viii black holes workshop, lisbon 22-12-2015 gravity’s rainbow
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Gravity’s Rainbow
and the
Tolman-Oppenheimer-Volkoff
Equation
Remo Garattini
Gianluca Mandanici
Università di Bergamo
I.N.F.N. - Sezione di Milano
VIII Black Holes Workshop,
Lisbon 22-12-2015
Plan of the Talk(Work in Progress!!!)
•Motivation and Introduction to Gravity’s Rainbow
•Gravity’s Rainbow and TOV
•Two simple solutions for the isotropic case (Dev-Gleiser energy
density potential)
•Possible origin of the Dev-Gleiser potential
•Conclusions and Outlooks
Ordinary TOV
Lisbon 3VIII Black Holes Workshop
3 2
2 2 2
2 2
4 / 2
1 2 /
4 ' 0
rr rt r
rr
Gr p c Gm rdp pp p
dr c rr Gm r rc
dpdmr r r c p r
dr dr
Why do we need to modify gravity at very high energies???
2
2
3 Misner-Zapolsky
56
cr
Gr
Two exact solutions for the isotropic case
constantr
The Core of a Compact Star
Lisbon 4VIII Black Holes Workshop
Credit: Dany P Page
GUP Quantum CorrectionsP. Wang, H.Yang and X. Zhang
PLB 718 (2012), 265
Asymptotically Safe Gravity
See F.Saueressig
Planck Stars
C.Rovelli and F.Vidotto
IJMPD 23 (2014) 12, 1442026
Examples of Stars with QG Modifications
Gravity’s Rainbow
Gravity’s Rainbow
2 2 2 2 21 2 1 2
/ 0 / 0/ / lim / lim / 1
P P
P P P PE E E E
E g E E p g E E m g E E g E E
Doubly Special Relativity
G. Amelino-Camelia, Int.J.Mod.Phys. D 11, 35 (2002); gr-qc/001205.
G. Amelino-Camelia, Phys.Lett. B 510, 255 (2001); hep-th/0012238.
Curved Space Proposal Gravity’s Rainbow[J. Magueijo and L. Smolin, Class. Quant. Grav. 21, 1725 (2004) arXiv:gr-qc/0305055].
2 2 2 22 2 2 2
2 2 21 2 22
22
exp 2sin
2/ / /1 /
is the redshift function is the mass function
P P PP
r dt dr r rds d d
Gm rg E E g E E g E Eg E E
rc
r m r
Lisbon 5VIII Black Holes Workshop
TOV+ Gravity’s Rainbow
Lisbon 6VIII Black Holes Workshop
3 2 2
2
2 2 2
2
2
2
4 / /
1 2 /
4 does not depend on
/
r Pr r
P
Gr p c g E E Gm rdp p
dr c r Gm r rc
r rdmE r
dr g E E
2 ' 0rr
dpr c p r
dr Invariant
Isotropic
Case
Compact Stars in Gravity’s Rainbow
Lisbon 7VIII Black Holes Workshop
r 2 / 1Pg E E
2 / 1Pg E E
2 / 1Pg E E Case A
Case B
Pl
610 to fix ideas!!!
TOV+ Gravity’s Rainbow
Lisbon 8VIII Black Holes Workshop
Simple case constantr
3 2 2
2
2 2 2
2
2
2
4 / /
1 2 /
4
/
r Pr r
P
Gr p c g E E Gm rdp p
dr c r Gm r rc
r rdm
dr g E E
Isotropic
Case
2 ' 0rr
dpr c p r
dr Invariant
Solving TOV+ Gravity’s Rainbow
Lisbon 9VIII Black Holes Workshop
Simple case constantr
P(R)=0
mass density in ordinary GRr
8 G
Solving TOV+ Gravity’s Rainbow
Lisbon 10VIII Black Holes Workshop
rp r
2
2 / P rg E E p r pressure in ordinary GRrp r
Or in terms of M and R
rp r
2
2 / P rg E E p r
Solving TOV+ Gravity’s Rainbow
Lisbon 11VIII Black Holes Workshop
is large but finitecp
The central pressure becomes
2 / 1Pg E E
PR l
Solving TOV+ Gravity’s Rainbow
Lisbon 12VIII Black Holes Workshop
Surface Redshift
1
0
/1 0
P
P E
dg E E
E dE
TOV+ Gravity’s Rainbow
Variable Energy
Density Case
EoS
2
2
3 Misner-Zapolsky
56
cr
Gr
21 g / 1
1/ 3 =3
r
P
p r r
c E E
TOV
2 / Pg E E
22
2 2
2
31
4 7 / 3P
c
c g E E
Lisbon 13VIII Black Holes Workshop
3 2 2 2
2
2 22 22
4 / / 4
/1 2 /
r Pr r
P
Gr p c g E E Gm r r rdp p dm
dr c dr g E Er Gm r rc
TOV+ Gravity’s Rainbow
Dark Energy Star???
Dust???
22
2
2
2
2
2
28/
3
/ 1
3
28 /
P
P
P
cg E E
g E E
c
g E E
Infinite Mass??Vanishing
Mass??
22
2 2
2
31
4 7 / 3P
c
c g E E
2 2 4 2
2 2 2
14 2/ 3 49 / 63 / 18
3 3P P Pc g E E g E E g E E
2
2
/ 1
3 2 2
Pg E E
c
Lisbon 14VIII Black Holes Workshop
Solve this infinite dimensional PDE with a Variational
Approach without matter fields contribution
is a trial wave functional of the gaussian type
Schrödinger Picture
Spectrum of L depending on the metric
Energy (Density) Levels
Possible Origin of the Dev-Gleiser Potential using
the Wheeler-De Witt Equation
3
1min
h
T
ij j
D h h d x h
V D h h h
D h D h D D h J
L
L
Induced
Cosmological
‘‘Constant’’
Lisbon
For a Minkowski or a de Sitter background
in a low energy limit 2
BA
Gr
L Dev-Gleiser
Energy
Density1 Loop
graviton contribution
3
ˆ 22
ij kl
ijkl
gG R
V d x g
L
15VIII Black Holes Workshop
Conclusions and Perspectives
Gravity’s Rainbow modifies TOV. Ordinary Compact Stars can turn to Dark Stars or
Dust??? Planck Stars??? Deeper Investigation!!
Investigation on the constant energy density term and on the complete Dev-Gleiser Energy density profile.
Zero Point Energy Calculations in Gravity’s Rainbow induce an Energy density profile of the Dev-Gleiser form.
Including corrections to the Dev-Gleiser Energy density profile.
Anisotropy and Polytropic relations must be included.
Lisbon 16VIII Black Holes Workshop
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