(Previous posts in this series: Part 1, Part 2, Part 3, Part 4, Part 5)

The fact that all masses accelerate at the same rate when subjected to purely gravitational forces but do not do so for any other type of force indicates the singular nature of gravity. Another peculiarity of gravity is that although we can shield bodies from the effects of other forces, we cannot shield them from gravitational forces. Although the concept of gravity has been around for a long time and we have used it to explain so much, from the motion of planets to terrestrial events, gravity remains a somewhat mysterious force, difficult to incorporate into more general schemes. Attempts to create unified theories of all the forces have been foiled by gravity. Trying to unify quantum mechanics with gravity has also proven to be extremely difficult, resisting the most determined efforts from some the best minds in physics, including Einstein. This is partly because gravity, unlike electric and magnetic forces, is a non-linear force and non-linear forces are notoriously difficult to handle mathematically.

A linear force like electricity can be understood by using the following example. Suppose I measure the force on a test electric charge Q at a location A due to the presence of another electric charge R that is located at some other point B. I then remove R and bring in a a different charge S and place it at yet a third point C and again measure the force on the test charge Q. If I now place both R at B and S at C, the total force on the test charge due to both will be just the sum of the two earlier forces as measured separately. If we have a complicated system of charges, we can use our powerful mathematical techniques, such as calculus, to add up all the forces due to each charge.

But in the case of gravity forces, if we replace the electric charges with masses in the above example, the resultant force due to both will not be the sum of the forces due to each separately. That is what we mean by non-linearity.

Why is gravity non-linear? The reason was hinted at earlier when we said that matter affects the nature of space. Using a crude metaphor, in the traditional view of space, we treat it as being like a stage on which all the events of nature, including the particles and forces, move around like actors. Whatever they do, the stage remains unchanged and so they can perform predictably and without much difficulty. But introduce the idea of matter affecting space and then the introduction of matter as actors makes the stage itself part of the performance as it starts to warp and shift and change in response to their presence, distorting and moving around as the matter moves, affecting even the behavior of the non-gravitational entities. The shifting stage affects the motion of matter which in turn makes the stage move even more. This makes it hard for the ensemble to perform together.

But don’t we routinely treat the gravitational force as if it is linear? Isn’t that is how we calculate the motion of pretty much everything including planets, etc.? For the most part, we get away with ignoring the non-linearity because the force of gravity is so extremely weak that the nonlinear effects do not have an appreciable effect, except when we have to deal with the large scale structure of space time or massive objects like black holes or are trying to use it exactly as part of a unifying scheme with other forces or quantum mechanics. It is only then that we have to invoke the nonlinear field equations of general relativity and it is that that causes difficulties. It is like how we can get away with treating the Earth as an inertial frame for many things because the non-inertial effects introduced by its rotations are small, except when we are doing high-precision work or are dealing with large-scale systems like air currents.

I said earlier that unlike with the other forces, we cannot shield ourselves from the force of gravity. That is not strictly true. Einstein said that the ‘happiest thought’ of his life came in 1907 when he was thinking about how to reconcile Newton’s theories with his own theory of special relativity and he realized that there was a way in which to make the force of gravity disappear. The basic idea is that if I have an object in my hand and let it go, it will fall away from me to the ground and we say that it is because of gravity. But if I happened to be in free fall and let go of the object, it will remain where it is relative to me. As far as I am concerned, the force of gravity has ‘disappeared’ while I am in free fall.

Einstein is a remarkably lucid writer and when he is at the top of his game, it is a waste of time to paraphrase his writings to try and make them clearer because you end up making things more obscure. In his collected works can be found a recounting of how he arrived at his idea that the gravity can be made to disappear.

In an example worth considering, the gravitational field has a relative existence only in a manner similar to the electric field generated by magneto-electric induction. Because for an observer in free-fall from the roof of a house there is during the fall—at least in his immediate vicinity—no gravitational field. Namely, if the observer lets go of any bodies, they remain relative to him, in a state of rest or uniform motion, independent of their special chemical or physical nature. The observer, therefore, is justified in interpreting his state as being “at rest.”

The extremely strange and confirmed experience that all bodies in the same gravitational field fall with the same acceleration immediately attains, through this idea, a deep physical meaning. Because if there were just one single thing to fall in a gravitational field in a manner different from all others, the observer could recognize from it that he is in a gravitational field and that he is falling. But if such a thing does not exist—as experience has shown with high precision—then there is no objective reason for the observer to consider himself as falling in a gravitational field. To the contrary, he has every right to consider himself in a state of rest and his vicinity as free of fields as far as gravitation is concerned. [Italics in original]

This realization led him to what we know as the Principle of Equivalence that is heavily used in addressing the radiation paradoxes.

(To be continued.)

1. Ketil Tveiten says

Can’t shield objects from the force of gravity? Bah, Cavorite does the trick just fine!

2. file thirteen says

But if such a thing does not exist—as experience has shown with high precision—then there is no objective reason for the observer to consider himself as falling in a gravitational field. To the contrary, he has every right to consider himself in a state of rest and his vicinity as free of fields as far as gravitation is concerned.

So you are weightless and subjectively at rest when falling, and only not weightless when hard up against some matter using electromagnetism to resist being compacted (and the strong force to resist being crushed)

3. Rob Grigjanis says

file thirteen @2:

only not weightless when hard up against some matter using electromagnetism to resist being compacted

At the risk of being pedantic, it’s not just electromagnetism. The Pauli Exclusion Principle is crucial to the stability of matter, and explaining why solid objects can’t pass through each other.

The strong force is extremely short range, so how would it resist crushing?

4. file thirteen says

The strong force is extremely short range, so how would it resist crushing?

Compacted, as in neutron star. Crushed, as in black hole.

5. file thirteen says

At the risk of being pedantic, it’s not just electromagnetism. The Pauli Exclusion Principle is crucial to the stability of matter, and explaining why solid objects can’t pass through each other.

Also, Pauli repulsion isn’t a force. There has to be a fundamental force resisting the gravitational attraction, right? If not electromagnetism, then what? (disclaimer, I am no physicist)

6. John Morales says

file thirteen, Rob did not claim it was a force, only that it’s not the only applicable consideration.

It is only then that we have to invoke the nonlinear <a href="https://en.wikipedia.org/wiki/Einstein_field_equations&quot; title="ObWikipedia"<field equations of general relativity and it is that that causes difficulties.

Mmmhmm. 🙂

7. John Morales says

[grr]

8. Rob Grigjanis says

Pauli repulsion isn’t a force

It’s not a fundamental force, but it can certainly give rise to short range repulsion between fermions (e.g. electrons). In astrophysics, it gives the Chandrasekhar limit;

White dwarfs resist gravitational collapse primarily through electron degeneracy pressure…

Electron degeneracy pressure is a quantum-mechanical effect arising from the Pauli exclusion principle. Since electrons are fermions, no two electrons can be in the same state, so not all electrons can be in the minimum-energy level. Rather, electrons must occupy a band of energy levels. Compression of the electron gas increases the number of electrons in a given volume and raises the maximum energy level in the occupied band. Therefore, the energy of the electrons increases on compression, so pressure must be exerted on the electron gas to compress it, producing electron degeneracy pressure,

Above the Chandrasekhar limit, nuclear repulsive forces do come into play.

There’s also the link I gave in #3.

9. file thirteen says

John @6:

Rob did not claim it was a force, only that it’s not the only applicable consideration.

I pointed out that it wasn’t a force because I was writing about forces in particular. I don’t see how there can be other considerations to be had when detailing that objects can only be stationary when forces are balanced by equal and opposite forces. Aren’t forces the only considerations there? Or, what did I miss?

Rob @8:

It’s not a fundamental force, but it can certainly give rise to short range repulsion between fermions (e.g. electrons).

Are you saying that the force providing that short range repulsion is not electromagnetism? If so, what is it? If not, what is incorrect about what I wrote? I thought that for an object to remain unmoving on, say, the surface of the Earth, the only forces opposing gravity and maintaining the strucuture of the Earth were electromagnetism (without which the Earth would collapse to something like neutron star material) and the strong force, without which it would collapse further to a… singularity? Black hole? You wrote, At the risk of being pedantic, it’s not just electromagnetism. What other forces are involved?

10. John Morales says

file thirteen,

I pointed out that it wasn’t a force because I was writing about forces in particular.

So… it repulses but it’s not a force.

Aren’t forces the only considerations there?

Maybe, I mean, the difference between a true force and a pseudo-force is in the definition, no?

Anyway, Rob has already clarified: “It’s not a fundamental force”.

Are you saying that the force providing that short range repulsion is not electromagnetism?

Obviously, since electromagnetism is a fundamental force.

11. Rob Grigjanis says

file thirteen @9:

Are you saying that the force providing that short range repulsion is not electromagnetism? If so, what is it?

It’s a strictly quantum mechanical effect which to a large extent determines the structure and stability of matter, as well as chemical properties. You might as well ask “what is the force keeping atomic electrons from all dropping into the lowest energy level?”. That is, in effect, a kind of repulsion. But it is not a result of electromagnetism. It’s because electrons have spin 1/2. It (the PEP) says that two electrons (or any two spin 1/2 particles of the same type, charged or not) with spin in the same direction cannot occupy the same spatial state.

For the purposes of this discussion, we’re ignoring the structure of matter (and chemistry!) and its quantum mechanical underpinning, simply assuming bodily integrity. For strong enough gravitational fields, we obviously couldn’t do that.

12. file thirteen says

You might as well ask “what is the force keeping atomic electrons from all dropping into the lowest energy level?”. That is, in effect, a kind of repulsion. But it is not a result of electromagnetism.

Right, I was indeed trying to ask that :). I had thought, assuming the structural integrity of matter could be explained as a “fundamental force vs fundamental force” kind of balancing, that the force endowing (incorporating? bestowing?) that repulsion was the “fundamental” force called electromagnetism. But from what you’ve explained, it sounds like “fundamental” is a misnomer as the PEP is even more fundamental than that. No ifs, buts or maybes!

But then what happens in a black hole, where gravity is so strong it overrides the structural integrity of matter? If the PEP says that electrons of the same spin can’t, just cannot, be mashed into the same space, what actually happens instead?

13. Rob Grigjanis says

file thirteen @12:

But then what happens in a black hole

That’s the 10,000,000 kronor question.

14. file thirteen says

Rob @13:

Bugger. So, pardon my naivety again, but I do have a followup question (and please excuse my lack of knowledge of the correct terminology for what I’m trying to describe here): did they ever try an experiment in the LHC where they launched electrons of the same spin at such speeds as would cause them to collide with such force that they would, theoretically (and I presume it can be theorised -- if not, how could black holes’ existence be theorised?), overcome the resistance of the structure of matter?

If they did, what happened? Or is the theorised speed that would have to be reached beyond the limits of what the LHC can achieve?

15. Rob Grigjanis says

file thirteen @14: The energies available at the LHC are far too low to produce black holes, at least according to the Standard Model. In fact, there are cosmic rays hitting the Earth every day with far more energy than the LHC can muster.

16. Mano Singham says

filethirteen @#14,

To add to Rob’s response, the LHC collides protons with protons, not electrons. The PEP still applies since protons are also spin-1/2 particles but a big difference from electrons is that protons are not point-like particles as electrons are theorized to be. They are extended objects. So the ‘location’ of a proton is not well defined, though we typically label the location as the ‘center’ of the proton for many purposes. The repulsion that had to be overcome in that case was just the Coulomb repulsion between two electric charges.

Think of the proton-proton collision in the LHC as the collisions of two little bags, each bag (proton) containing quarks and gluons. So the PEP would not come into play.

17. file thirteen says

Thanks Rob and Mano, and Merry… just Merry. Just be Merry. ‘Tis the season to be Merry. Cheers!! 🙂

(I have more naive questions about LHC activities now, but I’ll leave it open to you (Mano) to post about it some day, as you wish 🙂

18. Mano Singham says

file thirteen @#17,

You will need to let me know what the questions are!

19. file thirteen says

Mano @17:

Oh, I meant that rather than pester you with naive questions that take up your time patiently replying to each in a comment thread, you might decide to make a general post on the LHC activities one day, which would attract readers and maybe answer some of my questions before I asked them. 🙂

(I was wondering whether the LHC couldn’t accelerate some electrons into collisions rather than protons for a change)

20. Rob Grigjanis says

file thirteen @19: Actually, the tunnel used for the LHC was previously used for electron-positron collisions (when the system was called LEP). In either case, particles are accelerated by electric fields, then steered into the circle by magnetic fields.

The LHC uses much more more powerful magnets than the LEP did. You may recall from an earlier post that accelerated charges radiate energy*. If electrons were used in the LHC, they would radiate away all their kinetic energy very quickly, because they are much lighter than protons.

*The controversy was about linear acceleration, not circular.

21. file thirteen says

Rob @20:

Thanks, that was very enlightening. Difficult to accelerate electrons to really high speeds if the more energy you pump in the faster they radiate it away. Like trying to fill a bucket with a hole in it that becomes bigger the more water you fit in it.

22. file thirteen says

Rob @20, me @21:

Wait a minute. A proton is also a charged particle. Therefore, it too should radiate energy as it’s accelerated. So why is it any harder for the LHC to accelerate electrons rather than protons again?

23. John Morales says

“If electrons were used in the LHC, they would radiate away all their kinetic energy very quickly, because they are much lighter than protons.”

(The ratio of relativistic “mass” to rest mass)

24. Rob Grigjanis says

file thirteen @22: It’s much easier to accelerate the electron, which is the problem; for a given magnetic field, the initial acceleration of the electron is much greater than that of a proton. But the power radiated is proportional to a², where a is the acceleration. And, IIRC, relativistic effects magnify the power loss. So the electron loses energy much faster than the proton.

25. Rob Grigjanis says

I think it’s worthwhile to reiterate the point Mano made: bashing together two beams of electrons has nothing to do with the PEP. It’s not that two electrons can’t get near each other; with enough energy, that’s no problem. The PEP says two identical fermions can’t be in the same state. And two electrons with large relative velocities aren’t remotely in the same state.