Theory of relativity/General relativity/Einstein equations

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Einstein's Field EquationsEdit

 

  where this latter G is the gravitational constant of the universe, and   is the cosmological constant, sometimes referred to as the universe's dark energy, and is often taken to be zero.

Einstein's field equations couple the stress energy momentum tensor description of the matter contained by the spacetime to a constraint on the spacetime curvature through nonlinear coupled second order differential equations of 4 dimensional spacetime.

Bianchi IdentitiesEdit

Let us start with the Riemann spacetime curvature tensor in terms of the Christoffel symbols

 

First consider going to a local free fall frame so that the Christoffel symbols vanish even though their first derivatives do not where there is spacetime curvature

 

Taking a partial derivative

 

Permute the indeces and do the following sum

 

You can switch order of partial derivatives

 

And note now that all the Christoffel symbol terms add to zero

 

Going back to an arbitrary frame from free fall corresponds to merely replacing the partial derivative comma with the covariant derivative semicolin which derivative opperation involves the use of Christoffel symbols yeilding the Bianchi identities

 

Stress Energy Momentum TensorEdit

The stress-energy tensor for short of elements   contains information about the stress, pressure, energy density, and momentum density of the matter in the spacetime. For rectilinear inertial frame coordinates

 

is a statment of energy conservation. The closest thing to such a statment for general relativity where globally rectilinear inertial frames don't exist when there is Riemannian spacetime curvature present is

 

which is a statment of energy conservation for local free fall frames for which the Christoffel symbols vanish reducing it to the expression just above.

For such a rectilinear inertial frame the elements of the stress energy tensor have the following interpretations

  is the coordinate frame energy density.

  is a flow of momentum per area in the   direction or the pressure on a plane whose normal is in the   direction.

  is the   component of momentum per area in the   direction or describes a shearing from stresses.

  is the volume density of the ith component of momentum flow.

Coupling of Spacetime Curvature to Stress-EnergyEdit

Start with the Bianchi identities

 

Contract with the metric tensor over   and  , then contract again over   and   and use the antisymetry of the Riemann tensor in the last two indices to arrive at

 

Renaming the repeated index in common

 

combining like terms and dividing by -2

 

A chronecker delta may be inserted to yield the equivelent expression

 

Raise an index with the contravariant metric tensor   and opperate the chronecker delta on the metric tensor

 

Since the covariant divergence of the metric tensor is zero, the following is equivalent

 

Here we define the Einstein tensor  

 

So we find that the covariant divergence of the Einstein tensor is zero

 

As we want to retain a statement of energy conservation for general relativity which was

 

the simplest way to ensure this energy conservation statement in coupling the Einstein curvature tensor to the stress-energy is to write one proportional to the other

 

In a low speed weak field limit, for this to correspond to Newtonian gravitation we find that the coupling constant must be   where this G is the gravitational constant of the universe. Since the covariant divergence of the metric tensor is also zero, a more general possibility exists that the field equations should be

 

as this would also correspond to Newtonian gravitation in the low speed weak field limit in the case of a small enough cosmological constant  . Einstein originally included this in his field equations, but his motivation for doing so was an effort to seek static state universe solutions for it was not yet known that the universe was expanding. When the expansion was discovered, because of his motive, he thought including it was his "greatest blunder", but we have some evidence now that this cosmological constant, sometimes referred to as dark energy is not zero, so he was likely right to include it afterall. A positive cosmological constant yields accelerating universe solutions, and the evidence found is that the observed universe's expansion does look to have such an acceleration term.

ReferencesEdit