Newtonian Limit of the Geodesic Equations

Suppose that we have a weak gravitational field that leaves the spacetime approximately flat. We will represent the metric for this situation as follows:

 (12.1)

The quantity  represents the small part of the metric that gives gravitational field effects. We will show how this metric implies the standard Newtonian gravitational theory. We will assume that all motions are slow relative to the speed of light in free space:

 (12.2)

We are living in a gravitational field that is not changing on timescales relative to the timescales that we have been applying Newtonian gravitational theory to. The Sun has lasted for 4.5 billion years, and calculations of the nuclear processes that govern the Sun suggest that it will survive in a stable state for another 4.5 billion years. Hence, it makes some sense for us to make the assumption that although spatial changes in the gravitational field of our neighbourhood are significant, time-related changes of the gravitational field are not significant. This assumption, which we will call the quasi-static assumption, can be represented mathematically by the following expression:

 (12.3)

So as to get equations agreeing with the classical form of the gravitational equations, we will use coordinate time  as our curve parameter throughout this calculation, instead of proper time. The relation between the proper time and coordinate time is obtained from the following relation:

 (12.4)

This gives the following derivative relation when we introduce the coordinate time parameter

 (12.5)

Since the 00 terms will be much bigger than the terms having non-speed-of-light velocity factors, this equation reduces to the following

 (12.6)

where we keep only first order terms in h. This leads to the 2nd order derivative, which can be represented as

 (12.7)

The results (12.6) and (12.7) will become useful in converting the geodesic equation from proper time dependence to coordinate time dependence.

In General Relativity the geodesic equation determines how an object moves in the presence of a gravitational field. It is written as follows:

 (12.8)

We must hack away at this equation to get the Newtonian gravitational equation. Changing to coordinate time this equation becomes

 (12.9)

The function , a result of the conversion to coordinate time, has been introduced so as to show the pattern for the path equation more clearly. This function is defined as follows:

 (12.10)

Using the relation

 (12.11)

we can rewrite  as

 (12.12)

We are now ready to expand the geodesic equation out into its constituent terms. Doing so gives the following messy equation when we divide all terms by

(12.13)

Consistent with keeping small quantities only to first order, we can define the following relation for the raised  tensor.

 (12.14)

Then, with consistency, we can write

 (12.15) .

We will now calculate the metric connection given by

(12.16)

for our specific conditions. We get

(12.17)

Calculating individual components leads to the following equations.

(12.18)

Using the quasi-static assumption the time derivative terms are neglected to give

 (12.19)

Continuing with the other components we see that

(12.20)

where we again make use of the quasi-static assumption.

This last result is then simplified as follows:

 (12.21)

We don't need to calculate the components that go as  since these terms are coupled to velocity (v) squared terms, and the assumed small Newtonian velocities make these terms negligible relative to all of the other terms in the equation of motion. This reasoning is demonstrated qualitatively as follows

(12.22)

We now work on the right hand side

of the geodesic equation. Using (12.6) and (12.7) in the equation (12.12) gives us

(12.23)

Since the right hand side

goes as a term  that, as the previous result shows, is a time derivative, it will be smaller than any of the spatial derivative terms appearing on the left hand side of the geodesic equation. Furthermore, since  is multiplied by a velocity over c term then the overall effect is to make the right hand side of the geodesic equation totally negligible. We can therefore leave it out.

Filling in all of the previous results the geodesic equation for an object of rest mass m can now be written:

(12.24)

Usually terms that involve curl-like factors such as  have to do with concepts like the vorticity of fields (in this case the field that would have  vorticity is  ). The emergence of this type of term is not surprising here since the Principle of Equivalence asserts that forces of acceleration, such as the Coriolis force, are on the same footing as gravitational forces.  Our curl term represents a fictitious force introduced by us not taking a pure freefall frame as our reference frame. We will concentrate on a non-rotating frame and therefore insist that the 2nd term must be zero.

We therefore get that the equation of motion for a slowly moving particle in a weak and stable gravity field (from the perspective of a near-inertial nonrotating frame) is

 (12.25)

This can be rewritten as in the familiar Newtonian form

 (12.26)

We have used the definition

 (12.27)

where  is a constant. Hence, we see that equation (12.26) is just the Newtonian gravitational law. The gravitational field has gravitational potential given by .

We started this section with the assumption that the metric for a weak field could be represented as

 (12.28)

In terms of the Newtonian potential, we now see that the 00 component of the weak field metric can be written:

Since we are free to choose the constant to be zero, by doing so we can relate the metric directly to the Newtonian gravitational potential and get the relation

 (12.29)