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I hope this question is not a duplicate (it doesn't seem to be) and that it is appropriate for this site.

If we had a universe with only two bodies. One is ultra massive, and the other is very small. Why could a sufficient distance not mean that gravity could accelerate the smaller object to past the speed of light? If gravity affects all objects from all objects, gravity certainly should be acting on the smaller body (and also the smaller body on the larger).

I would think, in fact, that the relative sizes don't particularly matter. Why doesn't gravity make things accelerate past the speed of light if they are far enough away?

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    $\begingroup$ Hint: think about conservation of energy. (At least so long as neither object is a black hole) $\endgroup$
    – The Photon
    Commented Aug 16, 2019 at 17:24
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    $\begingroup$ There have been innumerable questions about whether any force applied for any duration can make any object go faster than the speed of light, and the answer always boils down the same thing: No, because the equations of Special Relativity make that impossible. Sometimes it is phrased as the geometry of spacetime making it impossible. $\endgroup$
    – G. Smith
    Commented Aug 17, 2019 at 5:05
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/43707/2451 and links therein. $\endgroup$
    – Qmechanic ♦
    Commented Aug 20, 2019 at 16:23

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I would think, in fact, that the relative sizes don't particularly matter. Why doesn't gravity make things accelerate past the speed of light if they are far enough away?

To be sure (just in case), please keep in mind that the (Newtonian) gravitational force is not constant with distance even though we usually approximate it as constant for elementary problems involving relatively small objects falling near the surface of Earth.

Consider a gravitating, spherical body with some non-zero radius, and consider a much smaller object that falls towards that body from rest a great distance away. We can calculate the speed with which the object impacts the surface. This speed is equal to the escape velocity at the surface of the body.

So, your question can be recast to something like this:

Why are the no bodies with escape velocity $v_e$ greater than the speed of light?

Note that such a body with $v_e \gt c$ would necessarily trap light. In the Newtonian context, where the speed $c$ isn't a speed limit, an object falling from a great distance would impact the surface of the body with speed greater than the speed of light. See, for example, Can a black hole be explained by newtonian gravity?

But in the relativistic context, no massive object can have relative speed $v \ge c$. So, if a massive body contracts to the radius such that the escape velocity at the surface is $c$, the body must continue to collapse leaving an event horizon from which the 'escape' velocity is precisely $c$

However, now we're in a highly curved spacetime where it's often very difficult if not impossible to properly define concepts that were straightforward in the Newtonian context.

For example, you might from the above conclude that an object falling from a great distance towards a black hole would have speed $c$ at the event horizon. However, in a curved spacetime, where the clocks and rods of different observers outside of and at rest relative to the black hole are not the same, it isn't at all clear what such a statement would mean. In fact, it turns out that no object actually reaches the event horizon according to these observers!

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  • $\begingroup$ This answer has completely changed my understanding of escape velocity. So, escape velocity is the velocity of impact as the distance between the objects approaches infinity and the only force acting between the two objects is the gravitational force? $\endgroup$
    – Bill W
    Commented Aug 16, 2019 at 20:27
  • $\begingroup$ @BillW, for example, see this: "It is often interesting to contemplate this reasoning in reverse. If we drop a rock onto the earth from a state of rest 'far away' (much farther than the radius of the earth, far enough away to be considered 'infinity'), it will REACH the earth with escape (kinetic) energy." $\endgroup$
    – Hal Hollis
    Commented Aug 20, 2019 at 15:33
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Let's compare the case of a gravitational acceleration to the case of acceleration of particles in a particle accelerator.

Do the laws of physics impose an upper limit to the amount of energy that the accelerator can impart to the accelerated particle? Well, no.

As an example of that: the OMG particle

When a cosmic ray hits the Earth atmosphere a cascade of collisions ensues, and by reconstructing the magnitude of the cascade the energy of the incoming cosmic ray can be inferred. In the case of the OMG particle: the inferred kinetic energy was the equivalent of the kinetic energy of a baseball travelling at about 26 m/s (94 km/h; 58 mph). That is way more kinitic energy that our particle accelerators can impart.

Gravitational acceleration:
The laws of physics do not impose an upper limit on the amount of kinetic energy that a gravitational field can impart to an object being accelerated by that gravitational field.

The upper limit of velocity is a property of spacetime. (As physics.stackexchange contributor 'S V' points out in the other answer, since this involves velocity close to lightspeed we must use relativistic physics.) In terms of relativistic physcis there is an upper limit to the relative velocity between two objects.

In a particle accelerator: as more kinetic energy is imparted to a particle the velocity of the particle with respect to the accelerator climbs ever closer to lightspeed. That relative velocity can only approach lightspeed, that is a property of spacetime. (As in: not a property of the accelerator)

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The acceleration you are thinking about is the one an observer inside the planet would measure with an accelerometer, assuming Newtonian mechanics. This acceleration can be arbitrarily large, but there is no problem as of the observer because the speed they measure in their own reference system is zero.

What about an innertial observer from outside would see? According to the laws of Special Relativity, they would measure a different acceleration, going rapidly to zero as the planets approach the speed of light, without reaching it. This is due to time dilation.

For a brief reading about this topic, I would suggest you the book 'General Relativity: An Introduction for Physicists', sections 1.13 & 1.14.

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