1. Strong Gravity and the BKL Conjecture
Introduction
Theory
Solutions
Closing Remarks
Strong Gravity and the BKL Conjecture
David Sloan, Penn State IGC
Work in collaboration with Abhay Ashtekar and Adam Henderson
Strong Gravity and the BKL Conjecture
David Sloan
Penn State IGC
August 9, 2007

2. Outline Introduction What is the BKL Conjecture? Our Interpretation
Theory
Solutions
Closing Remarks
Outline
What is the BKL Conjecture?
Our interpretation
Outline
Introduction
What is the BKL Conjecture?
Our Interpretation
Theory
Review of Ashtekar Variables
Constraints with Units
Solutions
Equations of Motion
Solution Metric
Adding Matter
Closing Remarks
Lessons
Summary
Strong Gravity and the BKL Conjecture
David Sloan
Penn State IGC
August 9, 2007

3. What is the BKL Conjecture?
Introduction
Theory
Solutions
Closing Remarks
Outline
What is the BKL Conjecture?
Our interpretation
What is the BKL Conjecture?
The BKL Conjecture (Belinsky, Khalatnikov, Lifshitz) is often thought of as the “Last Great Problem of Classical Relativity” - proposed in Recently become popular due to numerical work (Garfinkle, Uggla, Elst, Ellis, Wainwright, Curtis, Moncrief, Berger...)
Loosely speaking: In the Einstein Field equations some terms are dominant close to a singularity so we can attempt a solution ignoring the sub-dominant terms. BKL believe that the time derivative terms are dominant over spatial derivatives. This clearly has some problems:
Non Geometric
Only Numerical Evidence
Strong Gravity and the BKL Conjecture
David Sloan
Penn State IGC
August 9, 2007

4. Introduction
Theory
Solutions
Closing Remarks
Outline
What is the BKL Conjecture?
Our interpretation
Our Interpretation
We will take the BKL conjecture in a very simple form. As a singularity is approached:
The dynamics become local. Also known as “Asymptotically Velocity Dominated”.
∂a → 0
Solutions become vacuum dominated.
Tij → 0
The first of these conditions can in fact be somewhat relaxed, instead taking the limit:
Ea∂a → 0
This is much more appropriate for the BKL conjecture itself. But for the purposes of this talk the first limit is simpler. Likewise we will see that we can also relax the vacuum domination by adding some types of matter.
Strong Gravity and the BKL Conjecture
David Sloan
Penn State IGC
August 9, 2007

5. Review of Ashtekar Variables
Introduction
Theory
Solutions
Closing Remarks
Review of Ashtekar Variables
Constraints with Units
Review of Ashtekar Variables
We work in the first order formalism using a triad, E and a self dual connection, A. We will denote a positive density weight with bold letters and negative with italics. To relate to the usual second order formalism:
EaiEbi=qqab
Aai = (Γai-iKai)/GN
In a “lesson” from quantum mechanics, we will take the poisson bracket (equivalently symplectic structure or momentum conjugation) to be “fundamental” at least in terms of units:
{E,A}=ihδ3
Taking [E]=1 tells us that [A]= M/L2.
Strong Gravity and the BKL Conjecture
David Sloan
Penn State IGC
August 9, 2007

6. Constraints with Units
Introduction
Theory
Solutions
Closing Remarks
Review of Ashtekar Variables
Constraints with Units
Constraints with Units
The Scalar constraint is therefore:
C = N [Ea[iEbj](∂aAbkεijk+GAaiAbj)+8πρq]
Hence it is clear to see that at the level of constraints, the BKL conjecture (red → 0) and strong gravity (green → ∞) we have the same theory. This is mirrored in analysis done at the level of the equations of motion.
Likewise the Gauss/Vector constraints:
Gi = ∂a Eai +GεijkAajEak
Va=Ebi [2∂[aAbi +GAajAbkεijk].
Strong Gravity and the BKL Conjecture
David Sloan
Penn State IGC
August 9, 2007

7. Introduction
Theory
Solutions
Closing Remarks
Equations of Motion
Solution Metric
Adding Matter
Equations of Motion
We find the equations of motion for E and A from their poisson brackets with the scalar constraint:
{C,Eai} = iN Ea[iEbj]Abj
{C,Aia} = -iN EbjA[aiAb]j
From here (or from inspection of the constraints) we find a constant of the motion:
Æij = AaiEaj – AakEak δij
And this allows us to re-write the equation for E as:
{C,Eai} = iN ÆjiEaj.
Strong Gravity and the BKL Conjecture
David Sloan
Penn State IGC
August 9, 2007

8. Introduction
Theory
Solutions
Closing Remarks
Equations of Motion
Solution Metric
Adding Matter
Solution Metric
Solving the equations of motion for E (in a specific coordinate basis) we find that our space-time metric takes the form of a Bianchi I mode at every space-time point:
ds2 = -dt2 + t2Pxdx2 +t2Pydy2 +t2Pzdz2
Where Px, Py, Pz are functions of x,y,z subject to:
Px2 +Py2 +Pz2 = 1 = Px+Py+Pz
This is particularly exciting as Bojowald, and more recently Chiou and Vandersloot showed that Bianchi I spacetimes are singularity free, and in fact exhibit a bounce in LQC.
Strong Gravity and the BKL Conjecture
David Sloan
Penn State IGC
August 9, 2007

9. Introduction
Theory
Solutions
Closing Remarks
Equations of Motion
Solution Metric
Adding Matter
Adding Matter
We can add to our system matter in the form of dust (perfect fluid):
Tab = ρtatb
In the strong gravity limit, this of course is zero. In the BKL case, if we relax the restriction Tij = 0 we find that Æ is no longer a constant. However, through use of the continuity equation we find that its time derivative is, and solving for the metric we find corrections to our vacuum case of the form:
ds2 = -dt2 + t2Px(1+t)αx dx2 +t2Py(1+t)αy dy2 +t2Pz(1+t)αz dz2
for some positive constant functions αx,αy,αz. Hence introducing dust does not alter the behaviour close to the singularity.
Strong Gravity and the BKL Conjecture
David Sloan
Penn State IGC
August 9, 2007

10. Lessons This analysis does raise a few interesting issues:
Introduction
Theory
Solutions
Closing Remarks
Lessons
Summary
Lessons
This analysis does raise a few interesting issues:
The Hamiltonian formulation “knows more” than the geometric formulation
Poisson brackets are our route to quantizing
Access to more physics
Units can be important
In taking limits, what remains fixed is as important as the limit
Different parts of our constraints have different units
Is “strong gravity” more than just a linguistic feat?
Is strong gravity valid near a singularity?
Could this provide hints for proving the BKL conjecture itself?
Strong Gravity and the BKL Conjecture
David Sloan
Penn State IGC
August 9, 2007

11. Summary This work can be briefly summarized:
Introduction
Theory
Solutions
Closing Remarks
Lessons
Summary
Summary
This work can be briefly summarized:
The BKL conjecture is incredibly similar to strong gravity
The solutions to both are Bianchi I cosmologies
Strong Gravity and the BKL Conjecture
David Sloan
Penn State IGC
August 9, 2007