Tensors and Relativity: Chapter 1

Pre- relativistic physics

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The starting point for our work are Newtons laws  of motion. These can be stated as follows:

  • Free particles move with constant velocity.
  • The vector force tex2html_wrap_inline1114 is proportional to the rate of change of momentum i.e. tex2html_wrap_inline1116 .
  • To every action there is an equal and opposite reaction.

The first of these laws singles out inertial frames  as the non- accelerating ones.

Consider now a frame tex2html_wrap_inline1118 [ i.e. a set of spatial coordinates (x,y,z) and a time coordinate t ], and another frame tex2html_wrap_inline1124 with coordinates tex2html_wrap_inline1126 which moves in the x direction with uniform speed v relative to the frame tex2html_wrap_inline1118 .

Common sense suggests that the two sets of coordinates are related by

eqnarray691

These are the Galelian transformations  .

 

figure210


Figure 1.1: I do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the sea- shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. Isaac Newton.

If the particle has a velocity tex2html_wrap_inline1134 with components tex2html_wrap_inline1136 in tex2html_wrap_inline1118 , its velocity in tex2html_wrap_inline1124 is:

eqnarray697

or

equation699

where

equation707

More generally if the coordinate axes and the origins of tex2html_wrap_inline1118 and tex2html_wrap_inline1124 differ then:

equation711

where tex2html_wrap_inline1146 has components (x,y,z). Here tex2html_wrap_inline1150 is a rotation matrix aligning tex2html_wrap_inline1118 and tex2html_wrap_inline1124 , tex2html_wrap_inline1156 is the relative velocity of tex2html_wrap_inline1118 with respect to tex2html_wrap_inline1124 and tex2html_wrap_inline1162 is the displacement of the origin from tex2html_wrap_inline1124 .

Since the transformation is linear [ constant velocity in tex2html_wrap_inline1118 tex2html_wrap_inline1168 constant velocity in tex2html_wrap_inline1124 ], tex2html_wrap_inline1124 is inertial if tex2html_wrap_inline1118 is.

Thus there are an infinite set of inertial frames, all moving uniformly with respect to each other.

All of Newtons laws apply in any inertial frame since

equation736

and tex2html_wrap_inline1114 is invariant. Thus we have Newtonian [ Galelian ] Relativity.

The laws of mechanics do not allow measurement of absolute velocity, however one can measure absolute acceleration.

Newton explained inertial frames in terms of absolute space  identified with the center of mass of the solar system or a frame of ``fixed stars''. However this is unsatisfactory because:

  • There is no unique identification [ many inertial frames ].
  • Philosophically unappealing since absolute space affects everything but is affected by nothing.

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