in a D space time of coordinates [math]x^u,y^m[/math], with u=...p+1, m=...D-p-1, and A=.....D

if [math]R_{MN}=8 \pi G_D (T_{MN}-\frac{G_{MN}T^P_P}{D-2})[/math] (eq1)

 

I don't understand why

 

[math]R^u_u=\frac{8\pi G}{D-2}((D-p-3)T^u_u-(p+1)T^m_m)[/math] eq 2

[math]R^m_m=\frac{8\pi G}{D-2}((p-1)T^m_m-(D-p-1)T^u_u[/math] eq 3

 

i tried to multiplicate eq1 by [math]g^{MN}[/math] and then

 

[math]R=8\pi G_D(T-\frac{T}{D-2})[/math]

but, i don't go to the eqs 2 and 3

 

Substitute with 8*pi*Tmn instead of Gmn. And, then raise the index n. I think this will come very close to eq 2 and eq 3.

 

Multiplying with gmn will make the equation loose its nature tensor...... It will be scalar.