Tensors and Relativity: Chapter 4

The conservation equations

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We can now apply the conservation laws  tex2html_wrap_inline1185 and tex2html_wrap_inline1187 to find the conservation equations. We have:

eqnarray832

If we multiply by tex2html_wrap_inline1189 and use

equation834

and

equation836

we get

equation838

which gives

  equation840

where tex2html_wrap_inline1191 is the derivative along the world line of the fluid.

The i components of tex2html_wrap_inline1185 give

equation842

so in the MCRF  [ tex2html_wrap_inline1197 ] this equation is

  equation844

Equation (33) is the energy conservation equation   and equation (35) is the momentum conservation equation   [ or acceleration equation  ].

In the Newtonian limit tex2html_wrap_inline1199 and tex2html_wrap_inline1201 so in this case the conservations equations become [ Assignment 4 ]

equation846

and

equation848

where we have used

equation858