Slide 1 of 18 Tensors in Mathematica 9: Built-In Capabilities. George E. Hrabovsky MAST
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1 Slide of 8 Tensos in Mathematica 9: Built-In Capabilities eoge E. Habovsky MAST
2 This Talk I intend to cove fou main topics: How to make tensos in the newest vesion of Mathematica. The metic tenso and how to tansfom vectos into covectos. Catesian tenso opeations. R Opeations
3 How to Build a Tenso in Mathematica 9 Rank One Fo ank one tensos, we can wite them as tangent vectos, TableA9xi=, 8i, 8"", "", ""<<E MatixFom x x x o as covectos Table@xi, 8i, 8"", "", ""<<D 8x, x, x<
4 4 Rank Two and Highe We can use simila methods to develop ank two tensos, though Mathematica is not able to cope with abstact indices without help fom thid-paty softwae I like xact. TableAΣi j, 8i, 8x, y, z<<, 8j, 8x, y, z<<e MatixFom Σx Σx y Σx y Σx z Σy Σy z Σx z Σy z Σz TableAΣid@88i,j<<,Spacings.D, 8i, 8x, y, z<<, 8j, 8x, y, z<<e MatixFom Σx x Σx y Σx z Σy x Σy y Σy z Σz x Σz y Σz z You can poduce the individual tenso components, Σ@i_, j_, n_d := TableB Η xj viaxje + xi vj@xid - IfBi j, 8k,, n, <F MatixFom xk vk@xkd, F + If@i j, Ξ xk vk@xkd, D,
5 D, D, D< TaditionalFom Η v HxL + Ξ v HxL 4 : Η J v HxL Η J v HxL - v HxLN + Ξ v HxL, v HxLN + Ξ v HxL Η Hv HxL + v HxLL Η Hv HxL + v HxLL, Η Hv HxL + v HxLL Η Hv HxL + v HxLL Η Hv HxL + v HxLL > Η Hv HxL + v HxLL You can also wite a table to poduce the entie tenso st@n_d := Table@Σ@i, j, nd, 8i,, n, <, 8j,, n, <D MatixFom st@d TaditionalFom 4 Η v HxL + Ξ v HxL Η J v HxL Η J v HxL - v HxLN + Ξ v HxL v HxLN + Ξ v HxL Η Hv HxL + v HxLL Η Hv HxL + v HxLL Η Hv HxL + v HxLL Η Hv HxL + v HxLL Η Hv HxL + v HxLL Η Hv HxL + v HxLL Η Hv HxL + v HxLL Η Hv HxL + v HxLL Η Hv HxL + v HxLL Η J v HxL - Η J v HxL - 4 v HxLN + Ξ v HxL Η v HxL + Ξ v HxL v HxLN + Ξ v HxL Η Hv HxL + v HxLL Η Hv HxL + v HxLL Η Hv HxL + v HxLL Η Hv HxL + v HxLL Η Hv HxL + v HxLL Η Hv HxL + v HxLL Η Hv HxL + v HxLL Η Hv HxL + v HxLL Η Hv HxL + v HxLL Η J v HxL Η J v HxL 4 v HxLN + Ξ v HxL v HxLN + Ξ v HxL Η v HxL + Ξ v HxL 5
6 6 The Metic Tenso A specific example of a calculation is one whee we tansfom fom a tangent vecto to a covecto using the metic tenso, vi = gij v j () So, given the tangent vecto v j = H, cos Θ, - ΦL, and assuming we ae in spheical coodinates, we can find the metic. tv = 8, Cos@ΘD, - Φ<; met = CoodinateChatData@"Spheical", "Metic", 8, Θ, Φ<D TaditionalFom sinhθl We can even find the invese metic, im = CoodinateChatData@"Spheical", "InveseMetic", 8, Θ, Φ<D TaditionalFom cschθl
7 We take the poduct of the invese metic with the tangent vecto, tv. "InveseMetic", 8, Θ, Φ<D TaditionalFom giving us the covecto :, coshθl, Φ csc HΘL > 7
8 8 The Catesian Tenso We ceate a new tenso, newtenso = TableAxid@88i<<D yid@88j<<d zid@88k<<d, 8i,, <, 8j,, <, 8k,, <E; newtenso MatixFom TaditionalFom x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z We can see if its a tenso, AayQ@newtensoD Tue We can find its ank TensoRank@newtensoD
9 9 We can pefom a contaction contens = TableBâ newtenso@@i, a, add, 8i,, <F MatixFom TaditionalFom a= x y z + x y z + x y z x y z + x y z + x y z x y z + x y z + x y z We have the inne poduct in = Inne@Times, newtenso, newtenso, PlusD FullSimplify MatixFom x y z Ix z + x z + x z M x y z Ix z + x z + x z M x y z Ix z + x z + x z M x y+ z Ix z + x z + x z M x y+ z Ix z + x z + x z M x y+ z Ix z + x z + x z M x y + z Ix z + x z + x z M x y x y z Ix z + x z + x z M + z Ix z + x z + x z M x y + x z Ix z + x z + x z M x z Ix z + x z + x z M x x y z Ix z + x z + x z M x x y+ z Ix z + x z + x z M x y+ z Ix z + x z + x z M x y+ z Ix z + x z + x z M x x y+ z Ix z + x z + x z M x y+ z Ix z + x z + x z M x y+ z Ix z + x z + x z M x x y + x y z Ix z + x z + x z M x z Ix z + x z + x z M x y x y z Ix z + x z + x z M + z Ix z + x z + x z M x y x y z Ix z + x z + x z M + x y z Ix z + x z + x z M x x y+ z Ix z + x z + x z M x y+ z Ix z + x z + x z M x y+ z Ix z + x z + x z M x x y z Ix z + x z + x z M x y z Ix z + x z + x z M x x y+ z Ix z + x z + x z M x y+ z Ix z + x z + x z M x y+ z Ix z + x z + x z M x x y+ z Ix z + x z + x z M x y+ z Ix z + x z + x z M x y+ z Ix z + x z + x z M x newtenso. newtenso FullSimplify MatixFom x y z Ix z + x z + x z M x y+ z Ix z + x z + x z M x y+ z Ix z + x z + x z M x y+ z Ix z + x z + x z M x y z Ix z + x z + x z M x y + x y z Ix z + x z + x z M z Ix z + x z + x z M x y x y z Ix z + x z + x z M + x y z Ix z + x z + x z M z Ix z + x z + x z M x y + x z Ix z + x z + x z M x z Ix z + x z + x z M x x y z Ix z + x z + x z M x x y+ z Ix z + x z + x z M x y+ z Ix z + x z + x z M x y+ z Ix z + x z + x z M x x y + x y z Ix z + x z + x z M x z Ix z + x z + x z M x y + z Ix z + x z + x z M x y +
10 We can take the diect poduct, in = Oute@Times, newtenso, newtensod FullSimplify MatixFom A vey lage output was geneated. Hee is a sample of it: x y z x y z+ x y z+ H L H L x y+ z x y+ z+ x y z H L H L H L H L H L H L H L H L H L H L H L H L H L H L H L H L Show Less Show Moe Show Full Output Set Size Limit... H L H L H L H L H L H L H L H L H L H L H L
11 We can take the tace of the tenso, x y z + x y z + x y z We can tanspose, Tanspose@newtensoD MatixFom TaditionalFom x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z We can take a patial deivative x newtenso MatixFom TaditionalFom x - y z x - y z x - y z x - y z x - y z x - y z x - y z x - y z x - y z x - y z y z x - y z x - y z x - y z x - y z x - y z x - y z x - y z x - y z y z x x - y z x x - x - y z x - y z - x - y z x - - y z y z x - y z
12 we can detemine the diffeentials MatixFom TaditionalFom â x x - y z + â y x y - z + â z x y z - â x x - y z + â y x y - z + â z x y z - â x x - y z + â y x y - z + â z x y z - â x x - â x x - y z + â y x y - z + â z x y z - â x x - y z + â y x y - z + â z x y z - â x x - y z +ây x y - z + âz x y z â x x - â x x - y z + â y x y - z + â z x y z - â x x - y z + â y x y - z + â z x y z - â x x - âx x y z +ây x y - - z + âz x y z âx x - z + âz x y z â x x - y z + â y x y - z + â z x y z - âx x â x x - y z + â y x y - z + â z x y z - z + âz x y z - y z +ây x y - â x x - - â x x - y z + â y x y - z + â z x y z - â x x - y z + â y x y - z + â z x y z - y z +ây x y - âx x - â x x âx x - â x x â x x - - â x x - y z + â y x y - z + â z x y z - - âx x - y z +ây x y y z +ây x y - z + âz x y z - - z + âz x y z â x x -
13 We can look fo symmeties, 8< symten = Symmetize@newtenso, Antisymmetic@8, <DD StuctuedAay@SymmetizedAay, 8,, <, -Stuctued Data-D Nomal@symtenD MatixFom Ix y z - x y z M Ix y z - x y z M Ix y z - x y z M Ix y z - x y z M TensoSymmety@symtenD Antisymmetic@8, <D I- x y z + x y z M I- x y z + x y z M Ix y z - x y z M Ix y z - x y z M I- x y z + x y z M Ix y z - x y z M Ix y z - x y z M Ix y z - x y z M I- x y z + x y z M I- x y z + x y z M I- x y z + x y z M I- x y z + x y z M I- x y z + x y z M I- x y z + x y z M
14 4 R Opeations The Metic Hee we input the metic In[4]:= Metic@gi_D := Module@ 8n = Length@giD, g, ing<, g = Table@If@Μ ³ Ν, gi@@μ, ΝDD, gi@@ν, ΜDDD, 8Μ, n<, 8Ν, n<d; ing = Simplify@Invese@gDD; 8n, g, ing<d we assign labels to the coodinates In[4]:= Evaluate@Table@x@ΜD, 8Μ, 4<DD = 8t,, Θ, j<; Hee is the Schwazschild metic: In[4]:= MeticA98 - <, 8, - H - L<, 9,, - =, 9,,, - Sin@ΘD==E TaditionalFom - - Out[4]//TaditionalFom= :4, - - -, sinhθl - csc HΘL >
15 5 Hee we have the Chistoffel Symbols: In[5]:= g_, ing_<, := ModuleA 8, in<, = in = Table@, 8Λ, n<, 8Μ, n<, 8Ν, n<d; Μ, ΝDD = Simplify@HD@g@@Λ, ΝDD, x@μdd + D@g@@Λ, ΜDD, x@νdd - D@g@@Μ, ΝDD, x@λddl D; If@Μ ¹ Ν, ΜDD Μ, ΝDDD, 8Λ, n<, 8Μ, n<, 8Ν, Μ<D; Do@in@@Λ, Μ, ΝDD = Simplify@Sum@ing@@Λ, Μ, ΝDD, 8Ρ, n<dd; If@Μ ¹ Ν, in@@λ, Ν, ΜDD = in@@λ, Μ, ΝDDD, 8Λ, n<, 8Μ, n<, 8Ν, Μ<D; IfAOptionValue@PintNonZeoD, Do@If@@@Λ, Μ, ΝDD =!=, Pint@""id@88Λ-,Μ-,Ν-<<D, " Μ, ΝDDDD, 8Λ, n<, 8Μ, n<, 8Ν, Μ<D; DoAIfAin@@Λ, Μ, ΝDD =!=, PintA""id@88Λ-<<D id@88"",μ-,ν-<<d, " ", in@@λ, Μ, ΝDDEE, 8Λ, n<, 8Μ, n<, 8Ν, Μ< EE; In[49]:= 8, in<e Chistoffel@8n_, g_, ing_<, OptionsPatten@DD := Module@ 8, in<, = in = Table@, 8Λ, n<, 8Μ, n<, 8Ν, n<d; Μ, ΝDD = Simplify@HD@g@@Λ, ΝDD, x@μdd + D@g@@Λ, ΜDD, x@νdd - D@g@@Μ, ΝDD, x@λddl D; If@Μ ¹ Ν, ΜDD Μ, ΝDDD, 8Λ, n<, 8Μ, n<, 8Ν, Μ<D; Do@in@@Λ, Μ, ΝDD = Simplify@Sum@ing@@Λ, ΡDD Μ, ΝDD, 8Ρ, n<dd; If@Μ ¹ Ν, in@@λ, Ν, ΜDD = in@@λ, Μ, ΝDDD, 8Λ, n<, 8Μ, n<, 8Ν, Μ<D; Do@If@@@Λ, Μ, ΝDD =!=, Pint@"", Λ -, Μ -, Ν -, " Μ, ΝDDDD, 8Λ, n<, 8Μ, n<, 8Ν, Μ<D; Do@If@in@@Λ, Μ, ΝDD =!=, Pint@"", Λ -, Μ -, Ν -, " ", in@@λ, Μ, ΝDDDD, 8Λ, n<, 8Μ, n<, 8Ν, Μ< D; 8, in<d Hee we calculate the Chistoffel symbols fo the Schwazschild Metic In[5]:= ChistoffelA MeticA98 - <, 8, - H - L<, 9,, - =, 9,,, - Sin@ΘD==EE;
16 6 - Sin@ΘD - Sin@ΘD - Cos@ΘD Sin@ΘD H- + L - + H- + L - - -H- + L Sin@ΘD -Cos@ΘD Sin@ΘD Cot@ΘD
17 Hee we calculate the components of the Riemann tenso: In[54]:= g_, ing_<, := 8, in, R = Table@, 8Α, n<, 8Β, n<, 8Μ, n<, 8Ν, n<d, R = Table@, 8Μ, n<, 8Ν, n<d, R<, 8, in< = Chistoffel@8n, g, ing<d; Do@R@@Α, Β, Μ, ΝDD = R@@Β, Α, Ν, ΜDD = Simplify@Sum@g@@Α, ΛDD HD@in@@Λ, Β, ΝDD, x@μdd - D@in@@Λ, Β, ΜDD, x@νddl Λ, ΜDD in@@λ, Β, ΝDD Λ, ΝDD in@@λ, Β, ΜDD, 8Λ, n<dd; R@@Β, Α, Μ, ΝDD = R@@Α, Β, Ν, ΜDD = - R@@Α, Β, Μ, ΝDD; If@Μ ¹ Α, R@@Μ, Ν, Α, ΒDD = R@@Ν, Μ, Β, ΑDD = R@@Α, Β, Μ, ΝDD; R@@Ν, Μ, Α, ΒDD = R@@Μ, Ν, Β, ΑDD = - R@@Α, Β, Μ, ΝDDD, 8Α,, n<, 8Β, Α - <, 8Μ,, Α<, 8Ν, If@Μ === Α, Β, Μ - D<D; Do@R@@Μ, ΝDD = Simplify@Sum@ing@@Α, ΒDD * R@@Α, Μ, Β, ΝDD, 8Α, n<, 8Β, n<dd; If@Μ ¹ Ν, R@@Ν, ΜDD = R@@Μ, ΝDDD, 8Μ, n<, 8Ν, Μ<D; R = Simplify@Sum@ing@@Μ, ΝDD R@@Μ, ΝDD If@Μ ¹ Ν,, D, 8Μ, n<, 8Ν, Μ<DD; Do@If@R@@Α, Β, Μ, ΝDD =!=, Pint@"R"id@88Α-,Β-,Μ-,Ν-<<D, " ", R@@Α, Β, Μ, ΝDDDD, 8Α,, n<, 8Β, Α - <, 8Μ,, Α<, 8Ν, If@Μ === Α, Β, Μ - D<D; Do@If@R@@Μ, ΝDD =!=, Pint@"R"id@88Μ-,Ν-<<D, " ", R@@Μ, ΝDDDD, 8Μ, n<, 8Ν, Μ<D; If@R =!=, Pint@"R ", RDD; 8R, R, R<D 7
18 8 Hee is the output fo the Schwazschild metic, assuming S = : In[55]:= Riemann@Metic@88 - <, 8, - H - L<, 8,, - ^ <, 8,,, - ^ Sin@ΘD ^ <<DD; Sin@ΘD - Cos@ΘD Sin@ΘD - Sin@ΘD - Cos@ΘD Sin@ΘD H- + L - + H- + L - - -H- + L Sin@ΘD -Cos@ΘD Sin@ΘD Cot@ΘD R - + R R H- + L R - H- + L Sin@ΘD Sin@ΘD R - + R - Sin@ΘD
19 Thank You! eoge E. Habovsky MAST 9
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