Let 9 v=9v +g be an infinitesimal deformation of the metric tensor. Demon- strate that the...

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Let 9 →ğµv=9µv +ågμ be an infinitesimal deformation of the metric tensor. Demon- strate that the Christoffel symbols Ip and the Riemann tensor Ro change as follows: vp ST" = vp where 1 59° {V»89= +V89v -V489,p) , SRº = Vμľºvo - V₂¹°μ. αμοι να Additional problem for Daniel, Emmanouil, James and Michael Given a d-dimensional spacetime, consider a Weyl transformation of the metric tensor, 9μvμ = ²0 μ, with o and arbitrary smooth function. σμυ (a) Compute the Weyl-transformed Riemann tensor Rºgμv. (b) Consider the Weyl tensor defined by Сpoμ = Rp 9pu Pov + goμ Ppv +9pv Poμ - gov Ppp, Pμv 1 (Rp — 2(d - 179R) 1 ・d = 2 (Ruv d-2 σμυ is the so-called Schouten tensor. Demonstrate that Weyl tensor Co is invariant under the Weyl transformation. (c) Demonstrate that the Weyl tensor (i) has all the algebraic properties of the Riemann tensor; and (ii) it is traceless, CP apv=0. Let 9 →ğµv=9µv +ågμ be an infinitesimal deformation of the metric tensor. Demon- strate that the Christoffel symbols Ip and the Riemann tensor Ro change as follows: vp ST" = vp where 1 59° {V»89= +V89v -V489,p) , SRº = Vμľºvo - V₂¹°μ. αμοι να Additional problem for Daniel, Emmanouil, James and Michael Given a d-dimensional spacetime, consider a Weyl transformation of the metric tensor, 9μvμ = ²0 μ, with o and arbitrary smooth function. σμυ (a) Compute the Weyl-transformed Riemann tensor Rºgμv. (b) Consider the Weyl tensor defined by Сpoμ = Rp 9pu Pov + goμ Ppv +9pv Poμ - gov Ppp, Pμv 1 (Rp — 2(d - 179R) 1 ・d = 2 (Ruv d-2 σμυ is the so-called Schouten tensor. Demonstrate that Weyl tensor Co is invariant under the Weyl transformation. (c) Demonstrate that the Weyl tensor (i) has all the algebraic properties of the Riemann tensor; and (ii) it is traceless, CP apv=0.

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a The Weyltransformed Riemann tensor is given by R R 2g 2 2 where R ... View the full answer