Слово метрика здесь обозначает геометрию неевклидового простраства-времени. Если имеется однородное и постоянное во времени гравитационное поле, то под его действием частица с массой покоя m₀ начинает ускоряться относительно инерциальной системы отсчета и характер движения не зависит от m₀. Независимость формы траектории от массы является результатом криволинейной геометрии пространства. Но любому ли движению в плоском пространстве отвечает некоторая метрика в криволинейном?

3.What's an ugly smiling face?

It's the cat from curved space.

V. Komen, I. Tikhonenkov

Here we are discussing in general the majority of metrics for a stationary gravitation fields in one dimension. The only accepted approach so far to apply the equations of field (A. Einstein):

V. Komen, I. Tikhonenkov

In the previous post we've considered a model example of a motion of a free particle within a uniform gravitation field where a coupling to the field is defined by an observed inertion mass (see eq. (2) in https://habr.com/en/articles/739714/). The equation of motion was:

Uniform gravity, can it exist?

1. The motion of a free particle-like cat

V. Komen, I. Tikhonenkov

Good morning! You've woke up. Having prepared coffee and toasts of bread you are drifting from a kitchen to a table before a large wall TV. The left hand keeps a small plate with toasts and the right one controls coffee level within your beloved mug. The life is plotted out for ten seconds to come. But Ervin is already worried that you, as usual, made conspiracy and decided not to share your breakfast with him. So the damned cat thrusts himself across you pass, hits your legs. By the next moment the plate, toasts and the mug are falling. And – yes! All those precious things reached a floor level by the same time. Physics...It's how it shows up, unexpected. And we used to identify ourselves as physicists. The unexpected thing is that we still do. It's whence our motivation originates. We cannot pass by any falling objects quietly.