Terminal velocity of a body in a vacuum

Question

What velocity would a small body moving towards a large body reach under gravity alone? More importantly, is there a 'terminal velocity' for the small body due to the increasing energy required to accelerate the body as its velocity increases?

Assumptions: No other forces, gravity acts over infinite distances.

Parameters: Large mass=$10^{24}$kg Small mass=$100$kg, Distance between them=$10^{12}$m.

Answer 1

The answer is one of general relativity and there are a bunch of possible answers depending on how you interpret "the speed of light".

The usual interpretation is the same as the definition of speeds in our everyday world, i.e. the distance divided by the time. In GR we usually have (locally) a time coordinate and three space coordinates. So the speed of light means the maximum amount a space coordinate can change divided by a time interval in which the change can happen. For example, if the position can change at most by 1 light-year in a time period of one year, then we say that the speed of light is 1 light-year per year. In this sense, the maximum speed a particle can reach is the (local) speed of light.

The reason for calling it the "local" speed of light is that it doesn't make sense to compare speeds at two points that are far apart. In such a case, the relative speeds depend on our choice of coordinates.

Answer 2

is there a 'terminal velocity' for the small body due to the increasing energy required to accelerate the body as its velocity increases?

No there is not.

Under the conditions you specify, the only limiting force is the one when they literally collide. There is a maximum speed that will be reached, of course, right before they hit. In the absence of any other retarding force, however, the gravitational potential energy will be converted to kinetic energy with 100% efficiency. If it misses, for instance, then you enter the realm of orbital motion where the energy is constantly swapping back and forth between kinetic and gravitational potential energy.

Hope this helps. I think you have a solid intuition about the situation.

Limitations

The question can be formalized in the following way, there is a small mass $m$ and a large mass $M$ for which $m\ll M$. Say $M$ is at the origin and $m$ starts on the positive x-axis at a distance sufficiently large such that we can say the gravitational potential is zero. The acceleration of $m$ is written for the non-relativistic case as follows.

$$ a = - \frac{G M}{x^2}$$

It will follow that the kinetic energy as $m$ reaches any given point will be the following, then we can write the velocity. The conceptual question that follows is whether it can exceed the speed of light.

$$ KE = m \frac{G M}{x} = \frac{1}{2} m v^2$$ $$ v = \sqrt{ \frac{ 2 G M }{ x} }$$ $$ v \stackrel{?}{>} c$$

An obvious limitation is the case where $x$ reaches the value such that we are on the surface of $M$ if we take it to be a planet for instance. We can wave away this limitation by saying that M is a point. However, the formula for kinetic energy is invalid for when $v$ approaches $c$ (as well as the formula for potential energy!) so that's not very helpful.

In terms of common knowledge about general relativity, yes, it actually is the case that the speed exceeds the speed of light going inward by some formalizations in some reference frames. The equations above, however, do nothing more than give you and idea when when such concepts are going to become important. Any mass, no matter how small, can form the conditions of a black hole provided that the density is high enough (just decrease $x$ in that equation).

Answer 3

I think this will answer your question better: If you have a small body travelling in a single direction du to the gravity of a large body then as long as there is no other influence, and the two bodies never get any closer, then there is no limit to the speed that will be reached. The speed of light is not a limiting factor. Lets take an object obeying the gravitational pull of our planet. The speed is 9.8m/s/s (meters per second per second). What this means is that each second that passes the object will travel 9.8 meters per second faster than the previous second. The speed of light is 299,792,458 meters per second. this will take 30591067.14285714 seconds (509851.12 minutes, 8497.52 hours, 354.06 days, 50.58 weeks, 0.97 years) to achieve. If we add just a single second then the object will be travelling at 9.8 meters per second faster than light. After two years it will be more than twice the speed of light. The potential mass due to its velocity just means that it is virtually unstoppable. Hope this answers your question.