Quentin Bailey Department of Physics and Astronomy Embry-Riddle Aeronautical University, Prescott, AZ

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1 Quentin Bailey Department of Physics and Astronomy Embry-Riddle Aeronautical University, Prescott, AZ Third IUCSS Summer School and Workshop on the Lorentz- and CPT-violating Standard-Model Extension, Bloomington, Indiana, June 15-June 23

2 Outline 1) References 2) General framework 3) Symmetries and transformations of fields 4) Explicit versus spontaneous symmetry breaking 5) minimal SME gravity sector 6) Process of finding linearized field equations 7) Alternate routes 8) Brief description of nonminimal SME 9) Match to models of Lorentz violation in gravity 10) ongoing work... SME gravity phenomenology/experiment: Thursday morning talk (2 pm)

3 Acknowledgements SME school organizers Ralf Lehnert, Alan Kostelecký, helpers thank you! Support from: Collaborators/colleagues: Brett Altschul, Robert Bluhm, Alan Kostelecký, Matthew Mewes, Mike Seifert, Jay Tasson,, Embry-Riddle: Kellie Ault, Andri Gretarsson, Michele Zanolin,

4 Key references Gravitational Phenomenology in Higher-Dimensional Theories and Strings, Alan Kostelecký and Stuart Samuel, Phys. Rev. D 40, 1886 (1989). Gravity, Lorentz Violation, and the Standard Model, Alan Kostelecký, Phys. Rev. D 69, (2004). Signals for Lorentz Violation in Post-Newtonian Gravity, Q.G. Bailey and AK, Phys. Rev. D 74, (2006). Lorentz violation and gravity, PhD Thesis 2007, freely available through ProQuest. Short-range gravity and Lorentz violation, Q.G. Bailey, V.A. Kostelecky and Rui Xu, Phys. Rev. D 91, (2015). Testing Local Lorentz Invariance with Gravitational Waves, Alan Kostelecký and Matthew Mewes, Phys. Lett. B 757, 510 (2016)...and PLB 2018 Vector models of gravitational Lorentz-symmetry breaking, M.D. Seifert, Phys. Rev. D 79, (2009)....lots of papers on Lorentz violation in gravity... Review articles: A. Hees et al, Universe 2, 30 (2016), Yunes et al, PRD 2016 (GW), J. Tasson, Rept. Prog. Phys. 77, (2014),...

5 General Framework Review : field equations of General Relativity Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve -from Gravitation by Misner, Thorne, and Wheeler

6 General Framework Geometrical framework of General Relativity: pseudo-riemann spacetime (no Torsion) Manifold picture Spacetime described by metric, curvature Curved spacetime manifold Local Picture Metric Veirbein Spin connection Local Lorentz frame

7 Symmetries: Symmetries and Transformations of fields Local Lorentz symmetry (~rotations, 6) B 0a = a bb b! Diffeomorphism symmetry (P->Q, ~translations, 4) B! B (Also discrete C, P, T) Conservation laws associated with these symmetries T µ = T µ D µ T µ = 0 è Fun fact: GR can be formulated as a local gauge theory of gravity with gauge symmetries LLS and Diffeomorphism symmetry Weinberg, Kibble, Deser + others, 60s, 70s papers, see review by Hehl et al, Rev Mod Phys 48, 393 (1976)

8 Symmetry Breaking Local Lorentz frame Manifold b e a b a es x induce Result 1: Lorentz breaking diffeomorphism breaking Coefficients control Lorentz and diffeomorphism breaking Flat spacetime assumption b 0. In curved spacetime, integrability only satisfied for special geometries D b a b a b a b b 0 D b a 0 urn implies (Kostelecký PRD 04)

9 Symmetry breaking Explicit Lorentz breaking prescribed, nondynamical coefficients Produces modified conservation laws D T e J x D k x 0 T e T e e a e b k x X ab x y J y D µ G µ = 0 Conflicts with geometric Bianchi identities Result 2: Explicit Lorentz/diffeo breaking is generically incompatible with Riemann geometry* However Result 3: Spontaneous symmetry breaking saves geometry! *Can accomplish with solutions in some models (Kostelecký PRD 04, Bluhm & Kostelecký PRD 05, 08, Bluhm PRD 15)

10 Spontaneous Lorentz-symmetry breaking Tensor fields acquire vacuum expectation values e.g., vector field Potential Expand about minimum vev Fluctuations, includes Nambu- Goldstone and massive modes Key feature: Lorentz violation is dynamical Conservation laws are unaffected Bianchi identities are preserved Figure: Ralf Lehnert Other approaches Riemann-Finsler geometry Explicit breaking of diffeo symmetry (e.g., massive gravity )

11 Gravity and the SME Basic lagrangian expansion: Pure-gravity sector matter sector Dynamics for coefficients Coefficients can be treated dynamically assuming spontaneous symmetry breaking (avoids conflicts with geometric framework) Vacuum expectation value (~ similar to Higgs but not scalars) Matter gravity couplings see Jay Tasson s lecture

12 Pure-gravity sector of the SME Lagrangian: L = p g 2apple R +(k (4) ) R +(k (5) ) apple D apple R (k(6) 1 ) 126 apple {D apple,d }R +(k (6) 2 ) 210 apple µ R R apple µ + L 0 k terms=lorentz-violating coefficients Contains ordinary matter, dynamics for coefficients Key differences with flat spacetime SME Coefficients are not covariantly constant (geometry) Coefficients are dynamical if we assume spontaneous breaking Coefficient dynamics unknown (extra degrees of freedom/modes not seen in nature?) (Kostelecký PRD 04, Bailey & Kostelecký PRD 06 & Bailey et al PRD 15, Kostelecký & Mewes PLB 16)

13 Process of finding field equations Focus on mass dimension 4 coefficients (20) Alternate expression (u, s, t) Note: field refinitions, see Y. Bonder PRD 15 dynamics for coefficients Guidelines Linearized gravity limit Coefficients have dynamical terms Spontaneous symmetry breaking Action, field equations invariant/covariant Conservation laws hold Simplicity: minimize influence extra modes not seen in nature

14 Process of finding field equations Metric field equations (Mathematica xtensor output sample) Conservation law (with Bianchi identities ) (Kostelecký PRD 04, Bailey & Kostelecký PRD 06)

15 suggested exercise... Variation of action to find metric field equations G µ = apple(t g ) µ g µ k R +3k (µ R ) +2D D k µ( ).

16 Process of finding field equations Expand around a flat background: linearized limit: field equations are O(ε), Lagrangian is O(ε 2 ) Barred coefficient are the vacuum expectation values -assume constant µ k = 0

17 µ g µ µ (µ µ( ) ) (GL2)µ = (Tg )µ A.+ Stress-Energy (Tk )µ, and Decoupling 1 finding field R Process + 3kof R )dynamical + 2D equations D kµ( ).k 1 (µ µ 2 gµ k all + contributes from matter and the coefficients. k (R ) + 2k (R ) (T ) = k µ method L mass dimension L (µ The decoupling illustrated with the 4 coefficients (k R ) )µ = + 2 isµ 1 equations are (R ) field equations are ) = k + 2k (R ) + k R k µ L L ) equations (µ Full (µ ) 2 µ matter and the dynamical coefficients k. k µ(1 ) 2k 2h 4k (µ (µ ) G = (T ) + g k R + 3k R + 2D D kµ( ). (Tk )µ = 12 µ k (R(G ) + 2k (R ) + k R µ g µ µ L ) L )), (µ )(µ ) 2 ) (µ + (T = (T µ (µ ) g2h µ k h 4k. equations k µ( ) L 2k Linear the dynamical coefficients HereTg includes all contributes from matter and k. (µ ) (µ µ( ) ) The linearized field equations are 2h k 2k vation (G law )+2@ is (T L µ g )µ + (Tk )µ, (Tk )µ = 1 µ 2 µ k µ (Tµgwhere )µ = h. (GL )µ = (Tg )µ + (Tk )µ, (R ) = 2k µ (T )µ )µ L ) ) µµ(tk )µ ) + k (µ ) (T) k))µ 2h k µ (2k 1 (µ ) (µ+ (T = 2h g µ 1 (µ ) ) +µ2k (T kl(µ ) R (µ(rl ) ) + k(µ ) R k +(R ) + 2k )µ L = ) ) (µk(r 2 µ 2 µ k µ (R µgl µ µ (µ ) (µ Conservation linear (Tg )µ law = the (2k 2h h4k ) ) ) (µ ) (µ µ µ 2h k 2k (µ h ). µ( ) law is = R k µ( h 2k (µ )µ (µ µ µ) µ ). g(t )µ 4kµ Theµ is gconservation R µ = (T ) (T ) = (Tg )µ = n to this equation µ is 2h h (Tk µ (Tg )µ µ (Tk (2k 2h + ) to equation (non-unique) µ µ g )µ =µ( (µ ) (µ (Tg )µ (2k(µ ) 2h @ )h ) ) nheis unknown Solutionterm k µ µ. First record the linearized dieomor (Tg )µ k = (Tµk )µ µ R µ (T )µ = gµ (T ) = R (T ) = 4k R + µ, g µ µ (T 4k + +, µ g )µ = (µ ) R g µ (TM )µ (µ ) µ (Tg )µ (2k 2h + h ) ) The (µ ) solution to this equation for! thek unknown term k kµ( k veµ krecord k dieom, µ µ First linearized µ = 0. (T ) = 4k R +, =satisfies 0. g µ µ (µ ) (Tg )µ = 4kµ R fluctuations k µ : detofluctuations field where unknown µ in =the 0. field µ are µ term solveequations for the k µ( ). First record the lin that appear term k µ(. solve First record the dieomorphism We wish for the unknown termlinearized k µ( ). First record the linearized transf dieomor ) to

18 suggested exercise 2... Find linearized equations G µ = apple(t g ) µ g µ k R +3k (µ R ) +2D D k µ( ). (G L ) µ = apple(t g ) µ + apple(t k ) µ apple(t k ) µ = 1 2 µ k (R L ) +2k (µ (R L ) ) + k (µ ) R k µ( ) 2k (µ ) 2h h ).

19 Field equations so far Process of finding field equations (G L ) µ =(T M ) µ µ k (R L ) +2k (µ (R L ) ) + k (µ ) R k µ( ) 2k (µ ) 2h h ) 4k (µ ) R + apple µ Remaining unknown kµ =? µ =? Use Diffeomorphisms (derived from the underlying tensor component transformation rules Spontaneous breaking so STILL have underlying particle symmetry!) For the tilde terms

20 Expansion for the fluctuations Use flat metric, coefficients, partials, and metric fluctuations ( decoupling ) Express only the term appearing in the field equations (with derivatives) Process of finding field kµ =~ explicitly, the h match to convention Impose tensor symmetries and diffeomorphism symmetry Motivation: Knowledge of field equations allow solution in terms of metric Simplicity: avoiding extra modes affecting gravity Kostelecký Cutlass: One crazy idea at a time

21 Terms of the form: Process of finding field equations explicitly, the h match to convention Impose Obtain...

22 Pull results together, obtain effective linearized equations (G L ) µ =(T M ) µ Scaling 1: (3+2a 1 ) 2 +(3 + 2a 1 ) 1 2 µ k (R L ) +2k (µ (R L ) ) + k (µ ) R +apple µ. Key simplifying assumption: Decomposition: Scaling 2: 2s µ! s µ Process of finding field equations 0. to S k = t +2 [ [ s ] ] [ [ ] ] u, (G L ) µ = apple(t M ) µ s G (µ ) + u(g L ) µ Double-dual of curvature (Bailey & Kostelecký PRD 06 & Bailey PRD 10)

23 Process of finding field equations The t-puzzle : the t term (10 coefficients) vanishes due to tensor identity [...]=total antisymmetrization ure of the coefficie t 0. e coefficients of In an n-dimensional space any tensor expression T a1 a 2...a k T [a1 a 2...a k ] =0. (Edgar and Hogland, J Math Phys 2002) with k>nindices satisfies a tensor identity Possible solutions: redefinitions, role of cosmology (see papers by Y Bonder, ) Generalize dynamical terms kµ µ = µ t R Bailey, unpublished

24 suggested exercise 3... Follow same steps with s,t,u decomposition (G L ) µ = apple(t M ) µ s G (µ ) + u(g L ) µ

25 Alternate Route 1 Same equations for minimal SME gravity obtainable directly from quadratic action L K (d) = 1 4 h µν ˆK (d)µνρσ h ρσ, where ˆK (d)µνρσ = K (d)µνρσε 1ε 2...ε d 2 ε1 ε2... εd 2 The sbar term is a subset of the diffeomorphism invariant* general quadratic action: L (4) = 1 4apple sµapple h G µ apple *linearized (Bailey et al, PRD 15, Kostelecký and Mewes, PLB 16, 18)

26 Alternate Route 2 The effective equations produce a natural match to the simplest term in the photon sector V Gm 1m j~x 1 ~x 2 j 2 ^x ^j ^x ^k s ^j ^k q 1 x DE x 4 x 2x 2 j j x The simplest field equations producing this term, and consistent with spontaneous breaking is (G L ) µ = apple(t M ) µ s G (µ ) + u(g L ) µ (Kostelecký and Mewes, PRD 02, Bailey PRD 10)

27 Approach 1 L = p g 2apple Brief description on nonminimal SME R +(k (4) ) R +(k (5) ) apple D apple R (k(6) 1 ) apple {D apple,d }R +(k (6) 2 ) apple µ R R apple µ + L 0 Similar results linearized field equations obtained Approach 2 L K (d) = 1 4 h µν ˆK (d)µνρσ h ρσ, where ˆK (d)µνρσ = K (d)µνρσε 1ε 2...ε d 2 ε1 ε2... εd 2 See Matt Mewes talk, Josh Long ( short-range experiments) (Bailey et al PRD 15, Kostelecký and Mewes, PLB 16, 18, Bailey PRD 16)

28 Match to models of Lorentz violation Model Link to SME Lorentz- viola3ng fields General Test Framework PPN Yes None (α 1, α 2, w j ) Yes, metric SME, gravity sector Yes tensors, flavor dependent Yes, EFT Bumblebee Yes vector No Einstein- Aether ParMal vector No Horava gravity? vector No ATT model Yes AnM- symm. two- tensor No Cardinal Yes symm. two tensor No Massive gravity Yes* two- tensor? CS gravity? scalar No GW modified dispersion Yes None (parameters) Yes, Disp. Rel. NonComm gravity? Theta^ab No many pubs: C. Will 70s, Kostelecky & Samuel 89, Jacobson & Ma`ngly 01, Jackiw & Pi 03, Carroll & Lim 04, Bluhm et al 08, Yunes 09, Seifert 09, Altschul et al 10, Kostelecky Po`ng 05, Will & Yunes 12, Ciric et al PRD 2016, R Casana PRD 18,...

29 Match to PPN framework Decompose into trace and traceless pieces:,, Post Newtonian metric: Match to Parametrized Post-Newtonian formalism? Metric based framework (Will & Nordtvedt, ApJ 70s) Only in isotropic limit

30 Match to models of Lorentz violation Toy model: antisymmetric tensor model (ATT) Dynamical antisymmetric two tensor field Kinetic term Potential term Vacuum expectation value: SME-type nonminimal couplings Interesting features: -Renormalizable/stable for special potentials in flat spacetime -Generate Lorentz-violating background + scalar modes ( phon modes) (Altschul, Bailey, Kostelecký PRD 10)

31 Match to models of Lorentz violation Key feature of antisymmetric tensor model: Anisotropic Lorentz violation Effective s-type term: Isotropic constraints: Post-newtonian effects lie outside standard PPN analysis, but within SME analysis Other work on ATT models: Seifert PRL, PRD 10, PRD 17, C Hernaski PRD 16

32 Ongoing work Extend SME gravity to nonlinear terms...ongoing g µ = µ + h µ O(ε 3 ) in action, O(ε 2 ) in EOM Some results for mass dimension 8 published...

33 Cubic Curvature Coupling Most SME analysis (so far): linearized gravity g µ = µ + h µ 2 Gravity (e.g., EH action) has nonlinear interactions Nonminimal SME, (subset of) the mass dimension 8 coefficients involves cubic powers of curvature Lagrangian: p g LV = 2apple (k(8) ) apple µ R apple µ R R L (8) Field equations: G µ = apple(t M ) µ + 6(k (8) ) R (a) R (b) Elusive: lowest order is quadratic in h -> no effect on propagation no time-averaged effect on gravitational radiation from binary system Q.G. Bailey, PRD 16

34 Cubic Curvature Coupling Post-Newtonian metric g 00 = 1+2U +2 2U 2 +2 g 0j = 1 2 (7V j W j ), g jk = jk (1 + 2U), New potential at PNO(4) Z = 48(k (8) e ) jklmnp Z d 3 r jk [@0 lm U(r0 )@ npu(r 0 0 )] 4 r r 0 Depends on subset with 56 combinations of k (8) coefficients e.g., two-body acceleration correction a j 432 (GM)2 K kl n k n l n j 2 R 3 r 4 5 K jkn k Gravitational WEP or SEP broken Best tests from binary pulsars & short-range gravity Countershading, k~1 km 4 Q.G. Bailey, PRD 16

35 Summary Lorentz violation in gravity sector contains couplings to curvature L = p g 2apple R +(k (4) ) R +(k (5) ) apple D apple R (k(6) 1 ) apple {D apple,d }R +(k (6) 2 ) apple µ R R apple µ + L 0 Challenging process to find linearized equations for minimal, nmsme Match exists to some gravity models of LV Many Observations/experiments can probe Thursday talk More gravity talks: Seifert, Mewes, Tasson, Long, Shao Questions?