Conformal Einstein spaces and Bach tensor generalizations in n dimensions
|
|
- Jonah Lane
- 6 years ago
- Views:
Transcription
1 Linköping Studies in Science and Technology. Theses No Conformal Einstein spaces and Bach tensor generalizations in n dimensions Jonas Bergman Matematiska institutionen Linköpings universitet, SE Linköping, Sweden Linköping 004
2 ii Conformal Einstein spaces and Bach tensor generalizations in n dimensions c 004 Jonas Bergman Matematiska institutionen Linköpings universitet SE Linköping, Sweden jober@mai.liu.se LiU-TEK-LIC-004:4 ISBN ISSN Printed by UniTryck, Linköping 004
3 iii Abstract In this thesis we investigate necessary and sufficient conditions for an n- dimensional space, n 4, to be locally conformal to an Einstein space. After reviewing the classical results derived in tensors we consider the four-dimensional spinor result of Kozameh, Newman and Tod. The involvement of the four-dimensional Bach tensor which is divergence-free and conformally well-behaved in their result motivates a search for an n-dimensional generalization of the Bach tensor B ab with the same properties. We strengthen a theorem due to Belfagón and Jaén and give a basis U ab, V ab and W ab for all n-dimensional symmetric, divergence-free - index tensors quadratic in the Riemann curvature tensor. We discover the simple relationship B ab = 1 U ab V ab and show that the Bach tensor is the unique tensor with these properties in four dimensions. Unfortunately we have to conclude, in general that there is no direct analogue in higher dimension with all these properties. Nevertheless, we are able to generalize the four-dimensional results due to Kozameh, Newman and Tod to n dimensions. We show that a generic space is conformal to an Einstein space if and only if there exists a vector field satisfying two conditions. The explicit use of dimensionally dependent identities some of which are newly derived in this thesis is also exploited in order to make the two conditions as simple as possible; explicit examples are given in five and six dimensions using these tensor identities. For n dimensions, we define the tensors b abc and B ab, and we show that their vanishing is a conformal invariant property which guarantees that the space with non-degenerate Weyl tensor is a conformal Einstein space. Acknowledgments First and foremost, I would like to thank my supervisors Brian Edgar and Magnus Herberthson for their constant support, encouragement, and generous knowledge sharing, and for all the interesting discussions, and for giving me this chance to work on such an interesting topic. Thanks also goes to all my friends and colleagues at the Department of Mathematics, and especially Arne Enqvist for his support. I would also like to mention Anders Höglund, who let me use his fantastic program Tensign, and Ingemar and Göran, who read the manuscript and gave me valuable comments. Thanks guys! Finally, but not least, I would like to thank Pauline and my family for their support, encouragement, and understanding and for putting up with me, especially during the last month. Jonas Bergman, Linköping, 0 September 004
4 iv
5 Contents Abstract and Acknowledgments Contents iii v 1 Introduction and outline of the thesis 1 Preliminaries 5.1 Conventions and notation Conformal transformations Conformally flat spaces Conformal Einstein equations and classical results Einstein spaces Conformal Einstein spaces The classical results The Bach tensor in four dimensions and possible generalizations The Bach tensor in four dimensions Attempts to find an n-dimensional Bach tensor The tensors U ab, V ab and W ab Four-dimensional Bach tensor expressed in U ab, V ab and W ab An n-dimensional tensor expressed in U ab, V ab and W ab The Kozameh-Newman-Tod four-dimensional result and the Bach tensor Two useful lemmas C-spaces and conformal C-spaces Conformal Einstein spaces J = Listing s result in four dimensions Non-degenerate Weyl tensor Conformal C-spaces Conformal Einstein spaces
6 vi 7 Listing s result in n dimensions Non-degenerate Weyl tensor Conformal C-spaces Conformal Einstein spaces Edgar s result in n dimensions Using the Cayley-Hamilton Theorem Four dimensions Higher dimensions Using dimensionally dependent identities A five-dimensional example Six-dimensional examples Generalizing the Bach tensor in n dimensions A generic Weyl tensor The generalization of the KNT result n dimensions using generic results Five-dimensional spaces using dimensional dependent identities Six-dimensional spaces using dimensional dependent identities Conformal properties of different tensors The tensors b abc and B ac and their conformal properties in generic spaces The tensor L ab and its conformal properties Concluding remarks and future work 67 A The Cayley-Hamilton Theorem and the translation of the Weyl tensor/spinor to a matrix 69 A.1 The Cayley-Hamilton Theorem The case where n = 3 and the matrix is trace-free The case where n = 6 and the matrix is trace-free A. Translation of C ab cd to a matrix C A B A.3 Translation of Ψ AB CD to a matrix Ψ B Dimensionally dependent tensor identities 74 B.1 Four-dimensional identities B. Five-dimensional identities B.3 Six-dimensional identities B.4 Lovelock s quartic six-dimensional identity C Weyl scalar invariants 79 C.1 Weyl scalar invariants in 4 dimensions
7 vii D Computer tools 84 D.1 GRTensor II D. Tensign References 87
8 viii
9 Chapter 1 Introduction and outline of the thesis Within semi-riemannian geometry there are classes of spaces which have special significance from geometrical and/or physical viewpoints; e.g., flat spaces with zero Riemann curvature tensor, conformally flat spaces i.e., spaces conformal to flat spaces with Weyl tensor equal to zero, Einstein spaces with trace-free Ricci tensor equal to zero. There are a number of both physical and geometrical reasons to study conformally Einstein spaces i.e., spaces conformal to Einstein spaces, and it has been a long-standing classical problem to find simple characterizations of these spaces in terms of the Riemann curvature tensor. Therefore, in this thesis, we will investigate necessary and sufficient conditions for an n-dimensional space, n 4, to be locally conformal to an Einstein space, a subject studied since the 190s. Global properties will not be considered here. The first results in this field are due to Brinkmann [6], [7], but also Schouten [36] has contributed to the subject; they both considered the general n- dimensional case. Nevertheless, the set of conditions they found is large, and not useful in practice. Later, in 1964, Szekeres [39] introduced spinor tools into the problem and proposed a partial solution in four dimensions using spinors, restricting the space to be Lorentzian, i.e., to have signature. Nevertheless, the spinor conditions he found are hard to analyse and complicated to translate into tensors. Wünsch [43] pointed out a mistake in Szekeres s paper which means that his conditions are only necessary. However, in 1985, Kozameh, Newman and Tod [7] continued with the spinor approach and found a much simpler set consisting of only two independent necessary and sufficient conditions for four-dimensional spaces; however, the price they paid for this simplicity was that the result was restricted to a subspace of the most general class of spaces those for which
10 one of the scalar invariants of the Weyl tensor is non-zero, i.e., J = 1 C ab cdc cd ef C ef ab i C ab cd C cd ef C ef ab One of their conditions is the vanishing of the Bach tensor B ab ; in four dimensions this tensor has a number of nice properties. The condition J 0 in the result of Kozameh et al. [7] has been relaxed, also using spinor methods, by Wünsch [43], [44], by adding a third condition to the set found by Kozameh et al. This still leaves some spaces excluded; in particular the case when the space is of Petrov type N, although Czapor, McLenaghan and Wünsch [1] have some results in the right direction. The spinor formalism is the natural tool for general relativity in four dimensions in a Lorentzian space [3], [33] since it has built in both four dimensions and signature ; on the other hand, it gives little guidance on how to generalize to n-dimensional semi-riemannian spaces. However, using a more differential geometry point of view, Listing [8] recently generalized the result of Kozameh et al. [7] to n-dimensional semi-riemannian spaces having non-degenerate Weyl tensor. Listing s results have been extended by Edgar [13] using the Cayley-Hamilton Theorem and dimensionally dependent identities [14], [9]. There have been other approaches to this problem. For example, Kozameh, Newman and Nurowski [6] have interpreted and studied the necessary and sufficient condition for a space to be conformal to an Einstein space in terms of curvature restrictions for the corresponding Cartan conformal connection. Also Baston and Mason [3], [4], working with a twistorial formulation of the Einstein equations, found a different set of necessary and sufficient conditions. However, we shall restrict ourselves to a classical semi-riemannian geometry approach. In this thesis we are going to try and find n-dimensional tensors, n 4, generalizing the Bach tensor in such a way that as many good properties of the four-dimensional Bach tensor as possible are carried over to the n-dimensional generalization. We shall also investigate how these generalizations of the Bach tensor link up with conformal Einstein spaces. The first part of this thesis will review the classical tensor results; Chapters 5 to 8 will review and extend a number of the results in both spinors and tensors during the last 0 years. In the remaining chapters we will present some new results and also discuss the directions where this work can develop in the future. We have also included four appendices in which we have collected some old and developed some new results needed in the thesis, but to keep the presentation as clear as possible we have chosen to summarize these at the end. The outline of the thesis is as follows: We begin in Chapter by fixing the conventions and notation used in the thesis and giving some useful relations and identities. The chapter ends by
11 3 reviewing and proving the classical result that a space is conformally flat if and only if the Weyl tensor is identically zero. In Chapter 3 Einstein spaces and conformal Einstein spaces are introduced and the conformal Einstein equations are derived. Some of the earlier results in the field are also mentioned. In Chapter 4 the Bach tensor B ab in four dimensions is defined and various attempts to find an n-dimensional counterpart are investigated. We strengthen a theorem due to Belfagón and Jeán and give a basis U ab, V ab and W ab for all n-dimensional symmetric, divergence-free -index tensors quadratic in the Riemann curvature tensor. We discover the simple relationship B ab = 1 U ab V ab between the four-dimensional Bach tensor and these tensors, and show that this is the only -index tensor up to constant rescaling which in four dimensions is symmetric, divergence-free and quadratic in the Riemann curvature tensor. We also demonstrate that there is no useful analogue in higher dimensions. Chapter 5 deals with the four-dimensional result for spaces in which J 0 due to Kozameh, Newman and Tod, and both explicit and implicit results in their paper are proven and discussed. We also explore a little further the relationship between spinor and tensor results. In Chapter 6 and Chapter 7 the recent work of Listing in spaces with non-degenerate Weyl tensors is reviewed; Chapter 6 deals with the fourdimensional case and Chapter 7 with the n-dimensional case. In Chapter 8 we look at the extension of Listing s result due to Edgar using the Cayley-Hamilton Theorem and dimensionally dependent identities. In Chapter 9 the concept of a generic Weyl tensor and a generic space is defined. The results of Kozameh, Newman and Tod are generalized and generic results presented. We show that an n-dimensional generic space is conformal to an Einstein space if and only if there exists a vector field satisfying two conditions. The explicit use of dimensionally dependent identities is also exploited in order to make these two conditions as simple as possible; explicit examples are given in five and six dimensions. In Chapter 10, for n dimensions, we define the tensors b abc and B ab, whose vanishing guarantees a space with non-degenerate Weyl tensor being a conformal Einstein space. We show that b abc is conformally invariant in all spaces with non-degenerate Weyl tensor, and that B ab is conformally weighted with weight, but only in spaces with non-degenerate Weyl tensor where b abc = 0. We also show that the Listing tensor L ab is conformally invariant in all n-dimensional spaces with non-degenerate Wely tensor. In the final chapter we briefly summarize the thesis and discuss different ways of continuing this work and possible applications of the results given in the earlier chapters. Appendix A deals with the representations of the Weyl spinor/tensor as matrices and discusses the Cayley-Hamilton Theorem for matrices and tensors. In Appendix B dimensionally dependent identities are discussed. A number
12 4 of new tensor identities in five and six dimensions suitable for our purpose are derived; these identities are exploited in Chapter 9. In Appendix C, in four dimensions, we look at the Weyl scalar invariants and derive relations between the two complex invariants naturally arising from spinors and the standard four real tensor invariants. The last appendix briefly comments on the computer tools used for some of the calculations in this thesis. At an early stage of this investigation we became aware of the work of Listing [8], who had also been motivated to generalize the work of Kozameh et al. [7]. So, although we had already anticipated some of the Listing s generalizations independently, we have reviewed these generalizations as part of his work in Chapters 6 and 7. When we were writing up this thesis May 004 a preprint by Gover and Nurowski [16] appeared on the The first part of this preprint obtains some of the results which we have obtained in Chapter 9 in essentially the same manner; however, they do not make the link with dimensionally dependent identities, which we believe makes these results more useful. The second part of this preprint deals with conformally Einstein spaces in a different manner based on the tractor calculus associated with the normal Cartan bundle. Out of this treatment emerges the results on the conformal behavior of b abc and B ac, which we obtained in a more direct manner in Chapter 10. Due to the very recent appearance of [16] we have not referred to this preprint in our thesis, since all of our work was done completely independently of it.
13 Chapter Preliminaries In this chapter we will briefly describe the notation and conventions used in this thesis, but for a more detailed description we refer to [3] and [33]. We also review and prove the classical result that a space is conformally flat if and only if the Weyl tensor of the space is identically zero..1 Conventions and notation All manifolds we consider are assumed to be differentiable and equipped with a symmetric non-degenerate bilinear form g ab = g ba, i.e. a metric. No assumption is imposed on the signature of the metric unless explicitly stated, and we will be considering semi-riemannian or pseudo- Riemannian spaces in general; we will on occasions specialize to proper Riemannian spaces metrics with positive definite signature and Lorentzian spaces metrics with signature All connections,, are assumed to be Levi-Civita, i.e. metric compatible, and torsion-free, e.g. a g bc = 0, and a b b a f = 0 for all scalar fields f, respectively. Whenever tensors are used we will use the abstract index notation, see [3], and when spinors are used we again follow the conventions in [3]. The Riemann curvature tensor is constructed from second order derivatives of the metric but can equivalently be defined as the four-index tensor field R abcd satisfying [a b] ω c = a b ω c b a ω c = R abc d ω d.1 for all covector fields ω a, and it has the following algebraic properties, R abcd = R [ab][cd] = R cdab ; it satisfies the first Bianchi identity, R [abc]d = 0, and it also satisfies the second Bianchi identity, [a R bc]de = 0.
14 6 From the Riemann curvature tensor.1 we define the Ricci curvature tensor, R ab, by the contraction R ab = R acb c. and the Ricci scalar, R, from the contracted Ricci tensor R = R a a = R ab ab..3 For dimensions n 3 the Weyl curvature tensor or the Weyl conformal tensor, C abcd, is defined as the trace-free part of the Riemann curvature tensor, C abcd = R abcd g a[c R d]b g b[c R d]a + n n 1n Rg a[cg d]b.4 and as is obvious from above when R ab = 0 the Riemann curvature tensor reduces to the Weyl tensor. The Weyl tensor has all the algebraic properties of the Riemann curvature tensor, i.e. C abcd = C [ab][cd] = C cdab, C a[bcd] = 0, and in addition is trace-free, i.e. C abc a = 0. It is also well known that the Weyl tensor is identically zero in three dimensions. For clarity we note that due to our convention in defining the Riemann curvature tensor.1 we have for an arbitrary tensor field H b...d f...h that i j j i H b...d f...h = R ijb0 b H b0...d f...h R ijd0 d H b...d0 f...h R ijf f 0 H b...d f 0...h... R ijh h 0 H b...d f...h 0.5 and.5 is sometimes referred to as the Ricci identity. We will use both the conventions in the literature for denoting covariant derivatives, e.g., both the nabla and the semicolon, a v v ;a. Note however the difference in order of the indices in each case, a b v v ;ba. For future reference we write out the twice contracted second Bianchi identity, a R ab 1 br = 0,.6 the second Bianchi identity in terms of the Weyl tensor 0 = R ab[cd;e] = C ab[cd;e] + 1 n 3 g f 1 a[cc de]b ;f + n 3 g f b[cc ed]a ;f, the second contracted Bianchi identity for the Weyl tensor, d C abc d = n 3 1 [a R b]c + n n 1 g c[b a] R.7..8
15 7 Note that.8 also can be written d C d n 3 abc = n C cba.9 where C abc = [a R b]c + 1 n 1 g c[b a] R is the Cotton tensor. The Cotton tensor plays an important role in the study of thee-dimensional spaces [5],[15], [1]. We also have the divergence of.8 C db n 3 abcd; = n R ac; b n 3 b n 1 R n 3n ;ac + n R abr b c n 3 n 3 n Rbd C abcd n g acr bd R bd nn 3 n 1n RR n 3 ac n 1n g b acr ; b n 3 + n 1n g acr..10 Using the second Bianchi identity for the Weyl tensor, the Ricci identity and finally decomposing the Riemann tensor into the Weyl tensor, the Ricci tensor and the Ricci scalar we get the following identity [e d C ab]cd = n 3 n R [ab c f n 3 R e]f = n C [ab c f R e]f..11 In an n-dimensional space, letting H {Ω} a 1...a p = H {Ω} [a 1...a p] denote any tensor with an arbitrary number of indices schematically denoted by {Ω}, plus a set of p n completely antisymmetric indices a 1... a p, we define the Hodge dual H {Ω} a p+1...a n with respect to a 1... a p by H {Ω} a p+1...a n = 1 p! η a 1...a n H {Ω}a 1...a p.1 where η is the totally antisymmetric normalized tensor. Sometimes the is placed over the indices onto which the operation acts, e.g. H {Ω} a p+1...a n. In the special case of taking the dual of a double two-form H abcd = H [ab][cd] in four dimensions there are two ways to perform the dual operation; either acting on the first pair of indices, or the second pair. To separate the two we define the left dual and the the right dual as H ij cd = 1 η abijh abcd.13
16 8 and H ab ij = 1 η cdijh abcd.14 respectively. We are going to encounter highly structured products of Weyl tensors and to get a neater notation we follow [4] and make the following definition: Definition.1.1. For a trace-free, -form T abcd, i.e. for a tensor such that T a bad = 0, T abcd = T [ab]cd = T ab[cd].15 an expression of the form T ab c 1 d 1 T c1d1 c d... T cm dm c m 1 d m 1 T cm 1dm 1 ef }{{} m.16 where the indices a, b, e and f are free, is called a chain of the zeroth kind of length m and is written 1 T [m] ab ef. Hence, we have for instance that C[3] ab cd = C ab ijc ij klc kl cd. On occasions we will use matrices and these will always be written in bold capital letters, e.g. A. In this context O and I will denote the zero- and the identity matrix respectively and we will use square brackets to represent the operation of taking the trace of a matrix, e.g. [A] means the trace of the matrix A. Throughout this thesis we will only be considering spaces of dimension n 4. This is because in two dimensions all spaces are Einstein spaces and in three dimensions all spaces are conformally flat.. Conformal transformations Definition..1. Two metrics g ab and ĝ ab are said to be conformally related if there exists a smooth scalar field Ω > 0 such that holds. ĝ ab = Ω g ab.17 The metric ĝ ab is said to arise from a conformal transformation of g ab. Clearly we have ĝ ab = Ω g ab, since then ĝ ab ĝ bc = g ab g bc = δ a c. 1 In [4] this is denoted T 0 [m] ab ef.
17 9 To the rescaled metric ĝ ab there is a unique symmetric connection,, compatible with ĝ ab, i.e. c ĝ ab = 0. The relation between the two connections can be found in [3] and acting on an arbitrary tensor field H b...d f...h it is a H b...d f...h = a H b...d f...h + Q ab0 b H b 0...d f...h Q ad0 d H b...d 0 f...h Q af f 0 H b...d f 0...h... Q ah h 0 H b...d f...h 0.18 with Q ab c = Υ a δ c b g abυ c.19 where Υ a = Ω 1 a Ω = a ln Ω and Υ a = g ab Υ b. The relations between the tensors defined in and their hatted counterparts, i.e. the ones constructed from ĝ ab, are R abc d =R abc d δ d [a b]υ c + g c[a b] Υ d δ d [b Υ a]υ c + Υ [a g b]c Υ d + g c[a δ d b] Υ eυ e,.0 and R ab = R ab + n a Υ b + g ab c Υ c n Υ a Υ b + n g ab Υ c Υ c,.1 R = Ω R + n 1 c Υ c + n 1n Υ c Υ c,. Ĉ abc d = C abc d..3 Note that the positions of the indices in all equations are crucial since we raise and lower indices with different metrics, e.g. Ĉ abcd = ĝ de Ĉ abc e = Ω g de C abc e = Ω C abcd. Definition... A tensor field H b...d f...h is said to be conformally wellbehaved or conformally weighted with weight ω if under the conformal transformation.17, ĝ ab = Ω g ab, there is a real number w such that H b...d f...h Ĥb...d f...h = Ω ω H b...d f...h..4 If ω = 0 then H b...d f...h is said to be conformally invariant. Following [3] we introduce the tensor P ab = 1 n R 1 ab + n 1n Rg ab.5 These can be found, for instance, in [40], but note that Wald is using a different definition than.1 to define the Riemann curvature tensor.
18 10 and with the notation.5 we can express. -.4 as C ab cd = R ab cd + 4P [a [c g b] d],.6 and R ab = n P ab g ab P c c,.7 R = n 1P c c = n 1P..8 The contracted second Bianchi identity for the Weyl tensor.8 can now be written d C abc d = n 3 [a P b]c..9 Note that P ab is essentially R ab with a different trace term added and that P ab simply replaces R ab to make equations such as.6 and.9 simpler than.4 and.8, and hence to make calculations simpler. Furthermore, under a conformal transformation, P ab = P ab + Υ a Υ b a Υ b 1 g abυ c Υ c,.30 and P = Ω P c Υ c n Υ c Υ c,.31 which are simpler than the corresponding equations.1 and...3 Conformally flat spaces Definition.3.1. A space is called flat if its Riemann curvature tensor vanishes, R abcd = 0. Since flat spaces are well understood we would like a simple condition on the geometry telling us when there exists a conformal transformation making the space flat. Hence we define, Definition.3.. A space is called conformally flat if there exists a conformal transformation ĝ ab = Ω g ab such that the Riemann curvature tensor in the space with metric ĝ ab vanishes, i.e., R abcd = 0. From.0 we know that a space is conformally flat if and only if 0 =R abc d δ d [a b]υ c + g c[a b] Υ d δ d [bυ a] Υ c + Υ [a g b]c Υ d + g c[a δ d b] Υ eυ e.3 for some gradient vector field Υ a. Raising one index in equation.3 we can rewrite this as 0 = R ab cd + 4δ [c [a b]υ d] + 4Υ [a δ [c b] Υd] + δ [c [a δd] b] Υ eυ e = R ab cd + 4P [a [c δ d] b].33
19 11 where from.30 with P ab = 0 we have P ab = a Υ b Υ a Υ b + 1 g abυ c Υ c.34 and P [ab] = We will now prove the important classical theorem 3 Theorem.3.1. A space is conformally flat if and only if its Weyl tensor is zero. Proof. The necessary part follows immediately from the conformal properties of the Weyl tensor.3. To prove the sufficient part we break up the proof into two steps. First we shall show that if there exists some symmetric -tensor P ab in a space such that 0 = R ab cd + 4P [a [c δ d] b].36 P ab = a K b K a K b + 1 g abk c K c ; P [ab] = 0.37 for some vector field K a, then the space is conformally flat. From.37 it follows that [a K b] = 0 which means that locally K a is a gradient vector field, K a = a Φ.38 for some scalar field Φ, and substitution of P ab in.37 with this gradient expression for K a into.36 gives 0 =R abc d δ d [a b] c Φ + g c[a b] d Φ δ d [b a] Φ c Φ + [a Φ g b]c d Φ + g c[a δ d b] e Φ e Φ, i.e.,.0 with Υ a = a Φ, which implies that R abcd = 0 and so the space is conformally flat, ĝ = e Φ g ab. Secondly we shall show that if C abcd = 0 then.36 and.37 are satisfied. From.6 it follows immediately that if C abcd = 0, [37]. 0 = R ab cd + 4P [a [c δ d] b].39 3 The necessary part is originally due to Weyl [41], and the sufficient part to Schouten
20 1 i.e..36 is satisfied. Also, if C abcd = 0, we know from the second Bianchi identity.9 that [a P b]c = To check if.37 can be satisfied we calculate the integrability condition of.37 which is 0 = [c P a]b [c a] Υ b + Υ b [c Υ a] + Υ [a c] Υ b Υ e g b[a c] Υ e 1 = [c P a]b + R ca be + P [b [c δ e] a] Υ e.41 But from.39 and.40 it follows that this condition is identically satisfied. Hence we conclude that.37 is a consequence of.36. To summarize, we have shown that C abcd = 0 implies.36, which in turn implies.36 and.37 with the help of the Bianchi identities, meaning that the space is conformally flat.
21 Chapter 3 Conformal Einstein equations and classical results In this chapter we will define conformal Einstein spaces and derive the conformal Einstein equations in n dimensions. We will also give a short summary of the classical results due to Brinkmann [6] and Schouten [36]. 3.1 Einstein spaces Definition An n-dimensional space is said to be an Einstein space if the trace-free part of the Ricci tensor is identically zero, i.e. R ab 1 n g abr = Expressing this condition using the P ab tensor we get an expression having the same algebraic structure n P ab 1 n g abp = 0, 3. and we also note that in an Einstein space.5 becomes 1 P ab = nn 1 g abr. 3.3 From the contracted Bianchi identity.6 we find for an Einstein space that 0 = a n R ab b R = b R n, 3.4 i.e. the Ricci scalar must be constant.
22 14 3. Conformal Einstein spaces Definition An n-dimensional space with metric g ab is a conformal Einstein space or conformally Einstein if there exists a conformal transformation ĝ ab = Ω g ab such that in the conformal space with metric ĝ ab Rab 1 = 0 nĝab, 3.5 or equivalently P ab 1 P = 0 nĝab. 3.6 Note that from. and 3.4 we have that R = Ω R + n 1 c Υ c + n 1n Υ c Υ c = constant, 3.7 where we used the notation introduced in Chapter.. Further, equation 3.5 is equivalent to and 3.6 to R ab 1 n g n abr + n a Υ b g ab c Υ c n n n Υ a Υ b + g ab Υ c Υ c = 0, 3.8 n P ab 1 n g abp a Υ b + 1 n g ab c Υ c + Υ a Υ b 1 n g abυ c Υ c = respectively. 3.8 or 3.9 is often referred to as the n-dimensional conformal Einstein equations. Taking a derivative of 3.7 gives the relations 0 = a R RΥ a 4n 1Υ a c Υ c n 1n Υ a Υ c Υ c + n 1 a c Υ c + n 1n Υ c a Υ c = a P P Υ a + Υ a c Υ c + n Υ a Υ c Υ c a c Υ c n Υ c a Υ c 3.10 and using this, the first integrability condition of 3.9 is calculated to be or, using.9, [a P b]c + 1 C abcdυ d = 0, 3.11 d C abcd + n 3Υ d C abcd = Taking another derivative and using.9 again we have b [a P b]c + 1 P bd C abcd + n 4Υ b [a P b]c =
23 15 and clearly both 3.11 and 3.13 are necessary conditions for a space to be conformally Einstein. Obviously we could get additional necessary conditions by taking higher derivatives. The last equation 3.13 can also be written b d C abcd n 3 n Rbd C abcd n 3n 4Υ b Υ d C abcd = and in dimension n = 4 this condition 3.14 reduces to a condition only on the geometry, and is independent of Υ a. This condition, B ac b d C abcd 1 Rbd C abcd = 0, 3.15 is a necessary, but not a sufficient, condition for a four-dimensional space to be conformally Einstein. Note that if 3.8 holds for any vector field K a, R ab 1 n g n abr + n K a K b g ab c K c n n n K a K b + g ab K c K c = 0, 3.16 n and remembering that R ab is symmetric, then by antisymmetrising we get [a K b] = 0, i.e. that K a is locally a gradient. Hence we have that a space is locally a conformal Einstein space if and only if 3.16 holds for some vector field K a. Given that P ab is defined by.5 this same statement also holds for P ab 1 n g abp a K b + 1 n g ab c K c + K a K b 1 n g abk c K c = The classical results In 194 Brinkmann [6] found necessary and sufficient conditions for a space to be conformally Einstein. In his approach he derived a large set of differential equations involving Υ a and by exploiting both existence and compatibility of this derived set he was able to formulate necessary and sufficient conditions. However, from his results it is hard to get a constructive set of necessary and sufficient conditions, and his results are not very useful in practice. Brinkmann later also studied in detail some special cases of conformal Einstein spaces [7].
24 16 Schouten [36] used a slightly different approach and looked directly at the explicit form of integrability condition for the conformal Einstein equations, but he did not go beyond Brinkmann s results as regards sufficient conditions. Schouten found the necessary condition 3.13 which we will return to in the following chapters.
25 Chapter 4 The Bach tensor in four dimensions and possible generalizations In this chapter we will take a closer look at the four-dimensional version of the conformal Einstein equations introduced in the previous chapter. We will consider the Bach tensor B ab and derive and discuss its properties. A theorem stating that in n dimensions there only exists three independent symmetric, divergence-free -index tensors U ab, V ab and W ab quadratic in the Riemann curvature tensor is proven, extending a result due to Balfagón and Jaén []. The properties of these tensors are investigated, and we obtain the new result that B ab = 1 U ab V ab. We also seek possible generalizations of the Bach tensor in n dimensions. 4.1 The Bach tensor in four dimensions From 3.8, 3.9 in the previous chapter we know that in four dimensions the conformal Einstein equations are or R ab 1 4 g abr + a Υ b g ab c Υ c Υ a Υ b + g ab Υ c Υ c = P ab 1 4 g abp a Υ b g ab c Υ c + Υ a Υ b 1 4 g abυ c Υ c = 0. 4.
26 18 The necessary conditions 3.11 and 3.13 in four dimensions become [a P b]c + 1 C abcdυ d = and b [a P b]c + 1 P bd C abcd = respectively. The left hand side of this last equation 4.4 defines, as in 3.15, the tensor B ac, which can also be written as B ac = b [a P b]c + 1 P bd C abcd, 4.5 B ac = b d C abcd 1 Rbd C abcd, 4.6 and we see that the necessary condition 4.4 then can be formulated as B ac = 0. The tensor B ab is called the Bach tensor and was first discussed by Bach [1]. As seen above, the origin of the Bach tensor is in an integrability condition for a four-dimensional space to be conformal to an Einstein space. The Bach tensor is a tensor built up from pure geometry, and thereby captures necessary features of a space being conformally Einstein in an intrinsic way. It is obvious from the definition of B ab 4.6 that the Bach tensor is symmetric, trace-free and quadratic in the Riemann curvature tensor. Definition A tensor is said to be quadratic in the Riemann curvature tensor if it is a linear combination of products of two Riemann curvature tensors and/or a linear combination of second derivatives of the Riemann curvature tensor [11]. Calculating the divergence of B ab from 4.6 we get, after twice using.5 to switch the order of the derivatives, c B ac = 1 3 R a c c R + R bc c R ba 1 Rbc a R bc + 1 b c c R ba R aebd b R de 1 1 a c c R 1 6 c a c R =0 4.7 i.e. B ab is divergence-free 1. 1 This was first noted by Hesselbach [].
27 19 Under a conformal transformation, ĝ ab = Ω g ab, we see after some calculation that B ac = b d Ĉ abc d 1 R b dĉabc d = b d C abc d + Υ d C abc d 1 Ω R b d + b Υ d + δ b d c Υ c Υ b Υ d + δ b dυ c Υ c C abc d = Ω b d C abcd 1 Rbd C abcd = Ω B ac 4.8 so that B ac is conformally weighted with weight. We can also express the Bach tensor 4.6 in an alternative form in terms of the Weyl tensor, the Ricci tensor and the Ricci scalar, using the fourdimensional version of.10, b d C abcd = 1 R ac; b b 1 6 R ;ac 1 1 g acr ; b b + R ab R b c 1 Rbd C abcd 1 3 RR ac 1 4 g acr bd R bd g acr, 4.9 and we find B ac = 1 R ac;b b 1 1 g acr ;b b 1 6 R ;ac R bd C abcd + R ab R b c 1 4 g acr db R bd 1 3 RR ac g acr For completeness we also give the Bach tensor expressed in spinor language B ab = B AA BB = C A D B + ΦCD A B ΨABCD To summarize, the Bach tensor B ab in four dimensions given by 4.5, 4.6, 4.10 or 4.11 is symmetric, trace-free, quadratic in the Riemann curvature tensor, divergence-free, and is conformally weighted with weight. Although it is only in four dimensions that the Bach tensor has been defined and has these nice properties, it is natural to ask if there is an n-dimensional counterpart to the Bach tensor. Unfortunately as we shall see in the next section, it is easy to show that if we simply carry over the form of the Bach tensor given in 4.6 or 4.10 into n > 4 dimensions, it does not retain all these useful properties. So, in the subsequent sections we look to see if there is a generalization which retains as many as possible of the useful properties that the Bach tensor has in four dimensions. This result is originally due to Haantjes and Schouten [0].
28 0 4. Attempts to find an n-dimensional Bach tensor Before we begin looking for an n-dimensional Bach tensor we note that, using the notation from Chapter., we can derive the two useful relations b d Ĉ abcd = Ω b d C abcd + n 3C abcd b Υ d + n 4Υ d b C abcd +n 4Υ b d C abcd + n 3n 5Υ b Υ d C abcd and 4.1 R bd Ĉ abcd = R bd C abcd + n C abcd b Υ d n C abcd Υ b Υ d, 4.13 which are used extensively in this chapter. If we simply carry over the tensor in 4.6 to arbitrary n dimensions, and label this tensor B 1 ac, we find that Bac; a n 4 = 1 n B 1 ac = b d C abcd 1 Rbd C abcd, 4.14 C abcd R bd ; a n 1 Ra cr ;a + n 1 RR ;c n 3 R ab R bc;a R bd R bd;c n 4.15 and B 1 ac = Ω Bac + n 4 Υ b d C abcd + Υ d b C abcd C abcd b Υ d + n 7C abcd Υ b Υ d, 4.16 i.e., that its divergence-free and conformally well behaved properties do in general not carry over into n > 4 dimensions. Similarly the alternative form from 4.10, B ac = 1 R ac;b b 1 1 g acr ;b b 1 6 R ;ac R bd C abcd + R ab R b c 1 4 g acr db R bd 1 3 RR ac g acr, 4.17
29 1 also fails to have these properties in general in n > 4 dimensions since Bac; a = R ab R bc;a R bd R bd;c and n 4 n B bc =Ω B bc + n 4Ω n 3n C abcd Υ a Υ d + n 3 + 3n 5 Υ b c R + Υ c b R 4n 1 n + 1 6n 1 Ra cr ;a + n Υ a d C abcd 3n 10 ΘR bc 6 g bc R ad Θ ad Θ ab R a c R ab Θ a c + 3n 10n + 5 RΘ bc + 6n 1 n 3n + 3n 16n n 5 ΘΘ bc 3 3n 5 g bc a a Θ n 1 RR ;c 4.18 n Υ d a C abcd 8n n 1 RΥ bυ c n 73n 5 g bc Υ a a R 1n 1 3n 5n 7 g bc RΥ a Υ a 1n 1 Θ ab Θ a c 1 g bcθ ad Θ ad 3n 10n g bc RΘ 6 Υ c b Θ + Υ b c Θ 1 3 b c Θ n 73n 5 g bc ΘΥ a Υ a Υ a a Θ 6 + 3n 5ΘΥ b Υ c 6n 41n + 53 g bc Θ where Θ ab = a Υ b Υ a Υ b + 1 g abυ c Υ c and Θ = Θ a a = a Υ a + n Υ a Υ a. Going back to the origins of the Bach tensor as an integrability condition for conformal Einstein spaces 3.14, b d C abcd suggests considering the tensor n 3 n Rbd C abcd n 3n 4Υ b Υ d C abcd = Bac = b d n 3 C abcd 3 n Rbd C abcd = B 1 ac n 4 n Rbd C abcd. 4.1 But once again, for dimensions n > 4, we see that a n 4n 3 B c;a = 3 n R ab R bc;a R bd R bd;c 1 1 n 1 Ra cr ;a + n 1 RR ;c 4.
30 and [ B 3 ac = Ω Bac + n 4 Υ b d C abcd + Υ d b C abcd 3 + n 3C abcd Υ b Υ d] 4.3 so this tensor is neither divergence-free nor conformally well-behaved in general. So we need to look elsewhere for possible generalization of the Bach tensor to n > 4 dimensions. 4.3 The tensors U ab, V ab and W ab Many years ago Gregory [17] discovered two symmetric divergence-free tensors in four dimensions, and later Collinson [11] added a third. Recently Robinson [34] and Balfagón and Jaén [] have shown that these three tensors have direct counterparts in n > 4 dimension with the same properties. We shall first of all show that these three tensors, U ab, V ab and W ab, are the only three tensors with these properties and then also examine in more detail their structure and properties. Balfagón and Jaén [] have proven the following theorem 3 : Theorem In an arbitrary n-dimensional semi-riemannian manifold: a There exist 14 independent and quadratic in Riemann, four-index divergence-free tensors. b There are no totally symmetric, quadratic in Riemann, and divergencefree four-index tensors c The complete family of quadratic in Riemann, and divergence-free 3 Note that this is a quotation from their paper [], but here expressed using our conventions.
31 3 four-index tensors T abcd totally symmetric in bcd is T abcd = a S T abcd S + a R T abcd R ; 4.4 T abcd S = Q abcd ; 4.5 Q abcd = 1 3 gac R d ir ib R ab;dc Rbd;ac 4 3 gac R bd;i i + g ac R b i ;di Rb ir acid + R aibj R c i d j 1 gac R ij d k R ijbk 4.6 T abcd R = X ab g cd ; 4.7 X ab = KU ab + LV ab 1 4 W ab 4.8 U ab = G ab;s s G sb;a s + G a pr pb 1 gab G pq R pg 4.9 V ab = R ;ab + g ab R ;s s RS ab 4.30 W ab = G apqr R b pqr 1 4 gab G mpqr R mpqr 4.31 G a c = G ab cb 4.3 G ab cd = R ab cd 4δ [a [c Sb] d] 4.33 S ab = 1 4 gab R 4.34 where a S, a R, K and L are four independent constants 4. First note that there are some differences in signs in 4.6, 4.9 and 4.30 compared to the original definitions in []. This is due to our definition of the Riemann curvature tensor.1 which differs from the one in []. A change of convention makes the change R abcd R abcd, meaning that there is only going to be a difference in sign for the terms created from a odd number of Riemann tensors e.g. here exactly only one. However, our definition agrees with the one in [34] up to an overall sign. Secondly, by a divergence-free four index tensor Balfagón and Jaén mean a tensor T abcd such that a T abcd = 0, i.e. a tensor divergence-free on the first index. Hence, in Theorem a states that there exist only 14 independent such tensors, b states that none of these are totally symmetric, T abcd = T abcd and c gives all tensors T abcd such that T abcd = T abcd and a T abcd = 0. 4 T R is the tensor found by Robinson [34] and T S the tensor found by Sachs [35].
32 4 Note that Theorem implies that the tensors U ab, V ab and W ab are all divergence-free. This follows from the divergence-free property and the construction of the tensor T abcd R via 4.7 and 4.8. We are specially interested in the tensors U ab, V ab and W ab, and to see their inner structure and their properties we write them out in terms of the Riemann curvature tensor, the Ricci tensor and the Ricci scalar, U bc =n 3R abcd R ad n 3R a n 3 bc;a + g bc R ad R ad n 3 + n 3RR cb + g bc R a n 3 ;a g bc R, V bc = R ;bc + g bc R ;a a RR bc g bcr, 4.36 W bc =R b ade R cade 1 4 g bcr fade R fade + R ad R abcd + RR bc R ab R a c + g bc R ad R ad 1 4 g bcrr When we later study the conformal behavior of U ab, V ab and W ab it is useful to have them expressed in terms of the Weyl tensor, the Ricci tensor and the Ricci scalar, U bc =n 3R ad C abcd + n 3R a n 3 a bc;a g bc R ;a n 6n 3 + g bc R de R de 4n 3 + n n R abr a c + n 3n 5n + n 1n RR bc n 3n 3n 6 g bc R, 4n 1n 4.38 V bc = R ;bc + g bc R ;d d RR bc g bcr, 4.39 W bc = C ade b C cade 1 4 g bcc fade n 4 C fade + n C abcdr ad n 3n 4 n R ab R a c + n 3n 4 n g bc R ad R ad nn 3n 4 + n 1n RR n + n 3n 4 bc 4n 1n g bc R From or it is obvious that U ab, V ab and W ab are symmetric and quadratic in the Riemann curvature tensor in all
33 5 dimensions. We note from 4.40 that W bc = 0 in four dimensions because in four dimensions C b ade C cade 1 4 g bcc fade C fade = 0, see Appendix B. This fact was not noticed by Collinson [11] but was subsequently pointed out in [34] and []. By taking the trace of we have U a n 3n a = R a ;a + W a a = n 4n 3 R ad R ad n 4n 3 R, V a a = n 1R ;a a + n 4 C fade C fade + 4 n 4 R, n 3n 4 R ad R ad n nn 3n 4 4n 1n R A simple direct calculation would confirm that U ab, V ab and W ab are all divergence-free, but we have already noted that this can be deduced from Theorem It is easily checked that the three tensors U ab, V ab and W ab are independent and an obvious question is whether there are any more such tensors; we shall now show that there are not. Given any symmetric and divergence-free tensor, Y ab, quadratic in the Riemann curvature tensor we see that the tensor Y ab g cd is a four-index tensor which is totally symmetric over bcd, quadratic in the Riemann curvature tensor and divergence-free on the first index. Hence we know from Theorem c, that there exist constants a S, and a R such that Y ab g cd = a S T abcd S + a R T abcd R 4.44 holds. Taking the trace over c and d of 4.44 using the facts that g cd TR abcd = g cd X abcd = g cd KU ab + LV ab W ab g cd = n + KU ab + LV ab W ab, 4.45 where K and L are constants fixed by 4.44, and g cd T abcd S =g cd Q abcd = 4 3 Ra cr bc R ; ab 14 9 Rab ; c c 1 9 gab R ;d d R cdr adbc Radef R b def 1 9 gab R cd R cd 1 6 gab R cdef R cdef, 4.46
34 6 noting that 4.46 actually can be written as a linear combination of U ab, V ab and W ab, g cd T abcd S = 14 9n 3 U ab 8 9 V ab + 3 W ab, 4.47 we find that Y ab 14a S = a R K 9n n 3 ar a S 3n U ab + a R L 8a S V ab 9n W ab, 4.48 i.e., that Y ab is a linear combination of U ab, V ab and W ab. We summarize this result in the following theorem Theorem In an n-dimensional space there are only three independent symmetric and divergence-free -index tensors quadratic in the Riemann curvature tensor, e.g., U ab, V ab and W ab. Before we investigate the relations between the four-dimensional Bach tensor and U ab, V ab and W ab we note that under the conformal transformation ĝ ab = Ω g ab the tensors U ab, V ab and W ab transform according to [ Û bc =Ω U bc + Ω n 3 n C abcd Θ ad + n 6Υ a a R bc + Υ a c R ab + Υ a b R ac + R bc n 4Θ n 4Υ a Υ a + R ab 4Θ a c nυ a Υ c + R ac 4Θ a b nυ a Υ b + g bc R ef n 6Θ ef + Υ e Υ f Υ b c R Υ c b R n 6 + g bc R g bc Υ a a R + R n 5n + Θ + n 1 Υ b Υ c + n 5n + n 1 n 6 Υ a Υ a + g bc n Υ e Υ f Θ ef n a a Θ Θ bc + n n 6Υ a Υ a n n 11 Θ Θ n n 6 + Θ ad Θ ad n n 6Υ a a Θ + n a a Θ bc + n 6n Υ a a Θ bc n Υ b a Θ ac n Υ c a Θ ab + n Υ a b Θ ac + n Υ a c Θ ab + Θ ab 4n Θ a c nn Υ c Υ a nn Θ ac Υ a Υ b
35 7 + Θ bc n n 4Θ n n 4Υ a Υ a + n ΘΥ b Υ c ], 4.49 V bc =Ω V bc + Ω [ n 1ΘR bc 6RΥ c Υ b n 4RΘ bc n 7g bc RΥ a Υ a + n 4g bc RΘ + 3Υ c b R + 3Υ b c R + n 7g bc Υ a a R n 1 b c Θ + n 1g bc a a Θ + 6n 1Υ c b Θ + 6n 1Υ b c Θ + n 1n 7g bc Υ a a Θ + n 1n 7g bc Θ n 1n 7Θg bc Υ a Υ a 1n 1ΘΥ b Υ c n 1n 4ΘΘ bc ], 4.50 [ Ŵ bc =Ω W bc + Ω n 4 C abcd Θ ad n 3 n R abθ a c n 3 n R acθ a b n 3Θ ab Θ a c n 3 + n g bcθ ad R ad + n 3g bc Θ ad Θ ad nn 3 + n 1n RΘ n 3 bc + n R bcθ + n 3Θ bc Θ n 3g bc Θ nn 3 ] n 1n g bcrθ, 4.51 where Θ ab = a Υ b Υ a Υ b + 1 g abυ c Υ c and Θ = Θ a a = a Υ a + n Υ a Υ a. It is easily seen from that U bc and V bc are not conformally well-behaved in general in four dimensions where W bc = Four-dimensional Bach tensor expressed in U ab, V ab and W ab Since U ab, V ab and W ab constitute a basis for all -index symmetric divergence-free tensors quadratic in the Riemann curvature tensor, and the Bach tensor B ab has these properties, we must be able to express the Bach tensor 4.6 in four dimensions in terms of U ab and V ab, remembering W ab = 0 in four dimensions.
36 8 In four dimensions we know from U bc =R bc;a a 1 g bcr ;a a + C abcd R ad + R ab R c a 1 g bcr ad R ad 1 3 RR bc g bcr 4.5 V bc = R ;bc + g bc R ;a a RR bc g bcrr 4.53 Comparing these equations with 4.10 we conclude that we have the relation B bc = 1 U bc V bc The numerical relationship between the tensors V bc and U bc could also be found using the trace-free property of the Bach tensor. Making the ansatz B bc = αu bc + βv bc 4.55 we see from 4.41 and 4.4 in four dimensions that B b b = αr ;b b + 3βR ;b b = α 3βR ;b b = and hence in general we must have 3α = β. This link between the Bach tensor B ab and U ab and V ab in four dimensions does not seem to have been noted before. 4.5 An n-dimensional tensor expressed in U ab, V ab and W ab If we consider the tensor B bc = 1 U bc V bc 4.57 in n > 4 dimensions, it is clearly divergence-free due to the properties of U ab and V ab, but when we examine its conformal properties we find, after a lot of work and rearranging, Ûbc 6 V bc = Ω 1 U bc V bc + n 4Ω [ 1 n n 3C abcdυ a Υ d + n 3Υ a n 3 a R bc Υ a n 3 c R ab Υ a b R ac n 3 R ab Θ a n 3 c R ac Θ a n 3 b + g bc R ad Θ ad
37 9 3n 7 + R bc Θ 1 3 Υ b c R 1 Υ 3n 17 c b R g bc Υ a a R 1 3n 10 n 73n 5 g bc RΘ + RΥ b Υ c 6 n 1 n 73n 5 + g bc RΥ a Υ a n n 3 + g bc Θ ad Θ ad 1n 1 4 n n 3 n Θ ab Θ a c + 3n ΘΘ bc 3 3n 5 3n 5 + b c Θ g bc a 3n 5 a Θ 3 Υ b c Θ 3n 5 3n 5 3 Υ c b Θ n 7 g bc Υ a a Θ 6 3n 5 3n 5 + ΘΥ b Υ c + n 7 g bc ΘΥ a Υ a 6 6n 41n + 53 g bc Θ ], again getting a tensor that is not conformally well-behaved except in four dimensions. The n-dimensional analogous integrability condition that gave rise to the four-dimensional Bach tensor is 3.14, b d C abcd n 3 n Rbd C abcd n 3n 4Υ b Υ d C abcd = and taking only the terms built up from pure geometry and quadratic in the Riemann curvature tensor, e.g., the first and second terms, suggests that we study B 3 ac = b d C abcd n 3 n Rbd C abcd Although we have already shown that this tensor is neither conformally well-behaved nor divergence-free in n > 4 dimensions, it will be instructive to investigate its relationship to the tensors U ab, V ab and W ab. Using.10 we can equivalently express B 3 ac in the decomposed form n 3 Bac = 3 n R ac; b b n 3 n 1 R ;ac n 3 n 1n g acr ; b b n 3n + n R abr b n 3 c n Rbd C abcd n 3 n g acr bd R bd nn 3 n 1n RR ac n 3 + n 1n g acr. 4.61
38 30 This clearly reduces to the ordinary Bach tensor in four dimensions, but to investigate its other properties we first try to express B 3 ac in terms of U ac, V ac and W ac. Doing this we find Bac = 1 3 n U n 3 ac + n 1 V ac 1 W ac + 1 bde Ca C cbde 1 4 g acc fbde C fbde n 4 n Rbd C abcd, 4.6 and as we have already noted and is easily confirmed directly this tensor is only divergence-free in four dimensions, and not for n > 4 dimensions. However, defining the tensor B 4 ac, Bac = Bac 1 bde Ca C cbde g acc fbde n 4 C fbde + n Rbd C abcd = Bac 1 bde Ca C cbde g acc fbde n 4 C fbde + n Rbd C abcd 1 = n U n 3 ac + n 1 V ac 1 W ac, 4.63 we indeed get a tensor quadratic in the Riemann curvature tensor which is symmetric and divergence-free in all dimensions, and it collapses to the original Bach tensor in four dimensions. To investigate the conformal properties of B 4 ac and thereby also the fourdimensional B ac we use and find [ B ac =Ω B ac + n 4 Υ b d C abcd + Υ d b C abcd C abcd b Υ d + n 4C abcd Υ b Υ d] From 4.64 we see that, in general, it is only in four dimensions that Bac is conformally well-behaved. Hence, in general, there is no obvious 4 n-dimensional symmetric and divergence-free tensor which is quadratic in the Riemann curvature tensor and generalizes the Bach tensor in four dimensions, which also is of good conformal weight. We can now ask more generally if it is possible to construct any n-dimensional -index tensor of good conformal weight from U ab, V ab and W ab, i.e. a tensor which is symmetric, divergence-free, quadratic in the Riemann curvature tensor and of good conformal weight. To investigate this we look at
39 31 αu ab + βv ab + γw ab, where α, β and γ are arbitrary constants, αûbc+β V bc + γŵbc = Ω αu bc + βv bc + γw bc [ + Ω n 4 γc abcd Θ ad αn n 3C abcd Υ a Υ d + αn 3 Υ a a R bc Υ a c R ab Υ a b R ac 1 + n 3 α β n 3 + γ n RΘ bc n 1n n 1 + n 3 αn β n 3 + γ ΘΘ bc n 3 αn + γ Θ ab Θ a c n + n 3 α + γ g bc Θ ad Θ ad 1 + γ n 3 α + γ αn 3 β + γ n R bcθ R ab Θ a c + R ac Θ a b g bc R ad Θ ad n ] g bc RΘ + γn 3g bc Θ nn 3 n 1n + Ω [ 3 αn n 3 βn 1 n 1 RΥ bυ c n 7 + n 1 g bcrυ a Υ a + b c Θ + 6ΘΥ b Υ c n 7g bc Υ a a Θ ] 3 Υ b c Θ + Υ c b Θ g bc a a Θ + n 7g bc ΘΥ a Υ a + Ω [n n 1 3 αn 3 β R bc Θ n 3 n n 3n 11 α βn 1n 7 g bc Θ αn 3 3β Υ b c R αn 3 3β Υ c b R ] n 3n 6 α βn 7 g bc Υ a a R. 4.65
Self-dual conformal gravity
Self-dual conformal gravity Maciej Dunajski Department of Applied Mathematics and Theoretical Physics University of Cambridge MD, Paul Tod arxiv:1304.7772., Comm. Math. Phys. (2014). Dunajski (DAMTP, Cambridge)
More informationHow to recognise a conformally Einstein metric?
How to recognise a conformally Einstein metric? Maciej Dunajski Department of Applied Mathematics and Theoretical Physics University of Cambridge MD, Paul Tod arxiv:1304.7772., Comm. Math. Phys. (2014).
More informationNCORRECTED PROOF. Obstructions to conformally Einstein metrics in n dimensions. A. Rod Gover a, Paweł Nurowski b,1. Abstract
Journal of Geometry and Physics xxx (2005) xxx xxx 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Abstract Obstructions to conformally Einstein metrics in n dimensions A. Rod Gover a, Paweł
More informationHow to recognize a conformally Kähler metric
How to recognize a conformally Kähler metric Maciej Dunajski Department of Applied Mathematics and Theoretical Physics University of Cambridge MD, Paul Tod arxiv:0901.2261, Mathematical Proceedings of
More informationarxiv:gr-qc/ v1 19 Feb 2003
Conformal Einstein equations and Cartan conformal connection arxiv:gr-qc/0302080v1 19 Feb 2003 Carlos Kozameh FaMAF Universidad Nacional de Cordoba Ciudad Universitaria Cordoba 5000 Argentina Ezra T Newman
More informationWeek 6: Differential geometry I
Week 6: Differential geometry I Tensor algebra Covariant and contravariant tensors Consider two n dimensional coordinate systems x and x and assume that we can express the x i as functions of the x i,
More informationWeek 9: Einstein s field equations
Week 9: Einstein s field equations Riemann tensor and curvature We are looking for an invariant characterisation of an manifold curved by gravity. As the discussion of normal coordinates showed, the first
More informationIrreducible Killing Tensors from Conformal Killing Vectors
Proceedings of Institute of Mathematics of NAS of Ukraine 2004, Vol. 50, Part 2, 708 714 Irreducible Killing Tensors from Conformal Killing Vectors S. Brian EDGAR, Rafaelle RANI and Alan BARNES Department
More informationDifferential Geometry MTG 6257 Spring 2018 Problem Set 4 Due-date: Wednesday, 4/25/18
Differential Geometry MTG 6257 Spring 2018 Problem Set 4 Due-date: Wednesday, 4/25/18 Required problems (to be handed in): 2bc, 3, 5c, 5d(i). In doing any of these problems, you may assume the results
More information12 th Marcel Grossman Meeting Paris, 17 th July 2009
Department of Mathematical Analysis, Ghent University (Belgium) 12 th Marcel Grossman Meeting Paris, 17 th July 2009 Outline 1 2 The spin covariant derivative The curvature spinors Bianchi and Ricci identities
More informationSolutions of Penrose s equation
JOURNAL OF MATHEMATICAL PHYSICS VOLUME 40, NUMBER 1 JANUARY 1999 Solutions of Penrose s equation E. N. Glass a) Physics Department, University of Michigan, Ann Arbor, Michigan 48109 Jonathan Kress School
More informationarxiv: v3 [math.dg] 13 Mar 2011
GENERALIZED QUASI EINSTEIN MANIFOLDS WITH HARMONIC WEYL TENSOR GIOVANNI CATINO arxiv:02.5405v3 [math.dg] 3 Mar 20 Abstract. In this paper we introduce the notion of generalized quasi Einstein manifold,
More information7 Curvature of a connection
[under construction] 7 Curvature of a connection 7.1 Theorema Egregium Consider the derivation equations for a hypersurface in R n+1. We are mostly interested in the case n = 2, but shall start from the
More informationThe Einstein field equations
The Einstein field equations Part II: The Friedmann model of the Universe Atle Hahn GFM, Universidade de Lisboa Lisbon, 4th February 2010 Contents: 1 Geometric background 2 The Einstein field equations
More informationGeneral tensors. Three definitions of the term V V. q times. A j 1...j p. k 1...k q
General tensors Three definitions of the term Definition 1: A tensor of order (p,q) [hence of rank p+q] is a multilinear function A:V V }{{ V V R. }}{{} p times q times (Multilinear means linear in each
More information5 Constructions of connections
[under construction] 5 Constructions of connections 5.1 Connections on manifolds and the Levi-Civita theorem We start with a bit of terminology. A connection on the tangent bundle T M M of a manifold M
More informationRepresentation theory and the X-ray transform
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 1/18 Representation theory and the X-ray transform Differential geometry on real and complex projective
More informationCitation for published version (APA): Halbersma, R. S. (2002). Geometry of strings and branes. Groningen: s.n.
University of Groningen Geometry of strings and branes Halbersma, Reinder Simon IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please
More information2-Form Gravity of the Lorentzian Signature
2-Form Gravity of the Lorentzian Signature Jerzy Lewandowski 1 and Andrzej Oko lów 2 Instytut Fizyki Teoretycznej, Uniwersytet Warszawski, ul. Hoża 69, 00-681 Warszawa, Poland arxiv:gr-qc/9911121v1 30
More informationAXIOMATIC AND COORDINATE GEOMETRY
AXIOMATIC AND COORDINATE GEOMETRY KAPIL PARANJAPE 1. Introduction At some point between high school and college we first make the transition between Euclidean (or synthetic) geometry and co-ordinate (or
More informationHOMOGENEOUS AND INHOMOGENEOUS MAXWELL S EQUATIONS IN TERMS OF HODGE STAR OPERATOR
GANIT J. Bangladesh Math. Soc. (ISSN 166-3694) 37 (217) 15-27 HOMOGENEOUS AND INHOMOGENEOUS MAXWELL S EQUATIONS IN TERMS OF HODGE STAR OPERATOR Zakir Hossine 1,* and Md. Showkat Ali 2 1 Department of Mathematics,
More informationFrom holonomy reductions of Cartan geometries to geometric compactifications
From holonomy reductions of Cartan geometries to geometric compactifications 1 University of Vienna Faculty of Mathematics Berlin, November 11, 2016 1 supported by project P27072 N25 of the Austrian Science
More informationCurvature homogeneity of type (1, 3) in pseudo-riemannian manifolds
Curvature homogeneity of type (1, 3) in pseudo-riemannian manifolds Cullen McDonald August, 013 Abstract We construct two new families of pseudo-riemannian manifolds which are curvature homegeneous of
More informationclass # MATH 7711, AUTUMN 2017 M-W-F 3:00 p.m., BE 128 A DAY-BY-DAY LIST OF TOPICS
class # 34477 MATH 7711, AUTUMN 2017 M-W-F 3:00 p.m., BE 128 A DAY-BY-DAY LIST OF TOPICS [DG] stands for Differential Geometry at https://people.math.osu.edu/derdzinski.1/courses/851-852-notes.pdf [DFT]
More informationBondi mass of Einstein-Maxwell-Klein-Gordon spacetimes
of of Institute of Theoretical Physics, Charles University in Prague April 28th, 2014 scholtz@troja.mff.cuni.cz 1 / 45 Outline of 1 2 3 4 5 2 / 45 Energy-momentum in special Lie algebra of the Killing
More informationNewman-Penrose formalism in higher dimensions
Newman-Penrose formalism in higher dimensions V. Pravda various parts in collaboration with: A. Coley, R. Milson, M. Ortaggio and A. Pravdová Introduction - algebraic classification in four dimensions
More informationNOTE ON ASYMPTOTICALLY CONICAL EXPANDING RICCI SOLITONS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 NOTE ON ASYMPTOTICALLY CONICAL EXPANDING RICCI SOLITONS JOHN LOTT AND PATRICK WILSON (Communicated
More information1.13 The Levi-Civita Tensor and Hodge Dualisation
ν + + ν ν + + ν H + H S ( S ) dφ + ( dφ) 2π + 2π 4π. (.225) S ( S ) Here, we have split the volume integral over S 2 into the sum over the two hemispheres, and in each case we have replaced the volume-form
More informationNew Geometric Formalism for Gravity Equation in Empty Space
New Geometric Formalism for Gravity Equation in Empty Space Xin-Bing Huang Department of Physics, Peking University, arxiv:hep-th/0402139v2 23 Feb 2004 100871 Beijing, China Abstract In this paper, complex
More informationThe N = 2 Gauss-Bonnet invariant in and out of superspace
The N = 2 Gauss-Bonnet invariant in and out of superspace Daniel Butter NIKHEF College Station April 25, 2013 Based on work with B. de Wit, S. Kuzenko, and I. Lodato Daniel Butter (NIKHEF) Super GB 1 /
More informationStructure Groups of Pseudo-Riemannian Algebraic Curvature Tensors
Structure Groups of Pseudo-Riemannian Algebraic Curvature Tensors Joseph Palmer August 20, 2010 Abstract We study the group of endomorphisms which preserve given model spaces under precomposition, known
More informationPerelman s Dilaton. Annibale Magni (TU-Dortmund) April 26, joint work with M. Caldarelli, G. Catino, Z. Djadly and C.
Annibale Magni (TU-Dortmund) April 26, 2010 joint work with M. Caldarelli, G. Catino, Z. Djadly and C. Mantegazza Topics. Topics. Fundamentals of the Ricci flow. Topics. Fundamentals of the Ricci flow.
More informationAs always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing).
An Interlude on Curvature and Hermitian Yang Mills As always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing). Suppose we wanted
More informationNon-Abelian and gravitational Chern-Simons densities
Non-Abelian and gravitational Chern-Simons densities Tigran Tchrakian School of Theoretical Physics, Dublin nstitute for Advanced Studies (DAS) and Department of Computer Science, Maynooth University,
More informationFirst structure equation
First structure equation Spin connection Let us consider the differential of the vielbvein it is not a Lorentz vector. Introduce the spin connection connection one form The quantity transforms as a vector
More informationThe Divergence Myth in Gauss-Bonnet Gravity. William O. Straub Pasadena, California November 11, 2016
The Divergence Myth in Gauss-Bonnet Gravity William O. Straub Pasadena, California 91104 November 11, 2016 Abstract In Riemannian geometry there is a unique combination of the Riemann-Christoffel curvature
More informationCS 468 (Spring 2013) Discrete Differential Geometry
CS 468 (Spring 2013) Discrete Differential Geometry Lecture 13 13 May 2013 Tensors and Exterior Calculus Lecturer: Adrian Butscher Scribe: Cassidy Saenz 1 Vectors, Dual Vectors and Tensors 1.1 Inner Product
More informationGeneralized exterior forms, geometry and space-time
Generalized exterior forms, geometry and sace-time P Nurowski Instytut Fizyki Teoretycznej Uniwersytet Warszawski ul. Hoza 69, Warszawa nurowski@fuw.edu.l D C Robinson Mathematics Deartment King s College
More informationNew Geometric Formalism for Gravity Equation in Empty Space
New Geometric Formalism for Gravity Equation in Empty Space Xin-Bing Huang Department of Physics, Peking University, arxiv:hep-th/0402139v3 10 Mar 2004 100871 Beijing, China Abstract In this paper, complex
More informationBlack Holes, Thermodynamics, and Lagrangians. Robert M. Wald
Black Holes, Thermodynamics, and Lagrangians Robert M. Wald Lagrangians If you had asked me 25 years ago, I would have said that Lagrangians in classical field theory were mainly useful as nmemonic devices
More informationLECTURE 8: THE SECTIONAL AND RICCI CURVATURES
LECTURE 8: THE SECTIONAL AND RICCI CURVATURES 1. The Sectional Curvature We start with some simple linear algebra. As usual we denote by ( V ) the set of 4-tensors that is anti-symmetric with respect to
More informationCALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =
CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.
More informationThe Matrix Representation of a Three-Dimensional Rotation Revisited
Physics 116A Winter 2010 The Matrix Representation of a Three-Dimensional Rotation Revisited In a handout entitled The Matrix Representation of a Three-Dimensional Rotation, I provided a derivation of
More informationInvariant differential operators and the Karlhede classification of type N vacuum solutions
Class. Quantum Grav. 13 (1996) 1589 1599. Printed in the UK Invariant differential operators and the Karlhede classification of type N vacuum solutions M P Machado Ramos and J A G Vickers Faculty of Mathematical
More informationarxiv:gr-qc/ v3 5 Oct 2004
Unique characterization of the Bel-Robinson tensor arxiv:gr-qc/0403024v3 5 Oct 2004 G Bergqvist and P Lankinen 2 Matematiska institutionen Linköpings universitet SE-58 83 Linköping Sweden 2 Department
More informationSection 2. Basic formulas and identities in Riemannian geometry
Section 2. Basic formulas and identities in Riemannian geometry Weimin Sheng and 1. Bianchi identities The first and second Bianchi identities are R ijkl + R iklj + R iljk = 0 R ijkl,m + R ijlm,k + R ijmk,l
More informationIntrinsic Differential Geometry with Geometric Calculus
MM Research Preprints, 196 205 MMRC, AMSS, Academia Sinica No. 23, December 2004 Intrinsic Differential Geometry with Geometric Calculus Hongbo Li and Lina Cao Mathematics Mechanization Key Laboratory
More informationLecture II: Hamiltonian formulation of general relativity
Lecture II: Hamiltonian formulation of general relativity (Courses in canonical gravity) Yaser Tavakoli December 16, 2014 1 Space-time foliation The Hamiltonian formulation of ordinary mechanics is given
More informationFaraday Tensor & Maxwell Spinor (Part I)
February 2015 Volume 6 Issue 2 pp. 88-97 Faraday Tensor & Maxwell Spinor (Part I) A. Hernández-Galeana #, R. López-Vázquez #, J. López-Bonilla * & G. R. Pérez-Teruel & 88 Article # Escuela Superior de
More informationTensor Calculus, Part 2
Massachusetts Institute of Technology Department of Physics Physics 8.962 Spring 2002 Tensor Calculus, Part 2 c 2000, 2002 Edmund Bertschinger. 1 Introduction The first set of 8.962 notes, Introduction
More informationConformally Fedosov Manifolds
Workshop on Recent Developments in Conformal Geometry, Université de Nantes p. 1/14 Conformally Fedosov Manifolds Michael Eastwood [ joint work with Jan Slovák ] Australian National University Workshop
More informationUNIVERSITY OF DUBLIN
UNIVERSITY OF DUBLIN TRINITY COLLEGE JS & SS Mathematics SS Theoretical Physics SS TSM Mathematics Faculty of Engineering, Mathematics and Science school of mathematics Trinity Term 2015 Module MA3429
More informationOn the evolutionary form of the constraints in electrodynamics
On the evolutionary form of the constraints in electrodynamics István Rácz,1,2 arxiv:1811.06873v1 [gr-qc] 12 Nov 2018 1 Faculty of Physics, University of Warsaw, Ludwika Pasteura 5, 02-093 Warsaw, Poland
More informationConservation of energy and Gauss Bonnet gravity
Conservation of energy and Gauss Bonnet gravity Christophe Réal 22, rue de Pontoise, 75005 PARIS arxiv:gr-qc/0612139v2 7 Nov 2007 I dedicate this work to Odilia. November, 2007 Abstract In the present
More informationCONFORMAL CIRCLES AND PARAMETRIZATIONS OF CURVES IN CONFORMAL MANIFOLDS
proceedings of the american mathematical society Volume 108, Number I, January 1990 CONFORMAL CIRCLES AND PARAMETRIZATIONS OF CURVES IN CONFORMAL MANIFOLDS T. N. BAILEY AND M. G. EASTWOOD (Communicated
More informationLecture 3: Probability
Lecture 3: Probability 28th of October 2015 Lecture 3: Probability 28th of October 2015 1 / 36 Summary of previous lecture Define chance experiment, sample space and event Introduce the concept of the
More informationStress-energy tensor is the most important object in a field theory and have been studied
Chapter 1 Introduction Stress-energy tensor is the most important object in a field theory and have been studied extensively [1-6]. In particular, the finiteness of stress-energy tensor has received great
More informationTwistors and Conformal Higher-Spin. Theory. Tristan Mc Loughlin Trinity College Dublin
Twistors and Conformal Higher-Spin Tristan Mc Loughlin Trinity College Dublin Theory Based on work with Philipp Hähnel & Tim Adamo 1604.08209, 1611.06200. Given the deep connections between twistors, the
More informationSTRAND J: TRANSFORMATIONS, VECTORS and MATRICES
Mathematics SKE, Strand J STRAND J: TRANSFORMATIONS, VECTORS and MATRICES J4 Matrices Text Contents * * * * Section J4. Matrices: Addition and Subtraction J4.2 Matrices: Multiplication J4.3 Inverse Matrices:
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F ) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationElements of differential geometry
Elements of differential geometry R.Beig (Univ. Vienna) ESI-EMS-IAMP School on Mathematical GR, 28.7. - 1.8. 2014 1. tensor algebra 2. manifolds, vector and covector fields 3. actions under diffeos and
More information1.4 LECTURE 4. Tensors and Vector Identities
16 CHAPTER 1. VECTOR ALGEBRA 1.3.2 Triple Product The triple product of three vectors A, B and C is defined by In tensor notation it is A ( B C ) = [ A, B, C ] = A ( B C ) i, j,k=1 ε i jk A i B j C k =
More informationSolving Einstein s Equations: PDE Issues
Solving Einstein s Equations: PDE Issues Lee Lindblom Theoretical Astrophysics, Caltech Mathematical and Numerical General Relativity Seminar University of California at San Diego 22 September 2011 Lee
More informationA brief introduction to Semi-Riemannian geometry and general relativity. Hans Ringström
A brief introduction to Semi-Riemannian geometry and general relativity Hans Ringström May 5, 2015 2 Contents 1 Scalar product spaces 1 1.1 Scalar products...................................... 1 1.2 Orthonormal
More informationCharacterization of the Lovelock gravity
PRAMANA c Indian Academy of Sciences Vol. 74, No. 6 journal of June 2010 physics pp. 875 882 Characterization of the Lovelock gravity by Bianchi derivative NARESH DADHICH Inter-University Centre for Astronomy
More informationIs Every Invertible Linear Map in the Structure Group of some Algebraic Curvature Tensor?
Is Every Invertible Linear Map in the Structure Group of some Algebraic Curvature Tensor? Lisa Kaylor August 24, 2012 Abstract We study the elements in the structure group of an algebraic curvature tensor
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND First Printing, 99 Chapter LINEAR EQUATIONS Introduction to linear equations A linear equation in n unknowns x,
More informationMetrisability of Painleve equations and Hamiltonian systems of hydrodynamic type
Metrisability of Painleve equations and Hamiltonian systems of hydrodynamic type Felipe Contatto Department of Applied Mathematics and Theoretical Physics University of Cambridge felipe.contatto@damtp.cam.ac.uk
More informationOn Indefinite Almost Paracontact Metric Manifold
International Mathematical Forum, Vol. 6, 2011, no. 22, 1071-1078 On Indefinite Almost Paracontact Metric Manifold K. P. Pandey Department of Applied Mathematics Madhav Proudyogiki Mahavidyalaya Bhopal,
More informationRANDERS METRICS WITH SPECIAL CURVATURE PROPERTIES
Chen, X. and Shen, Z. Osaka J. Math. 40 (003), 87 101 RANDERS METRICS WITH SPECIAL CURVATURE PROPERTIES XINYUE CHEN* and ZHONGMIN SHEN (Received July 19, 001) 1. Introduction A Finsler metric on a manifold
More informationTHE INITIAL VALUE FORMULATION OF GENERAL RELATIVITY
THE INITIAL VALUE FORMULATION OF GENERAL RELATIVITY SAM KAUFMAN Abstract. The (Cauchy) initial value formulation of General Relativity is developed, and the maximal vacuum Cauchy development theorem is
More informationGravitation: Tensor Calculus
An Introduction to General Relativity Center for Relativistic Astrophysics School of Physics Georgia Institute of Technology Notes based on textbook: Spacetime and Geometry by S.M. Carroll Spring 2013
More informationChoice of Riemannian Metrics for Rigid Body Kinematics
Choice of Riemannian Metrics for Rigid Body Kinematics Miloš Žefran1, Vijay Kumar 1 and Christopher Croke 2 1 General Robotics and Active Sensory Perception (GRASP) Laboratory 2 Department of Mathematics
More informationSpectral Graph Theory Lecture 2. The Laplacian. Daniel A. Spielman September 4, x T M x. ψ i = arg min
Spectral Graph Theory Lecture 2 The Laplacian Daniel A. Spielman September 4, 2015 Disclaimer These notes are not necessarily an accurate representation of what happened in class. The notes written before
More informationa 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 11 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,, a n, b are given real
More informationMatrix Approach to Petrov Classification
Matrix Approach to Petrov Classification B. E. Carvajal-Gámez 1, J. López-Bonilla *2 & R. López-Vázquez 2 151 Article 1 SEPI-ESCOM, Instituto Politécnico Nacional (IPN), Av. Bátiz S/N, 07738, México DF
More informationFoundations of tensor algebra and analysis (composed by Dr.-Ing. Olaf Kintzel, September 2007, reviewed June 2011.) Web-site:
Foundations of tensor algebra and analysis (composed by Dr.-Ing. Olaf Kintzel, September 2007, reviewed June 2011.) Web-site: http://www.kintzel.net 1 Tensor algebra Indices: Kronecker delta: δ i = δ i
More informationTensors, and differential forms - Lecture 2
Tensors, and differential forms - Lecture 2 1 Introduction The concept of a tensor is derived from considering the properties of a function under a transformation of the coordinate system. A description
More informationSome Variations on Ricci Flow. Some Variations on Ricci Flow CARLO MANTEGAZZA
Some Variations on Ricci Flow CARLO MANTEGAZZA Ricci Solitons and other Einstein Type Manifolds A Weak Flow Tangent to Ricci Flow The Ricci flow At the end of 70s beginning of 80s the study of Ricci and
More informationClassical differential geometry of two-dimensional surfaces
Classical differential geometry of two-dimensional surfaces 1 Basic definitions This section gives an overview of the basic notions of differential geometry for twodimensional surfaces. It follows mainly
More informationQuintic Quasitopological Gravity
Quintic Quasitopological Gravity Luis Guajardo 1 in collaboration with Adolfo Cisterna 2 Mokthar Hassaïne 1 Julio Oliva 3 1 Universidad de Talca 2 Universidad Central de Chile 3 Universidad de Concepción
More informationPhysics 772 Peskin and Schroeder Problem 3.4.! R R (!,! ) = 1 ı!!
Physics 77 Peskin and Schroeder Problem 3.4 Problem 3.4 a) We start with the equation ı @ ım = 0. Define R L (!,! ) = ı!!!! R R (!,! ) = ı!! +!! Remember we showed in class (and it is shown in the text)
More informationChapter 2. Linear Algebra. rather simple and learning them will eventually allow us to explain the strange results of
Chapter 2 Linear Algebra In this chapter, we study the formal structure that provides the background for quantum mechanics. The basic ideas of the mathematical machinery, linear algebra, are rather simple
More informationGauge Fixing and Constrained Dynamics in Numerical Relativity
Gauge Fixing and Constrained Dynamics in Numerical Relativity Jon Allen The Dirac formalism for dealing with constraints in a canonical Hamiltonian formulation is reviewed. Gauge freedom is discussed and
More informationVector and Tensor Calculus
Appendices 58 A Vector and Tensor Calculus In relativistic theory one often encounters vector and tensor expressions in both three- and four-dimensional form. The most important of these expressions are
More informationChapter 7 Curved Spacetime and General Covariance
Chapter 7 Curved Spacetime and General Covariance In this chapter we generalize the discussion of preceding chapters to extend covariance to more general curved spacetimes. 145 146 CHAPTER 7. CURVED SPACETIME
More informationChapter XI Novanion rings
Chapter XI Novanion rings 11.1 Introduction. In this chapter we continue to provide general structures for theories in physics. J. F. Adams proved in 1960 that the only possible division algebras are at
More informationParticles I, Tutorial notes Sessions I-III: Roots & Weights
Particles I, Tutorial notes Sessions I-III: Roots & Weights Kfir Blum June, 008 Comments/corrections regarding these notes will be appreciated. My Email address is: kf ir.blum@weizmann.ac.il Contents 1
More informationFaculty of Engineering, Mathematics and Science School of Mathematics
Faculty of Engineering, Mathematics and Science School of Mathematics JS & SS Mathematics SS Theoretical Physics SS TSM Mathematics Module MA3429: Differential Geometry I Trinity Term 2018???, May??? SPORTS
More information,, rectilinear,, spherical,, cylindrical. (6.1)
Lecture 6 Review of Vectors Physics in more than one dimension (See Chapter 3 in Boas, but we try to take a more general approach and in a slightly different order) Recall that in the previous two lectures
More informationη = (e 1 (e 2 φ)) # = e 3
Research Statement My research interests lie in differential geometry and geometric analysis. My work has concentrated according to two themes. The first is the study of submanifolds of spaces with riemannian
More informationarxiv: v2 [math.dg] 1 Aug 2015
AN INTRODUCTION TO CONFORMAL GEOMETRY AND TRACTOR CALCULUS, WITH A VIEW TO APPLICATIONS IN GENERAL RELATIVITY SEAN CURRY & A. ROD GOVER arxiv:1412.7559v2 [math.dg] 1 Aug 2015 Abstract. The following are
More informationCHAPTER 1 PRELIMINARIES
CHAPTER 1 PRELIMINARIES 1.1 Introduction The aim of this chapter is to give basic concepts, preliminary notions and some results which we shall use in the subsequent chapters of the thesis. 1.2 Differentiable
More informationA solution in Weyl gravity with planar symmetry
Utah State University From the SelectedWorks of James Thomas Wheeler Spring May 23, 205 A solution in Weyl gravity with planar symmetry James Thomas Wheeler, Utah State University Available at: https://works.bepress.com/james_wheeler/7/
More informationVectors. January 13, 2013
Vectors January 13, 2013 The simplest tensors are scalars, which are the measurable quantities of a theory, left invariant by symmetry transformations. By far the most common non-scalars are the vectors,
More informationMultilinear (tensor) algebra
Multilinear (tensor) algebra In these notes, V will denote a fixed, finite dimensional vector space over R. Elements of V will be denoted by boldface Roman letters: v, w,.... Bookkeeping: We are going
More informationSpin(10,1)-metrics with a parallel null spinor and maximal holonomy
Spin(10,1)-metrics with a parallel null spinor and maximal holonomy 0. Introduction. The purpose of this addendum to the earlier notes on spinors is to outline the construction of Lorentzian metrics in
More informationDifferential Geometry of Warped Product. and Submanifolds. Bang-Yen Chen. Differential Geometry of Warped Product Manifolds. and Submanifolds.
Differential Geometry of Warped Product Manifolds and Submanifolds A warped product manifold is a Riemannian or pseudo- Riemannian manifold whose metric tensor can be decomposes into a Cartesian product
More informationModuli spaces of Type A geometries EGEO 2016 La Falda, Argentina. Peter B Gilkey
EGEO 2016 La Falda, Argentina Mathematics Department, University of Oregon, Eugene OR USA email: gilkey@uoregon.edu a Joint work with M. Brozos-Vázquez, E. García-Río, and J.H. Park a Partially supported
More information