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Tensors and Relativity: Chapter 6

# Manifolds, tangent spaces and local inertial frames

A manifold is a continuous space whose points can be assigned coordinates, the number of coordinates being the dimension  of the manifold [ for example a surface of a sphere is 2D, spacetime is 4D ].

A manifold is differentiable if we can define a scalar field at each point which can be differentiated everywhere. This is always true in Special Relativity and General Relativity.

We can then define one- forms as having components and vectors as linear functions which take into the derivative of along a curve with tangent : Tensors  can then be defined as maps from one- forms and vectors into the reals [ see chapter 3].

A Riemannian manifold  is a differentiable manifold with a symmetric metric tensor  g at each point such that for any vector for example Euclidian 3D space .

If however is of indefinite sign as it is in Special and General Relativity it is called Pseudo- Riemannian.  .

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 such that Note that the sum of the diagonal elements is conserved; this is the signature of the metric  [ +2 ].

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 .

In summary we can define a local inertial frame  to be one where 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.