tokens: the presence of perturbations is at (for details about that, see Pereira et al. 2018 ). From Equations ( 7 ) and ( 12 ), one has thatwhere use has been made ofas well as Equation ( 4 ) and . The perturbed surface tension , with the help of Equations ( 8 ) and ( 11 ), can be written asTherefore, likewise to the slow phase conversion case, no thin shell surface degrees of freedom are induced in perfect fluids when the phase conversion is rapid. Now we investigate solid hybrid stars. In this case, as commented in the previous section, . Besides, now changes with respect to its perfect fluid counterpart, Equation ( 17 ), and it is given by (a mere replacement of by )where summarizes the contribution to the radial shear stresses in the presence of perturbations (see Penner et al. 2011 and Krüger et al. 2015 for realizations of them). For slow phase conversions in solid hybrid stars, due to Equations ( 11 ) and ( 13 ), it still follows that and . However, nontrivial results hold for solid hybrid stars under rapid phase conversions. Indeed, for , we haveBefore calculating the thin shell surface tension for this case, let us expand Equation ( 13 ), because . From our assumption to elastic aspects in hybrid stars, one can generically write . Thus, after some simple calculations it follows from Equations ( 4 ), ( 13 ), and ( 23 ) that (recall that )which turns out to be exactly the same as that for slow reactions in solid stars (because there and ). From the above equation, one can clearly see that indeed in general. (Equation ( 24 ) also makes it clear that the Lagrangian change of the pressure is null for perfect fluids, as already known from other methods; Pereira Rueda 2015 ; Pereira et al. 2018 .) Now, for the thin shell surface tension induced by rapid reactions in solid hybrid stars, from Equations ( 12 ), ( 22 ), and ( 24 ), one obtains that The fact that rapid conversions in solid stars induce thin shell surface degrees of freedom could have interesting consequences; for instance, the induction of a surface mass (general relativistic energy) on a phase-splitting surface. Since the induced thin shell surface energy density is a local quantity, so is the shell’s proper mass (note that for first-order quantities, such as those associated with surface degrees of freedom, it is irrelevant to use R or R 0 for functions multiplying them)Total mass measurements at infinity will have to take into account the correction factor (related to the gravitational redshift), as we shall show in Equation ( 31 ). From Equations ( 7 ), ( 26 ), and ( 5 ), it follows thatGiven that , when only linear terms in ξ are kept, the above equation can be simplified to . Thus, from Equations ( 26 ) and ( 27 ),where it has been assumed that , m 0 being the background mass function ( because the background has been assumed not to have surface degrees of freedom—see the Appendix ) and m 1 being its first-order correction in the presence of perturbations. In summary, among other
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