Spontaneous Lorentz-Symmetry Breaking in Gravity Erik Lentz Embry-Riddle Aeronautical University

Lorentz-Symmetry Invariance of reference frames under rotations and boosts Rotations: rotate about one of three axis A boost is giving a reference frame a constant velocity Boosts and rotations of coordinate systems are observer Lorentz transformations

Lorentz-Symmetry Breaking Implies a change in the properties of matter when matter is rotated relative to a fixed background Background (red arrows)

Why Test It? General Relativity and the Standard Model are based on it. The two theories are not unified.

Testing Model: Standard-Model Extension (SME) (Colladay and Kostelecký PRD 97, 98; Kostelecký PRD 04) Gravity Sector: Leading Lorentz-violating couplings

SME experiments testing Lorentz symmetry to date: meson oscillations (BABAR, BELLE, DELPHI, FOCUS, KTeV, OPAL, …) neutrino oscillations (MiniBooNE, LSND, Minos, Super K,… ) muon tests (Hughes, BNL g-2) Yale, … spin-polarized torsion pendulum tests (Adelberger, Hou, …) U. of Washington tests with resonant cavities (Lipa, Mueller, Peters, Schiller, Wolf, …) Stanford, Institut fur Physik, Univ. West. Aust. clock-comparison tests (Hunter, Walsworth, Wolf, …) Harvard-Smithsonian Penning-trap tests (Dehmelt, Gabrielse, …) U. of Washington Lunar laser ranging (Battat, Stubbs, Chandler) Harvard Atom interferometric gravimeters (Chu, Mueller, …) Stanford cosmological birefringence (Carroll, Jackiw, Mewes, Kostelecky) MIT, IU pulsar timing (Altschul) South Carolina synchrotron radiation (Altschul) South Carolina Cosmic Microwave Background (Mewes, Kostelecky) Marquette U., IU

Spontaneous Breaking Two varieties Why study a specific model? Explicit: conflicts with Riemann geometry Spontaneous: Lorentz symmetry holds in the underlying theory but the solutions break it OK with Riemann geometry√ Why study a specific model? To investigate Lorentz-symmetry breaking in strong-field gravity Want to find meaning of these terms Note: This model already being tested in weak-field gravity with lunar laser ranging, atom interferometry, etc.

We consider a two-tensor model Einstein Equation Two-tensor field equation Key feature: The potential V has a minimum for some constant bμν0 V B

Flat spacetime limit (Btz only) Results Flat spacetime limit (Btz only) Functions seem to be independent of their inherent directions. (x2 +y2) Btz = 4λ (-2B2tz+b2)Btz Graph shows oscillations about two minimum values (+ or – b/√2) for dependence on one variable Minkowski metric

Results (cont.) Complications: Dependence on x and y Complications: Differential equations are nonlinear ellipticals which present difficulty in both analytical and numerical analysis

What’s next? Curved spacetime Toy Metric: made to explore curved space-time Future Metrics (generalized black hole solution?) Figure: http://einstein.stanford.edu/ Ultimate goal: understand Lorentz violation in strong-field gravity