The Einstein Field Equations from a Tautology.

Let's try and guess the  Einstein Field Equations from the Bianchi Identity which is a tautology in Differential geometry. Now let us consider the curvature two form "R" and see what the Bianchi Identity looks like now: 

Now let's prove it. This as we have seen earlier is the covariant exterior derivative on the Tangent Bundle of the Riemannian manifold we know as spacetime. Let us work with an arbitrary Bundle having a connection 'D' then we consider:

Now lets switch over to the Tangent Bundle whose sections are tensors, (actually you could also call these as ttensor bundles, and the sections as tensor values forms). Here the Bianchi Identity Tells us this:

We could rewrite the term on the L.H.S as:

What this means is the the Einstein Tensor is divergence free. We know from the locacl conservation law of energy momentum that the Stress energy tensor too is divergence free!, thus we could equate the two upto a constant factor like(considering units such that c=1):

And this is the Einstein Field Equation!, now since we are playing this game, why dont we also consider the fact that the metric, is also divergence free i.e:

now following what we did previously we could include this into the einstein Field equations but it must be scaled by a constant factor, i.e:

and the Lambda there is Einstein's cosmological Constant!.

This may note be a rigourous derivation but it is fun to ponder over..!!

By Vasudev Shyam (Admin at PDEP)