pacific 2018 massive and moorea, sept the 3rd …...pacific 2018 moorea, sept the 3rd 2018. massive...
TRANSCRIPT
![Page 1: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/1.jpg)
PACIFIC 2018
Moorea, Sept the 3rd 2018.
Massive and
Partially Massless (PM)
gravitons
on curved space-times
1. A short review on Massive
Gravity
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: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/2.jpg)
3 (good ?) reasons to give this talk here
![Page 3: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/3.jpg)
3 (good ?) reasons to give this talk here
1. Arkady is in the room !
![Page 4: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/4.jpg)
3 (good ?) reasons to give this talk here
1. Arkady is in the room !
2. Massive gravity is cool and 2/3 of you were
not there in the PACIFIC.2 Japanese edition !
![Page 5: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/5.jpg)
3 (good ?) reasons to give this talk here
1. Arkady is in the room !
2. Massive gravity is cool and 2/3 of you were
not there in the PACIFIC.2 Japanese edition !
L. Bernard, C.D., K. Hinterbichler and M. von Strauss
arXiv:1703.02538 (PRD)
3. Erratum necessary for some technicalities in
(thanks to Charles Mazuet)
![Page 6: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/6.jpg)
1. A short review on Massive Gravity
1.1. Why massive gravity ?
![Page 7: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/7.jpg)
One way to modify gravity at « large distances »
… and get rid of dark energy (or dark matter) ?
Changing the dynamics
of gravity ?
Dark matter
dark energy ?
1. A short review on Massive Gravity
1.1. Why massive gravity ?
![Page 8: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/8.jpg)
One way to modify gravity at « large distances »
… and get rid of dark energy (or dark matter) ?
Changing the dynamics
of gravity ?
Historical example the
success/failure of both
approaches: Le Verrier and
• The discovery of Neptune
• The non discovery of Vulcan…
but that of General Relativity
Dark matter
dark energy ?
1. A short review on Massive Gravity
1.1. Why massive gravity ?
![Page 9: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/9.jpg)
for this idea to work…
I.e. to « replace » the cosmological constant by a
non vanishing graviton mass…
![Page 10: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/10.jpg)
for this idea to work…
One obviously needs
a very light graviton
(of Compton length
of order of the size of
the Universe)
I.e. to « replace » the cosmological constant by a
non vanishing graviton mass…
![Page 11: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/11.jpg)
for this idea to work…
I.e. to « replace » the cosmological constant by a
non vanishing graviton mass…
NB: It seems one of the
Einstein’s motivations to
introduce the cosmological
constant was to try to « give a
mass to the graviton »
(see « Einstein’s mistake and the
cosmological constant »
by A. Harvey and E. Schucking,
Am. J. of Phys. Vol. 68, Issue 8 (2000))
![Page 12: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/12.jpg)
1. A short review on Massive Gravity
1.1. Why massive gravity ?
Theoretical challenge to give a mass to the graviton
(here we will rather stay on this « abstract » side
rather than discussing real world applications)
One way to modify gravity at « large distances »
… and get rid of dark energy (or dark matter) ?
Changing the dynamics
of gravity ?
Dark matter
dark energy ?
![Page 13: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/13.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. A short review on Massive Gravity
1.2. Fierz-Pauli theory on Einstein space-times
![Page 14: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/14.jpg)
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
1. A short review on Massive Gravity
1.2. Fierz-Pauli theory on Einstein space-times
![Page 15: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/15.jpg)
Kinetic
operator
Mass
term
Cosmological
constant
Comes from expanding the
Einstein-Hilbert action
![Page 16: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/16.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 17: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/17.jpg)
Kinetic
operator
Mass
term
Cosmological
constant
Comes from expanding the
Einstein-Hilbert action
Interest: Cosmology ?
The mass term leads to « self
acceleration » (C.D., Dvali, Gabadadze 2001)
![Page 18: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/18.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 19: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/19.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 20: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/20.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 21: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/21.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 22: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/22.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 23: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/23.jpg)
How to count DOF ?
![Page 24: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/24.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 25: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/25.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 26: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/26.jpg)
Results in an the off-shell identity
![Page 27: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/27.jpg)
Results in an the off-shell identity
And the on-shell relation
4 vector
constraints
Kills 4 out of 10 DOF of
![Page 28: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/28.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 29: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/29.jpg)
Taking an extra derivative of the field equation
operator yields (off shell)
![Page 30: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/30.jpg)
Taking an extra derivative of the field equation
operator yields (off shell)
While tracing it with the metric gives
![Page 31: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/31.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 32: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/32.jpg)
Generically: yields
i.e. a “scalar constraint”
reducing from 6 to 5 the
number of propagating DOF
![Page 33: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/33.jpg)
However, if
Then this vanishes identically
![Page 34: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/34.jpg)
However, if
Then this vanishes identically
As is
![Page 35: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/35.jpg)
However, if
Then this vanishes identically
As is
Shows the existence of a gauge symmetry
![Page 36: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/36.jpg)
Hence, if
one has 6 - 2 = 4 DOF
The massive graviton is
said to be
“Partially massless” (PM)
![Page 37: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/37.jpg)
1. A short review on Massive Gravity
1.3. Non linear completions of Fierz-Pauli theory
In the generic case (with 5 DOF) the scalar polarization
of the Fierz-Pauli graviton is an obstacle to real world
applications.
![Page 38: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/38.jpg)
1. A short review on Massive Gravity
1.3. Non linear completions of Fierz-Pauli theory
In the generic case (with 5 DOF) the scalar polarization
of the Fierz-Pauli graviton is an obstacle to real world
applications.
Wrong light bending
![Page 39: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/39.jpg)
1. A short review on Massive Gravity
1.3. Non linear completions of Fierz-Pauli theory
In the generic case (with 5 DOF) the scalar polarization
of the Fierz-Pauli graviton is an obstacle to real world
applications.
Wrong light bending
Stays sending the mass of the
graviton smoothly to zero (van Dam,
Veltman, Zakharov, Iwasaki 1970)
![Page 40: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/40.jpg)
1. A short review on Massive Gravity
1.3. Non linear completions of Fierz-Pauli theory
In the generic case (with 5 DOF) the scalar polarization
of the Fierz-Pauli graviton is an obstacle to real world
applications.
Wrong light bending
Stays sending the mass of the
graviton smoothly to zero (van Dam,
Veltman, Zakharov, Iwasaki 1970)
A way out relying on non linearities
was suggested by Arkady in 1972
(« Vainshtein mechanism »)
![Page 41: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/41.jpg)
Arkady’s suggestion was criticized by
Boulware and Deser (BD) in 1972
![Page 42: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/42.jpg)
BD pointed out that one needed to find non singular
solution of an existing massive gravity theory
featuring the behaviour conjectured by Arkady
Arkady’s suggestion was criticized by
Boulware and Deser (BD) in 1972
![Page 43: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/43.jpg)
BD pointed out that one needed to find non singular
solution of an existing massive gravity theory
featuring the behaviour conjectured by Arkady
It was only recently that such solutions were found
(C.D., Dvali, Gabadadze, Vainshtein 2002,
Babichev, C.D., Ziour, 2009-2010)
Arkady’s suggestion was criticized by
Boulware and Deser (BD) in 1972
![Page 44: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/44.jpg)
BD pointed out that one needed to find non singular
solution of an existing massive gravity theory
featuring the behaviour conjectured by Arkady
It was only recently that such solutions were found
(C.D., Dvali, Gabadadze, Vainshtein 2002,
Babichev, C.D., Ziour, 2009-2010)
In the same paper BD also pointed out another generic (they
think) pathology of non linear massive gravity: the « BD ghost »
Arkady’s suggestion was criticized by
Boulware and Deser (BD) in 1972
![Page 45: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/45.jpg)
BD pointed out that one needed to find non singular
solution of an existing massive gravity theory
featuring the behaviour conjectured by Arkady
It was only recently that such solutions were found
(C.D., Dvali, Gabadadze, Vainshtein 2002,
Babichev, C.D., Ziour, 2009-2010)
In the same paper BD also pointed out another generic (they
think) pathology of non linear massive gravity: the « BD ghost »
This was overcome in 2010-2011 by de Rham,
Gababadze, Tolley (« dRGT theory »)
Arkady’s suggestion was criticized by
Boulware and Deser (BD) in 1972
![Page 46: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/46.jpg)
Where
with
Has been shown to propagate 5 (or less d.o.f.)
in a fully non linear way …
… evading Boulware-Deser no-go « theorem »
dRGT theory de Rahm, Gabadadze; de Rham, Gababadze, Tolley;
Hassan, Rosen 2010, 2011
![Page 47: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/47.jpg)
1. A short review on Massive Gravity
1.4. Non linear completions of Partially Massless theory ?
Finding a non linear completion of the PM theory would
be very interesting
![Page 48: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/48.jpg)
1. A short review on Massive Gravity
1.4. Non linear completions of Partially Massless theory ?
Finding a non linear completion of the PM theory would
be very interesting
Open question with some attempts and
no-go results (de Rham, Hinterbichler, Rosen, Tolley; Deser, Waldron…)
![Page 49: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/49.jpg)
1. A short review on Massive Gravity
1.4. Non linear completions of Partially Massless theory ?
Finding a non linear completion of the PM theory would
be very interesting
Open question with some attempts and
no-go results
A related question: can a PM graviton
exist on a non Einstein space-time ?
Addressed here…
(de Rham, Hinterbichler, Rosen, Tolley; Deser, Waldron…)
![Page 50: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/50.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 51: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/51.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)
(see also C. Mazuet, M. Volkov 2015)
Can be compared with Buchbinder, Gitman,
Krykhtin, Pershin (2000)
![Page 52: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/52.jpg)
Our massive graviton theory is defined by
![Page 53: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/53.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 54: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/54.jpg)
2. The following (linear) field equations
![Page 55: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/55.jpg)
2. The following (linear) field equations
Linearized Einstein operator
![Page 56: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/56.jpg)
2. The following (linear) field equations
![Page 57: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/57.jpg)
We follow the analysis for the case of Einstein
space-times:
DOF counting ?
![Page 58: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/58.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 59: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/59.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 60: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/60.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 61: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/61.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 62: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/62.jpg)
The scalar constraint reads
With
![Page 63: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/63.jpg)
The scalar constraint reads
With
![Page 64: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/64.jpg)
The scalar constraint reads
With
Missing terms in v1
![Page 65: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/65.jpg)
The scalar constraint reads
With
![Page 66: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/66.jpg)
The scalar constraint reads
With
![Page 67: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/67.jpg)
To get a PM theory we need to look for space-times
where and vanish identically.
![Page 68: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/68.jpg)
To get a PM theory we need to look for space-times
where and vanish identically.
Most general solution ?
![Page 69: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/69.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 70: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/70.jpg)
Space-times possessing a covariantly constant
tensor H¹ º are severely restricted…
Non trivial integrability conditions
![Page 71: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/71.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 72: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/72.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 73: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/73.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 74: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/74.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 75: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/75.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 76: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/76.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 77: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/77.jpg)
Explicit solutions (I)
with
dimensionless dimensionful
Analogous
to
![Page 78: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/78.jpg)
For all these solutions one has
There also exist another set of Petrov
type D solutions for which
![Page 79: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/79.jpg)
Explicit solutions (II)
with
No type O solutions
(Einstein static Universe is gone as a
solution in our v2)
![Page 80: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/80.jpg)
Explicit solutions (II)
with
No type O solutions (if one chooses the simplest kind of matrix square root
solving for S¹º)
(Einstein static Universe is gone as a
solution in our v2)
![Page 81: PACIFIC 2018 Massive and Moorea, Sept the 3rd …...PACIFIC 2018 Moorea, Sept the 3rd 2018. Massive and Partially Massless (PM) gravitons on curved space-times 1. A short review on](https://reader033.vdocuments.mx/reader033/viewer/2022060303/5f08b2a37e708231d4234925/html5/thumbnails/81.jpg)
Conclusions
PM exists on non Einstein spacetimes ! (in contrast with previous no-go claim by Deser, Joung, Waldron
In 1208.1307 [hep-th]…)
Solution for the vanishing
Of and
are not known in full generality !