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