Slides of 5 lectures at XV Brazilian School on Cosmology and ...

Zalaletdinov’s macroscopic gravityDefine bilocal averaging oper<strong>at</strong>ors, A µ α(x,x ′ ), withsupport <strong>at</strong> two points x ∈ M and x ′ ∈ MConstruct a bitensor extension, T µ ν(x,x ′ ), <strong>of</strong> tensorT µ ν(x) according toT µ ν(x,x ′ ) = A µ α ′ (x,x ′ )T α′ β ′(x′ )A β′ ν(x ′ ,x).Integr<strong>at</strong>es bitensor extension over a 4-dimensionalspacetime region, Σ ⊂ M, with volume V Σ , to obtainregional averageT µ ν (x) = 1V Σ∫Σd 4 x ′ √ −g(x ′ )T µ ν(x,x ′ ),Bitensor transforms as a tensor <strong>at</strong> each point but is ascalar when integr<strong>at</strong>ed for regional average.15th <strong>Brazilian</strong> <strong>School</strong> on Cosmology and Gravit<strong>at</strong>ion, August 2012 – p.29/143