Tensors and Relativity: Chapter 7


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In order to have a complete theory of gravity, we need to know

  • How particles behave in curved spacetime.
  • How matter curves spacetime.

The first question is answered by postulating that free particles [ i.e. no force other than gravity ] follow timelike or null geodesics. We will see later that this is equivalent to Newton's law   tex2html_wrap_inline825 in the week field limit [ tex2html_wrap_inline827 ].

The second requires the analogue  of tex2html_wrap_inline829 . We first consider the vacuum case [ tex2html_wrap_inline831 ] tex2html_wrap_inline833 tex2html_wrap_inline835 .

The easiest way to do this is to compare the geodesic deviation  equation derived in the last section with its Newtonian analogue. In Newtonian theory the acceleration of two neighboring particles with position vectors tex2html_wrap_inline837 and tex2html_wrap_inline839 are:


so the separation evolves according to:


This gives us




This clearly is analogous to the geodesic deviation equation


provided we relate the quantities tex2html_wrap_inline841 and tex2html_wrap_inline843

Both quantities have two free indices, although the Newtonian index runs from 1 to 3 while in the General Relativity case it runs from 0 to 3.

The Newtonian vacuum equation is tex2html_wrap_inline835 which implies that


so we can write


Since tex2html_wrap_inline855 is arbitrary we end up with


These are the vacuum field equations .