The Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria A Nonlinear Theory of Tensor Distributions J.A. Vickers J.P. Wilson Vienna, Preprint ESI 566 (1998) June 17, 1998 Supported by Federal Ministry of Science and Transport, Austria Available via

2 A NONLINEAR THEORY OF TENSOR DISTRIBUTIONS J.A. Vickers and J.P. Wilson Abstract. The coordinate invariant theory of generalised functions of Colombeau and Meril is reviewed and etended to enable the construction of multi-inde generalised tensor functions whose transformation laws coincide with their counterparts in classical distribution theory. 1. Introduction Colombeau's theory of new generalised functions (Colombeau, 1984) has increasingly had an important role to play in General Relativity, enabling a distributional interpretation to be given to products of distributions which would otherwise be undened in the framework of Classical distribution theory. Recent applications of Colombeau's theory to Relativity have included the calculation of distributional curvatures which correspond to metrics of low dierentiability, such as those which occur in space-times with thin cosmic strings (Clarke et al, 1996; Wilson 1997) and Kerr singularities (Balasin, 1997), and the electromagnetic eld tensor of ultra-relativistic Riessner-Nordstrm solution (Steinbauer, 1997). The theory of General Relativity is built upon the Principle of General Covariance which means that the measurement of all physical quantities do not depend on what local coordinate system the observer is using; therefore they must transform locally as tensors under C 1 dieomorphisms. Unfortunately, Colombeau's theory is not naturally suited for representing such a covariant physical theory because it is heavily built upon the linear structure. Colombeau's algebra G() is a quotient algebra on the algebra E M () of smooth functionals A 0!R, one denes the kernel spaces A k as being the subspaces of functions 2 D such that ()d 1 (1) () d 0 2Nn ; 1 6 jj 6 k; (2) These spaces are not invariant under a smooth dieomorphism 0 (), unless is linear. Since these spaces are used in both the denitions of the spaces E M () (moderate functions) and N () (null functions) it makes the denition of a mapping ~ : E( 0 )! E() which preserves both E M () and N () very problematic. As a consequence of this, one must apply Colombeau's theory with great caution in that coordinate transformations should be carried out at the distributional level and Colombeau's algebra is to be used as a tool for assigning a distributional interpretation to nonlinear operations involving distributions. It was shown by Vickers and Wilson (1998) that the calculation of Clarke et al. (1996), which shows that the distributional Ricci scalar density of a conical singularity with a decit angle of 2(1?A) is 4(1?A) (2), is invariant under C 1 transformations in the sense that the following diagram commutes g 0 ab? y g ab????! f 0 g 0 ab????! R ~ 0p ~g 0????! 4(1? A) (2)????! fg ab????! ~ R p ~g??y????! 4(1? A) (2) the maps denote the relevant tensor distribution transformations. It should be noted that Colombeau's theory is applied to both coordinate systems and that no attempt is made to compare PACS: 0240, Keywords: Colombeau algebras, Dieomorphic invariance.. 1

3 2 J.A. VICKERS AND J.P. WILSON generalised functions in one coordinate system with their counterparts until the distributional curvature has been recovered via weak equivalence. The problem resulting from the inability of the spaces A k to transform invariantly has been overcome by the use of the `simplied' algebra G s () which is formed from using a base algebra E s () consisting of smooth functionals on (0; 1] rather than on A k (Biagioni, 1990; Colombeau 1992), so one does not need to worry about how A k () transforms under, and hence is able to dene a natural map ~ : E s ( 0 )! E s () which preserves its moderate subalgebra E M;s ( 0 ) and its null ideal N s ( 0 ). However there is still a problem which undermines the suitability of using G() as a coordinate invariant representation of distribution theory, which itself is a coordinate invariant construction, in that the embedding : D 0 ()! E M () provided by the smoothing convolution integral does not commute with the dieomorphism. A solution to this problem was proposed by Colombeau and Meril (1994), in which they constructed a coordinate invariant G() together with a scalar transformation that commutes with the embedding. It essentially involves weakening the moment conditions used in dening the smoothing kernel space A k which is then dened in a coordinate invariant manner. In this paper we shall begin by reviewing the algebra of Colombeau and Meril and etend the denitions so that multi-inde tensor transformations which commute with the embedding may also be dened. 2. Scalars Let and 0 be open subsets ofrn that are dieomorphic to each other by :! 0. In general the map will induce a map : C( 0 )! C(), between the spaces of continuous functions on and omega, given by (f 0 ) f 0, enabling f 0 to transform as a scalar between the two coordinate systems. Functions from C() may be embedded into Colombeau's generalised function algebra G() by the smoothing convolution : C()! E M () ~f(; ) (f) ~ f f( + )() d 2 A k We would like to be able to construct a map ~ : E M ( 0 )! E M (), an analogue of, such that C( 0 )? y C() 0????! E M ( 0 )??y~????! E M () commutes, and furthermore we would like ~ to map N ( 0 ) to N () so that we can construct G( 0 ) and G() consistently. Given ~ f 0 2 E M ( 0 ), the obvious way to construct ~ ~ f 0 is by dening ~ ~ f(; ) ~ f(^ ; ()) ^ is some mapping on the kernel space A k. If we take ^ ()?1 then ^ () will no longer satisfy the normalisation condition ^ () d 1 However if we take ^ () 1 jj j?1 then the normalisation is preserved. With the choice above, ~ ~ f 0 (; ) f 0 1 (() + ) J (?1 ()) (?1 ()) d f 0 (() + ())() d

4 A NONLINEAR THEORY OF TENSOR DISTRIBUTIONS 3 On the other hand ^f0 (; ) f 0 (( + ))() d so ~ f ~ 0 and^f0 are dierent. In order for ~ f ~ 0 and^f0 to represent the same generalised function, their dierence must be null; however, the problem is that there is no obvious relation between ^ (A k ) and A p. This prevents us from having a coordinate invariant denition in the framework of Colombeau's original theory. This problem was resolved by Colombeau and Meril (1994) in which they gave a new denition of the space of smoothing kernels, Ak ~ together with a map ^ in such a way that ~ f ~ 0 ^f0 and the notion of a null function was preserved. It is convenient to start by considering generalised functions onrn together with dieomorphisms :Rn!Rn. We will later restrict the algebra to open sets and consider dieomorphisms between? open sets. We begin by considering the embedding of f using the -dependent kernel () 1 n ^f 0 ( ; ) 1 n f 0 (( + ))() d f 0 (( + ))() d f 0 (() + ) jj (?1 (() + ))j?1 (() + )? is dened by ( + ) () +. If we now dene 1 ^ () jj (?1 (() + ))j?1 (() + )? then ^f0 ( ; ) f 0 (() + )) (^ ()) d 1 n f 0 (() + )) ^ ~ f ((^ ) ; ()) However ^ now depends upon (although unlike it is bounded as! 0) so we replace A 0 by the space ^A0 of bounded paths in D(Rn ). Denition 1. The space ^A0 is the set of all smooth maps (0; 1]! D(Rn ) 7! d d such that : 2 (0; 1] is bounded in D(Rn ) and () d (0; 1] However as well as an -dependence we see that ^ also introduces an -dependence. We therefore modify the denition of ^A0 as follows: Denition 2. The space A ~ (Rn 0 ) is dened to be the set of all 2 D(Rn ) ^ for some 2 ^A0 (Rn ) and some 2 Di(Rn ;Rn ) Note that elements of ~ A0 (Rn ) satisfy the normalisation condition () d 1 8(; ) 2 (0; 1] Rn and that this condition is preserved by ^. Unfortunately the moment condition (2) is not preserved by ^. We therefore weaken the moment condition as follows: (3)

5 4 J.A. VICKERS AND J.P. WILSON Denition 3. The space ~ A k (Rn ) is the space of all elements of ~ A0 () which have the property The dieomorphism :Rn!Rn induces a map () d O( k ) 8 2Nn ; 1 6 jj 6 k (4) ^ : ~ A0 (Rn )! ~ A0 (Rn ) ^ () 1 jj (?1 (() + ))j Theorem 1. ^ ehibits the following properties 1. 2 ~ A 0 (Rn ) ) ^ 2 ~ A 0 (Rn ) 2. 2 ~ A k (Rn ) ) ^ 2 ~ A bk2c (Rn ) 3. If :Rn!Rn and :Rn!Rn are dieomorphisms then ( ). Proof. Refer to Colombeau and Meril (1994). Denition 4. The unbounded path ( ; ) corresponding to ; () 1 n?1 (() + )? (5) () 2 ~ A 0 (Rn ) is dened by The unbounded path may be regarded as a regularisation for the delta distribution since 8 2 D(Rn ); lim!0 ; () () d (0) hence its alternative name, the delta-net. Here the unbounded path ( ; ) 2 A ~ (Rn 0 ), unlike its counterpart in Colombeau's original theory. A transformation law for these unbounded paths may be formed from (5); Writing 0 ^ () we have that which we shall also epress as 0 (); () 1 n 0 () () ^ ; () 1 1 n jj (?1 (() + ))j 1 jj (?1 (() + ))j ;?1 (() + )???1 (() + )? 1? jj (?1 (() + ))j ;?1 (() + )? It should be noted that in this contet ^ is a map acting on the space of unbounded paths and not on elements of ~ A0. Generalised functions are then dened to make use of the new ~ A k s; Denition 5. The space E(Rn ) is the set of all maps which are C 1 as functions of ~f : ~ A0 Rn!R ( ; ) 7! ~ f( ; ) Since the smoothing kernels now depend upon, it is no longer sucient to require smoothness with respect to the second argument of ~ f. Instead we also require some kind of smooth dependence upon the smoothing kernels. Colombeau and Meril require Silva-dierentiability of ~ f as a function of the mollier (Colombeau, 1982). An alternative, and possibly simpler requirement, is to demand dierentiability in the sense of calculus on `convenient vector spaces' (Kriegl and Michor, 1997). We net dene a subalgebra of moderate functions

6 A NONLINEAR THEORY OF TENSOR DISTRIBUTIONS 5 2N Denition 6. E (Rn M ) is the space of f ~ 2 E(Rn ) such that 8K Rn, 8 2Nn, 9N such that 8 2 AN ~ (Rn ), sup D f(; ~ ; ) O(?N ) (as! 0) 2K Note that this diers from the denition in Colombeau and Meril because our ~ A 0 have an -dependence. We now dene an ideal of negligible (null) functions Denition 7. N (Rn ) is the space of functions f ~ 2 E (Rn M ) such that 8K Rn, 8 2Nn, 9N 2N, 9 2 S such that 8 2 A ~ (Rn k ) (k > N), sup D f(; ~ ; ) O( (k) ) (as! 0) 2K S f :N!R+ j (k + 1) > (k); (k)! 1 g The space of generalised functions is then dened as the quotient We may then dene a map G(Rn ) E M(Rn ) N (Rn ) : ~ : E(Rn )! E(Rn ) ~ f ~ ( ; ) f ~ (^ ; ()) The fact that ^ maps the kernel spaces ~ Ak (Rn ) into ~ Abk2c (Rn ) will imply that ~ ~ f 2 EM (Rn ) () ~ f 2 E M (Rn ) ~ ~ f 2 N (Rn ) () ~ f 2 N (Rn ) which will induce a well dened map [~ ] : G(Rn )! G(Rn ). Furthermore if we dene an embedding by : C(Rn )! E M (Rn ) (f) ~ f ~f( ; ) f( + ) () d (6) then if f 2 C(Rn ) we have ~ ~ f( ; ) f(() + ) f(( + )) () d?1 (() + )? d jj (?1 (() + ))j giving us that ~ (f) (f) as we required. We now dene the algebra G() on open sets Rn. E M () and N () are dened in the obvious way by restricting to be in. Then G() E M ()N () If we now only have a dieomorphism between open sets :! 0 rather than the whole ofrn we see that ^ : A0 ~ (Rn )! A ~ (Rn 0 ) is not always well dened by (5). However it is always well dened for 2 and suciently small. Since this is all that is needed for the theory we may dene ^ for other

7 6 J.A. VICKERS AND J.P. WILSON values in some arbitrary manner without eecting the denition of ~ f( ~ ; ). Thus a dieomorphism :! 0 induces a well dened map ~ : G( 0 )! G(). We now consider the embedding. Given f 2 D() we etend f to some function g 2 D(Rn ) such that gj f. We then embed g into E (Rn M ) using (6). Again one can show that for all 2 and suciently small the answer does not depend upon the etension. We therefore have a well dened embedding : D()! G(). Furthermore we still have the result that ~ (f) (f) (7) as the previous result is valid for suciently small. We are nally in a position to consider generalised functions on manifolds. Let M be a manifold with!r atlas A f( ; V ) : 2 Ag. Given a function f 2 D(M) we dene f : (V ) by f f?1. The set ff g 2A then satises f j (V \V ) f j (V \V )?1 (?1 ) f j (V \V ) 8; 2 A with V \ V 6 ; (8) Conversely any set of functions ff 2 D( (V ))g 2A which satises (8) denes an element of D(M). In eactly the same way we make the following denition Denition 8. A generalised function ~ f 2 G(M) is a set of generalised functions f ~ f 2 G( (V ))g 2A which satises ~f j (V \V ) ( ~?1 ) ~ ~ f j (V \V ) 8; 2 A with V \ V 6 ; If we dene ~ f f is the embedding : D( (V ))! G( (V )), then one has an embedding : D(M)! G(M) 3. Smoothing of vector and covector fields We now have a coordinate invariant theory of generalised functions at the level of scalars. We would like to etend this theory to enable vectors, covectors and, ultimately, multi-inde tensors to be dened as generalised functions whose transformation laws will coincide with those of distributions. We shall rst consider the smoothing of covector eld!, which may be represented in both coordinate systems 2 and as!! a ()d a! 0 a( 0 )d 0a ; with the components! a and! 0 a related by! a! 0 a We could use a componentwise smoothing, : C 0 1( 0 )! C 0 1();! 0 a() b a()! 0 b(()): : C 0 1 ()! E 0 1 () (!) ~! ~! a (; ) and relate the smoothings by ~ : E 0 1 ( 0 )! E 0 1()! a ( + ) () d (9) ~ ~! 0 a(; ) b a()~! 0 b(^ ; ) in which case we would have that ~! a (; ) b a( + )! b? 0 ( + ) () d?1? () +! b? 0 () + ^ b a () d

8 A NONLINEAR THEORY OF TENSOR DISTRIBUTIONS 7 and ~ ~! 0 a(; ) b a()! b(() 0 + )^ () d If it is the case that! 0 a is C 1, we can epand b a and! 0 a in powers of which will reveal that ~! a (; )? ~ ~! 0 a(; ) O( k ) for ^ 2 N ( 0 ) so ~! a (; ) and ~ ~! a 0 may be regarded as representing the same element of G(). On the other hand if! a 0 admitted only a nite level of dierentiability, then ~! a and ~ ~! a 0 would not be equivalent as elements of G() although they may well be equivalent at the level of association, given that the level of dierentiability was high enough. A possible remedy for this problem is to introduce smoothing kernels with indices, thus we replace (9) by and dene ~! a () a b () d b a + O( k )! b ( + ) a b () d a b () d O( k ) 2Nn ; 1 6 jj 6 k ^ a b () c a ()b d (?1 (() + )) jj (?1 (() + ))j a b ()?1a b(()) c d?1 (() + )? Proposition 2. The moment conditions (10) are preserved under ^ in that ^ a b () d b a + O( k ) ^ a b () d O( bk2c ) 2Nn ; 1 6 jj 6 bk2c (10) (11) Proof. The rst condition holds since a b () d a() c b d( + ) c d () d c a() b d() + X jjk+1 b a + O( k+1 ) a b () d + kx jj1 D c a( + ) b d() jj jj! D a() c b d() jj jj! c d () d c d () d A similar calculation (See Colombeau and Meril, 1994) will show that the second condition is preserved with k replaced by bk2c. Proposition 3. ~ ~! 0 a ~! a Proof. ~! a (; ) c b ( + )!0 c (( + )) a b () d c b(?1 (() + ))! c(() 0 + ) a b?1 (() + )? b a()! 0 c(() + )^ b c (; (); ) d ~ ~! 0 a(; ) d

9 8 J.A. VICKERS AND J.P. WILSON The need for smoothing kernels with indices is clear, because we are integrating the coecients of the covector eld at the point + to give a covector at the point. The kernel a b therefore has the eect of transporting a covector at the point + to the point. Rather than considering general kernels with indices a b we shall consider those which have the form a b ()? b a (; + ) ()? :!T () T() (; y) 7!? a b (; y) is a smooth map with? b a (; ) b a and 2 A ~ (Rn 0 ). The respective transformation laws (5) and (11) for and b a will imply that? b a (; y) has to transform as ^? a b ((); (y)) c a() b d(y)? c d (; y) so? a b may be regarded as components of a covector (with inde a) located at and a vector (with inde b) located at y. A similar procedure could be applied to smooth a vector X X a 0b with the components X a and X 0a related by X a X 0a : C 1 0( 0 )! C 1 0(); X 0a () a b ()X0b (()): In this case we use a smoothing of the form ~X a () X b ( + ) a b() d By allowing a b to transform as ^ we are able to dene a map ~ by which will satisfy a b() a c()b d(?1 (() + )) jj (?1 (() + ))j c d ~ ~! 0a (; ) a b ()~! 0b (^ ; ()) ~ ~ X 0a ~ X a?1 (() + )? This time a b transports a vector at + to a vector at, we may therefore consider kernels a b which admit the form a b()? a b(; + ) () 2 ~ A0 (Rn ) and the map? :!T() T () (; y) 7!? a b(; y) is smooth with? a b(; ) a b and transforms as ^? a b((); (y)) a c() d b(y)? c d(; y) (12) We shall formalise our transport operators? a b and? a b by dening the following space of transport operators;

10 A NONLINEAR THEORY OF TENSOR DISTRIBUTIONS 9 Denition 9. The space T () is dened to be the set of maps? :!T() T () (; y) 7!? a b(; y) (13) which are smooth and are such that? a b(; ) a b. In this way we may simultaneously write down smoothings of vectors and covectors as functions ~A 0 (R) T ()!R ~X a (;?; ) ~! a (;?; ) X b ( + )? a b(; + )() d! b ( + )? a b (; + )() d (14)? a c? b c c b: (15) An important consequence of this arrangement is that? will preserve contractions of covectors with vectors; suppose that X a ()? a b(; y)x 0b (y) then! a ()? a b (; y)! 0 b(y)! a ()X a ()? a b (; y)? a c(; y)! 0 a(y)x 0a (y)! 0 a (y)x0a (y) 4. Remarks on the transport operator We now show that the derivative of the transport operator denes a connection and that conversely a connection denes a transport operator (in a normal neighbourhood). We rst consider rules for dierentiating?. We may b c (; y) d ca()? d b (; y) using the fact that? a b(; y)? b c(y; ) a c we also have? a c(; y)? b c (; y) a a c (; y)?b cd(y)? a d (; a c (; y)?a cd()? d b(; a c (; y) d cb(y)? d a(; y) Proposition 4. a bc transforms as a connection Proof. The connector? a b transforms as? 0 a b ( 0 ; y d? c d (; y) so this implies b d? e d f 0d ca? 0 d d? e d g fe? g d

11 10 J.A. VICKERS AND J.P. WILSON and hence 0b d? g d? g d g fe? g d on multiplying this epression throughout q?q p one obtains the standard connection transformation law 0c d 2 0c If is a connection then one can always choose a neighbourhood U of any point 2 such that the normal coordinates at are well dened. Moreover one can choose U to be simply conve; that is that U is also a normal neighbourhood for any point y 2 U. In a simply conve neighbourhood one can dene? a b(; y) by parallel propagation with respect to along the unique geodesic connecting and y, so in geodesic coordinates? a b(; y) a b In general if we allow any connection, the sets in the atlas will not be normal neighbourhoods. However since has compact support, the integration will be conned to a normal neighbourhood for suciently small. Since we are only interested in results for suciently small we could also work with in place of?. However once we consider tensor elds on manifolds, rather than open sets, it is much less restrictive to work with a connection rather than a global connector?. Even when working with subsets ofrn it is preferable to work with connections onrn rather than connectors on, since the set of algebras then has the structure of a (ne) sheaf. If one does this, then one denes the embedding by (14),? is dened by parallel transport along geodesics in a normal neighbourhood and arbitrarily else. Since one is in a normal neighbourhood for suciently small epsilon, the embedding into the algebra will be well dened. 5. General tensors Having dened procedures for smoothing covectors and vectors in such a way that ~ 0, it is now clear how to go about etending this smoothing operation to more general tensors; Suppose T is a tensor eld of type (p; q) whose components in the two coordinate systems 2 and are related by T 0a:::b c:::d T a:::b c:::d then we may dene a smooth tensor eld by and a map ~ by which will satisfy ~T a:::b c:::d(;?; ) T 0a:::b c:::d () a e () : : : b f ()g c () : : :h 0e:::f d ()T g:::h (()) T e:::f g:::h ( + ) a:::b c:::de:::g g:::h () d c:::de:::g g:::h ()? a e( + ) : : :? b f( + )? g c ( + ) : : :? h d ( + ) () (16) ~ ~ T 0a:::b c:::d(;?; ) a e() : : : b f () g c() : : : h d() ~ T 0e:::f g:::h (^ ; ^?; ()) (17) ~T a:::b c:::d ~ ~ T 0a:::b c:::d (18) We are now in a position to formally dene our generalised tensor elds on. The kernel space ~ A0 (Rn ) is dened as in Denition 2 and K(Rn ) is the space of smooth connections onrn. We rst dene the base algebra E p q() Denition 10. Let E p q() be the set of all maps ~T : ~ A0 K!Rp+q (; ; ) 7! ~ T a:::b c:::d(; ; ) such that 7! ~ T a:::b c:::d (; ; ) is C1. As in the case of scalar elds we require some smooth dependence upon the smoothing kernel and connection to ensure smooth dependence in. As usual we restrict to those of moderate growth

12 A NONLINEAR THEORY OF TENSOR DISTRIBUTIONS 11 Denition 11. E p q;m () is the space of tensors ~ T 2 E p q() such that 8K, 8 2Nn, 9N 2Nsuch that 8 2 ~ A N (Rn ), 8 2 K, sup D T ~ (; ; ; ) O(?N ) (as! 0) 2K We dene an ideal of negligible (null) functions Denition 12. N p q() is the space of tensors ~ T 2 E p q;m () such that 8K, 8 2Nn, 9N 2N, 9 2 S such that 8 2 Ak ~ (Rn ) (k > N), 8 2 K, sup D T ~ (; ; ; ) O( (k) ) (as! 0) 2K S f :N!R+ j (k + 1) > (k); (k)! 1 g We then dene our generalised function space as a quotient Denition 13. G p q() Ep q;m () N p q() One may now dene generalised tensor elds on manifolds as a collection of generalised tensor elds on (V ) which transform in the appropriate way under ( ~?1 ) ( ~ ) using (17). Because of (18) we also have a well dened embedding of Dq(M) p into G p q(m). Throughout the paper we have been using distributions that may be dened as locally integrable functions. It is possible to etend our results to cover the smoothing of more general tensors. We rst recall how distribution theory is formulated in a coordinate invariant manner; Suppose :! 0 is a C 1 dieomorphism, then we may dene D ~ p() q to be the space of C 1 multi-inde functions with compact support which transform as type (q; p) densities with weight +1 under ^ c:::d a:::b (()) 1 jj ()j c g () : : : d h ()e a ()f c:::d b () a:::b () We simply dene type (p; q) tensor distributions as elements of the dual space ~ D 0p q() which then admit a transformation law : ~ D 0p q ( 0 )! ~ D 0p q () of the form h T 0 ; i ht; i This immediately becomes evident in case of T being a locally integrable tensor eld, for the integral is coordinate invariant. We now dene ht; i T a:::b c:::d() c:::d a:::b() d : D 0 ()! E M () (T ) ~ T ~T a:::b c:::d(; ; ) T; a:::b c:::d ; ( a:::b c:::d) g:::h e:::f (y)?a e(; y) : : :? b f (; y)? c g (; y) : : :? d h(; y) (y)? y It should be noted that the test function a:::b c:::d ; behaves as a type (p; q) tensor with respect to and that the translated object ; will transform as ()^ ; () 1 jj (?1 ())j ; (?1 ())

13 12 J.A. VICKERS ] AND J.P. WILSON Proposition 5. If T 2 D 0p q (0 ) then ~ T ~ 0 T 0. Proof. ~ T ~ 0a:::b c:::d (; ; ) T ~ 0a:::b c:::d (^ ; ; ()) T 0 ; ^ ( ; ) a:::b c:::d D T 0 ; e:::f g:::h ; E e() a : : :f b () g c() : : : d()] h e() a : : :f b () g c() : : : h T 0 e:::f g:::h (; ; ) This result enables us to dene an embedding of D 0p q(m) into G p q(m). Since elements of the generalised function space G p q() are represented by smooth functions, we may dene tensor operations on these generalised functions in the usual way. In particular we are able to dene the following; 1. Tensor products; Suppose [ ~ S 0 ] 2 G p q( 0 ) and [ ~ T 0 ] 2 G r s( 0 ) are type (p; q) and type (r; s) tensors respectively then we may dene [ ~ S 0 ~ T 0 ] 2 G p+r q+s( 0 ) by ( ~ S 0 ~ T 0 ) a:::b c:::d e:::f g:::h (0 ; 0 ; 0 ) ~ S 0a:::b c:::d( 0 ; 0 ; ) ~ T 0e:::f g:::h (0 ; 0 ; 0 ) the resulting object [ ~ S 0 ~ T 0 ] will transform as a type (p + r; q + s) tensor in that ~ ( ~ S 0 ~ T 0 ) ~ ~ S 0 ~ ~ T 0 2. Contractions; Suppose ~ T 0 2 G p+1 q+1 (0 ) is a type (p; q) tensor then we may dene ~ S 0 2 G p q( 0 ) by [ ~ S 0 ] will transform as a type (p; q) tensor because ~S 0a:::b c:::d( 0 ; 0 ; 0 ) ~ T 0a:::be c:::de( 0 ; 0 ; 0 ) ~ S ~ 0a:::b c:::d ~ T ~ 0a:::be c:::be 3. Dierentiation; Suppose that [ ~ T 0 ] 2 G p q (0 ) then we may dene [@ ~ T 0 ] 2 G p q+1 (0 ) 0 T ~ 0a:::b c:::de (0 ; 0 ; 0 T ~ 0a:::b 0e ( 0 ; 0 ; 0 ) In general [@ 0 ~ T 0 ] will not transform as a tensor for we do not necessarily have ~ (@ 0 ~ T 0 0 (~ ~ T 0 ) However, if we are able to dene a connection [~? 0a bc ] 2 G1 2( 0 ) that transforms as ~ ~? 0a bc (; ; ) a d() e b() f c () ~? 0d ef(^ ; ; ()) + a d() e b;c() then we are able to dene a covariant derivative [r 0 ~ T 0 ] 2 G p q+1 (0 ) by r ~ T 0a:::b c:::de which will 0e T 0a:::b c:::d +? ~ 0a ~ ef T 0f:::b c:::d + +? ~ 0b ~ ef T 0a:::f c:::d?? ~ 0f ~ ec T 0a:::b f:::d??? ~ 0f ~ ed T 0a:::b c:::f ~ (r 0 ~ T 0 ) r 0 (~ ~ T 0 ) 4. Lie derivatives. For [ X ~ 0 ] 2 G1( 0 0 ) and [ T ~ 0 ] 2 G p q(), the Lie derivative [L 0 ~X T ~ ] 2 G p q may be dened as 0 L 0 ~X T ~ 0a:::b 0 c:::d X ~ 0e T ~ 0a:::b c:::de? X ~ 0a ;f T ~ 0f:::b c:::d?? X ~ 0b ;f T ~ a:::f c:::d + X ~ 0f ;c T ~ 0a:::b f:::d + + X ~ 0f ~ ;d T 0a:::b c:::f which will satisfy ~ L 0 ~X 0 ~ T 0 L ~ X 0(~ ~ T 0 ) The linear tensor operations such as addition, symmetrisation and antisymmetrisation of indices also etend to our generalised function valued tensors in a natural way.

14 A NONLINEAR THEORY OF TENSOR DISTRIBUTIONS Differentiation A C 1 tensor eld T may be embedded in to G p q () in two ways; 1. By smoothing; T 7! T ~ 2. By dening an element of E p q;m () which is independent of and ; S(; ~ ; ) T () By dening G p q() as a quotient group we are able to guarantee that S ~ and T ~ represent the same generalised function. Here we shall verify that this is also the case for derivatives in that if T 2 C p;1 q () then it is the case that T ~ a:::b c:::d;e and T a:::b c:::d;e represent the same generalised function. We begin with scalars; suppose f 2 C 1 () then we have, ~f ;a () ff ;a () f ;a ( + ) () d + f ;a ( + ) () d f( + ) ;a() d ;a is the partial derivative of the operator with respect to a. We now epand the dierence ~ f ;a? f f;a as a Taylor series (for some 2 [0; 1]); ~f ;a? f f;a kx jj jj! D f() jj0 X + k+1 jjk+1 ;a() d (k + 1)! D f( + ) ;a() d The rst term on the right-hand side may be simplied by dierentiating the moment conditions (3) and (4). ;a() d 0 ;a() d O( k ); 1 6 jj 6 q (19) which shows that it is O( k ), as the second term is clearly O( k+1 ). This shows that ~ f ;a? f f;a 2 N (), so that ~ f ;a and f f;a represent the same generalised function. If f had only a nite level of dierentiability then f f;a and ~ f ;a would represent dierent generalised functions. (The dierence would be O( m+1 ) for f 2 C m, which would then make f f 0 ;a and ~ f ;a equivalent at the level of association). This highlights an important dierence from Colombeau's original theory in which dierentiation of distributions manifestly commutes with smoothing. This is a price we have to pay for introducing smoothing kernels that are dependent on. The net step is to etend this to multi-inde tensors; an added complication being that the transport operator? will depend on, so dierentiating it will introduce etra terms. We shall use the notation T je to denote covariant dierentiation with respect to the connection a bc Without loss of generality we shall consider the dierentiation of vectors (the dierentiation of covectors and more general tensors is achieved by adding the relevant terms), so for vectors we have X a jb X a ;b + a bcx c Lemma 6. [ ~ X a jb(; ; )] X c jb( + )? a c(; + ) () d

15 14 J.A. VICKERS AND J.P. WILSON Proof. ~X a ;b(; ; ) + X c ;b( + )? a c(; + ) + () d X c ( + )?? a bd()? d c(; + ) + d bc ( + )?a d(; + ) () d X c ( + )? a c(; + ) ;b() d X c jb( + )? a c(; + )? a bd() + () d X c ( + )? d c(; + ) () d X c ( + )? a c(; + ) ;b() d By Taylor epanding and using the moment conditions (19) it may be shown that for 2 ~ A k (), X c ( + )? a c(; + ) ;b() d O( k ) therefore [ ~ X a jb (; ; )] X c jb( + )? a c(; + ) () d Proposition 7. Let X a and Y a be smooth vector elds then covariant dierentiation of X a with respect to a bc in the Y a direction commutes with the ^embedding in the sense that; Y b X a jb Y ~ b X ~ a jb : Proof. This follows from the above lemma together with the fact that we may identify Y a and ~ Y a for a smooth vector eld. Suppose? a bc is another connection, then covariant dierentiation with respect to it may be epressed as X a ;b X a ;b +? a bc Xc this may also be epressed as X a ;b X a jb + ^? a bcx c ^? a bc?a bc? a bc. The symbol ^? a bc will transform as a type (1; 2) tensor because it is the dierence of two connections. If X a, Y a and? a bc are all smooth then Y b X a jb and ^? a bc Xc Y b commute with the embedding. This implies that the covariant derivative Y b X a ;b also commutes with the embedding. We may dene a torsion free connection ^ a bc by taking the symmetric part of the background connection X a :b X a jb? a [bc] Xc X a ;b + ^ a bc Xc For smooth tensor elds covariant dierentiation with respect to the torsion-free connection ^ a bc will commute with the embedding. Rather than using the partial derivative to construct invariant objects, we shall use the covariant derivative of the background torsion free connection: a. The Lie bracket [X; Y ] of two vectors X and Y is independent of a torsion free connection so one may write [X; Y ] a X b Y a :b? Y b X a :a since the covariant derivative : a commutes with the embedding we have [(X); (Y )] ([X; Y ])

16 A NONLINEAR THEORY OF TENSOR DISTRIBUTIONS 15 As a second eample we consider a smooth metric g ab and the Levi-Civita connection Using implies that therefore we may write? a bc 1 2 gad (g bd;c + g cd;b? g cd;b ) g ab:c g ab;c? ^ d acg bd? ^ d bcg ad? a bc ^a bc gad (g bd:c + g cd:b? g bc:d ) X a ;b X a :b + ^? a bc Xc ^? a bc 1 2 g ad(g bd:c + g cd:b? g bc:d ) Thus for a smooth metric the covariant derivative with respect to the Levi-Civita connection commutes with the embedding. 7. Association The relation of association or weak equivalence for generalised functions is much the same as the normal denition; Denition 14. We say that [ T ~ ] 2 G p q () is associated to zero (written as [ T] ~ 0) if 8 2 D ~ q p (), 9k > 0 such that ~T a:::b b:::d( ; ; ; ) c:::d a:::b() d A ~ (Rn 0 ); 2 K: lim!0 Two generalised functions [ ~ S] and [ ~ T ] are associated to each other if ~ S? ~ T 0 Denition 15. We say that [ T ~ ] 2 G p q() is associated to the distribution S 2 D 0p q() (written as [ T ~ ] S) if 8 2 D(), ~ 9k > 0 such that lim!0 ~T a:::b b:::d( ; ; ; ) c:::d a:::b() d hs; i 8 2 A0 ~ (R); 2 K We nally verify that this assignment of a distributional interpretation to a generalised function also commutes with Proposition 8. [ ~ T 0 ] S 0 implies [~ ~ T 0 ] S 0 Proof. Letting 2 ~ D 0q p (), lim!0 ~ ~ T 0a:::b c:::d( ; ; ) c:::d a:::b d lim!0 lim!0 hs 0 ; i h S 0 ; i a e() : : : b f () g c() : : : h d ~ T 0e:::f g:::h (^ ; ; ()) c:::d a:::b() d ~T 0e:::f g:::h (^ ; ; 0 ) c:::d a:::b () d0 8. Conclusion We now have a covariant theory of generalised functions in which we are able to dene tensors whose transformation laws coincide with those of distributions, both in the senses of convolution embedding and association. This will mean that we may carry out calculations in General Relativity that involve the evaluation of products of distributions, such as the evaluation of the distributional curvature at the verte of a cone with a decit angle 2(1? A) by Clarke et al. (1996) in a coordinate independent way. One would need to rst choose a convenient coordinate system, then embed the metric into the space G 0 2(R2 ), which would transform as a type (0; 2) tensor using the denitions in this paper, calculate the distributional curvature [ ~ R p ~g] as a generalised function, which will transform as a scalar density of weight +1 and show that it is associated to the distribution 4(1? A) (2) obtained in that paper.

17 16 J.A. VICKERS AND J.P. WILSON Acknowledgement The authors wish to thank ESI for supporting their visit to the institute as part of the project on Nonlinear Theory of Generalised functions. The authors thank M. Kunzinger, M. Oberguggenberger and R. Steinbauer for helpful discussions. J. Wilson acknowledges the support of EPSRC. grant No. GR/ References H. Balasin, Distributional energy-momentum tensor of the etended Kerr geometry, Class. Quantum Grav. 14 (1997), 3353{3362. H. A. Biagioni, A nonlinear theory of generalised functions, Lecture Notes in Mathematics 1421, Springer, C. J. S. Clarke, J. A. Vickers and J. P. Wilson, Generalised functions and distributional curvature of cosmic strings, Class. Quantum Grav. 13 (1996), 2485{2498. J. F. Colombeau, Dierential Calculus and Holomorphy, Real and Comple analysis in Locally Conve Spaces, North- Holland Mathematics Studies 64, North-Holland, J. F. Colombeau, New generalised functions and multiplication of distributions, North-Holland Mathematics Studies 84, North-Holland, J. F. Colombeau, Multiplication of distributions, Lecture Notes in Mathematics 1532, Springer, J. F. Colombeau and A. Meril, Generalised functions and multiplication of distributions on C1 manifolds, J. Math. Anal. Appl. 186 (1994), 357{364. A. Kriegl and P. W. Michor, The convenient setting of global analysis, Mathematical Surveys and Monographs 53, American Mathematical Society, R. Steinbauer, The ultrarelativistic Riessner-Nordstrm eld in the Colombeau algebra, J. Math. Phys. 38 (1997), 1614{ J. A. Vickers and J. P. Wilson, Invariance of the distributional curvature of the cone under smooth dieomorphisms, Preprint (1998). J. P. Wilson, Distributional curvature of time-dependent cosmic strings, Class. Quantum Grav. 14 (1997), 3337{3351. Faculty of Mathematical Studies, University of Southampton, Southampton SO17 1BJ, UK. Addresses: jav@maths.soton.ac.uk, jpw@maths.soton.ac.uk