Fractional boundary for the Gott-Hiscock string

Transcription

1 JOURNAL OF MATHEMATICAL PHYSICS 46, Fractinal bundary fr the Gtt-Hiscck string J. P. Krisch Department f Physics, University f Michigan, Ann Arbr, Michigan Received 2 Octber 2004; accepted 11 January 2005; published nline 4 April 2005 A fractinal bundary cnditin is used t jin the Gtt-Hiscck string t a Levi- Civita vacuum. The use f a fractinal derivative generates Israel bundary layers whse density depends n the rder f the fractinal derivative. Variable bundary layers fr the same tw bunding space times can be studied. The string angular deficit depends n the rder f the fractinal deficit American Institute f Physics. DOI: / I. INTRODUCTION The Gtt-Hiscck 1,2 slutin describes a cnstant density string matched t a vacuum Levi- Civita space time with angular deficit. Tw f the mst cmmn bundary matching cnditins fr nn-null bundaries are due t Lichnerwicz and Darmis. These cnditins have been discussed by Bnnr and Vickers. 3 Bth cnditins invlve matching a metric and sme metric derivatives acrss the bundary. The Darmis cnditin matches the metric and its secnd fundamental frms n the bundary while the Lichnerwicz cnditin matches the metric and its first derivatives n the bundary. The resulting space time exterir t the string has an angular deficit related t the mass per unit length f the string interir. Fr general space times, if a derivative, r extrinsic curvature, match is nt pssible, the mst cmmn prcedure is t use the Israel 4 frmalism, defining a surface bundary layer stress energy thrugh the jump in the extrinsic curvature between space times n either side f the bundary. Other space times, whse extrinsic curvature was nt cntiniuus acrss the bundary, culd have been used fr the exterir match, creating an Israel layer n the string bundary. The Israel technique prvides infrmatin abut the stress energy cntent f the bunding surface layer. While the Israel surface layer can be used t describe the jump in the extrinsic curvature acrss a space time bundary, it will nt distinguish surface layers f different densities r structures that might bund the same tw spacetimes. Within the Israel frmalism, the nly way f varying the bundary layer prperties is t vary the bunding space times. One way f mdifying the bundary cnditins t include different kinds f bundary surfaces fr the same bunding space times is t generalize the Lichnerwicz bundary cnditins t a fractinal derivative matching and then t use the Israel frmalism t describe the resulting surface layer. This creates bundary layers whse stress energy cntent depends n the rder f the fractinal derivative and allws the study f variable bundary structures between the same tw space times. In the next sectin, we apply this fractinal matching methd t the Gtt- Hiscck string. The hallmark f string behavir, the angular deficit, als depends n the rder f the fractinal derivative. The change in the angular deficit intrduced by the fractinal match is discussed. II. FRACTIONAL BOUNDARY CONDITION A. The metrics Cnsider a string riented alng the z axis with cnstant stress energy cntent Frm the field equatins, the interir metric 2 is T t t = T z z = /2005/46 4 /042506/7/$ , American Institute f Physics

2 J. P. Krisch J. Math. Phys. 46, ds 2 = dt 2 + d 2 + dz 2 + *2 sin 2 * d 2, 2 where *= 8 1/2 and the string is assumed t be flat at =0. The exterir vacuum space time is ds 2 = dt 2 + dr 2 + dz 2 + a 2 r 2 d 2. 3 B. The nnfractinal bundary match The metric match at =, r=r prvides the cnditin ar = * sin *. The extrinsic curvature is defined as 5 K ij = n ; e i e j, where the e i are the tangents t the bundary hypersurface with nrmal vectr n. Calculating the extrinsic curvatures ne finds in the interir K = sin * cs * *. In the exterir the extrinsic curvatures are 4 K tt =0, K = a 2 r, 5 K zz =0. The difference in the extrinsic curvatures acrss the bundary is K = K vacuum K interir = a 2 r * sin * cs *, K = K vacuum K interir = 1 1 cs / * r * sin / * = 1 1 cs / *. r a r In the Israel frmalism, 4 the stress energy f the bundary layer is 8 S ab = K ab h ab K c c, 8 S tt = K h = cs / * a ar, 8 S zz = a cs / * ar, 8 S =0. 6

3 Fractinal bundary J. Math. Phys. 46, If the extrinsic curvatures match, there is n stress energy in the bundary. The match f the extrinsic curvature prvides the same cnditin as a match f the g derivatives acrss the string bundary, a = cs *. C. The fractinal match In the fractinal bundary match, the metric matching cnditins are the same as in the smth bundary match: ar = * sin *. 7 Instead f a match f the extrinsic curvatures, a fractinal derivative match is used with the regular partial derivative, i replaced by an rder Caput fractinal derivative as described in the Appendix. 1. Fractinal derivatives fr the interir metric In the interir, since the Caput derivative f a cnstant is zer, the nly functin t cnsider is f = *2 sin 2 / *. Using as the fractinal index we have that the th fractinal derivative is 2 * D,L *2 sin = * S, *,. The frm f the functin S, *, depends n the range f the fractinal derivative rder. Fr 1 we have S, *, = Fr 1 the functin is S, *, = */2 1 1 sin 2 2 / * z cs z dz cs 2 2 / * z sin z dz. 8 */2 1 2 sin 2 2 / * z 1 sin z dz + cs 2 2 / * z 1 cs z dz. 9 The tw can be related by an integratin by parts and the use f a gamma functin recursin relatin. Fr =1, the usual first derivative is recvered. 2. Fractinal derivatives fr the exterir metric In the exterir it is the right-handed derivative which shuld be applied. The exterir fractinal derivative f the metric functin is D r,r a 2 r 2 = 3 3 a2 r 2. 10

4 J. P. Krisch J. Math. Phys. 46, TABLE I. Values f the functin h, / * =cs / * a as and / * vary. is the radius f the string in the internal metric. * = 8 1/2 where is the cnstant internal energy density f the string. Cmplex values are indicated by C. When =1, h, / * =0, prviding the usual derivative matching cnditin. / * C C C 3. The bundary match The equatin fr the bundary match is 3 3 a2 r 2 = * S, *,. Fr =1 the usual derivative is btained, leading t the riginal derivative matching result a = cs *. 11 Fr 1 there is a bundary layer which can be described by the Israel 4 frmalism. III. BOUNDARY STRESS ENERGY The bundary stress energy, Eq. 6, invlves the functin h, = * cs a. * The tw fractinal matching equatins, Eqs. 7 and 10, can be cmbined t give an equatin fr the parameter a, 1:a =, / * 1 sin 2 2 / * cs z dz cs 2 2 / * sin z dz, 12 1:a =, / * 2 sin 2 2 / * 1 sin z dz + cs 2 2 / * 1 cs z dz, 13 where, / * = 3 sin 2 / * /2 2. Fr a smth match, h, / * shuld be zer. Using Eq. 12, Eq. 13, and MAPLE, h, / * was evaluated fr a range f / * and fractinal indices. The results are in Table I. Examining Table I, ne sees in all cases that, as =1 is apprached, the stress energy f the layer appraches zer. Fr 1 the bundary layer has negative density and psitive z-stress. As / * mves abve /2, the values becme cmplex. Fr 1, the bundary layer density is psitive and the stress is a z-tensin. Fr 1 cmplex values ccur fr values f / * /2.

5 Fractinal bundary J. Math. Phys. 46, TABLE II. Values f the angular deficit factr a as a functin f and / *. is the radius f the string in the internal metric. * = 8 1/2 where is the cnstant internal energy density f the string. Cmplex values are indicated by C. / * C C The string angular deficit, the cut in the range f the crdinate due t the presence f the parameter a in the exterir metric, is a hallmark f csmic string behavir. The angular deficit depends n the energy density f the string. 2 When a fractinal bundary cnditin is used, the internal stress energy f the string des nt change but the energy density f the bundary layer affects the angular deficit since it must be included in the ttal energy density f the string. The angular deficit can be defined by =2 1 a. The values f a fr varius and / * were calculated and are fund in Table II. In Table II, the =1 clumn values are calculated frm the standard nnfractinal match. The linear density f the string is nt affected by the bundary layer but the angular deficit is, increasing in value as increases. Fr 1, the angular deficit is less than the zer-bundary-layer, =1, value because the negative bundary layer density is decreasing the verall mass density f the string. Fr 1 the deficit is larger than the zer-bundary-layer value because the psitive bundary layer mass increases the string density. One pssible way t interpret the variatin f the parameter a and the surface energy density is thrugh a packing fractin. The packing fractin is nrmally defined as the rati f the area cvered by a tiling t the ttal area being tiled. Adapting this t the surface layer we can cnsider the rati f the 1 surface area t the =1 surface area which is just the rati f the a values in each rw f Table II t the a value fr =1: pf = 2 a 1 Lr a 1 = 2 a =1 Lr a =1. Fr 1, the layer with its negative density matter seems verpacked. Fr 1 this rati can be used t make sme qualitative cmments abut the energy distributin. Frm Table II, generally ne ntes that the packing fractin decreases as increases fr a given / * and that the packing fractin decreases as the / * increases. The latter effect can be simply explained by examining cylinders f cnstant radius and varying internal energy density. As the internal energy density decreases and the rati / * ges up, the packing fractin decreases as there is less energy t distribute ver the surface. The first effect is mre difficult t mtivate. One pssible explanatin cmes frm assuming a specific tiling mechanism fr the bundary layer and then using the packing fractin variatin t mtivate differences within the tiling. Fr example, if the surface layer is tiled with an Apllnian packing, 6,7 circles with increasing smaller circles packed in the interstitial regins, the packing fractin is given rughly by pf =1 r min /r max 2 d where r max is the radius f the largest circle in the packing, r min is the smallest circle in the tiling, and d 1.3 is the fractal dimensin 8 f the tiling in flat space. Using this, the variatin in 14

6 J. P. Krisch J. Math. Phys. 46, the packing fractin as a functin f can be explained as a variatin in the range f granule sizes making up the surface layer. IV. DISCUSSION In this nte, we have suggested using a Caput space fractinal derivative t generate variable density Israel surface layers n the Gtt-Hiscck string bunded by a Levi-Civita vacuum. The methd generates, fr 1, a family f psitive energy, increasingly dense bundary layers between the cnstant density string interir and the vacuum exterir. Fr 1, the energy density is negative. The Caput derivatives can be mre restrictive than the Riemann-Liuville frms since the derivatives f the matching metric functins must exist t use the Caput definitin. The Riemann-Liuville frm wuld give an additinal nnzer matching cnditin fr the cnstant metric functins which wuld identify r and. When jining many hypersurfaces, it is cmmn practice t match extrinsic curvatures. Since the extrinsic curvature is a Lie derivative, it can be calculated frm simple partial derivatives and can be fractinally generalized. Because the string bundary, as defined in the bunding space time, had a well-defined unit nrmal and derivatives, a simple fractinal extrinsic curvature matching wuld have given the same result as the fractinal derivative matching. Hwever, null surfaces, fr example, d nt have a well-defined extrinsic curvature and junctin cnditins utside f an extrinsic curvature matching have been develped Mars and Senvilla 12 have discussed junctin cnditins using a rigged metric cnnectin fr general hypersurfaces. Hayward 13 has discussed an actin fr nnsmth bundaries. Using fractinal derivatives as a methd fr generating variable density bundary layers apart frm the secnd fundamental frm might instead be a way f giving physical meaning t pssible fractinal extrinsic curvature definitins. A smth bundary with well-defined derivatives is nt a necessary cnditin fr the use f a fractinal derivative matching. The Riemann-Liuville derivatives can be applied t functins which themselves d nt have a well defined derivatives. Fr example, using the Riemann- Liuville frm, Rcc and West 14 shwed that the cntinuus but nndifferentiable Wierstrass functin had a well-defined fractinal derivative. Klwankar and Gangal 15 als discussed the fractinal derivatives f the Weierstrass functin. Fractinal derivatives have seen increasing use in the develpment f fractinal kinetics, 16 particularly in discussins f anmalus transprt prcesses. 17 While the methd presented in this nte generates a family f surface layers f varying densities between the same tw bunding static space times, the use f fractinal derivatives in bundary matching culd reflect an underlying bundary matter structure that was built by an anmalus transprt prcess. A single tiling was used as an example f the relatin between pssible matter distributins and the rder f the fractinal derivative. The results frm ther tilings r sums ver tilings is an interesting questin fr investigatin. APPENDIX: FRACTIONAL OPERATIONS The idea f a fractinal derivative arse in 1695 when L Hpital 18 asked Leibniz abut the meaning f d n y/dx n fr n= 1 2. The ideas f fractinal calculus have attracted the attentin f many f the same peple that are assciated with the develpment f physics: Lagrange, Laplace, Furier, Liuville, Riemann, and Weyl, amng thers. 19 Since the first cnference n fractinal calculus in 1974, the applicatins t physics have grwn enrmusly, describing phenmena such as the mdeling f viscelastic phenmena 20 and fractinal matter transprt. 17 There have been many definitins f fractinal peratins because f the diverse array f applicatins. Sme f the fractinal differentiatin definitins are a the left-handed Riemann-Liuville frm, 18 m D a f x = dm x 1 f y x y dx m dy m 1, a and b the right-handed Riemann-Liuville frm, 18

7 Fractinal bundary J. Math. Phys. 46, m b D f x = dm b 1 m f y y x dx m dy m 1, x where, in bth cases, m is the smallest integer bigger than. One very interesting thing abut this frm f the fractinal derivative is that the derivative f a cnstant is nt zer. Fr example, the left-handed derivative f 1 is where the definitin f the beta functin, 2 D a 1 = d2 x 1 x y dx dy 1 = x , B z,w = 0 1 x z 1 1 x w 1 dx = z w z + w, and the recursin relatin fr the gamma functin have been used. There are ther fractinal frms, with mdified definitins. The Caput 19 fractinal derivatives are similar t the Riemann- Liuville derivatives except that the derivative appears inside f the integral; f m y is the mth derivative f f x. The left-hand Caput 19 derivative is x C D a f x = 1 x y m 1 f m y dy. m a The right-hand derivative is similar. Butzer and Westphal 21 credit Liuville with the definitin. In this paper we shall use the Caput frm. The ntatin that we will use is i D il, i D ir, where the fractinal rder, has been placed in parentheses t distinguish it frm a tensr index, i is the index f the crdinate derivative, and L, R dente the left- r right-hand derivatives. 1 J. R. Gtt III, Astrphys. J. 288, W. A. Hiscck, Phys. Rev. D 31, W. B. Bnnr and P. A. Vickers, Gen. Relativ. Gravit. 13, W. Israel, Nuv Ciment B 44, E. Pissn, A Relativists Tlkit Cambridge U. P., Cambridge, T. Aste and D. Weaire, In Pursuit f Perfect Packing IOP, Bristl, H. J. Hermann, G. Mantica, and D Bessis, Phys. Rev. Lett. 65, S. S. Manna and J. Hermann, J. Phys. A 24, L A. H. Taub, J. Math. Phys. 21, C. Barrabes, Class. Quantum Grav. 6, C. J. S. Clarke and T. Dray, Class. Quantum Grav. 4, M. Mars and J. M. Senvilla, Class. Quantum Grav. 10, G. Hayward, Phys. Rev. D 47, A. Rcc and B. J. West, Physica A 265, K. M. Klwankar and A. D. Gangal, Chas 6, I. M. Sklv, J. Klafter, and A. Blumen, Phys. Tday 55, G. M. Zaslavsky, Phys. Rep. 371, L. Pdlubny, Fractinal Differential Equatins Academic, New Yrk, S. G. Samk, A. A. Kilbas, and O. I Marichev, Fractinal Integrals and Derivatives: Thery and Applicatins Grdn and Breach Science, New Yrk, F. Mainardi, Fractals and Fractinal Calculus in Cntinium Mechanics, edited by A. Carpinteri and F. Mainardi, Springer, New Yrk, P. L. Butzer and U. Westphal, Applicatins f Fractinal Calculus in Physics, edited by R. Hilfer Wrld Scientific, Singapre, 2000.