Thought Experiment: An object falling indefinitely under gravity?

Question

So the other day, I devised this thought experiment:

Consider an infinite tunnel, and you drop a coin of mass $m$ into it. Considering the effect of gravity to be applicable and neglecting air drag and other viscous forces, it seems that from Newton's second law the coin would accelerate indefinitely since force ($mg$) is constantly being applied. But on the other hand I also know nothing can go faster than speed of light, as stated by Einstein's theory of relativity so how can I describe the above experiment where it seems like the coin would approach the speed of light?

I read somewhere else that when a object made to accelerate indefinitely then as per Mass-Energy Equivalence the mass of object begins to increase exponentially and hence greater and greater force is required to increase object's speed.

But somehow the above explanation doesn't seem to explain much in my thought experiment.

I would like to know the actual reason why the above scenario can't happen.

Answer 1

Suppose you are an observer stationary in this infinite gravitational field, so you feel a gravitational force $g$ just as you do standing on the surface of the Earth. The only difference is that this acceleration $g$ is constant and doesn't change as you go higher or lower.

In that case your spacetime is described by the Rindler metric (as Phoenix87 mentions in a comment). You'll see the metric written in a variety of coordinates. The most physically intuitive, if not the easiest to work with is:

$$ ds^2 = -\left(1 + \frac{g}{c^2}z \right)^2 c^2 dt^2 + dz^2 $$

where I've left in the factors of $c$ that we traditionally set to unity. The variable $z$ is the distance relative to your stationary position i.e. at your position $z = 0$. There is a coordinate singularity at $z = - \tfrac{c^2}{g}$, i.e. where $1 + \tfrac{g}{c^2}z$ becomes zero, which is akin to a black hole event horizon.

If you drop a coin you will observe it accelerate downwards away from you towards the coordinate singularity. However as the coin approaches the horizon it will slow asymptotically and never reach the horizon even if you wait an infinite time. This is closely analogous to what happens if you dropped your coin into a black hole, where you would observe the coin to slow and freeze as it approached the horizon.

There is another singularity at $z = + \tfrac{c^2}{g}$ that is akin to a white hole. If you throw your coin upwards (to positive $z$) then no matter how fast you throw it, i.e. no matter how closely its velocity approaches the speed of light, the coin will never reach $z = + \tfrac{c^2}{g}$. So in effect your entire universe consists of the range $- \tfrac{c^2}{g} \le z \le \tfrac{c^2}{g}$