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In Sean M Carroll's Introduction to General Relativity: Spacetime and Geometry, after deriving the Tolman-Oppenheimer-Volkoff equation on page 233, Chapter 5, he says:

"To get a closed system of equations, we need one more relation: the equation of state."

What does 'closed system of equations' mean? Does it mean that the differential equation has a closed form solution? In that case, how does that happen upon knowing the equation of state?

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    $\begingroup$ Roughly speaking (the term "closed" is quite vague here), when the number of unknowns equals the number of independent equations. The hope is to prove an existence and uniqueness theorem. Details depend on the specific system of (differential) equations. $\endgroup$ May 19, 2021 at 8:43
  • $\begingroup$ The slightly vague term closed might refer to an also-called complete system which is solvable (doesn't need any further equations). $\endgroup$
    – Steeven
    May 19, 2021 at 8:48

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"Closed system of equations" means the number of unknown variables is equal to the number of equations supplied with boundary and initial conditions sufficient to unambiguously find the searched variables.

In case of "the equation of state", it is often a solution (an expression rather than a differential equation) for some variable. In othe words, for this variable the approximate solution is considered good in a given context.

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In this context, a system of equations is said to be closed when it constrains fully the dynamics of the system. Heuristically, it means that there are as many equations as there are independent degrees of freedom.

For example, the Maxwell equations : $$\begin{array}{ccc} \nabla \cdot E = \frac{\rho}{\epsilon_0} && \nabla \times E = -\partial_t B\\ \nabla\cdot B = 0 && \nabla\times B = \mu_0 j + \mu_0\epsilon_0\partial_tE \end{array}$$

are a closed set of equations if the source $\rho$ and $j$ are considered as a fixed background. If they are considered as independent variables, you would need to add equations to describe how the dynamics of $\rho$ and $j$ is determined by the electromagnetic field.

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    $\begingroup$ But Maxwell's equations in this form have 6 unknown functions ($\mathbf E$ and $\mathbf B$) and 8 equations (2 vector and 2 scalar equations)! $\endgroup$ May 19, 2021 at 10:11
  • $\begingroup$ Actually the equations regarding the rot operator just constraint two (transverse) components of the two vector fields (that is evident passing to Fourier representation), hence we have $2+2+2$ equations for $6$ components. $\endgroup$ May 19, 2021 at 11:11

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