Mathematical Definition
The stress-energy tensor of a relativistic fluid can be written in the form
Here
- the world lines of the fluid elements are the integral curves of the velocity vector ,
- the projection tensor projects other tensors onto hyperplane elements orthogonal to ,
- the matter density is given by the scalar function ,
- the pressure is given by the scalar function ,
- the heat flux vector is given by ,
- the viscous shear tensor is given by .
The heat flux vector and viscous shear tensor are transverse to the world lines, in the sense that
This means that they are effectively three-dimensional quantities, and since the viscous stress tensor is symmetric and traceless, they have respectively 3 and 5 linearly independent components. Together with the density and pressure, this makes a total of 10 linearly independent components, which is the number of linearly independent components in a four-dimensional symmetric rank two tensor.
Read more about this topic: Fluid Solution
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