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Tensors and Relativity: Chapter 6

Manifolds, tangent spaces and local inertial frames

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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 tex2html_wrap_inline3392 at each point which can be differentiated everywhere. This is always true in Special Relativity and General Relativity.

We can then define one- forms tex2html_wrap_inline3394 as having components tex2html_wrap_inline3396 and vectors  tex2html_wrap_inline3398 as linear functions which take tex2html_wrap_inline3394 into the derivative of tex2html_wrap_inline3392 along a curve with tangent tex2html_wrap_inline3398 :

equation2655

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

equation2661

for any vector tex2html_wrap_inline3398 for example Euclidian 3D space .

If however tex2html_wrap_inline3410 is of indefinite sign as it is in Special and General Relativity it is called Pseudo- Riemannian.  .

For a general spacetime with coordinates tex2html_wrap_inline3412 , the interval between two neighboring points is

equation2673

In Special Relativity we can choose Minkowski coordinates such that tex2html_wrap_inline3414 everywhere. This will not be true for a general curved manifold. Since tex2html_wrap_inline3416 is a symmetric matrix, we can always choose a coordinate system at each point tex2html_wrap_inline3418 in which it is transformed to the diagonal Minkowski form, i.e. there is a transformation

equation2675

such that

eqnarray2679

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

In general tex2html_wrap_inline3420 will not diagonalize tex2html_wrap_inline3416 at every point since there are ten functions tex2html_wrap_inline3424 and only four transformation functions tex2html_wrap_inline3426 .

We can also choose tex2html_wrap_inline3420 so that the first derivatives of the metric vanishes at tex2html_wrap_inline3418 i.e.

equation2688

for all tex2html_wrap_inline3432 , tex2html_wrap_inline3434 and tex2html_wrap_inline3436 . This implies

equation2690

That is, the metric near tex2html_wrap_inline3418 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

equation2693

for all tex2html_wrap_inline3432 , tex2html_wrap_inline3434 ;

equation2695

for all tex2html_wrap_inline3432 , tex2html_wrap_inline3434 , tex2html_wrap_inline3436 ; however

equation2697

for at least some values of tex2html_wrap_inline3432 , tex2html_wrap_inline3434 , tex2html_wrap_inline3436 and tex2html_wrap_inline3456 .

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.


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Next: Covariant derivatives and Christoffel Up: Title page Previous: Title page