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In the diagonal terms of the energy-momentum tensor, the flow of $x$-momentum in the $x$-direction is the $x$-pressure. Why the flow of momentum is pressure?

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Consider some basic unit analysis. Pressure is defined as force/area which is the same as momentum/area/time since F=dp/dt. Momentum flow would be the momentum passing through a unit area per unit time so it's the same units. More physically, think of a gas at constant pressure in a box. If you popped a little hole of unit area in the side of the box, the pressure would be the amount of momementum escaping per unit time. Hope that helps!

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