Torsion free biconformal spaces: Reducing the torsion field equations

Transcription

1 Uth Stte University From the SelectedWorks of Jmes Thoms Wheeler Winter Jnury 20, 2015 Torsion free iconforml spces: Reducing the torsion field equtions Jmes Thoms Wheeler, Uth Stte University Aville t:

2 Reducing the torsion field equtions Jnury 1, 2015 Astrct Our gol is to solve the full set of torsion nd co-torsion field equtions of Eucliden iconforml spce, with only the ssumption of vnishing torsion. Here we egin y resolving the involution constrints, symmetry conditions nd torsion field eqution into single eqution for further study. In two preceding studies, we hve looked t whether certin solutions flt for the connection of flt iconforml spce lso solve the curved spce field equtions. Now we tke more mitious pproch nd try to solve the full set of torsion nd co-torsion field equtions, with only the ssumption of vnishing torsion. Here we egin y resolving the involution constrints, symmetry conditions nd torsion field eqution into single eqution for further study. 1 Torsion-free field equtions Beginning with the field equtions, 0 = β T T + S 0 = β T + S S 0 = p T c δt c e e δs c e e 12 η c + 12 δc f η f η cd µ d + δ cη d µ fdf + W η c δ cη f W f 0 = p S nd the involution conditions 1 2 ηcd η d δc η d e η ed + ρ c δρ c e e c δcs e e + δct e e 1 2 η c + δc 1 2 e η e + η ec ρ e 1 2 η cd η d 1 2 δ cη d f η df + η cd η de µ e δcη d η de µ f ef T c = η d ρ d c η d ρ c d + η c u η u c = ρ c ρ c + η c u η u c δ cη d ρ d e e η c W + δ cη e W e S c = kη d µ d c kη d µ d c kη v c + kη c v = k µ c η v c µ c η c v we wish to determine the six independent prts of the torsion nd co-torsion, T c, T c, T c, S c, S c, S c 1

3 Antisymmetrizing the involution conditions fully nd noting the symmetry of ρ c nd µ c, we immeditely see tht nd we hve the trces T [c] = 0 S [c] = 0 η T c = ρ c ρ c n 1 u c η S c = k µ c µ c n 1 v c ut further progress is difficult without some ssumption. In keeping with ides from Riemnnin geometry, we ssume vnishing torsion, T c = 0 T c = 0 T c = 0 Choosing the metric conformlly orthonorml, η = e 2φ η 0, the field equtions reduce to 0 = βs 0 = β S 0 = p = p 0 = p = p S δ c Se e η c φ δη c e e φ η cd µ d + δη c d µ fdf + W η c δη c f W f δ c φ + δ c φ + ρ c δρ c e e ρ c δρ c e e η cd µ d + δη c d µ f df + ηc W + φ δη c e W e + e φ S c δcs e e ηc φ + δcη e e φ + η ec ρ e δcη d ρ d e e η c W + δcη e W e δc φ + δc φ + η cd η de µ e δcη d η de µ f ef S c δcs e e η cd η de µ e δcη d η de µ f ef + η ecρ e δcη d ρ d e e η c W + φ + δcη e W e + e φ We recognize u = W + φ nd v = W + φ. Collecting everything, we hve two equtions which re independent of the co-torsion, 0 = η d ρ c 0 = p together with the symmetry of ρ c p S d η d ρ c d + η c u η u c ρ c δρ c e e η cd µ d + δη c d µ f df + ηc v δη c e v e nd µ c nd four equtions involving co-torsion, S c = k η d µ d c η d µ d c η v c + η c v S = 0 S = 0 S c δ cs e e = p η cd η de µ e δcη d η de µ f ef + η ecρ e δcη d ρ d e e η c u + δcη e u e 2

4 For now, we focus on the following six of these conditions, with the remining three ove tken s constrining the co-torsion. From this point on, since we re deling only with single form ech of Q c, ρ e c, µ ec nd S c, it will e fr simpler to llow rising nd lowering of indices using η, η. At the end we will return the indices to their proper positions. Rising ll ut the co-torsion indices nd the e d eqution, we hve 0 = ρ c ρ c + η c u η u c 1 ks c = µ c µ c η v c + η c v 2 Q c µ c η c µ e e ρ c + η c ρ e e η c v + η c v e dq c = 0 4 ρ c = ρ c 5 µ c = µ c 6 These equtions relte components of the connection. Notice tht the only co-torsion term is the spce-time piece, S c = S c. 2 Solving the connection conditions 2.1 Expnding e d First, notice tht if Q c = 0, these equtions cnnot determine the reltion etween the metrics, δ nd η. However, the ritrriness of Q c keeps us from solving the finl eqution for the this reltionship. Since other rguments hve shown us tht the metric must e Lorentzin, we strt y writing the inverse metric reltion in the mnifestly Lorentzin form δ = A η 2w 2 w w for some vector w nd mplitude A, nd deriving the properties of Q c. It follows tht the metric is nd we hve: δ = 1 A η 2w 2 η cη d w c w d W 2 δ w w = 1 η w w 2w A 2 η cη d w c w d w w = 1 A w2 where w 2 η w w. Therefore, we hve δ = η w2 W 2 2w 2 w w δ = W 2 η 2w 2 η cη d w c w d Now expnd eq.4: 0 = 2 e dq c w 2

5 = δdδ e δ e δ d Q c = Q c d e η w2 e 2w W 2 w w e 2 = Q c e d W 2 w 2 η d 2w 2 η f η dg w f w g Q c η e η d Q c 2 w 2 w w e η d Q c 2 w 2 ηe η f η dg w f w g Q c w 2 w 2 w w e w f w g Q c η f η dg Rising d, Check: Q cde Q ced = 2 w 2 we w Q cd 2 w 2 wd w Q ce + 4 w 2 2 we w d Q c w w 7 0 = 2 e dq c = δdδ e δ e δ d Q c η e 2w 2 w w η e d 2w 2 η df η g w f w g = Q c e d = Q c e d Q c η e η d Q c 2 w 2 ηe η df η g w f w g Q c 2 w 2 w w e η d Q c + 2 w 2 η 2 df η g w 2 w w e w f w g Q c 0 = Q cde Q ced + 2 w 2 wd w Q ce + 2 w 2 we w Q cd w 2 2 wd w e w w Q c Q cde Q ced = 2 w 2 wd w Q ce 2 w 2 we w Q cd + 4 w 2 2 wd w e w w Q c 2.2 Symmetric nd ntisymmetric prts of the field eqution Antisymmetrize If we ntisymmetrize ed, 2 Q cde Q ced = 2 w 2 wd w Q ce 2 w 2 we w Q cd + 2 w 2 we w Q cd + 2 w 2 wd w Q ce = 2 w 2 wd w Q ce Q ce 2 w 2 we w Q cd Q cd = 2 w 2 wd w δ e Q c Q c 2 w 2 we w δ d Q c Q c Q cde Q ced = 1 w d w 2 w δ e w e w δ d Q c Q c Therefore, the ntisymmetric prt of Q c is mixed projection: Q cde Q ced = 1 w 2 w d w δ e w e w δ d Q c Q c Symmetrize Symmetrizing ed, 0 = 2 w 2 wd w Q ce 2 w 2 we w Q cd + 4 w 2 2 wd w e w w Q c 2 w 2 we w Q cd 2 w 2 wd w Q ce + 4 w 2 2 wd w e w w Q c 0 = 1 w 2 wd w δ e Q c + Q c 1 w 2 we w δ d Q c + Q c + 2 w 2 2 wd w e w w Q c + Q c 4

6 This gives the symmetric prt s 0 = 1 w 2 wd w δ e Q c + Q c 1 w 2 we w δ d Q c + Q c + 2 w 2 2 wd w e w w Q c + Q c 9 Now comine the results to check. Adding the symmetric nd ntisymmetric prts, Q cde Q ced = 1 w 2 w d w δ e w e w δ d Q c Q c we recover the field eqution: 0 = 1 w d w 2 w δ e + w e w δ d Q c + Q c + 2 w 2 2 wd w e w w Q c + Q c Q cde Q ced = 1 w 2 w d w δ e w e w δ d Q c Q c Solving for ρ c nd µ c We hve Consider the first nd fifth: Therefore, the symmetric prt is nd therefore, 1 w d w 2 w δ e + w e w δ d Q c + Q c + 2 w 2 2 wd w e w w Q c + Q c = 2 w 2 w d w Q ce + w e w Q cd + 4 w 2 2 wd w e w w Q c 0 = ρ c ρ c + η c u η u c ks c = µ c µ c η v c + η c v Q c µ c η c µ e e ρ c + η c ρ e e η c v + η c v e dq c = 0 ρ c = ρ c µ c = µ c 0 = ρ c ρ c + η c u η u c 0 = ρ c ρ c ρ c = ρ c = ρ c η c u + η u c = ρ c η c u + η u c ρ c ρ c = η u c η c u p c ρ c + ρ c + ρ c = ρ c + ρ c + η c u η u c + ρ c + η c u η u c = ρ c + η c u η u c + η c u η u c ρ c = p c + 1 2η u c η c u η c u 5

7 Similrly, ks c = µ c µ c η v c + η c v µ c µ c = ks c + η v c η c v µ c µ c = ks c + η v c η c v µ c µ c = 0 Then with Therefore, we hve: with trces u c = µ c + µ c + µ c = µ c + µ c ks c + η v c η c v + µ c + ks c + η c v η v c = µ c + ks c ks c 2η v c + η c v + η c v µ c = u c + 1 ks c + ks c + 2η v c η c v η c v ρ c = p c + 1 2η u c η c u η c u µ c = u c + 1 2η v c η c v η c v + 1 ks c + ks c ρ e e = p e e 1 n 1 u nd so Q c µ c η c µ e e ρ c + η c ρ e e = u c + 1 µ e e = u e e 1 n 1 v + 1 kse e η c v + η c v 2η c v η v c η c v + 1 p c + 1 2η c u η c u η u c + η p c e e 1 n 1 u η c v + η c v ks c + ks c η c u e e 1 n 1 v + 1 kse e = u c p c η c u e e + η c p e e + 1 ks c + ks c 1 kηc S e e + 1 n + 1 η c v η c v η v c n + 1 η c u + η c u + η u c = U c η c U e e + k S c + S c η c S e e + 1 n + 1 η c v u η c v u η v c u c Define k v u. Then Q c is Q c = U c η c U e e + k where the totlly symmetric prt of Q c is S c + S c η c S e 1 e + n + 1 η c k η c k η k c U c = u c p c This Q c is the only comintion of µ c, ρ c nd S c tht is constrined y the field eqution, = 0. e d Qc 6

8 4 Conclusion: the field eqution Expnding the field eqution, we require 0 = Q c Q c + 2 w 2 w w e Q ce + 2 w 2 w w e Q ce w 2 2 w w Q cde w d w e = U c η c U e e + k S c + S c η c S e 1 e + n + 1 η c k η c k η k c U c + η c U e e k S c + S c η c S e 1 e n + 1 η c k η c k η k c + 2 w 2 w w e U ce w c U f f + k w e S ce + w e S ce w c S f f + 2 w 2 w w 2 2 w w + 1 n + 1 wc k η c k e w e w k c w e S ec + w e S ce η c w e S fe f + 1 n + 1 η c k e w e w c k w k c w e U ec η c w e U fe f + k w d w e U cde w c w e U fe f + k w d w e S ecd w c w e S fe f + 1 n + 1 w c k e w e w c k d w d w 2 k c = η c U e e η c U e e + 2 w 2 w w e U ce w c U f f + 2 w 2 w w e U ec η c w e U fe f w 2 2 w w w d w e U cde + 4 w 2 2 w w w c w e U fe f + k S c + S c η c S e k e S c + S c η c S e 2 e + w w 2 w e S ce + w w e S ce w w c S f f + 2 w w 2 w e S ec + w w e S ce w η c w e S fe 4 f w 2 2 w w w d w e ks ecd 4 + w 2 2 w w w c w e ks fe f + 1 n + 1 η c k η c k η k c 1 n + 1 η c k η c k η k c + 2 n + 1 w w 2 w c k w η c k e w e w w k + 2 n + 1 w w 2 η c k e w e w w c k w w k c 4 w 2 2 nw w w c k e w e + 4 w 2 w w k c = η c U e e η c U e e + 2 w 2 w w e U ce + 2 w 2 w w e U ec 2 w 2 w w c U f f 2 w 2 w η c w e U fe f w 2 2 w w w d w e U cde + 4 w 2 2 w w w c w e U fe f ks c + 2 w w 2 w e ks ce + ks ce + w w e ks ec + ks ce 2 w 2 w w c ks f f + 1 η c ks e e η c ks e e w 2 2 w w w d w e ks ecd 4 + w 2 2 w w w c w e ks fe f 2 w 2 w η c kw e S fe f + 1 n + 2 η c k n + 2 η c k + 2 n + 1 w w 2 w c k w w c k + n + 1 w η c k e w e w η c k e w e w 2 2 nw w w c k e w e In the next report, we therefore consider the eqution 0 = η c U e e η c U e e + 2 w 2 w w e U ce + 2 w 2 w w e U ec 2 w 2 w w c U f f 2 w 2 w η c w e U fe f w 2 2 w w w d w e U cde + 4 w 2 2 w w w c w e U fe f ks c + 2 w 2 w w e ks ce + ks ce + w w e ks ec + ks ce 2 w 2 w w c ks f f + 1 η c ks e e η c ks e e 7

9 w 2 2 w w w d w e ks ecd 4 + w 2 2 w w w c w e ks fe f 2 w 2 w η c kw e S fe f + 1 n + 2 η c k n + 2 η c k + 2 n + 1 w w 2 w c k w w c k + n + 1 w η c k e w e w η c k e w e w 2 2 nw w w c k e w e This, mong other vriles, reltes the time vector w to the connection vectors u nd v. Once the consequences of this eqution re quntified, we will consider the remining field equtions for the cross co-torsion nd the momentum co-torsion. References [1] Jeffrey S Hzoun nd Jmes T Wheeler, Time nd drk mtter from the conforml symmetries of Eucliden spce, Clss. Quntum Grv pp, doi: / /1/21/ [2] Spencer J A nd Wheeler J T 2011 The existence of time Int. J. Geom. Method Mod. Phys [] Jmes Thoms Wheeler, Vrition of the liner iconforml ction, Digitl Commons 2015, [4] Jmes Thoms Wheeler, Guge trnsformtions of the iconforml connection, Digitl Commons 2014, [5] Jmes Thoms Wheeler, Studies in torsion free iconforml spces. Cse 2: γ = 0, Digitl Commons 2015, [6] Jmes Thoms Wheeler, Studies in torsion free iconforml spces. Cse 1: γ + = 0, Digitl Commons 2015, [7] Jeffrey S Hzoun nd Jmes T Wheeler, Curved iconforml spce, in preprtion. [8] Jmes T. Wheeler, Weyl grvity s generl reltivity, Physicl Review D 90, , DOI: /PhysRevD [9] S. Koyshi nd K. Nomizu, Foundtions of Differentil Geometry John Wiley nd Sons, New York, 196. [10] E. A. Ivnov nd J. Niederle, Phys. Rev. D 25, [11] J. T. Wheeler, New conforml guging nd the electromgnetic theory of Weyl, J. Mth. Phys. 9, [12] A. Wehner nd J. T. Wheeler, Conforml ctions in ny dimension, Nucl. Phys. B557, [1] Anderson L B nd Wheeler J T 2004 Biconforml supergrvity nd the AdS/CFT conjecture Nucl. Phys. B [14] Anderson L B nd Wheeler J T 2007 Yng Mills grvity in iconforml spce Clss. Quntum Grv [15] Wheeler J T 2007 Guging Newton s lw Cn. J. Phys [16] Anderson L B nd Wheeler J T 2006 Quntum mechnics s mesurement theory on iconforml spce Int. J. Geom. Mth. Mod. Phys