Proof of Existence of Locally Inertial Frames

This is my own version of the proof. There is a version in Bernard Schutz's book, "A First Course in General Relativity" (Cambridge University Press) but the proof there seems overly complicated and also has some gaps relating to consistency of the the systems of linear equations he sets up.

First, we need a fact from linear algebra: if -,- is an inner product on the vector space L, then there exists a basis {V(1), V(2), . . . , V(n)} for L such that

V(i), V(j) = ±ij

(To prove this, use the fact that any symmetric matrix can be diagonalized using a P-PT type operation.)

To start the proof, fix any chart xi near m, with xi(m) = 0 for all i, and choose a basis {V(i)} of the tangent space at m such that they satisfy the above condition. With our bare hands, we are now going to specify a new coordinate system be i = i(xj) such that

ij = V(i), V(j)     (showing part (a))

The functions i = i(xj) will be specified by constructing their inverse xi = xi(j) using a quadratic expression of the form:

 xi = j A(i,j) + 12 jk B(i,j,k)

where A(i,j) and B(i,j,k) are constants. It will follow from Taylor's theorem (and the fact that xi (m) = 0 ) that

 A(i,j) = xij m and B(i,j,k) = 2xijk m

so that

 xi = j xij m + 12 jk 2xijk m

where all the partial derivatives are evaluated at m.

Note These partial derivatives are just (yet to be determined) numbers which, if we differentiate the above quadratic expression, turn out to be its actual partial derivatives evaluated at m.

In order to specify this inverse, all we need to do is specify the terms A(i,j) and B(i,j,k) above.. In order to make the map invertible, we must also guarantee that the Jacobean (xi/j)m is invertible, and this we shall do.

We also have the transformation equations

 ij = xki xlj gkl ...   (I)

and we want these to be specified and equal to V(i), V(j) when evaluated at m. This is easy enough to do: Just set

 A(i, j) = xij m = V(j)i.

For then, no matter how we choose the B(i,j,k) we have

ij(m) =  xki m xlj m gkl
=  V(i)k V(j)lgkl
=  V(i), V(j),

as desired. Notice also that, since the {V(i)} are a basis for the tangent space, the change-of-coordinates Jacobean, whose columns are the V(i), is automatically invertible. Also, the V(i) are the coordinate axes of the new system.

(An Aside This is not the only choice we can make: We are solving the system of equations (I) for the n2 unknowns xi/j|m. The number of equations in (I) is not the expected n2, since switching i and j results in the same equation (due to symmetry of the g's). The number of distinct equations is

 n + n2 = n(n+1)2 ,

leaving us with a total of

 n2 - n(n-1)2

of the partial derivatives xi/j that we can choose arbitrarily. * )

* In the real world, where n = 4, this is interpreted as saying that we are left with 6 degrees of freedom in choosing local coordinates to be in an inertial frame. Three of these correspond to changing the coordinates by a constant velocity (3 degrees of freedom) or rotating about some axis (3 degrees of freedom: two angles to specify the axis, and a third to specify the rotation).

Next, we want to kill the partial derivatives ij/a by choosing appropriate values for the B(i, j, k) (that is, the second order partial derivatives 2xi/jk).By the lemma, it suffices to arrange that

hpk(m) = 0.
But
hpk(m) =
 rti(m) pxt xrh xik + pxt 2xthk (m)
=
 pxt rti(m) xrh xik + 2xthk (m)
so it suffices to arrange that
 2xthk (m) = -rti xrh xik (m).
That is, all we need to do is to define
 B(t, h, k) = -rti xrh xik (m).
and we are done.