Tensors and Relativity: Chapter 7

The weak field approximation

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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 tex2html_wrap_inline925 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

equation732

where tex2html_wrap_inline927 is small. Since we require that tex2html_wrap_inline929 , the inverse metric is given by

equation734

To work out the geodesic equations we need to work out what the components of the Christoffel symbols are:

equation736

Substituting for tex2html_wrap_inline931 etc. in terms of tex2html_wrap_inline933 we obtain

equation738

The geodesic equations are

equation740

But for a slowly moving particle tex2html_wrap_inline935 so

equation742

Also tex2html_wrap_inline937 , so we can neglect terms like tex2html_wrap_inline939 . The geodesic equation reduces to

equation744

so the ``space'' equation (three- acceleration) is

equation746

Since tex2html_wrap_inline941 we get

equation748

Now

eqnarray750

where we have neglected time derivatives over space derivatives. The spatial geodesic equation then becomes

equation752

But Newtonian theory has

equation754

where tex2html_wrap_inline943 is the gravitational potential. So we make the identification

equation756

This is equivalent to having spacetime with the line element

equation758

This is what we deduced using the Equivalence Principle.

Let's now look at the field equations [ with tex2html_wrap_inline915 ]:

equation760

Taking the trace we get

eqnarray762

This allows us to write the field equations as

equation764

Let us assume that the matter takes the form of a perfect fluid, so the stress- energy tensor takes the form:

equation766

Taking the trace gives

equation768

so the field equations become

equation770

The Newtonian limit is tex2html_wrap_inline947 . This gives

equation772

Look at the 00 component of these equations:

eqnarray774

to first order in tex2html_wrap_inline951 . Now

equation776

to first order in tex2html_wrap_inline951 . The (0,0) component of this equation is

equation778

and since spatial derivatives dominate over time derivatives, we get

equation780

So the field equations are

equation782

This is just

equation784

Comparing this with Poisson's equation:

equation786

we see that we get the same result if the constant tex2html_wrap_inline889 is

equation788

We can now use this result to write down the full Einstein field equations: 

equation790


next up previous index
Next: Index Up: Title page Previous: The non- vacuum field