Tensors and Relativity: Chapter 6

The principle of equivalence again

The principle of equivalence again

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In Special Relativity, in a coordinate system adapted for an inertial frame, namely Minkowski coordinates, the equation for a test particle is:

equation3065

If we use a non- inertial frame of reference, then this is equivalent to using a more general coordinate system tex2html_wrap_inline3665 . In this case, the equation becomes

equation3069

where tex2html_wrap_inline3384 is the metric connection of tex2html_wrap_inline3416 , which is still a flat metric but not the Minkowski metric tex2html_wrap_inline3671 . The additional terms involving tex2html_wrap_inline3384 which appear, are inertial forces.

The principle of equivalence requires that gravitational forces, a well as inertial forces, should be given by an appropriate tex2html_wrap_inline3384 . In this case we can no longer take the spacetime to be flat. The simplest generalization is to keep tex2html_wrap_inline3384 as the metric connection, but now take it to be the metric connection of a non- flat metric. If we are to interpret the tex2html_wrap_inline3384 as force terms, then it follows that we should regard the tex2html_wrap_inline3416 as   potentials. The field equations of Newtonian gravitation  consist of second- order partial differential equations in the potential tex2html_wrap_inline3683 . In an analogous manner, we would expect that General Relativity also to involve second order partial differential equations in the potentials tex2html_wrap_inline3416 . The remaining task which will allow us to build a relativistic theory of gravitation is to construct this set of partial differential equations. We will do this shortly but first we must define a quantity that quantifies spacetime curvature.