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

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 $1 \, m/s$.

• 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. That is incorrect statement. Even when close to each other if the lab moves with nonzero velocity they would disagree on what $u_l$ and $u_t$ is, in accordance to rules of special relativity. – A.V.S. May 10 '18 at 9:15
• @A.V.S. At the beginning both are close and at rest. And then the observer gives o push (initial velocity of 1 m/s). Remember, the observer built the lab ;-) – J. Manuel May 10 '18 at 9:22
• Yes, and when the lab acquires this velocity all the distances as seen by it become different from those seen by stationary observer, and all other clocks now count time at different rate. – A.V.S. May 10 '18 at 9:31
• @A.V.S. Let's say initial push is of $1 \,m/s$ and after some time the observer notices that the speed changed to $4 \,m/s$. Therefore, GR affects (gravity) really stands out ($4$ times velocity change), but at $4 \,m/s$ special relativity effects really don't count much. Meaning that space-time curvature account for most of the acceleration observed which goes back the initial problem. To have acceleration (as we observe falling bodies accelerating) unit distance "should" get bigger and time unit "should" get smaller. – J. Manuel May 10 '18 at 9:52
• @StéphaneRollandin. Yes it was. Thanks and sorry for that. I'm removing all mentions of "intuition" in order to make it non-personal. By the way, I removed my comment as well since it doesn't contributed to the question. – J. Manuel May 10 '18 at 15:05