- #1
Ron19932017
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Hi all, I am reading Bernard Schutz's a first course in general relativity. In Chapter 4 it introduced the energy stress tensor in two ways: 1.) Dust grain 2.) Perfect fluid.
The book defined the energy stress tensor for dust grain to be ## p⊗N ##, where ##p## is the 4 momentum for a single dust grain, which writes ##(m,0,0,0)## in the rest frame (m is mass of a dust grain), and ##N## is the number density flux, which writes ##(n,0,0,0)## (n is scalar number density) in the rest frame. Thus
##T## writes ##((mn,0,0,0),(0,0,0,0),(0,0,0,0),(0,0,0,0))## in the rest frame.
Such definition of energy stress tensor ##T## has a physical meaning that: ##T^{ij}## is the i-th component of 4 momentum density flux into j-th surface.
When we are dealing with fluid, which has random motion thus pressure, the book claim the energy stress tensor in the rest frame to be ##((\rho,0,0,0),(0,P,0,0),(0,0,P,0),(0,0,0,P))## where ##\rho## is energy density and P is the pressure. I understand the ##\rho## term but not the ##P## term.
I understand that the unit of a component of energy stress tensor is same as pressure, and the idea of momentum density flux matches the idea of pressure. However, using such expression, when we try to compute the x-momentum in x-direction, i.e. ##T^{1,1}=P##, it does not have the meaning of "x-momentum density flux in x-direction", because such pressure will not lead to net bulk flow of momentum. The net flow of momentum is 0 as we are assuming the fluid in its equilibrium.
Can someone explain the "inconsistency"? Or the energy stress tensor has simply different physical definition for dust grain and fluid?
The book defined the energy stress tensor for dust grain to be ## p⊗N ##, where ##p## is the 4 momentum for a single dust grain, which writes ##(m,0,0,0)## in the rest frame (m is mass of a dust grain), and ##N## is the number density flux, which writes ##(n,0,0,0)## (n is scalar number density) in the rest frame. Thus
##T## writes ##((mn,0,0,0),(0,0,0,0),(0,0,0,0),(0,0,0,0))## in the rest frame.
Such definition of energy stress tensor ##T## has a physical meaning that: ##T^{ij}## is the i-th component of 4 momentum density flux into j-th surface.
When we are dealing with fluid, which has random motion thus pressure, the book claim the energy stress tensor in the rest frame to be ##((\rho,0,0,0),(0,P,0,0),(0,0,P,0),(0,0,0,P))## where ##\rho## is energy density and P is the pressure. I understand the ##\rho## term but not the ##P## term.
I understand that the unit of a component of energy stress tensor is same as pressure, and the idea of momentum density flux matches the idea of pressure. However, using such expression, when we try to compute the x-momentum in x-direction, i.e. ##T^{1,1}=P##, it does not have the meaning of "x-momentum density flux in x-direction", because such pressure will not lead to net bulk flow of momentum. The net flow of momentum is 0 as we are assuming the fluid in its equilibrium.
Can someone explain the "inconsistency"? Or the energy stress tensor has simply different physical definition for dust grain and fluid?