Relativistic Euler equations

In fluid mechanics and astrophysics, the relativistic Euler equations are a generalization of the Euler equations that account for the effects of special relativity.

Related Topics:
Fluid mechanics - Astrophysics - Euler equations - Special relativity

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The equations of motion are contained in the continuity equation of the stress-energy tensor T^{mu u}:

Related Topics:
Continuity equation - Stress-energy tensor

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:

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abla_mu

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T^{mu u}=0

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For a fluid,

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:T^{mu u}=(e+p)u_mu u_ u+pg_{mu u}.

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Here e is the relativistic rest energy of the fluid, p is the pressure, u is the four-velocity of the fluid, and g_{mu u} is the metric tensor.

Related Topics:
Pressure - Four-velocity - Metric tensor

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To the above equations, a statement of conservation is usually added, usually conservation of baryon number. If n is the number density of baryons this may be stated

Related Topics:
Conservation - Baryon number

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:

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abla_mu

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(nu_mu)=0.

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These equations reduce to the classical Euler equations if u

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The relativistic Euler equations may be applied to calculate the speed of sound in a fluid with a relativistic equation of state (that is, one in which the pressure is comparable with the internal energy density e, including the rest energy; e= ho c^2+ ho e^C where e^C is the classical internal energy).

Related Topics:
Speed of sound - Equation of state - Internal energy - Rest energy

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Under these circumstances, the speed of sound S is given by

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:

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S^2=c^2

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left.

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rac{partial p}{partial e}

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ight|_{ m adiabatic}.

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(note that

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:e= ho (c^2+e^C)

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is the relativistic internal energy density). This formula differs from the classical case in that ho has been replaced by e/c^2.

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