1. Tina A. Harriott Mount Saint Vincent University Nova Scotia, Canada J. G. Williams Brandon University Manitoba, Canada & The Winnipeg Institute For Theoretical Physics Non-singular solution for the null-surface formulation of general relativity in 2+1 dimensions CCGRRA XV: Winnipeg, Manitoba. May 2014.

2. Introduction CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 2 Null-surface formulation of general relativity (NSF). Solve the NSF field equations of the (2+1)-dimensional model and present an exact solution. Discussion of the solution properties.

3. Null Surface Formulation CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 3 An alternative formulation of general relativity. Introduced by Frittelli, Kozameh and Newman in 1995. (1-3) The central role is played by families of null surfaces rather than the metric tensor. (1) Frittelli, S., Kozameh, C.N., Newman, E.T.: J. Math. Phys. 36, 4975 (1995) (2) Frittelli, S., Kozameh, C.N., Newman, E.T.: J. Math. Phys. 36, 4984 (1995) (3) Frittelli, S., Kozameh, C.N., Newman, E.T.: J. Math. Phys. 36, 5005 (1995)

4. Goals CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 4 To provide new insight into issues such as quantization. To find new spacetime solutions.  NSF has proven very difficult to solve: solutions found to date have used the light cone cut approach. Null geodesics of known solutions, such as Schwarzschild and Kerr, are followed out to null infinity I + (4-6),(7). (4) Joshi, P.S., Kozameh, C.N., Newman, E.T.: J. Math. Phys. 24, 2490 (1983) (5) Joshi, P.S., Newman, E.T.: Gen. Relativ. Gravit. 16, 1157 (1984) (6) Kling, T.P., Newman, E.T.: Phys. Rev. D 59, 124002–1 (1999)

5. Method of Solution CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 5 Use the (2+1)  dimensional formulation of the theory developed by  Forni, Iriondo, Kozameh and Parisi (7,8,)  Tanimoto (9)  Silva Ortigoza (10) Some light cone cut solutions found. (11) Solve the NSF field equations directly to find an exact solution. (7) Forni, D.M., Iriondo, M., Kozameh, C.N.: J. Math. Phys. 41, 5517 (2000) (8) Forni, D.M., Iriondo, M., Kozameh, C.N., Parisi, M.F.: J. Math. Phys. 43, 1584 (2002) (9) Tanimoto, M.: arXiv:gr-qc/9703003v1 (10) Silva-Ortigoza, G.: Gen. Relativ. Gravit. 32, 2243 (2000) (11) Harriott, T.A.,Williams, J.G.: Light cone cut solution in the 2+1 null surface formulation. In: Damour, T., Jantzen, R.T., Ruffini, R. (eds.) Proceedings of the Twelfth Marcel Grossmann Meeting on General Relativity. World Scientific, Singapore (2012)

6. 2+1 Version of NSF CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 6 On a manifold, M, assume the existence of one parameter functions denoted by Z(x a ;  ) where  The spacetime coordinates x a (a = 1, 2, 3) arise as constants of integration.  The parameter, , is angular in nature, and is of particular importance. The Z(x a ;  ) is the principal variable of the NSF. Assume that level surfaces of Z: Z(x a ;  ) = constant locally foliate the manifold and seek solutions that define null surfaces with respect to some spacetime metric g ab (x a ).

7. 2+1 Version of NSF CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 7 Since the principal requirement of the NSF is that these surfaces Z(x a ;  ) = constant are null surfaces with respect to some spacetime metric g ab (x a ) for arbitrary values of the parameter , the gradient of Z must satisfy However conditions must be imposed on Z to ensure a solution exists.

8. 2+1 Version of NSF CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 8 It is convenient to introduce new coordinates called intrinsic coordinates because they are naturally adapted to the surfaces where the ∂ or denotes differentiation with respect to , with the x a held fixed. The equations for u, ω, ρ can be inverted, x a = x a (u, ω, ρ,  ).

9. 2+1 Version of NSF CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 9 In addition a function  (u, ω, ρ,  ) can now be introduced, defined as  (u, ω, ρ,  ) := ∂ 3 Z(x a (u, ω, ρ,  );  ). However, ∂ 3 Z(x a (u, ω, ρ,  );  )≡ u , and so, the above equation can also be written as a differential equation for u u  =  (u, u, u ,  ).

10. 2+1 Version of NSF CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 10 This third-order nonlinear ordinary differential equation u  =  (u, u, u ,  ), can in principle be solved for u := Z and is the central equation of (2+1)  NSF.  To solve it we must find the so called metricity conditions that  must satisfy to ensure the solutions u := Z(x a ;  ) define null surfaces with respect to some spacetime metric g ab (x a ).

11. 2+1 Version of NSF CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 11 In what follows Indices a, b, c,... will refer to space-time coordinates x a. Indices i, j, k,... will refer to intrinsic coordinates u, , . The action of the differential operator  acting on any f (u, , ,  ) is (7) (7) Forni, D.M., Iriondo, M., Kozameh, C.N.: J. Math. Phys. 41, 5517 (2000)

12. 2+1 Version of NSF CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 12 Since we seek the family of surfaces u = Z(x a ;  ) that are null surfaces with respect to some spacetime metric g ab (x a ), for arbitrary values of the parameter , the gradient of Z satisfies Applying the  operator several times to this equation for the gradient of Z yields the metric components in intrinsic coordinates (7) ( 7) Forni, D.M., Iriondo, M., Kozameh, C.N.: J. Math. Phys. 41, 5517 (2000)

13. 2+1 Version of NSF CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 13 Specifically one finds that Also the remaining components are all proportional to and so this can be extracted as an overall multiplicative factor, which will be written as  2. That is

14. 2+1 Version of NSF CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 14 and inverting gives

15. 2+1 Version of NSF CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 15 Further application of the  operator to the equation for the gradient of Z that ensures the level surfaces are null yields the metricity conditions and

16. 2+1 Version of NSF CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 16 It can also be shown (7) that if the metricity conditions are satisfied then so is provided (7) Forni, D.M., Iriondo, M., Kozameh, C.N.: J. Math. Phys. 41, 5517 (2000)

17. 2+1 Version of NSF: Summary CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 17 To summarize: in order to find solutions that represent (general relativistic) null surfaces u := Z(x a ;  ) in the (2+1)  NSF approach one must solve the following coupled equations: (I) (II) (III) for  and  and use the resulting  in u  =  (u, u, u ,  ), which must then be solved for u.

18. Solution CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 18 Assume that  =  (  ) and  =  (  ) only. Then 3  =    shows that Also the main metricity condition reduces to the third order non linear ordinary differential equation

19. Solution CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 19 This main metricity condition can be shown to have the general solution (12) (12) Polyanin, A.D., Zaitsev, V.F.: Handbook of Exact Solutions for Ordinary Differential Equations. CRC Press, Boca Raton (1995)

20. Solution CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 20 The solution is To ensure that  and  are real everywhere and at least the third order derivatives with respect to  exist we require

21. Solution CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 21 This can now be used in u  =  (u, u, u ,  ) to solve for Recall that  = u  where the prime denotes differentiation with respect to . The equation to solve for u is or in terms of 

22. Solution CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 22 This equation for  can be integrated three times to find u(x a,  ). The first integration results in the following expression after solving for 

23. Solution CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 23 Recalling that  =  can be integrated with respect to  to find

24. Solution CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 24 Since  = u a final integration with respect to  leads to

25. Solution CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 25 This expression is the desired solution for u = Z(x a ;  ) representing a family of null surfaces It is well defined for all values because it can be shown that

26. Solution Properties: Metric CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 26 The general form of metric in intrinsic coordinates was shown to be For this solution it is

27. Solution Properties: Curvature Scalars CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 27 The three independent curvature scalars in 2+1 dimensions are (13) These are non zero and so the spacetime is not flat. (13) Weinberg, S.: Gravitation and Cosmology. Wiley, New York (1972)

28. Solution Properties: Cotton-York Tensor CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 28 NSF does not distinguish between conformally related spacetimes, so one needs to check that the spacetime is not conformally flat. The Cotton-York tensor (14–16) is identically zero if and only if the 2+1 spacetime is conformally flat. (14) Cotton, E.: Ann. Fac. Sci. Toulouse (Sér. 2) 1, 385 (1899), online version: http://www.numdam.org/item?id=AFST_1899_2_1_4_385_0 (15) York Jr, J.W.: Phys. Rev. Lett. 26, 1656 (1971) (16) Deser, S., Jackiw, R., Templeton, S.: Ann. Phys. (NY) 140, 372 (1982)

29. Solution Properties: Cotton-York Tensor CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 29 There are six non-zero components of the Cotton-York tensor and so this spacetime is not conformally flat.

30. 2+1 Version of NSF: Summary CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 30 To find solutions that represent null surfaces u := Z(x a ;  ) in the (2+1) - NSF approach one must solve the coupled equations: (I) (II) (III) for  and  and use the resulting  in u  =  (u, u, u ,  ) which must then be solved for u.

31. Solution Properties: Source CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 31 Recalling that this equation shows that

32. Solution Properties: Perfect Fluid Source CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 32 This value of T  is consistent with a perfect fluid source where and the mass energy density and isotropic pressure are with an equation of state:  = 3p.

33. Solution Properties: Perfect Fluid Source CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 33 The expansion scalar  and the shear tensor  ij are zero. The vorticity tensor The non zero components are and the scalar vorticity is

34. Alternative Interpretation of the Solution CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 34 Recall that our solution has and we chose a > 0 and 4ac > b 2 to ensure that  and  were real everywhere and at least the third order derivatives with respect to  existed for all . In the case when 4ac < b 2 we can restrict  to ensure that this solution satisfies the topologically massive gravity field equations.

35. Alternative Interpretation of the Solution CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 35 The TMG field equations are (19) where is the cosmological constant and m is a constant. Choosing the velocity vector : then we have the following perfect fluid solution (19) Deser, S., Jackiw, R., Templeton, S.: Phys. Rev. Lett. 48, 975 (1982 )

36. Alternative Interpretations of the Solution CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 36 For the TMG field equations with the particular choice gives a vacuum solution analogous to the regular de Sitter solution.

37. A New Solution ? CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 37 The metric in intrinsic coordinates is Recall, x a = x a (u, ω, ρ,  ) where x a can be interpreted as spacetime coordinates for any choice of . Selecting  = 0 a coordinate transformation gives

38. Vuorio’s Solutions CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 38 The family of (2+1)  dimensional ‘Gödel-type’ universes that was discovered by Vuorio (18) is where k is a constant. For Vuorio’s spacetime, the three independent curvature scalars are (18) Vuorio, I.: Phys. Lett. B 163, 91 (1985)

39. Future Work CCGRRA, May, 2014. Tina. A. Harriott & J. G. Williams 39 Finding further (2+1)  dimensional solutions by allowing  (u, ω, ρ,  ) and  (u, ω, ρ,  ) to be more general functions of the intrinsic variables. Generalizing the solutions to (3+1)  dimensional NSF.