Tensors and Relativity: Chapter 4

Conservation of energy- momentum

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Since tex2html_wrap_inline1076 represents the energy and momentum content of the fluid, there must be some way of using it to express the law of conservation of energy and momentum. In fact it is reasonably easy.

Consider a cubical fluid element  [ see Figure 4.1 ] of side a, seen only in cross section [ z direction suppressed ].

  figure285
Figure 4.1: Energy flow across a fluid element.

Energy can flow  across all sides. The rate of flow across face (4) is tex2html_wrap_inline1134 , and across (2) is tex2html_wrap_inline1136 ; the second term has a minus sign because tex2html_wrap_inline1138 represents energy flowing in the positive x- direction, which is out of the volume across face (2). Similarly, the energy flowing in the y direction is tex2html_wrap_inline1144 .

The sum of these rates across each face must be equal to the rate of increase of energy inside the cube tex2html_wrap_inline1146 : This is the statement of conservation of energy.  Therefore we have:

eqnarray810

Dividing by tex2html_wrap_inline1148 and taking the limit tex2html_wrap_inline1150 gives

equation812

Dividing by c we get

equation814

Since tex2html_wrap_inline1154 , tex2html_wrap_inline1156 , tex2html_wrap_inline1158 and tex2html_wrap_inline1160 , we can write this as

equation816

or

equation818

This is the statement of the law of conservation of energy.

Similarly momentum is conserved.  The same mathematics applies, with the index 0 changed to i [ the spatial components ] i.e.

equation820

The general conservation law of energy and momentum is therefore 

equation822

This applies to any material in Special Relativity.