Strong Gravity and the BKL Conjecture David Sloan Penn State IGC August 9, 2007

1. 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 What is the BKL Conjecture? Our interpretation Introduction Theory Solutions Closing Remarks • Outline • Introduction • Outline • 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. Outline What is the BKL Conjecture? Our interpretation Introduction Theory Solutions Closing Remarks • 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 1971. 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. Outline What is the BKL Conjecture? Our interpretation Introduction Theory Solutions Closing Remarks • 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 Constraints with Units Introduction Theory Solutions Closing Remarks • 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. Review of Ashtekar Variables Constraints with Units Introduction Theory Solutions Closing Remarks • 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= ∂aEai +GεijkAajEak • Va=Ebi [2∂[aAbi +GAajAbkεijk]. Strong Gravity and the BKL Conjecture David Sloan Penn State IGC August 9, 2007

7. Equations of Motion Solution Metric Adding Matter Introduction Theory Solutions Closing Remarks • 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. Equations of Motion Solution Metric Adding Matter Introduction Theory Solutions Closing Remarks • 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. Equations of Motion Solution Metric Adding Matter Introduction Theory Solutions Closing Remarks • 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 Summary Introduction Theory Solutions Closing Remarks • 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. Lessons Summary Introduction Theory Solutions Closing Remarks • 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