Spherical collapse in f(R) gravity

Jun-Qi GuoSimon Fraser University

with Andrei V. Frolov (SFU)Daoyan Wang(U. of British Columbia)

27th Texas Symposium on Relativistic Astrophysics, Dallas, Dec. 8--13, 2013

What have been done:1) Static black hole solution in f(R) gravity.2) Quasi-static collapse: simulation of halo formation. 3) Dynamical collapse in f(R) gravity: FRW dust collapse.4) Usually, the potential is set to zero.

This talk is on spherical collapse:1) fully dynamical. 2) general: source fields have non-constant profiles.3) the potential is not zero.

What have been done:1) Static black hole solution in f(R) gravity.2) Quasi-static collapse: simulation of halo formation. 3) Dynamical collapse in f(R) gravity: FRW dust collapse.4) Usually, the potential is set to zero.

This talk is on spherical collapse:1) fully dynamical. 2) general: source fields have non-constant profiles.3) the potential is not zero.

Dynamics of the metric components and scalar fields

1) during the whole collapse process

2) near the singularity

Outline

Introduction

Motivation 1: Black hole physics: interface between gravity and cosmology

Motivation 1: Black hole physics: interface between gravity and cosmology

BH solution: static vs. collapse

Static solution:“No-hair” theorem (Ruffini et al.,1971)Hawking theorem (Hawking,1972)A novel “no-hair” theorem (Bekenstein,1995)

A scalar will be constant if1) the scalar is minimally coupled to gravity,2) the scalar has a non-negative energy density,3) the global minimum of the potential is zero.

GR: Oppenheimer-Snyder solution (1939)

Brans-Dicke theory: Shibata et al. (1994) ,Scheel et al. (1995).

Numerical confirmation of the novel “no-hair” theorem: T. Hertog, (2006).

Collapse:

GR: Oppenheimer-Snyder solution (1939)

Brans-Dicke theory: Shibata et al. (1994),Scheel et al. (1995).

Numerical confirmation of the novel “no-hair” theorem: T. Hertog, (2006).

Collapse in scalar-tensor theory or f(R) theory?

Collapse:

Motivation 2: Dynamics near the singularity

BKL(Belinsky-Khalatnikov-Lifshitz) conjecture:

Near the singularity, in the equations of motion,1) compared to the time derivative terms, the spatial

derivative terms can be negligible. 2) compared to the gravity, the contribution from the

matter fields can be neglected. 3) metric components and scalar fields are expressed

by Kasner solutions (Kasner 1921)

Motivation 2: Dynamics near the singularity

BKL(Belinsky-Khalatnikov-Lifshitz) conjecture:

Near the singularity, in the equations of motion,1) compared to the time derivative terms, the spatial

derivative terms can be negligible. 2) compared to the gravity, the contribution from the

matter fields can be neglected. 3) metric components and scalar fields are expressed

by Kasner solutions (Kasner 1921)

Motivation 2: Dynamics near the singularity

BKL(Belinsky-Khalatnikov-Lifshitz) conjecture:

Near the singularity, in the equations of motion,1) compared to the time derivative terms, the spatial

derivative terms can be negligible. 2) compared to gravity, the contribution from the matter

fields can be neglected. 3) metric components and scalar fields are expressed

by Kasner solutions (Kasner 1921)

Motivation 2: Dynamics near the singularity

BKL(Belinsky-Khalatnikov-Lifshitz) conjecture:

Near the singularity, in the equations of motion,1) compared to the time derivative terms, the spatial

derivative terms can be negligible. 2) compared to gravity, the contribution from the matter

fields can be neglected. 3) metric components and scalar fields are expressed

by Kasner solutions (Kasner 1921)

1. Cosmology (Berger and Garfinkle, 1998)

Vacuum Gowdy universe

2. Black hole in GR (Garfinkle, 2010)

A test massive scalar field in FRW spacetime

3. Loop quantum gravity (Ashtekar et al. 2011)

Simulations of dynamics near singularitiesReviews, see Berger, arXiv:gr-qc/0201056

Rendall, arXiv:gr-qc/0505133

The f(R) theory is a unique scalar-tensor theory

The potential depends on f’ and R.

Spherical collapse in f(R) theory

The f(R) theory is a unique scalar-tensor theory

The potential depends on f’ and R.

Spherical collapse in f(R) theory

Couplings between:

metric components; source fields; and potential term.

Framework

Source of instability!Jordan frame:

Jordan frame Einstein frame

Einstein frame:

Source of instability!Jordan frame:

Jordan frame Einstein frame

Source fields:

Einstein frame:

1)

2)

Source of instability!Jordan frame:

Jordan frame Einstein frame

Kruskal coordinate:

Coordinate

Advantages: 1) conformally flat

2) horizon-penetrating

3) global spacetime structure

Kruskal coordinate:

Coordinate

Advantages: 1) conformally flat

2) horizon-penetrating

3) global spacetime structure

Double-null coordinate: (Christodoulou, 1993)

f(R) model: (Hu and Sawicki, 2007)

f(R) model: (Hu and Sawicki, 2007)

Equations:For the metric components: Einstein field equations

For the scalar fields: Lagrange equations

f(R) model: (Hu and Sawicki, 2007)

Equations:For the metric components: Einstein field equations

For the scalar fields: Lagrange equations

Numerical tests:1. Residual of constraint equation

2. Residuals of E.o.M.

3. Convergence test: first-order convergent

Results

1: A black hole can be formed

r=0

Horizon

Kruskal coordinate1: A black hole can be formed

r=0

Horizon

Kruskal coordinate

Initial slice

Double-null coordinate1: A black hole can be formed

Initial value

Evolutions at consecutive time slices

Evolutions at consecutive time slices

Evolutions at consecutive time slices

Evolutions at consecutive time slices

Evolutions at consecutive time slices

Initial value

Final value

Evolutions at consecutive time slices

Initial value

Final value

Evolution at consecutive time slices

Initial value

Final value

Evolution at consecutive time slices

At the horizon:

2. Evolutions of metric components and scalar fields

Initial value

Final value Initial value

Final value

Initial value

Final value

In the beginning of the collapse, the curvature scalar R is high, GR is restored.

However, in the late stage, R becomes very low. Consequently, although gravity

is very strong near/inside the black hole,

1) the modification term in f(R) function is important,

2) f’ is light,

3) dynamical solution is very different from de Sitter-Schwarzschild solution.

Initial value

Final value

Sample slice we study:

3. Dynamics near the singularity

Metric components:

Scalar fields:

Variables we study:

1)

2)

2)

3)

2)

3)

4)

Definition of :

Kasner solutions!

Kasner solutions!

Kasner solutions!

Future plan

Potential in Einstein frame

Future plan

• Collapse in Jordan frame

• Final state of collapse

Potential in Einstein frame

Summary

• Simulations of fully dynamical collapse in f(R) theory are obtained.

• A black hole can be formed in f(R) collapse.

• During the collapse process, near /inside the black hole , although gravity is very strong, the curvature R is low. As a result, 1) the modification term is important.2) f’ is decoupled from matter density, and becomes light. 3) the dynamics black hole solution is very different from de Sitter-Schwarzschild solution. Evolution of f’. f’ 0 near the singularity

• Numerical results strongly support the BKL conjecture.

Summary

• Simulations of fully dynamical collapse in f(R) theory are obtained.

• A black hole can be formed in f(R) collapse.

• During the collapse process, near /inside the black hole , although gravity is very strong, the curvature R is low. As a result, 1) the modification term is important.2) f’ is decoupled from matter density, and becomes light. 3) the dynamics black hole solution is very different from de Sitter-Schwarzschild solution. Evolution of f’: f’ 0 near the singularity.

• Numerical results strongly support the BKL conjecture.

Summary

• Simulations of fully dynamical collapse in f(R) theory are obtained.

• A black hole can be formed in f(R) collapse.

• During the collapse process, near /inside the black hole , although gravity is very strong, the curvature R is low. As a result, 1) the modification term is important.2) f’ is decoupled from matter density, and becomes light. 3) the dynamics black hole solution is very different from de Sitter-Schwarzschild solution. Evolution of f’: f’ 0 near the singularity.

• Numerical results strongly support the BKL conjecture.

Summary

• Simulations of fully dynamical collapse in f(R) theory are obtained.

• A black hole can be formed in f(R) collapse.

• During the collapse process, near /inside the black hole , although gravity is very strong, the curvature R is low. As a result, 1) the modification term is important.2) f’ is decoupled from matter density, and becomes light. 3) the dynamics black hole solution is very different from de Sitter-Schwarzschild solution. Evolution of f’: f’ 0 near the singularity.

• Numerical results strongly support the BKL conjecture.

Summary

• Simulations of fully dynamical collapse in f(R) theory are obtained.

• A black hole can be formed in f(R) collapse.

• During the collapse process, near /inside the black hole , although gravity is very strong, the curvature R is low. As a result, 1) the modification term is important.2) f’ is decoupled from matter density, and becomes light. 3) the dynamics black hole solution is very different from de Sitter-Schwarzschild solution. Evolution of f’: f’ 0 near the singularity.

• Numerical results strongly support the BKL conjecture.