Manifolds, tangent spaces and local inertial frames
Tensors can then be defined as maps from one- forms and vectors into the reals [ see chapter 3].
For a general spacetime with coordinates , the interval between two neighboring points is
In Special Relativity we can choose Minkowski coordinates such that everywhere. This will not be true for a general curved manifold. Since is a symmetric matrix, we can always choose a coordinate system at each point in which it is transformed to the diagonal Minkowski form, i.e. there is a transformation
In general will not diagonalize at every point since there are ten functions and only four transformation functions .
We can also choose so that the first derivatives of the metric vanishes at i.e.
for all , and . This implies
That is, the metric near is approximately that of Special Relativity, differences being of second order in the coordinates. This corresponds to the local inertial frame whose existence was deduced from the equivalence principle .
for all , ;
for all , , ; however
for at least some values of , , and .
It reflects the fact that any curved space has a flat tangent space at every point, although these tangent spaces cannot be meshed together into a global flat space.
Recall that straight lines in a flat spacetime are the worldlines of free particles; the absence of first derivative terms in the metric of a curved spacetime will mean that free particles are moving on lines that are locally straight in that coordinate system. This makes such coordinates very useful for us, since the equations of physics will be nearly as simple as they are in flat spacetime, and if they are tensor equations they will be valid in every coordinate system.