Page 12: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/12.jpg)
How to count DOF ?
![Page 13: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/13.jpg)
Kinetic
operator
Mass
term
Cosmological
constant
Comes from expanding the
Einstein-Hilbert action
This implies the (Bianchi) offshell
identities
How to count DOF ?
![Page 14: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/14.jpg)
Kinetic
operator
Mass
term
Cosmological
constant
Comes from expanding the
Einstein-Hilbert action
This implies the (Bianchi) offshell
identities
How to count DOF ?
Analogous to
In Maxwell and Proca theories
![Page 15: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/15.jpg)
Results in an the off-shell identity
![Page 16: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/16.jpg)
Results in an the off-shell identity
And the on-shell relation
4 vector
constraints
Kills 4 out of 10 DOF of
![Page 17: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/17.jpg)
Results in an the off-shell identity
And the on-shell relation
4 vector
constraints
Kills 4 out of 10 DOF of
Analogous to the constraint
of Proca theory
![Page 18: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/18.jpg)
Taking an extra derivative of the field equation
operator yields (off shell)
![Page 19: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/19.jpg)
Taking an extra derivative of the field equation
operator yields (off shell)
While tracing it with the metric gives
![Page 20: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/20.jpg)
Taking an extra derivative of the field equation
operator yields (off shell)
While tracing it with the metric gives
Hence we have the identity
Yielding on shell
![Page 21: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/21.jpg)
Generically: yields
i.e. a “scalar constraint”
reducing from 6 to 5 the
number of propagating DOF
![Page 22: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/22.jpg)
However, if
Then this vanishes identically
![Page 23: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/23.jpg)
However, if
Then this vanishes identically
As is
![Page 24: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/24.jpg)
However, if
Then this vanishes identically
As is
Shows the existence of a gauge symmetry
![Page 25: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/25.jpg)
Hence, if
one has 6 - 2 = 4 DOF
The massive graviton is
said to be
“Partially massless” (PM)
![Page 26: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/26.jpg)
Two questions:
Can a fully non linear PM
theory exist ?
Can a PM graviton exist on
non Einstein space-times ?
![Page 27: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/27.jpg)
Two questions:
Can a fully non linear PM
theory exist ?
Can a PM graviton exist on
non Einstein space-times ?
Here we address the second one…
![Page 28: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/28.jpg)
2. Consistent massive graviton on
arbitrary backgrounds
Einstein-Hilbert
kinetic operator
Mass term
First, we need to introduce the theory of a
massive graviton on arbitrary backgrounds
![Page 29: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/29.jpg)
The theory has been obtained in
out of the dRGT theory
L.Bernard, CD, M. von Strauss
1410.8302 + 1504.04382
+ 1512.03620 (with A. Schmidt-May)
de Rahm, Gabadadze;
de Rham, Gababadze, Tolley 2010, 2011
Fully non linear theory for a dynamical metric + a non
dynamical one which has been shown to propagate 5
(or less d.o.f.) in a fully non linear way …
… evading Boulware-Deser no-go « theorem »
![Page 30: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/30.jpg)
Our massive graviton theory is defined by
![Page 31: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/31.jpg)
Our massive graviton theory is defined by
1. A symmetric tensor obtained from the
background curvature solving
with ¯0, ¯1 and ¯2 dimensionless parameters
and m the graviton mass, ei the symmetric polynomials
![Page 32: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/32.jpg)
2. The following (linear) field equations
![Page 33: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/33.jpg)
2. The following (linear) field equations
Linearized Einstein operator
![Page 34: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/34.jpg)
2. The following (linear) field equations
![Page 35: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/35.jpg)
We follow the analysis for the case of Einstein
space-times:
DOF counting ?
![Page 36: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/36.jpg)
We follow the analysis for the case of Einstein
space-times:
The linearized Bianchi identity
Yields 4 vector constaints reducing from 10 to 6
the number of propagating DOF
DOF counting ?
![Page 37: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/37.jpg)
We follow the analysis for the case of Einstein
space-times:
The linearized Bianchi identity
Yields 4 vector constaints reducing from 10 to 6
the number of propagating DOF
A scalar constraint reduces to 5 the
number of DOF
DOF counting ?
![Page 38: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/38.jpg)
We look for (non Einstein) space-times where the
scalar constraint
Identically vanishes
3. PM graviton on non Einstein space-times ?
Yielding the gauge symmetry with
![Page 39: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/39.jpg)
We look for (non Einstein) space-times where the
scalar constraint
Identically vanishes
3. PM graviton on non Einstein space-times ?
Yielding the gauge symmetry with
We need to look in detail at the
structure of the constraint
![Page 40: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/40.jpg)
The scalar constraint reads
With
![Page 41: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/41.jpg)
The scalar constraint reads
With
![Page 42: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/42.jpg)
The scalar constraint reads
With
![Page 43: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/43.jpg)
The scalar constraint reads
With
![Page 44: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/44.jpg)
To get a PM theory we need to look for space-times
where and vanish identically.
![Page 45: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/45.jpg)
To get a PM theory we need to look for space-times
where and vanish identically.
Most general solution ?
![Page 46: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/46.jpg)
To get a PM theory we need to look for space-times
where and vanish identically.
Most general solution ?
Assume
(i.e. covariantly constant)
makes and vanish.
![Page 47: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/47.jpg)
Space-times possessing a covariantly constant
tensor are classified as (provided
is not proportional to the metric)
1. Spacetime is decomposable
and
2. The spacetime admits a covariantly constant
vector and
![Page 48: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/48.jpg)
Note further that the definition
Imposes that
Hence the space-time must be “Ricci Symmetric”
…i.e. the covariantly constant tensor is the Ricci
tensor…
![Page 49: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/49.jpg)
The spacetimes of interest for us here will all have
with and constant
as a consequence of the integrability conditions
And have to solve (in order to get a PM graviton)
![Page 50: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/50.jpg)
The spacetimes of interest for us here will all have
with and constant
as a consequence of the integrability conditions
And have to solve (in order to get a PM graviton)
In order to get a vanishing
![Page 51: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/51.jpg)
The spacetimes of interest for us here will all have
with and constant
as a consequence of the integrability conditions
And have to solve (in order to get a PM graviton)
From the definition of
![Page 52: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/52.jpg)
The spacetimes of interest for us here will all have
with and constant
as a consequence of the integrability conditions
And have to solve (in order to get a PM graviton)
NB: this implies that the Ricci tensor
obeyes indeed the required relation
![Page 53: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/53.jpg)
Explicit solutions (I)
with
dimensionless dimensionful
Analogous
to
![Page 54: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/54.jpg)
Explicit solutions (II)
with
dimensionless dimensionful
Analogous
to
![Page 55: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/55.jpg)
One simple example of the last kind is
Einstein static Universe !
with
![Page 56: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/56.jpg)
One simple example of the last kind is
Einstein static Universe !
with
The gauge symmetry reads
With e.g. one solution being
![Page 57: Partially Massless (PM) Cédric Deffayet Graviton on Curved ... · Kiroro, Feb the 18th 2018. From a Massive to a Partially Massless (PM) Graviton on Curved Space-times 1. Massive](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55a39af7a30323ef1753c6/html5/thumbnails/57.jpg)
Conclusions
PM exists on non Einstein spacetimes ! (in contrast with previous no-go claim by Deser, Joung, Waldron
In 1208.1307 [hep-th]…
… in particular some of our solution have a vanishing Bach tensor)
Solution for the vanishing
Of and
are not known in full generality !