5 The Transformations of Density and Pressure
The Avogadro constant NA jdefines the amount of particles in a mole. It describes a certain number of atoms or molecules, and it is directly related to the rest mass of the amount of a substance. This rest mass and the Avogadro constant are relativistically invariant.
Generally, absolute numbers such as the number N jof gas molecules enclosed in a vessel are independent of the relative speed of an observer. But since volumes have to be transformed the particle density, i.e. the number of particles per unit volume, is not invariant. For an observer flying past, the density increases, since the same number of particles is compressed into a smaller volume. The root factor is in the denominator. We refrain from introducing a symbol for the particle density.
Pressure is a scalar quantity. In a gas or in a liquid it is (in the absence of a gravitational field) the same everywhere. It is isotropic, that is, it generates in all directions the same force per square centimeter on the walls of the enclosing vessel.
Therefore we can concentrate on the motion of the particles in the y-direction against the vessel's wall as the vessel is moving towards us in the x-direction with relative velocity vj. Momentum in the y-direction transforms according to the formula jpy' j= py . The movement is slowed in the transverse direction by the root factor, however this is (according to traditional terminology) compensated by the relativistic increase in the mass of the particle.
Because the transverse velocity u' jof the particle in the y-direction appears slower to us it should follow that per unit time fewer particles will strike the vessel wall - if the particle density were unchanged. But the particle density increases due to the length contraction by exactly the same factor as the transverse velocity is decreased. So overall the same number of particles per unit time encounter the vessel's wall with the same momentum, generating the same pressure that an observer would measured in the rest frame. Pressure P jtherefore transforms as follows:
So far, almost all authors agree ...