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- I'm struggling with understanding the meaning of an extra "term" in the motion equations for a relativistic fluid.. wether it's in SR or GR. More info below.

Hello. Could anyone help me with some insight in an extra term appearing in the motion equations of a relativistic fluid? I say extra term, because it's not present on the motion for a test particle, as it follows:

Let's propose Minkowski space-time, the motion equations for a fluid with zero pressure, with four-velocity U, mass density ¶ and subject to an external force density f is:

¶ a= f + <f,U> U (with a being the four-acceleration). This is called the relativistic Euler equation and can be easely deduced from the energy momentum conservation of the tensor T generated by said force.

While the same equation, when written for a test particle submitted to the same force , would follows the usual equation m.a=F , written in four-vectors.

What is that extra term <f,U> U ?

Let's propose Minkowski space-time, the motion equations for a fluid with zero pressure, with four-velocity U, mass density ¶ and subject to an external force density f is:

¶ a= f + <f,U> U (with a being the four-acceleration). This is called the relativistic Euler equation and can be easely deduced from the energy momentum conservation of the tensor T generated by said force.

While the same equation, when written for a test particle submitted to the same force , would follows the usual equation m.a=F , written in four-vectors.

What is that extra term <f,U> U ?