Fluid Solution - Mathematical Definition

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.