Collapsing solutions in f(R) gravity are restricted due to junction conditions that demand continuity of the Ricci scalar and its normal derivative across the time-like collapsing hypersurface. These are obtained via the method of R-matching, which is ubiquitous in f(R) collapse scenarios. In this paper, we study spherically symmetric collapse with the modification term \(\alpha R^2\), and use R-matching to exemplify a class of new solutions. After discussing some mathematical preliminaries by which we obtain an algebraic relation between the shear and the anisotropy in these theories, we consider two metric ansatzes. In the first, the collapsing metric is considered to be a separable function of the co-moving radius and time, and the collapse is shear-free, and in the second, a non-separable interior solution is considered, that represents gravitational collapse with non-zero shear viscosity. We arrive at novel solutions that indicate the formation of black holes or locally naked singularities, while obeying all the necessary energy conditions. The separable case allows for a simple analytic expression of the energy-momentum tensor, that indicates the positivity of the pressures throughout collapse, and is further used to study the heat flux evolution of the collapsing matter, whose analytic solutions are presented under certain approximations. These clearly highlight the role of modified gravity in the examples that we consider.

由于结条件要求Ricci标量及其法向导数在类似时间的塌陷超曲面上具有连续性，因此在f（R）重力下的塌陷解受到限制。这些是通过R匹配的方法获得的，该方法在f（R）崩溃场景中无处不在。在本文中，我们使用修正项\（\ alpha R ^ 2 \）研究球对称塌陷，并使用R-matching举例说明一类新的解决方案。在讨论了一些数学上的初步知识之后，我们通过这些数学上的知识在这些理论中获得了剪切力与各向异性之间的代数关系，我们考虑了两种度量解析。在第一种情况下，塌缩度量被认为是共同移动半径和时间的可分离函数，并且塌陷是无剪切的；在第二种情况下，考虑了不可分离的内部解，它表示了重力塌陷。非零剪切粘度。我们得出了新颖的解决方案，这些方案表明了黑洞或局部裸露的奇点的形成，同时遵循了所有必要的能量条件。可分离的情况允许能量动量张量的简单解析表达式，表明整个塌陷中压力的正值，并进一步用于研究坍塌物质的热通量演化，其解析解在一定近似下给出。这些清楚地突出了我们所考虑的示例中重力修正的作用。