How The Einstein Tensor Was Reconciled With The Ricci Tensor (On The Basis Of Being Divergence-Free)

                     Einstein: Showed his tensor G_mn was also divergence free.

 Recall the Riemann curvature tensor was defined in an earlier blog post, e.g.

A Look At General Relativity and Tensors (Part 3)

 as:   R ⁿ _b m n

This tensor , as noted in that post, is of key importance in general relativity. The Ricci tensor of the first kind is defined simply as a contraction of the Riemann tensor of the 2nd kind, i.e. the above  curvature tensor.  Thus:

R  _b m =   R ⁿ _b m n

Raising an index yields  the Ricci tensor of the second kind, e.g.  R ^b _m 

It is completely determined by knowing the quantity  R ^b  m   for all vectors V _i  of unit length. The tensor is obtained by defining a Ricci tensor of the 2^nd kind thus:

  R ^b _m   =  g ^bn  R  _nm

The number of independent components of this tensor  in a space of N- dimensions is:  ½ N (N + 1 ).  Where the  g ^bn  denote the  g- tensor components with indices raised.

Hence, there will be three components if N = 2,  six components if N = 3 and ten  components if N = 4.

We note here that the Ricci curvature tensor:  Rmn

  is derived from the Riemann curvature tensor.

The contracted Bianchi identities (after Luigi Bianchi)  can be written:

These relate to the divergence of the Ricci tensor (or   Ñ Ric).  'Divergence' (symbol Ñ) as we learned in Advanced Calculus, is associated with how much 'stuff' is flowing into or out of a particular volume or space.  For example, the Maxwell equation:

  Ñ · D  =  r    

With divergence D, of the displacement current, it pertains to the density of charge r in a volume of space. Specifically how much is flowing into or out of the unit volume at any time.

Moving on, it turns out that the divergence of the Ricci tensor, Ñ Ric, turns out to be exactly equal to the divergence of  ½ Rg _mn.  Which equivalence in turn implies the divergence of:

R_mn  - ½ Rg _mn   must be zero.

But please note that  is also the Einstein tensor  G_mn .  So we have:

R_mn  - ½ Rg _mn   =  G_mn    = 0

Which is another way of affirming that the Einstein tensor satisfies the law of energy conservation. I.e. with zero divergence, no net energy can enter or leave the system.

But we aren't done yet!   Einstein had later showed that:

G_mn    = T_mn 

 the stress -energy tensor, which we had earlier met back in a post from Dec. 27, 2014, i.e. see the link posted at top.  In that post we saw the elements for the Einstein tensor for flat space time (for which  G _mn   =  0 ), as well as the elements for the stress-energy (or energy-momentum) tensor, T_mn.  

Energy-momentum because the standard elements for T_mn:

T ₁₁      T ₁₂      T ₁₃        T ₁₄
             T ₂₂     T ₂₃         T ₂₄
                         T ₃₃         T ₃₄
                                         T ₄₄

=

p ₁₁  + r u², p ₁₂ + uv,    p ₁₃    +r uw,  - ru    

                    p ₂₂  + r v²,   p  ₂₃  +r vw,    - rv   

                                             p ₃₃ + r w² ,      - rw 

      
                                                                            r

can be translated into densities r  and pressures p.

Again, the densities r  expressed in the standard way, e.g.  (g/ cm ³), and the elements would be zero in a matter-free region (flat space-time).  In a vacuum, of course, G_mn    = T_mn = 0.

The end result being that since  G_mn    = T_mn    then  the stress-energy tensor T_mn   must likewise be divergence-free.  In other words, energy conservation prevails on both sides of the equation.

See Also:

• A Look at General Relativity and Tensors (Part 1)

And:

• A Look At General Relativity and Tensors (Part 2)

And:

Brane Space: A Closer Look At The Ricci Tensor