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Cosmic strings in $f\left(R,L_m\right)$ gravity

We consider Kasner-type static, cylindrically symmetric interior string solutions in the $f\left(R,L_m\right)$ theory of modified gravity. The physical properties of the string are described by an anisotropic energy-momentum tensor satisfying the condition $T_t^t=T_z^z$; that is, the energy density of the string along the $z$-axis is equal to minus the string tension. As a first step in our study we obtain the gravitational field equations in the $f\left(R,L_m\right)$ theory for a general static, cylindrically symmetric metric, and then for a Kasner-type metric, in which the metric tensor components have a power law dependence on the radial coordinate $r$. String solutions in two particular modified gravity models are investigated in detail. The first is the so-called "exponential" modified gravity, in which the gravitational action is proportional to the exponential of the sum of the Ricci scalar and matter Lagrangian, and the second is the "self-consistent model", obtained by explicitly determining the gravitational action from the field equations under the assumption of a power law dependent matter Lagrangian. In each case, the thermodynamic parameters of the string, as well as the precise form of the matter Lagrangian, are explicitly obtained.