Active Gravitational Mass

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The term active gravitational mass refers to the source of gravity. In Newtonian gravity the density of this quantity, r, is related to the gravitational potential by Poisson's equation 

In general relativity the equation that relates the gravitational potentials, g_mn, to the source is Einstein’s field equations

The potentials are buried in the left hand side while the right hand side T^mn describes the active gravitational mass. For an relativistic fluid  Einstein’s equations yield in the Newtonian limit [1]  

where u₀ is the proper energy density of the fluid and p the fluid pressure. This equation looks odd at first because one assumes that mass-energy, u/c², should be the source of gravity. A more detailed explanation is required which emphasizes the limitations of the mass-energy relationship.
     Consider a fluid element at rest in frame S. Let the element be immersed in a fluid whose pressure is p. In frame S’, moving in the +x direction relative to S, the momentum, p, of the fluid element is given by 

The mass, m, of the fluid element is defined by m = p/v and therefore has the value 

The mass density, r = m/V, is then given by 

If we now let v ® 0 the mass density becomes 

If we plug this value into Poisson's equation we obtain 

Since there are three independent directions we must add the pressure that contributes to the mass relating to the other two dimensions. This means we multiply the fluid pressure by three to obtain 

Eq. (9) is identical to Eq. (2), which was obtained through Einstein’s equations.

References:

[1] Cosmological Principles, John A. Peacock, Cambridge University Press (1999), page 25.

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