Einstein's Field Equations

We will now discuss the set of nonlinear, coupled differential equations that Einstein put forward in 1916 as the gravitational field equations. Solving these equations gives the strength of the gravitational field at any location in spacetime.

 

 

The metric tensor contains two types of information:

 

(1) It contains relatively unimportant information about the coordinate system that has been chosen for the description of  the gravitational field. (Examples - spherical polar coordinates, Cartesian coordinates, etc.)

 

(2) It contains the important information regarding the existence of gravitational potentials at all points in spacetime. In the Newtonian limit we have seen that

 

(14.1)                                                              

 

 

 

As a first stab at guessing what the gravitational field equations with matter sources present might look like, one could consider the equation

 

(14.2)

 

 

where we are using the tensor  to represent the energy-momentum tensor that is causing the spacetime curvature. The energy momentum tensor must be a covariantly conserved quantity in spacetime. This conservation is expressed mathematically as follows:

 

 

(14.3)

.

We will see later that this equation gives rise to a continuity equation and the equation of motion when we take the energy-momentum tensor to be some general type of fluid filling up the spacetime manifold. Going back to equation (14.2) we see that our guessed at field equation is consistent with the energy-momentum conservation law (14.3) in that metric compatibility guarantees the relation

 

(14.4)

.

 

Einstein considered this equation but rejected it since it does not reduce down to the Newtonian gravitational equations in the form of the Poisson's equation, the necessary 2nd order form when matter is present in Newton's theory.

 

(14.5)                                                              

 

 

To get this Newtonian equation one would need 2nd order derivatives of the metric. This implies that the left-hand side of the field equation should be a curvature time term since the Riemann, Ricci, or Einstein curvature tensors involve 2nd order derivatives. In 1915 Einstein made a stab at this idea by publishing a paper using the Ricci tensor. He published the following equation as the gravitational field equation.

 

 

(14.6)

 

 

The problem with this equation is that it does not satisfy the conservation law requirement (14.3) since

 

(14.7)

 

 

 

Later in 1915 Einstein changed his field equations such that the Einstein tensor combination of curvature terms was involved. He put forward the final form as follows:

 

 

(14.8)

 

or equivalently, in terms of the Einstein tensor we get

 

 

(14.9)

 

Since the Einstein tensor is specifically constructed such that

 

(14.10)

.

 

we now have consistency with the conservation equation of energy momentum (14.3). The field equations (14.9) reduce properly to the Newtonian Poisson equation. From this reduction we can get what the constant that couples to energy momentum is. We get

 

 

(14.11)

 

In terms of all of the constants we can write the field equations as

 

(14.12)

 

This equation involves 10 nonlinear differential equations (and not 16) since the tensors are symmetric.

The Einstein field equations can be written in the alternative form (exercise)

 

(14.13)

 

 

When one works a lot with these equations it is convenient to adopt units where  and . In this case the field equations are written

 

 

(14.14)

 

or

 

(14.15)

 

In this equation we have introduced the energy-momentum scalar T, which is defined as

 

(14.16)

 

 

The vacuum Einstein equations are just the above field equations with the energy-momentum tensor set equal to zero.

They are written as follows:

 

(14.17)

or equivalently

 

(14.18)

.

 

All of the field equations expressed in contravariant-index form above can be trivially rewritten in covariant-index form as well. The differential equations in each case will reduce down to the same set of nonlinear equations.

 

 

 

 

When it was pointed out to Einstein (largely by the Russian scientist Friedman) that when the field equations were applied to cosmological models the solutions always predicted that the universe itself was dynamic, either expanding or contracting with space, Einstein tried to alter this situation by forcing the field equations into a new form with an added constant . This way he could get at least one static cosmological solution. The new field equations were as follows:

 

 

(14.19)

or

 

(14.20)

 

 

Due to its origins the constant is given the name Cosmological Constant. In 1931, it was pointed out to Einstein by Hubble, that the universe was indeed expanding. Einstein then referred to the introduction of the cosmological constant as the ‘biggest blunder of his life’. Today the universe is thought to have a residual cosmological constant that is making the expansion of the universe accelerate.