Conservation of energy- momentum
Since 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.
Energy can flow across all sides. The rate of flow across face (4) is , and across (2) is ; the second term has a minus sign because 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 .
Dividing by and taking the limit gives
Dividing by c we get
Since , , and , we can write this as
This is the statement of the law of conservation of energy.
This applies to any material in Special Relativity.