In the traditional frame of classical electrodynamics, a hydrogen atom would emit electromagnetic waves and thus constantly lose energy (radiative damping), resulting in the fall of the electron on the proton over a finite period of time. The corresponding results were derived under the assumption of the instantaneous interaction between the proton and the electron. In 2004, Raju published a paper where he removed the assumption of the instantaneous interaction and studied the role of a time delay (retardation) in the classical hydrogen atom. He introduced a model of this effect that can be solved analytically. He calculated the radius r_sel of a selected orbit, for which this effect (the retardation) and the radiation reaction force would cancel each other with respect to the tangential acceleration. It turned out that r_sel is by an order of magnitude smaller than the Bohr radius. In the present paper we use Raju’s model for the corresponding gravitational (rather than electromagnetic) situation. In frames of Newton’s gravity, we calculate analytically the radius and the energy E_sel of a selected orbit, for which the retardation effect and the radiation reaction force (caused by the emission of gravitational waves) would cancel each other out with respect to the tangential acceleration. Then we compare our outcome with some results of Kaplan’s solution for the two-body gravitational problem based on the Schwarzschild’s field. We show that E_sel is very close to the minimum energy of bound states obtained by Kaplan in Einstein’s gravity. We emphasize that the selected orbit from the present paper represents the second only physical situation (the first being a uniformly accelerated charge) where the radiation reaction force is (effectively) zero, but the radiation-caused energy loss is not zero.

在传统的经典电动力学框架中，氢原子会发出电磁波，从而不断失去能量（辐射阻尼），从而导致电子在有限的时间内掉落在质子上。在质子和电子之间瞬时相互作用的假设下得出了相应的结果。2004年，Raju发表了一篇论文，其中他删除了瞬时相互作用的假设，并研究了时间延​​迟（延迟）在经典氢原子中的作用。他介绍了这种影响的模型，可以通过解析来解决。他计算出半径r _sel在所选轨道上，该效应（延迟）和辐射反作用力将相对于切向加速度相互抵消。事实证明，r _sel比玻尔半径小一个数量级。在本文中，我们将Raju模型用于相应的重力（而非电磁）情况。在牛顿引力的框架中，我们通过解析计算半径和能量E _sel在选定轨道上，延迟效应和辐射反作用力（由引力波的发射引起）相对于切向加速度会相互抵消。然后，我们将结果与Kaplan基于Schwarzschild场的两体重力问题解的一些结果进行比较。我们表明，E _sel非常接近卡普兰在爱因斯坦引力下获得的束缚态的最小能量。我们强调，从本文中选择的轨道代表了第二个唯一的物理情况（第一个是均匀加速的电荷），其中辐射反作用力（有效）为零，但辐射引起的能量损失不为零。