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Tensors and Relativity: Chapter 7

# The weak field approximation

We have to check that the appropriate limit, General Relativity leads to Newton's theory. The limit we shall use will be that of small velocities and that time derivatives are much smaller than spatial derivatives.

There are two things we must do:

• We have to relate the geodesic equation to Newton's law of motion.

and

• Relate Einstein's field equations to the Newton- Poisson equation.

Let's assume that we can find a coordinate system which is locally Minkowski [ as demanded by the Equivalence Principle ] and that deviations from flat spacetime are small. This means we can write where is small. Since we require that , the inverse metric is given by To work out the geodesic equations we need to work out what the components of the Christoffel symbols are: Substituting for etc. in terms of we obtain The geodesic equations are But for a slowly moving particle so Also , so we can neglect terms like . The geodesic equation reduces to so the ``space'' equation (three- acceleration) is Since we get Now where we have neglected time derivatives over space derivatives. The spatial geodesic equation then becomes But Newtonian theory has where is the gravitational potential. So we make the identification This is equivalent to having spacetime with the line element This is what we deduced using the Equivalence Principle.

Let's now look at the field equations [ with ]: Taking the trace we get This allows us to write the field equations as Let us assume that the matter takes the form of a perfect fluid, so the stress- energy tensor takes the form: Taking the trace gives so the field equations become The Newtonian limit is . This gives Look at the 00 component of these equations: to first order in . Now to first order in . The (0,0) component of this equation is and since spatial derivatives dominate over time derivatives, we get So the field equations are This is just Comparing this with Poisson's equation: we see that we get the same result if the constant is We can now use this result to write down the full Einstein field equations: 