Big Mass, Little Mass

Although Newton’s universal law of gravitation tells us any two bodies have an attractive pull on each other (mGM/R^2), stiil we know the effect is negligible for small masses.

At what threshold mass can we no longer ignore the effects of this?

What matters, surely, is not the absolute mass of the less massive body, but the ratio between the two masses, and the degree of precision which we need for the purposes for which we are carrying out the calculation?

The question is based on a false assumption. The effect does not have a threshold. Your own gravitational field is weak, and may well be overcome by so small a factor as friction. The proportions remain the same as every other body. Every grain of sand orbiting the sun pulls on each other grain of sand. While the effect is small, it is proportional in the same way, directly with mass, and inversely with the square of distance for every single pair of masses in the universe.

The effect of the gravitational field of S. Doradus on my body is negligible, but it is one of the larger visible objects in the sky. It is very distant, though, and that matters a lot more than its size.


Seems to me that the answer is: It depends.

It depends on how much force is needed to make a difference larger than the margin of error on whatever system you’re measuring.

If you’re measuring the path of a proton, then a very small mass might make the difference. If you’re measuring the path of a bullet, then you need a lot more.

On the other hand, if you’re measuring the path of a bulldozer or a planet, then the effects are negligible except for extremely large objects.

Nope. It takes less force to cause a given acceleration for a proton, but a proton will also feel less gravitational force to begin with. The mass of your test body cancels out. What really matters is how well you can measure acceleration, and how much you’re willing to consider negligable. When you’re bowling, you ignore the gravitational force of the ball on the pins, but Henry Cavendish measured the gravitational attraction between bowling-ball sized objects, so for him, a bowling ball’s mass wasn’t negligable.

Chronos, usually I shut up and accept it when you correct me, but it looks like you’re saying a proton will feel the same small gravitational force whether it’s near a bowling ball or a planet.


Am I missing something here, or was I just really unclear?

FTR, what I was saying is that the path of a proton will be deflected more significantly than a bulldozer as each one passes near a given object, and the path of a planetary mass will be deflected less than the path of a boulder as each one passes by a given object.

Weren’t you the one who started the hammer and feather thread, oh so many (well, two) years ago? Isn’t this more-or-less the same thing? The proton feels much less force than the bulldozer, but it takes much less force for the same deflection.

Hmm, your FTR explanation appears to be flat out wrong, so I’m guessing that you’re missing something. What, I’m not sure. Are you aware that the acceleration (“deflection”) due to gravity at the Earth’s surface is essentially (“significantly”) the same for all objects?

Good pickup, ZenBeam.

I submit that the ultimate threshold of mass that can generate a gravitational field is the Planck Mass.

I have no proof…just a reasonably valid opinion.

That’s a bit problematic… A proton, for instance, is much less than the Planck mass. While we’ve never measured the gravitational field of a single proton, we have measured the field due to a great number of protons. If a single proton can’t gravitate, one must explain how a group of them can.

There are also problems with Newton’s third law. We’ve measured the gravitational force on neutrons, and it’s exactly what we thought it should be. But if the Earth is putting a force on the neutron, then the neutron is also putting a force on the Earth.

I have to ask, Enola Straight, what do you mean when you say “reasonably valid opinion” in this context?

To expand on Chronos’ post a bit, one might try to argue that when atoms are bound together, e.g. in the Earth or Sun, they act as a single particle, and so produce gravity, but this doesn’t solve the problem. The Planck mass is 1.3 * 10[sup]19[/sup] the mass of a proton. How did any stars or galaxies ever form in the first place from the gas of independent atoms of Hydrogen and Helium following the Big Bang?

I submit that statement on the grounds that, as Chronos has stated, a proton (and pretty much all subatomic particles) are less massive than Pl. The Pl is the lower limit for a Black Hole…
any quantum singularity smaller than that basically explode into radiation, so that tells me that the Pl is the limit for General Relativity and Gravity, and where Quantum Mechanics picks up.

The easiest way to see how this works is to stick with Newton’s gravitation formula.

F = GmM/r[sup]2[/sup]

F = ma

F = ma = GmM/r[sup]2[/sup]

a = GmM/r[sup]2[/sup]/m = GM/r[sup]2[/sup]

As you can see the acceleration of the test particle is independent of its mass.

So how do you solve the problems Chronos mentioned? Specifically, if a single proton doesn’t have gravitation, how does a group of them have gravitation, and how does Newton’s third law hold.

I assume that, although masses smaller than Planck Mass can be affected by gravity and follow the geodesic of warped spacetime, any mass smaller than Pl is insufficient to create a spacetime warp.

Let’s try again: If a single proton doesn’t create a spacetime warp, how do a group of them create a spacetime warp, and how does Newton’s third law hold.

Well, I’m reaching the end of my physics knowledge, so I’ll try a different analogy.

Let’s suppose spacetime is a wet napkin, and protons are marbles.

The Napkin can hold one or two marbles just fine, but place enough of them on the wet napkin, and they fall through.

This critical mass which breaks the camel’s back is the Planck Mass.
I assume.

(yes, I’m making an ass of u and me.)

Enola, I’m not trying to make you out to be foolish or wrong. The point is, what you say may, in fact, be right, and there may be some sort of quantum weirdness which allows groups of protons to gravitate, without individual ones gravitatating, or some such. The point is, though, that nobody as yet knows just what form that quantum weirdness would take. I’m just trying to illustrate why this is not considered a solved problem.

The comment about Planck-mass black holes, by the way, is also speculation. There may or may not be black holes smaller than the Planck mass, and we really don’t know enough to be able to say.