Why this intuitive idea on unit length and unit time in general relativity fails so miserably

Question

Let’s imagine a massive object and an observer. In time $t_0$ the observer pushes an automated test lab having clocks and rulers towards the massive object. Let’s says, it gives $1 \,u_l/u_t$ push to the test object ($u_l$, $u_t$ are unit length and time). In the beginning, since both the observer and the lab are far away from the massive object and close to each other, both agree on the initial speed and units.

Let’s admit that the distant observer don’t know about GR as well. After moving for a while, nothing happens to the test lab as seen by itself. From its own point of view, it is just moving constantly at $1 \,u_l/u_t$. This is in agreement with the correspondence principle.

However, at this new position $1 \,u_l$ and $1 \,u_t$ are no longer equivalent to $1 \,u_l$ and $1 \,u_t$ of the distant observer. Let’s admit that $1 \,u_l$ in the new point corresponds $2 \,u_l$ in the distant observers reference frame and that time is cut to half. The distant observer -admitting that units are the same everywhere in universe- now measures the speed of the test lab and realize that it is moving double units of length at half unit time, therefore he concludes that lab speed passed from $1$ to $4$, i.e., the lab is accelerating towards the massive object.

So, “curved” spaces will easily been understood as Newtonian acceleration. However, to do so unit length should become bigger and unit time smaller, as the test object approaches the massive object, or at least one of them should behave this way if the test object was to accelerate towards the massive object.

The thing is, it is exactly the opposite that happens, unit length will be perceived smaller and unit time bigger, like what can be seen is this question and many other in this site, to the point that a distant object will need an eternity to see the test lab crossing the event horizon.

To this guy, since he gave a pull to the lab, and now he is Seeing it braking, he may conclude that gravity may sometimes act as an opposing force and not always as an attractive one.

I know this comes from the solutions to the GR field equations like Schwarzschild’s. So, the answer may just be: "Because it is what you get from the solutions". However, what I really wanted to know is:

Why my intuition was so wrong to the point of getting the opposite result? Is other then the result of the solutions, reason to why this is so? for example: Conservation of A demands this; Principle B would be violated if this didn’t happen this way, and so on…

EDIT: At the beginning the lab is at rest in the observers frame, and then the former gave the lab a push of say $1 \, m/s$.

Answer 1

I try to follow your question step by step.

At the beginning when both the observer and the lab are far away from the massive object: 1. They agree on the relative velocity according to the first principle of SR (special relativity), that is each inertial reference frame is equivalent, but 2. They disagree on the units of length and time, as each of the frames measures the length contraction and the time dilation against the other.

The lab in its reference frame is stationary.

The correspondence principle states that systems described by quantum mechanics behave in accordance to classical physics in the limit of large quantum numbers. Here we are arguing about SR and GR (general relativity), no reason to ask for that principle.

As the lab proceeds towards the massive object, the distant observer measures a progressively increasing radial velocity up to a certain radial distance from which the radial velocity progressively decreases until becoming negligible and eventually stopping. Of course, the slowing down occurs if an event horizon exists, otherwise the distant observer measures roughly an acceleration as per Newtonian gravity.

Your intuition is wrong because you apply the ordinary experience, on which Newtonian law was built, to configurations which are not ordinary. The genius of Einstein consisted in figuring out general requirements to the physical laws that allowed to construct a theory to be applied to non ordinary systems as well.