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The EnergyMomentum Tensor in .NET framework
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Relativistic Effects on Number Density
We now take a slight digression to investigate the effects of motion on the density of particles within the context of special relativity. Consider a rectangular volume V containing a set of particles. We can de ne a number density of particles which is simply the number of particles per unit volume. If we call the total number of particles in the volume N , then the number density is given by n= N V In relativity, this is true only if we are in a frame that is at rest with respect to the volume. If we are not, then length contraction effects will change the number density that the observer sees. Suppose that we have two frames F and F in the standard con guration, with F moving at velocity v along the xaxis. The number of particles in the volume is a scalar, and so this does not change when viewed from a different frame. However, length contraction along the direction of motion means that the volume will change. In Fig. 72, we show motion along the xaxis. Lengths along the y and z axes are unchanged under a Lorentz transformation under these conditions. If the volume of the box in a comoving rest frame is V, then the volume of the box as seen by a stationary observer is V = 1 v2V = 1 V Therefore, the number density in a volume moving at speed v as seen by a stationary observer is given by n = N = n V EXAMPLE 73 Consider a box of particles. In the rest frame of the box, the volume V = 1 m3 and the total number of particles is N = 2.5 1025 . Compare the number density of particles in the rest frame of the box and in a rest frame where the box has velocity v = 0.9. The box moves in the xdirection with respect to the stationary observer. The EnergyMomentum Tensor
F' y' x x' z z' Fig. 72. A volume V, which we take in this example to be a rectangular box, is shortened along the direction of motion by the length contraction effect. This will change the number density of particles contained in V. SOLUTION 73 In the rest frame of the box, the number density n = 2.5 1025 particles per cubic meter. Now 1 = = 1 v2 1 1 (0.9) 2.29 A stationary observer who sees the box moving at velocity v sees the number density of particles in the box as n = n = (2.3) 2.5 1025 = 5.75 1025 particles per cubic meter. Along the xdirection, the length of the box is x= 1 1 x = m 0.43 m 2.3

