Connection Variables in General Relativity

Transcription

1 Connection Variables in General Relativity Mauricio Bustamante Londoño Instituto de Matemáticas UNAM Morelia 28/06/2008 Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

2 Contents 1 Motivation 2 The frame fields 3 Palatini s Action 4 Self-dual formalism 5 Constraints in terms of the new variables Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

3 Motivation Motivation Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

4 Motivation Motivation Hamiltonian formulation of general relativity in terms of the ADM variables give us a very complicated system of constraints and a infinite-dimensional configuration space which does not seems a measureable space. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

5 Motivation Motivation Hamiltonian formulation of general relativity in terms of the ADM variables give us a very complicated system of constraints and a infinite-dimensional configuration space which does not seems a measureable space. We can think the connection as an object which give us information about curvature (by mean of the parallel transport), it suggests to look for a formulation of general relativity, not in metric variables, but in connection variables. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

6 Motivation Motivation Hamiltonian formulation of general relativity in terms of the ADM variables give us a very complicated system of constraints and a infinite-dimensional configuration space which does not seems a measureable space. We can think the connection as an object which give us information about curvature (by mean of the parallel transport), it suggests to look for a formulation of general relativity, not in metric variables, but in connection variables. Connection variables are fundamentals variables in the gauge theories. Maybe we could take some tools of gauge theories in understanding general relativity. As we will see, it will happen. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

7 The frame fields The frame fields Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

8 The frame fields The frame fields On the tangent space in a point x of the spacetime M we can choose a usual basis of partial derivatives {( µ ) x } and an orthonormal basis of tetrads {e I (x)} such that e I (x) = e µ I (x) µ and e µ I eν J g µν = η IJ. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

9 The frame fields The frame fields On the tangent space in a point x of the spacetime M we can choose a usual basis of partial derivatives {( µ ) x } and an orthonormal basis of tetrads {e I (x)} such that e I (x) = e µ I (x) µ and e µ I eν J g µν = η IJ. On the cotangent space we have a set of one-forms with values in a Minkowski space: e I (x) = e I µ(x)dx µ, and by the orthonormality condition it will be interpreted as the gravitational field. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

10 The frame fields The frame fields On the tangent space in a point x of the spacetime M we can choose a usual basis of partial derivatives {( µ ) x } and an orthonormal basis of tetrads {e I (x)} such that e I (x) = e µ I (x) µ and e µ I eν J g µν = η IJ. On the cotangent space we have a set of one-forms with values in a Minkowski space: e I (x) = e I µ(x)dx µ, and by the orthonormality condition it will be interpreted as the gravitational field. It is easy to check that the frame fields (tetrads) transforms, locally, under the Lorentz group, that is to say, the symmetry of such vectors is SO(4) (Euclidean case). Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

11 The frame fields Note that more technically we are dealing with the frame bundle on M with structure group SO(4) and then we can define an one-form conncetion ω as: ω I J (x) = ωi J (x) so(4) Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

12 The frame fields Note that more technically we are dealing with the frame bundle on M with structure group SO(4) and then we can define an one-form conncetion ω as: ω I J (x) = ωi J (x) so(4) and the two-form curvature as R IJ [ω] = dω IJ + ω I K ωkj Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

13 The frame fields Note that more technically we are dealing with the frame bundle on M with structure group SO(4) and then we can define an one-form conncetion ω as: ω I J (x) = ωi J (x) so(4) and the two-form curvature as R IJ [ω] = dω IJ + ω I K ωkj A volume element can be written as ǫ IJKL e I e J e K e L where ǫ IJKL is a totally antisymmetric symbol with ǫ 0123 = 1. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

14 Palatini s action Palatini s Action Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

15 Palatini s action Palatini s Action Now we have all the elements needed to write the Hilbert-Einstein action in terms of frame fields. A simple calculation gives us S[g] = d 4 x gr S[e, ω] = 1 ǫ IJKL e I e J R[ω] KL 2 This is the Palatini s action. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

16 Palatini s action Palatini s Action Now we have all the elements needed to write the Hilbert-Einstein action in terms of frame fields. A simple calculation gives us S[g] = d 4 x gr S[e, ω] = 1 ǫ IJKL e I e J R[ω] KL 2 This is the Palatini s action. By taking variations of the action we obtain the equations of motion: ǫ IJKL e J e K R KL = 0 Eisntein s equations D(e I e J ) = 0 Free torsion Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

17 Palatini s Action Some comments about Palatini s action Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

18 Palatini s Action Some comments about Palatini s action This action splits the degrees of freedom contained in the H-E action in two sets, the frame fields and the connection!. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

19 Palatini s Action Some comments about Palatini s action This action splits the degrees of freedom contained in the H-E action in two sets, the frame fields and the connection!. The Palatini s formalism is a first order formalism. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

20 Palatini s Action Some comments about Palatini s action This action splits the degrees of freedom contained in the H-E action in two sets, the frame fields and the connection!. The Palatini s formalism is a first order formalism. We can recover the usual Einstein s equation. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

21 Palatini s Action Some comments about Palatini s action This action splits the degrees of freedom contained in the H-E action in two sets, the frame fields and the connection!. The Palatini s formalism is a first order formalism. We can recover the usual Einstein s equation. As the torsion is defined as T I = De I is easy to check that the second equation is equivalent to the free torsion condition. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

22 Self-dual formalism Self-dual formalism Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

23 Self-dual formalism Self-dual formalism A little bit of SO(4): The dual operator Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

24 Self-dual formalism Self-dual formalism A little bit of SO(4): The dual operator Over a four dimensional manifold we define the dual operator by acting on antisymmetric tensors T µν as T µν = ǫ ρσ µν T ρσ Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

25 Self-dual formalism Self-dual formalism A little bit of SO(4): The dual operator Over a four dimensional manifold we define the dual operator by acting on antisymmetric tensors T µν as T µν = ǫ ρσ µν T ρσ If we note that 2 = 1 then the eigenvectors of such operator are: T + µν = 1 2 (T µν + T µν ) }{{} self dual and T µν = 1 2 (T µν T µν ) }{{} antiself dual Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

26 Self-dual formalism Now define T i = 1 2 (T 0i + T 0i ), i = 1, 2, 3 and note that Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

27 Self-dual formalism Now define T i = 1 2 (T 0i + T 0i ), i = 1, 2, 3 and note that [ ] T + i, T + j = [ ] T i, T j = [ ] T + i, T j 3 ǫ ijk T + k i=1 3 ǫ ijk T k i=1 = 0. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

28 Self-dual formalism Now define T i = 1 2 (T 0i + T 0i ), i = 1, 2, 3 and note that [ ] T + i, T + j = [ ] T i, T j = [ ] T + i, T j 3 ǫ ijk T + k i=1 3 ǫ ijk T k i=1 = 0. But the generators of the Lie algebra of SO(4) are precisely antisymmetric matrices, then all that means, in this context, that so(4) = so(3) so(3) Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

29 Self-dual formalism The last fact let us to write the connection as ω I J }{{ (x) = ω +I } J (x) + ω I J }{{} (x). }{{} so(4) so(3) so(3) Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

30 Self-dual formalism The last fact let us to write the connection as And then one prove that ω I J }{{ (x) = ω +I } J (x) + ω I J }{{} (x). }{{} so(4) so(3) so(3) R IJ [ω] = R IJ [ω + + ω ] = R IJ [ω + ] + R IJ [ω ], Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

31 Self-dual formalism The last fact let us to write the connection as And then one prove that ω I J }{{ (x) = ω +I } J (x) + ω I J }{{} (x). }{{} so(4) so(3) so(3) R IJ [ω] = R IJ [ω + + ω ] = R IJ [ω + ] + R IJ [ω ], hence, S[e, ω] = S[e, ω + ] + S[e, ω ]. }{{}}{{} S + S Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

32 Self-dual formalism The main result: The stationary values of S and S + are over the same frame fields. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

33 Self-dual formalism The main result: The stationary values of S and S + are over the same frame fields. That means that the equations of motion obtained with S are the same that the obtained with S +, in other words, we just need the information contained in the half of the Palatini s action to describe the dynamics of the gravitational field!. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

34 e0 i The self-dual action Self-dual formalism As in ADM formalism, consider a foliation of spacetime M given by the diffeomorphism R Σ, where Σ is a compact, orientable 3-manifold. t+dt Ndt T(X) e t Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

35 e0 i The self-dual action Self-dual formalism As in ADM formalism, consider a foliation of spacetime M given by the diffeomorphism R Σ, where Σ is a compact, orientable 3-manifold. t+dt Ndt T(X) e t The foliation vector T(X) can be written in terms of the shift vector and lapse function as Ne 0 + N i e i. Before to write the action in terms of N and N i it is better for our proposes to take into account the following relations: Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

36 Self-dual formalism If q ab = δ ij eae i j b g = N q. is the induced metric on Σ then is easy to see that Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

37 Self-dual formalism If q ab = δ ij eae i j b g = N q. is the induced metric on Σ then is easy to see that Define E µ I = qe µ I. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

38 Self-dual formalism If q ab = δ ij eae i j b g = N q. is the induced metric on Σ then is easy to see that Define E µ I = qe µ I. Define Ñ = N/ q. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

39 Self-dual formalism If q ab = δ ij eae i j b g = N q. is the induced metric on Σ then is easy to see that Define E µ I = qe µ I. Define Ñ = N/ q. Putting all together in the Palatini s action we get: S = dx 0 d 3 xñr IJ µνe µ I Eν J, Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

40 Self-dual formalism If q ab = δ ij eae i j b g = N q. is the induced metric on Σ then is easy to see that Define E µ I = qe µ I. Define Ñ = N/ q. Putting all together in the Palatini s action we get: S = dx 0 d 3 xñr IJ µνe µ I Eν J, or S = dx 0 d 3 x ) ij (ÑR ab Ea i Ej b 2N a R 0i ab Eb i + 2R 0i 0aEi a Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

41 Self-dual formalism We have two options: Going on as in the ADM formalism (calculating momenta and all that), or using what we know about self-duality. Obviously we choose the second. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

42 Self-dual formalism We have two options: Going on as in the ADM formalism (calculating momenta and all that), or using what we know about self-duality. Obviously we choose the second. Since 1 2 ǫi jk R+jk = R +0k, the last action becomes (without tildes and plus signs) S = dx 0 d 3 x ) ij (ÑR ab Ea i Ej b 2N a R 0i ab Eb i + 2R 0i 0aEi a, Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

43 Self-dual formalism We have two options: Going on as in the ADM formalism (calculating momenta and all that), or using what we know about self-duality. Obviously we choose the second. Since 1 2 ǫi jk R+jk = R +0k, the last action becomes (without tildes and plus signs) S = dx 0 d 3 x ) ij (ÑR ab Ea i Ej b 2N a R 0i ab Eb i + 2R 0i 0aEi a, and let s make-up her a little: Define a su(2)- valued one form connection as A i a = 2ω i a and note that the two-form curvature associated with this connection is F i ab = ( da i) ab + [A, A]i ab = 2R0i ab Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

44 Self-dual formalism We have two options: Going on as in the ADM formalism (calculating momenta and all that), or using what we know about self-duality. Obviously we choose the second. Since 1 2 ǫi jk R+jk = R +0k, the last action becomes (without tildes and plus signs) S = dx 0 d 3 x ) ij (ÑR ab Ea i Ej b 2N a R 0i ab Eb i + 2R 0i 0aEi a, and let s make-up her a little: Define a su(2)- valued one form connection as A i a = 2ω i a and note that the two-form curvature associated with this connection is F i ab = ( da i) ab + [A, A]i ab = 2R0i ab Also 2R 0i 0a = 0A i a + D a λ i, where λ i = 2ω i 0. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

45 Self-dual formalism Finally, the action for general general relativity in terms of connection variables or Ashtekar variables is [ ( ( 0 S = dt d 3 x A i ) a E a i N a F k ab Eb k + λi D a Ei a 1 )] 2 NF ij ab Ea i Ej b. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

46 Self-dual formalism Finally, the action for general general relativity in terms of connection variables or Ashtekar variables is [ ( ( 0 S = dt d 3 x A i ) a E a i N a F k ab Eb k + λi D a Ei a 1 )] 2 NF ij ab Ea i Ej b. and the Hamiltonian of the theory is H = d 3 x N a F i ab Eb i Σ }{{} D[N a ] + λ i D a E a i }{{} G[λ i ] 1 2 NF ij ab Ea i Ej b } {{ } S[N] Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

47 Gauss constraint Constraints in terms of the new variables Variations on λ produce the Gauss constraint D a E a i = 0. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

48 Gauss constraint Constraints in terms of the new variables Variations on λ produce the Gauss constraint D a E a i = 0. Following the Dirac s prespription to deal with constraints systems, we may interpret the Gauss constraint in terms of gauge symmetries. To do that we calculate the Poisson brackets of the constraint with each one of the phase space variables (A i a, E a i ): Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

49 Gauss constraint Constraints in terms of the new variables Variations on λ produce the Gauss constraint D a E a i = 0. Following the Dirac s prespription to deal with constraints systems, we may interpret the Gauss constraint in terms of gauge symmetries. To do that we calculate the Poisson brackets of the constraint with each one of the phase space variables { (A i a, Ei a): } Taking into account that A i a(x), Ej b(y) = δj iδb aδ 3 (x, y) we have { A i a (x), G[λ j ] } = D a λ i { E a i (x), G[λ j ] } = ǫ k ij λ j (x)e a k (x) Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

50 Gauss constraint Constraints in terms of the new variables Variations on λ produce the Gauss constraint D a E a i = 0. Following the Dirac s prespription to deal with constraints systems, we may interpret the Gauss constraint in terms of gauge symmetries. To do that we calculate the Poisson brackets of the constraint with each one of the phase space variables { (A i a, Ei a): } Taking into account that A i a(x), Ej b(y) = δj iδb aδ 3 (x, y) we have { A i a (x), G[λ j ] } = D a λ i { E a i (x), G[λ j ] } = ǫ k ij λ j (x)e a k (x) Gauss constraint generate SU(2) gauge transformations. General relativity looks like a Yang-Mills theory!. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

51 Constraints in terms of the new variables Diffeomorphisms constraint Variations on N a produce the diffeomorphisms constraint F i ab Eb i = 0. Defining D[N b ] = D[N b ] G[N b A j b ], we find { } A i a, D[N b ] = L N ba i a { } Ei a, D[N b ] = L N bei a Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

52 Constraints in terms of the new variables Diffeomorphisms constraint Variations on N a produce the diffeomorphisms constraint F i ab Eb i = 0. Defining D[N b ] = D[N b ] G[N b A j b ], we find { } A i a, D[N b ] = L N ba i a { } Ei a, D[N b ] = L N bei a This constraint generate the diffeomorphisms on the 3-manifold Σ. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

53 Scalar constraint Constraints in terms of the new variables Variations on N produce the Scalar constraint NF ij ab Ea i Ej b = 0. Its Poisson brackets with the phase space variables is: { A i a (x), S[N] } = Nǫ ij k F ab k (x)eb j (x) ( ) {Ei a (x), S[N]} = D b Nǫ jk i Eb j Ek a. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

54 Scalar constraint Constraints in terms of the new variables Variations on N produce the Scalar constraint NF ij ab Ea i Ej b = 0. Its Poisson brackets with the phase space variables is: { A i a (x), S[N] } = Nǫ ij k F ab k (x)eb j (x) ( ) {Ei a (x), S[N]} = D b Nǫ jk i Eb j Ek a. Although is not evident, this constraint generate transverse moves of Σ and together with the last constraint we obtain the diffeomorphisms on M. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

55 Constraints in terms of the new variables Some comments about constraints Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

56 Constraints in terms of the new variables Some comments about constraints The phase space of general relativity is now coordinated by the connection A and the gravitational field E. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

57 Constraints in terms of the new variables Some comments about constraints The phase space of general relativity is now coordinated by the connection A and the gravitational field E. The constraints are in polynomial form. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

58 Constraints in terms of the new variables Some comments about constraints The phase space of general relativity is now coordinated by the connection A and the gravitational field E. The constraints are in polynomial form. In the Dirac s terminology the set of constraints is a set of first class constraints (I will not prove it here). Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

59 Constraints in terms of the new variables Some comments about constraints The phase space of general relativity is now coordinated by the connection A and the gravitational field E. The constraints are in polynomial form. In the Dirac s terminology the set of constraints is a set of first class constraints (I will not prove it here). Now let s go to the quantization... Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20

60 References References J.Baez, Spin networks in gauge theory, Advances in Mathematics 117, , J.Baez, Spin networks in non perturbative quantum gravity, arxiv:gr-qc/ A.Perez, Introduction to loop quantum gravity and spin foams, arxiv:gr-qc/ C.Rovelli, Quantum Gravity, Cambridge University Press, Jose Antonio Zapata s and Robert Oeckl s comments. Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/ / 20