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- What is the meaning of the coordinates of R with a spherical Schwarzschild metric?

First of all, I want to note that geometry is being discussed, which in fact is the General Theory of Relativity. And in any geometry, there are infinitely thin, weightless, etc. lines, rulers, and so on. In the future I will remind you about this.

The system of units is meters.

There is a Schwarzschild metric, with a horizon radius of r (meters).

The observer is at rest at the point R (meters). This is the coordinate.

Next, the observer wants to "knock on the horizon."

He takes a ruler (weightless, infinitely thin, etc., etc.) with a length of 1 meter and applies it from himself in the opposite direction from the origin (center).

Shifts it 1 meter towards the center.

At the nth step, the ruler N is applied and shifted together with the N-1 rulers placed earlier.

Mathematically, this corresponds to adding 2 vectors (-1, 0) and (0, N-1) and shifting the total vector (-1, N-1) by 1 meter to the center and obtaining the final vector (0, N), where N is at least the number of rulers.

The procedure is repeated only R-r times, and after R-r steps the total ruler (vector) should hit the firmament, the event horizon.

If it does not work out to "reach out", due to the fact that the far end of the vector moves in a "special" way during the shift (for example, at a shorter distance, when "looking" from point R), and moreover, it may never reach the horizon at all, then what is the meaning of the coordinate R (meters)?

I note that the described procedure has a deep physical meaning.

If you start "walking" with a step length of 1 meter according to such a constructed ruler, then no matter how the position of EACH gluing of meter rulers (we will call them serifs) is transformed, the "steps" will fall into these serifs, since the step length is transformed in the same way as the length of the rulers at each N step.

That is, it is a clear and BASIC concept of the distance between the beginning and the end of the vector

The system of units is meters.

There is a Schwarzschild metric, with a horizon radius of r (meters).

The observer is at rest at the point R (meters). This is the coordinate.

Next, the observer wants to "knock on the horizon."

He takes a ruler (weightless, infinitely thin, etc., etc.) with a length of 1 meter and applies it from himself in the opposite direction from the origin (center).

Shifts it 1 meter towards the center.

At the nth step, the ruler N is applied and shifted together with the N-1 rulers placed earlier.

Mathematically, this corresponds to adding 2 vectors (-1, 0) and (0, N-1) and shifting the total vector (-1, N-1) by 1 meter to the center and obtaining the final vector (0, N), where N is at least the number of rulers.

The procedure is repeated only R-r times, and after R-r steps the total ruler (vector) should hit the firmament, the event horizon.

If it does not work out to "reach out", due to the fact that the far end of the vector moves in a "special" way during the shift (for example, at a shorter distance, when "looking" from point R), and moreover, it may never reach the horizon at all, then what is the meaning of the coordinate R (meters)?

I note that the described procedure has a deep physical meaning.

If you start "walking" with a step length of 1 meter according to such a constructed ruler, then no matter how the position of EACH gluing of meter rulers (we will call them serifs) is transformed, the "steps" will fall into these serifs, since the step length is transformed in the same way as the length of the rulers at each N step.

That is, it is a clear and BASIC concept of the distance between the beginning and the end of the vector