Interior Solution

 

 

 

 

We will assume that we are dealing with a static and spherically symmetric configuration of mass-energy, in which the density   and the pressure   are functions only of a radial coordinate .

 

 

(25.1)

 

(25.2)

 

 

A local relation is assumed to exist between these quantities. This relation is called the equation of state.  It is written in generic form as:

 

 

 

(25.3)

 

 

We will assume that the spherically symmetric static metric has the following general form:

 

 

(25.4)

 

 

The energy momentum tensor for the interior of the spherical object will be assumed to be of the form of the Perfect Fluid Tensor given as follows:

 

 

 

(25.5)

 

NOTE: In the following calculations we will have units where c=1.

 

The velocity in this equation is a 4 dimensional velocity given by

 

 

(25.6)

 

 

 

We will assume that the matter is at rest at each point in the interior of the spherical object (hereafter called 'the star'). This means that the following condition holds in our calculation:

(25.7)

.

In general Newtonian physics is able to handle the interior description of most ordinary stars. The type of general relativistic model that we are discussing here is necessary for very dense stars such as white dwarves and neutron stars where the densities are extremely high. A neutron star model diagram is shown below.

 

 

 

On the trajectory of each particle of matter in the star fluid, the line element gives the relation between proper time and coordinate time:

 

(25.8)

 

It will be convenient to work with the energy-momentum tensor in covariant-index form. Hence we lower the velocity index as follows:

 

(25.9)  

 

The covariant form of the perfect fluid tensor can be written as

 

(25.10)

 

 

 

 

In matrix form this tensor can now be decomposed as follows:

 

(25.11)                                                            

 

 

 

Substituting in the metric components we arrive at the following equation for the energy-momentum equations:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(25.12)  

We will use the Ricci tensor form for the Einstein field equations

 

(25.13)

 

to find the relations between the metric functions  and the phenomenological fluid parameters . We have suppressed all constants in this equation.

 

 

The scalar trace

 

is very easily obtained from (25.10):

 

 

(25.14)

 

In getting this trace we have made use of the proper velocity normalization condition

 

 

(25.15)

 

 

and the 4-D metric trace relation

 

(25.16)

.

 

It is now possible to calculate the right-hand-side of the field equations (25.13). We get the following results.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(25.17)

 

 

 

The corresponding left-hand-side of the field equations are following Ricci components:

 

 

(25.18)                                                            

 

 

Using these equations we get the following set of coupled nonlinear differential equations:

 

 

(25.19)  

 

 

(25.20)  

 

 

 

(25.21)  

 

Combining (25.19) and (25.20) we get the following result

 

 

(25.22)

 

This relation then implies that .  We now solve (25.22) and (25.21) for . We get

 

(25.23)

 

 

 

(25.24)

 

 

We get a third equation by combining (25.20) and (25.21) :

 

 

 

(25.25)  

 

 

We now have a system of 3 coupled differential equations. We have four unknown functions to solve for: . The extra equation that is needed for a complete solution is the phenomenologically derived equation of state

 

 

(25.26)

 

 

 

We now define a mass function that is meant to be consistent with the mass constant in the exterior Schwarzschild solution.

 

 

 

(25.27)

 

 

Using equation (25.23) we can solve for  to get

 

 

(25.28)

 

 

Equation (25.24) can be rewritten as

 

 

 

(25.29)

 

This equation can be integrated to give the  function. We now relate . Differentiate (25.24) and then use (25.25) to obtain the following

 

 

(25.30)

 

 

 

Using (25.29) in this equation then gives:

 

 

(25.31)

 

 

 

This equation is called the Tolman-Oppenheimer-Volkoff (TOV) equation for hydrodynamic equilibrium. When units are put back in this equation is as follows:

 

(25.32)  

Note that when the radius value goes up, the pressure goes down. The boundary of a star will be defined to be the radius value where the pressure goes to zero. The mass of the star will then be the integral of equation (25.28) integrated up to the boundary radius.

 

 

We will now summarize the set of equations necessary for the general relativistic description of the interior of a star.

 

 

 

 

 

 

 

Summary

 

 

(25.33)

 

 

(25.34)                                                                       

 

 

(25.35)  

 

 

These equations can be integrated to give values for the gravitational metric at any location r within the star provided that some initial data is given for the point . Assuming that  we need to know the value of  The densities associated with the core of white dwarf stars would be of the order of  and with neutron stars .

 

The metric components are obtained from the relations:

 

 

(25.36)

 

 

 

and

(25.37)  

 

 

These equations are matched at the star boundary to the exterior Schwarzschild solution.