As for this question, Column by Mr. EMAN whom I respect as my teacher without permission explains very carefully. And this is, I think, widely accepted common view. But, I feel uneasy about this. So, I make the point of discussion clearer.
!! CAUTION !! This is “MY QUESTION”. This can be in discord with standard view.
At first, we will review Column by Mr. EMAN.
(Sorry, this is a Japanese page. But following summary will do sufficiently.)
Schwarzschild solution is described as follows,
In eq. (1), we neglect the third and the fourth term on the right hand side because we will study only radial direction for spherically symmetric spacetime now. And we neglect the first term because we will study stationary geometry now. Then, the rest is only the second term. With it, we can derive the length of the line element ds which is to the radial direction at the same time.
Integrating eq. (2), we can get the distance from a point outside the black hole to the surface of the black hole. Letting rt , rs denote position of the point and the surface of the black hole respectively, distance l between them can be obtained as follows.
Solving eq. (3), we can obtain following result.
Adding condition of rt >> rs to eq. (4),
Eq. (5) shows that l is finite.
We will study this result further using a concrete instance studied in “Explicit description of Schwarzschild spacetime with Excel“. Fig. 2 in this page describes the situation that a test mass m is free-falling from the infinity to a large gravity source M whose Schwarzschild radius is a with three coordinates. The test mass is now falling at r0 : r0 / a = 1/0.36 = 2.777. In this example, notation of Mr. EMAN is interpreted as rt → r0 = a / 0.36、 rs → a 、 ds → dp. Letting the coefficient of eq. (2) γS, γS = 1.25 at this place. Units of vertical and horizontal axes are both [m] in geometrized unit system. Time unit [m] means the time span in which light travels distance of 1 m.
As shown in Fig. 2, when the trailing end of a rod whose length at infinity was 4 m is fixed still at t = o at point O, leading end becomes at point S. For a stationary observer there, as length of the rod is measured with Stationary coordinates Δp, it is 4 m. On the other hand, for a Eulerian observer at infinity, as it is measured with Eulerian coordinates Δr, it is contracted to 3.2 m for him. This contraction rate becomes larger when an object nears Schwarzschild radius a, and becomes infinite at Schwarzschild radius. For this reason, there was a suspect that even if how many rods you join, it could be short for reaching the surface of the black hole. What Mr. EMAN showed with eq. (5) is the answer for this question. The distance to a black hole is finite.
No objection till now.
From here, I will show my concern. Please note that following is my own idea which can be in discord with common view.
In general, I believe that when we are working with general relativity (GR). both distance and time should be measured with light. Thus, I define the light whose wave length λ = 1 μm as the primary standard for time and length. Time span in which this wave beats 106 times is defined 1 m. Distance which contains 106 beats is defined 1 m here.
By the way,
speed of light c is constant for everyone. But, this is only true for the space near the observer. Letting the speed of light for observer i, at position j, to the direction k ckji, speed of light for each observer at each position are ckLL = ckSS = ckEE = c （k is to the all direction) where L, S, E represent Lagrangian, Stationary and Eulerian respectively. But, speed of light of r direction at r0 observed by Eulerian observer cr0E = dr/dt is not c (=1) but obtained letting ds = dθ = dϕ = 0 in eq. (1) as follows.
At position r0 = a / 0.36, this value becomes 0.64（non-dimentional).
When a rod whose length is 4 m was dropped from infinity and the leading tip arrived at point P, for a Eulerian observer at t = 3.75 m, this rod exists between points P and R. From the figure, you can read that the length of the rod is contracted to 0.64 times as large as the original as speed of light does.
Now, suppose this light was emitted from infinity to r0 for time span of 4 m. This light pulse flies as a light arrow whose length is 4 m. And suppose this light arrow arrived at position r0 at time t = q = 0. This event is plotted as O in Fig. 2. Δｔ = 4 m later, the trailing tip of the light arrow arrives at position r0. This event is plotted as U. Time span of OU is Δｔ = 4 m observed from Eulerian observer at infinity. But observed from stationary observer at r0, it is Δq = 3.2 m. Light propagates 1 m for a stationary observer there in the time span of 1 m for the stationary observer. Thus, the leading end of this light arrow proceeds on red dotted line connecting O and Q. And trailing end of it proceeds on red dotted line which pass over point U. At time t = 4 m ≡ q = 3.2 m, this light arrow exists the region between T and U which contains 4 × 106 waves. Fig. 2 shows that length of the rod PR is the same as TU. So, using the primary distance standard defined above, the length of the rod is measured correctly as 4 m.
Time T needed for this light arrow to reach a from r0 is calculated as follows.
This value goes to infinite.
From eq. (5), distance turned out to be finite, but from eq. (7), traveling this finite distance at the speed of light, it takes infinite time. Emitting this light continuously, we can send infinite number of waves before the leading end reaches the surface of a black hole. Therefore, measuring the distance of r0 - a with the primary standard for distance defined above, it is infinite. How shall we understand this?
I express my understanding.
The meaning of the distance ｌ calculated with eq. (4) is; “If you could put some object between r0 and black hole, and successfully collect it back to the place far from the black hole, the length of the object measured there is ｌ. And what eq. (7) states is, in the first place, it is impossible to place any object touching a black hole. At the first step of eq. (2), we abbreviated the term of time. But even when we discuss only about distance, I think, the term of time is essential.
Due to this, though eq. (4) is valid for virtual drawing on a sheet of paper, I want to be in the position to say, “Distance to a black hole is infinite.” because I want to study substantial being in this “down-to-earth physics” site.
The reason PR is shorter than OS is because of Lorentz contraction due to the falling speed as explained in the Excel file Sorao_Schwarzschild.xls. As it is taken for granted that the object should be stationary when we measure the size of the object, eq. (4) seems to be proper. But as stated above, it is impossible in reality, and free-falling is the natural state of an object. Moreover, it is unchallenged fact that even if we have a mutton dagger which extends infinitely at the speed of light, it can not reach the surface of the black hole within the life span of our universe.
To be continued to Simple questions “Is a black hole crammed full of stuff?”