University of Groningen Field perturbations in general relativity and infinite derivative gravity Harmsen, Gerhard Erwin

Field perturbations in general relativity and infinite derivative gravity

Harmsen, Gerhard Erwin

DOI:

10.33612/diss.99355803

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Publication date: 2019

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Harmsen, G. E. (2019). Field perturbations in general relativity and infinite derivative gravity. University of Groningen. https://doi.org/10.33612/diss.99355803

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91

Appendix A

Gamma Matrices

During the derivation of the Super covariant derivative and subsequently the deriva-tion of the radial equaderiva-tions we have used the Dirac gamma matrices some of their identities. As such we will briefly go over these matrices and show some useful iden-tities. The gamma matrices are a set of γi, where i =0, 1, 2, ..., N−1, N×N matrices with a specific anti-commutation relation that ensures they generate theCl_1,N−1(R) Clifford algerbra, where N represents the total number of space time dimensions. That is they must obey the relation [119]

γµ, γν =2ηµνIN. (A.1)

Furthermore if N is even we can construct another matrix as follows[119]

γN =iN/2−1 N−1

∏

i=0

γi, (A.2)

in some literature this is called the Hermitian chiral matrix.

For our work the identities we will need involve the anti-symmetric gamma matrix relations given as follows

γaγb=γab+gab γaγbγc =γabc+γagbc−γbgac+γcgab γaγbγcγd =γabcd+  γabgcd−γacgbd+γadgbc  +γbcgad−γbdgac+γcdgab  +gabgcd−gacgbd+gadgbc γaγbγcγdγe =γabcde+ 

γabcgde−γabdgce+γabegcd+γacdgbe−γacegbd

+γadegbc−γbcdgae+γbcegad−γbdegac+γcdegab

 + gbcgde−gbdgce+gbegcdγa−  gcdgea−gcegda+gcagdeγb +gdegab−gdageb+gdbgeaγc−  geagbc−gebgac+gecgabγd +gabgcd−gacgbd+gadgbcγe  (A.3) Using the above identities we can easily, although tedious, obtain the following re-sults

γaγa = D (A.4)

γabγ_b= (D−1)γa (A.7) γabγbc= (D−2)γa_c+ (D−1)ga_c (A.8) γabγ_bcd= (D−3)γa_cd− (D−2)(γcgad−γdgac) (A.9) γabcγc = (D−2)γab (A.10) γabcγcd = (D−3)γ_dab+ (D−2)(γagb_d−γbga_d) (A.11) γabcγce f =(D−4)γab_{e f}− (D−3)  γa_egb_f −γa_fgb_e−γb_ega_f +γb_fga_e  − (D−2)(ga_egb_f −ga_fgb_e). (A.12) γge f aγa = (D−3)γge f (A.13) γgacbγ_be = (D−4)γgace+ (D−3) γgagc_e−γgcga_e+γacgeg
(A.14) γgacbγ_{be f} = (D−5)γgac_{e f} − (D−4)γgaegcf −γ ga fg c e+γ gc fg a e−γ gc egaf +γaceg g f −γ ac fg g e  − (D−3)  γg  ga_egc_f −ga_fgc_e−γa  gc_fgeg−gceg g f  +γc  geggaf −gaeg g f   (A.15) There are further relations we may need to consider which are nessecary for the calculation of the super covariant derivative. We have given there results below.

γλµν[D, γνρσFρσ] =2(D−3)γνλ∇µFµν− (D−3)γµν∇λFµν+2(D−2)∇µFµλ. γλµν[γµ, γnu] = −2(D−2)(D−1)γλ. γλµν[γµ, γρFνρ] = −2(D−3)γλνρFνρ+2(D−2)γνFνλ. γλµν= −2(D−4)(D−3)γλ_ρσFρσ+4(D−3)(D−2)γρFρλ γλµν[γρFµρ, γσFνσ] = −2γλµνρσFµνFρσ−4γµνρFµρFνλ−2γλFρσFρσ−4γµFµνFνλ. γλµν[γαFµα, γνρσF ρσ_{] = −2(D−5)} γλµνρσFµνF ρ_+6(D−4) γµρσF ρσ_Fλ µ . γλµν[γµαβF αβ_{, γ} νρσFρσ] = −2(D2−11D+26)γλαβρσF αβ_Fρσ_{+8(D−6)(D−3)} γραβF αβ_Fρλ +4(D−4)(D−3)γλFρσFρσ−16(D−3)γρFρσFσλ.

93

Appendix B

Full expressions of the curvature

tensors

In this appendix we show the full expressions for all curvature tensors and invari-ants for the IDG linearized static metric for a charged point-source derived in Eqs. (6.19). Furthermore we recall that F(x)is the Dawson function and Erf(x) is the error function.

The Ricci scalar is given by[110]:

R = GmM 3 se− 1 4r2M2s √ π ; (B.1)

the non-zero components of the Riemann tensor:

R₀₁₀₁= 1 8G " M_s3   2Q2_FrMs 2  r + 4me−14r2M2s √ π  + 8Ms  Q2_FrMs 2  + 2mre− 1 4r2 M2s √ π  r3 +Q2rM5_sF rMs 2  − 16mErf  rMs 2  r3 − 4Q2M2 s r2 −Q 2_M4 s # , R₀₃₀₃= R₀₂₀₂sin2(θ) = 1 4G sin 2₍ θ) h Ms  − 2Q2FrMs 2  r − 4me−14r2M2s √ π   −Q2_rM3 sF  rMs 2  + 4mErf(rMs2 ) r +Q2M2s i , R₁₃₁₃= R₁₂₁₂sin2(θ) = 1 16G sin 2₍ θ) " Ms   4Q2F  rMs 2  r + 16me−14r2M2s √ π  +Q2r3M5_sF  rMs 2  −16mErf(rMs2 ) r + 8mr2M3 se− 1 4r2M2s √ π −Q2r2M4_s −2Q2M2_s # , R₂₃₂₃= 1 4Gr sin 2₍ θ) " Ms −2Q2F rMs 2  − 8mre −1 4r2Ms2 √ π ! −Q2r2M3_sF rMs 2  +8mErf rMs 2  +Q2rM2_s # ; (B.2)

R00= 1 8GM 3 s s 2 r + 4me √ π −Q2Ms, R₁₁= 1 4GMs  Ms 2mMse −1 4r2M2s √ π + Q 2 r2 ! − Q 2 _r2_M2 s+2
F  rMs 2  r3  , R₂₂= 1 16GMs   Q2 r4M4_s+4
FrMs 2  r +Ms 8mr2Mse− 1 4r2M2s √ π −Q2r2M2_s −2Q2 ! , R₃₃= 1 16G sin 2₍ θ)Ms " Q2 r4M4_s+4
FrMs 2  r +Ms  8mr2Mse− 1 4r2 M2s √ π −Q 2_r2_M2 s−2Q2 # ; (B.3) the non-zero components of the Weyl tensor:

C₀₁₀₁ = 1 48G " M3_s   12Q2FrMs 2  r + 16me−14r2M2s √ π  + Ms  36Q2FrMs 2  +96mre− 1 4r2 M2s √ π  r3 +3Q2rM_s5F rMs 2  −96mErf  rMs 2  r3 − 18Q2M2_s r2 −3Q 2_M4 s # , C₀₃₀₃ = −C₁₃₁₃ = C₀₂₀₂sin2(θ) = −C1212sin2(θ) = 1 96G sin 2₍ θ) " 4rM3_s −3Q2F rMs 2  − 4mre −1 4r2M2s √ π ! +Ms  −36Q2F(rMs2 ) r −96me −1₄r2 M2_s √ π  −3Q2r3M5_sF rMs 2  + 96mErfrMs 2  r +3Q 2_r2_M4 s +18Q2M2s # , C₂₃₂₃ = 1 48Gr sin 2₍ θ) " 4r2M3_s −3Q2F rMs 2  −4mre −1 4r2M2s √ π ! +Ms  −36Q2FrMs 2  −96mre−14r2 M2s √ π  −3Q2r4M5_sF rMs 2  +96mErf rMs 2  +3Q2r3M4_s+18Q2rM_s2 # . (B.4) We will now show the expressions for the curvature invariants. The Ricci scalar squared is given by:

R2 = G 2_m2_M6 se− 1 2r2M2s π ; (B.5)

Appendix B. Full expressions of the curvature tensors 95

the Ricci tensor squared:

RµνRµν = G2_M2 s 128πr6 " F rMs 2  32√πmQ2r7M6_se− 1 4r2M2s ₋₆₄√_πmQ2_r5_M4 se− 1 4r2M2s −6πQ4r7M7_s +4πQ4r5M5_s −24πQ4r3M3_s−48πQ4rMs ! +πQ4  3r8M8_s−8r6M6_s +24r4M4_s+32r2M2_s+48  F rMs 2 2 +M4_s128m2r6e−12r2Ms2_+4πQ4_r4  −32√πmQ2r6M_s5e− 1 4r2Ms2 +3πQ4r6M6_s +12πQ4r2M_s2 # ; (B.6) the Weyl tensor squared:

CµνρσCµνρσ= G2_e−1 2r2M2s 192πr6 " −4r2M3_s  3√πQ2e14r2M2s_F rMs 2  +4mr  −12Ms  3√πQ2e 1 4r2M2s_F rMs 2  +8mr  −3√πQ2r4M5_se 1 4r2M2s_F rMs 2  +96√πme 1 4r2M2s_Erf rMs 2  +18√πQ2rM_s2e 1 4r2Ms2₊₃√_πQ2_r3_M4 se 1 4r2M2s #2 ; (B.7)

K = 32πr6 3e πQ r F ₂ Ms −6e πQ r F ₂ Ms +32e14r2M2s_m√_πQ2_F rMs 2  r7+3e12r2M2s_πQ4_r6 +8e12r2M2s_πQ4_F  rMs 2  2_r6_M8 s −4e14r2M2s√_πQ2_r5  7e14r2M2s√_πF rMs 2  Q2+8mr  M7_s +4r418e12r2M2s_πF rMs 2  2_Q4_+5e1 2r2M2s_πQ4_+32e41r2M2s_m√_πrF rMs 2  Q2 +24m2r2M6_s −24e14r2M2s√_πQ2_r3  8mr+e14r2Ms2√_πF rMs 2   5Q2+4mrErf rMs 2  M5_s +4r2  40e12r2M2s_πF rMs 2  2_Q4_+15e1 2r2M2s_πQ4_+144e41r2M2s_m√_πrF rMs 2  Q2 +24e12r2M2s_mπrErf rMs 2  Q2+128m2r2  M_s4 −16e14r2M2s√_πr  3e14r2M2s√_πF rMs 2   5Q2+8mrErf rMs 2  Q2 +4mr  9Q2+8mrErf rMs 2   M3_s+48  5e12r2M2s_πF rMs 2  2_Q4 +24e14r2M2s_m√_πrF rMs 2  Q2+4mr  3e12r2M2s_πErf rMs 2  Q2+8mr   M2_s −384e14r2M2s_m√_π  3e14r2M2s√_πF rMs 2  Q2+8mr  Erf rMs 2  Ms +1536e12r2M2s_m2_πErf  rMs 2 2 # . (B.8) In the case Q=0, we would recover all curvature tensors and invariants for the case of a neutral point-source obtained in Ref. [120], as expected.