Γttt
factor out index raise
gtaΓatt
substitute definition of connection
12gta(gtt,a+2gat,t)
split the index a into t and j
12(gtjgtt,j+gttgtt,t+2gtjgjt,t)
replace ADM metric definitions
12(βjα2(α2+βkβlγkl),j+(α2+βkβlγkl),t1α2+2βjα2βj,t)
raise βj
12(βjα2(α2+βkβlγkl),j+(α2+βkβlγkl),t1α2+2βjα2(γjkβk),t)
simplify...
α,t1α+112βlβk,tγkl1α2+112βkβlγkl,t1α2+112βkβl,tγkl1α2+α,jβj1α+βjβkγjk,t1α2+βjβk,tγjk1α2+112βjβlβk,jγkl1α2+112βjβkβlγkl,j1α2+112βjβkβl,jγkl1α2
relabel...
α,t1α+112βkβlγkl,t1α2+112βkβl,tγkl1α2+112βlβk,tγkl1α2+α,jβj1α+βjβkγjk,t1α2+βjβk,tγjk1α2+112βjβkβlγkl,j1α2+112βjβlβk,jγkl1α2+112βjβkβl,jγkl1α2
for βi=0
1αα,t

Γtti
factor out index raise
gtaΓati
substitute definition of connection
12gta(gat,i+gai,tgti,a)
split the index a into t and j
12(gtjgjt,i+gtjgji,tgtjgti,j+gttgtt,i)
replace ADM metric definitions
12(βjα2βj,i+βjα2γji,tβjα2γti,j+(α2+βkβlγkl),i1α2)
raise βj
12(βjα2(γjkβk),i+βjα2γji,tβjα2γti,j+(α2+βkβlγkl),i1α2)
simplify...
12βjγji,t1α2+112βjγti,j1α2+α,i1α+12βjβkγjk,i1α2+112βkβlγkl,i1α2+112βlβk,iγkl1α2+112βkβl,iγkl1α2+12βjβk,iγjk1α2
relabel...
12βjγji,t1α2+112βjγti,j1α2+α,i1α+112βkβl,iγkl1α2+12βjβkγjk,i1α2+12βjβk,iγjk1α2+112βkβlγkl,i1α2+112βlβk,iγkl1α2
for βi=0
1αα,i

Γitt
factor out index raise
giaΓatt
substitute definition of connection
12gia(gtt,a+2gat,t)
split the index a into t and j
12(gijgtt,j+gitgtt,t+2gijgjt,t)
replace ADM metric definitions
12((γijβiβjα2)(α2+βkβlγkl),j+βiα2(α2+βkβlγkl),t+2(γijβiβjα2)βj,t)
raise βj
12((γijβiβjα2)(α2+βkβlγkl),j+βiα2(α2+βkβlγkl),t+2(γijβiβjα2)(γjkβk),t)
simplify...
1α,tβi1α+αα,jγij+12βiβkβl,tγkl1α2+12βiβlβk,tγkl1α2+12βiβkβlγkl,t1α2+1α,jβiβj1α+1βiβjβk,tγjk1α2+βkγijγjk,t+1βiβjβkγjk,t1α2+βk,tγijγjk+112βlβk,jγijγkl+12βiβjβlβk,jγkl1α2+112βkβlγijγkl,j+112βkβl,jγijγkl+12βiβjβkβlγkl,j1α2+12βiβjβkβl,jγkl1α2
relabel...
1α,tβi1α+αα,jγij+12βiβkβl,tγkl1α2+12βiβlβk,tγkl1α2+12βiβkβlγkl,t1α2+1α,jβiβj1α+βkγijγjk,t+1βiβjβkγjk,t1α2+1βiβjβk,tγjk1α2+βk,tγijγjk+12βiβjβlβk,jγkl1α2+112βkβlγijγkl,j+12βiβjβkβlγkl,j1α2+112βlβk,jγijγkl+12βiβjβkβl,jγkl1α2+112βkβl,jγijγkl
for βi=0
αα,jγij

Γimt
factor out index raise
giaΓamt
substitute definition of connection
12gia(gam,t+gat,mgmt,a)
split the index a into t and j
12(gijgjm,tgijgmt,j+gijgjt,m+gitgtt,m+gitgtm,tgitgmt,t)
replace ADM metric definitions
12((γijβiβjα2)γjm,t(γijβiβjα2)γmt,j+(γijβiβjα2)βj,m+βiα2(α2+βkβlγkl),m+βiα2γtm,tβiα2γmt,t)
raise βj
12((γijβiβjα2)γjm,t(γijβiβjα2)γmt,j+(γijβiβjα2)(γjkβk),m+βiα2(α2+βkβlγkl),m+βiα2γtm,tβiα2γmt,t)
simplify...
112βiγmt,t1α2+12βiγtm,t1α2+12βiβjγmt,j1α2+112βiβjγjm,t1α2+12γijγjm,t+112γijγmt,j+1α,mβi1α+12βiβkβlγkl,m1α2+112βiβjβkγjk,m1α2+112βiβjβk,mγjk1α2+12βiβkβl,mγkl1α2+12βiβlβk,mγkl1α2+12βkγijγjk,m+12βk,mγijγjk
relabel...
112βiγmt,t1α2+12βiγtm,t1α2+112βiβjγjm,t1α2+12βiβjγmt,j1α2+12γijγjm,t+112γijγmt,j+1α,mβi1α+12βiβlβk,mγkl1α2+12βkγijγjk,m+12βiβkβl,mγkl1α2+112βiβjβkγjk,m1α2+112βiβjβk,mγjk1α2+12βiβkβlγkl,m1α2+12βk,mγijγjk
for βi=0
12γij(γmt,j+γjm,t)

Γtim
factor out index raise
gtaΓaim
substitute definition of connection
12gta(gai,m+gam,igim,a)
split the index a into t and j
12(gtjgji,mgtjgim,j+gtjgjm,i+gttgtm,i+gttgti,mgttgim,t)
replace ADM metric definitions
12(βjα2γji,mβjα2γim,j+βjα2γjm,i+γtm,i1α2+γti,m1α2γim,t1α2)
raise βj
12(βjα2γji,mβjα2γim,j+βjα2γjm,i+γtm,i1α2+γti,m1α2γim,t1α2)
simplify...
12γim,t1α2+112γti,m1α2+112γtm,i1α2+12βjγjm,i1α2+112βjγim,j1α2+12βjγji,m1α2
relabel...
12γim,t1α2+112γtm,i1α2+112γti,m1α2+12βjγjm,i1α2+112βjγim,j1α2+12βjγji,m1α2
for βi=0
γti,mγtm,i+γim,t2α2

Γimn
factor out index raise
giaΓamn
substitute definition of connection
12gia(gam,n+gan,mgmn,a)
split the index a into t and j
12(gijgjm,ngijgmn,j+gijgjn,m+gitgtn,m+gitgtm,ngitgmn,t)
replace ADM metric definitions
12((γijβiβjα2)γjm,n(γijβiβjα2)γmn,j+(γijβiβjα2)γjn,m+βiα2γtn,m+βiα2γtm,nβiα2γmn,t)
raise βj
12((γijβiβjα2)γjm,n(γijβiβjα2)γmn,j+(γijβiβjα2)γjn,m+βiα2γtn,m+βiα2γtm,nβiα2γmn,t)
simplify...
112βiγmn,t1α2+12βiγtn,m1α2+12βiγtm,n1α2+12βiβjγmn,j1α2+112βiβjγjn,m1α2+112βiβjγjm,n1α2+12γijγjm,n+12γijγjn,m+112γijγmn,j
relabel...
112βiγmn,t1α2+12βiγtm,n1α2+12βiγtn,m1α2+12βiβjγmn,j1α2+112βiβjγjn,m1α2+112βiβjγjm,n1α2+12γijγjm,n+12γijγjn,m+112γijγmn,j
for βi=0
12γij(γmn,j+γjn,m+γjm,n)

Rttti
substitute definition of Riemann curvature
Γtti,tΓttt,i+ΓtetΓetiΓteiΓett
split the index e into t and m
Γtti,tΓttt,i+(Γttt+Γtmt)(Γtti+Γmti)(Γtti+Γtni)(Γttt+Γntt)
Γtti,tΓttt,i+ΓtmtΓtti+ΓmtiΓttt+ΓmtiΓtmtΓtniΓtttΓnttΓttiΓnttΓtni

Rttij
substitute definition of Riemann curvature
Γttj,iΓtti,j+ΓteiΓetjΓtejΓeti
split the index e into t and m
Γttj,iΓtti,j+(Γtti+Γtmi)(Γttj+Γmtj)(Γttj+Γtnj)(Γtti+Γnti)
Γttj,iΓtti,j+ΓtmiΓttj+ΓmtjΓtti+ΓmtjΓtmiΓtnjΓttiΓntiΓttjΓntiΓtnj

Rtitj
substitute definition of Riemann curvature
Γtij,tΓtit,j+ΓtetΓeijΓtejΓeit
split the index e into t and m
Γtij,tΓtit,j+(Γttt+Γtmt)(Γtij+Γmij)(Γttj+Γtnj)(Γtit+Γnit)
Γtij,tΓtit,j+ΓtijΓtmtΓnitΓttj+ΓmijΓttt+ΓmijΓtmtΓtitΓtnj+ΓtijΓtttΓtitΓttjΓnitΓtnj

Rittj
substitute definition of Riemann curvature
Γitj,tΓitt,j+ΓietΓetjΓiejΓett
split the index e into t and m
Γitj,tΓitt,j+(Γitt+Γimt)(Γttj+Γmtj)(Γitj+Γinj)(Γttt+Γntt)
Γitj,tΓitt,j+ΓimtΓttjΓitjΓntt+ΓittΓmtj+ΓimtΓmtjΓinjΓttt+ΓittΓttjΓitjΓtttΓinjΓntt

Rtijk
substitute definition of Riemann curvature
Γtik,jΓtij,k+ΓtejΓeikΓtekΓeij
split the index e into t and m
Γtik,jΓtij,k+(Γttj+Γtmj)(Γtik+Γmik)(Γttk+Γtnk)(Γtij+Γnij)
Γtik,jΓtij,k+ΓtikΓtmjΓnijΓttk+ΓmikΓttj+ΓmikΓtmjΓtijΓtnk+ΓtikΓttjΓtijΓttkΓnijΓtnk

Ritjk
substitute definition of Riemann curvature
Γitk,jΓitj,k+ΓiejΓetkΓiekΓetj
split the index e into t and m
Γitk,jΓitj,k+(Γitj+Γimj)(Γttk+Γmtk)(Γitk+Γink)(Γttj+Γntj)
Γitk,jΓitj,k+ΓimjΓttkΓitkΓntj+ΓitjΓmtk+ΓimjΓmtkΓinkΓttj+ΓitjΓttkΓitkΓttjΓinkΓntj

Rijtk
substitute definition of Riemann curvature
Γijk,tΓijt,k+ΓietΓejkΓiekΓejt
split the index e into t and m
Γijk,tΓijt,k+(Γitt+Γimt)(Γtjk+Γmjk)(Γitk+Γink)(Γtjt+Γnjt)
Γijk,tΓijt,k+ΓimtΓtjkΓitkΓnjt+ΓittΓmjk+ΓimtΓmjkΓinkΓtjt+ΓittΓtjkΓitkΓtjtΓinkΓnjt

Rijkl
substitute definition of Riemann curvature
Γijl,kΓijk,l+ΓiekΓejlΓielΓejk
split the index e into t and m
Γijl,kΓijk,l+(Γitk+Γimk)(Γtjl+Γmjl)(Γitl+Γinl)(Γtjk+Γnjk)
Γijl,kΓijk,l+ΓimkΓtjlΓitlΓnjk+ΓitkΓmjl+ΓimkΓmjlΓinlΓtjk+ΓitkΓtjlΓitlΓtjkΓinlΓnjk