Simple Quantum Gravitation

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(original: April 28, 2009, as a Google ‘Knol’; converted to blog article November 24, 2001; converted to HTML post March 10, 2016)

The reader is assumed to know a fair amount of background information, involving a number of concepts (if we want to keep it simple, then concepts are more important than heavy-duty math –and if the concepts are combined in a way that stays sensible, appropriate high-density math will eventually follow). Here’s a minimal list of relevant concepts:

Mass ( );
Force ( );
Momentum ( );
Conservation Laws ( );
Curved Space ( );
Gravity ( );
Gravity Waves ( );
Mass-Energy Equivalence ( );
Reference Frames ( );
Tests of General Relativity ( );
Uncertainty Principle ( );
–especially the Energy-Time version of Uncertainty (same reference)–
Wave-Particle Duality ( );
and Virtual Particles ( ).

In any notion about Quantized Gravitation, two objects must be described as exchanging virtual particles between them. The rate at which virtual particles are exchanged must be directly proportionate not just to the masses of the two objects, but to their total mass-energy. That’s because some things have zero mass (like photons) but nevertheless can interact gravitationally (they have energy which is equivalent to mass).

In the same way that the electromagnetism is associated with ordinary photons, and virtual photons are the exchange particles for the electromagnetic force, it is expected that a virtual type of gravitational wave (called a “graviton”) will be the exchange particle for the gravitational force. Part of the problem with developing a theory of Quantized Gravitation is that ordinary gravitational waves have not yet been indisputably detected, which means they haven’t been studied enough for anyone to be confident about how a virtual version of them would work as an exchange particle.

There is also a possibility that certain expected aspects of gravity waves might actually differ between one wave and another. For example, it is necessary that some of them must move at the speed of light (required for predictions about binary pulsars to match observations –which they do), but it is not known that all gravity waves must move at light-speed. It may even fit certain lack-of-data, regarding gamma-ray-burst events that have not been associated with any detection of gravity waves (the detectors were thought to be good enough, but if the waves travel slower than gamma rays, they wouldn’t have arrived yet). Here is a link: The possibility that the expected gravity waves might not travel at light-speed is not discussed at that linked article. Nevertheless, it is widely accepted that gravity waves from a binary pulsar are not generated in the same way as gravity waves from a gamma-ray-burst event. And it is only an assumption that they all travel at light-speed, an assumption not yet supported by any evidence.

The gravity waves that Einstein described are generated when a mass experiences an acceleration (a change in its velocity). These are expected to move at light-speed. Other workers have found that gravity waves may also be generated when a mass experiences a change in its rate of acceleration. A link:

It is not known whether or not these gravity waves must move at lightspeed. Also, more than one line of reasoning can be associated with this variety of gravity wave. One of them is controversial because it has a kind of “missing link”. That is, it references matter that is subjected to certain stresses, not unlike those in the above paper, and the Universe as a whole needs to be part of the interaction, but the link between the stressed matter and the Universe is not specified (it could be these gravity waves). Here are more details:

Another line of reasoning involves a type of gravity-wave detector known as a “Weber bar”, and also the Time Reversal Symmetry of Quantum Mechanics:

Basically, if a passing gravity wave can cause a Weber bar to experience stress that can be detected, then T-Symmetry implies that if a Weber bar is subjected to stress by other means, it should emit a gravity wave. Note the “Weber bar” article indicates that the results are controversial. This is because Einstein’s gravity waves are not expected to be so easily detectable. However, the other kind of gravity waves, emitted by matter that is stressed by experiencing a change in acceleration, are expected by some researchers to be much more easily detectable. Weber could thus have been detecting waves from matter stressed by such mundane things as the explosions of war, train wrecks, avalanches, etc. These more-detectable gravity waves have also been described as having other differences related to the way they are generated, similar to the way a “solitary wave” in water differs from an ordinary wave in water. A link:

At first glance, a solitary wave has significant particle-like properties, and could be a good candidate as a virtual exchange particle. This speculation will thus assume that gravitons are solitons.

Next, the momentum of any object is closely related to its mass-energy (and relative velocity). The wave-particle duality associates the momentum of any object with a rate of vibration. If we pick a particular reference frame in which all the objects are moving at the same speed (but not the same direction/velocity), then we can closely relate the mass-energy of any object in that frame with a rate of vibration. And since one aspect of vibration is that it involves changing accelerations, it is an easy step to speculate that virtual gravitons could be emitted as a result. (Later on, the math can be developed to take different speeds into account.) There’s also another way to get to this conclusion, as will be explained later.

If we do have a rate-of-production of virtual gravitons that is directly proportional to the total mass-energy of any two objects that interact gravitationally, then that’s only half of the speculative answer, since to fully qualify as exchange particles, the virtual gravitons must also be absorbed. Furthermore, in this particular situation, the rate of production is, frankly, too-incredibly-high for gravitation to be observed to be the weakest-by-far force in Nature, if all that pass between two objects are absorbed. But are all of them absorbed?

Let’s approach that from a new starting point. There is a hypothesized type of matter that is called “negative”. None is known to exist, but neither has it been shown to be unable to exist. If it did exist, it might exist, as does anti-matter, in the form of particles that are quite similar to ordinary elementary particles such as electrons, protons, neutrons, and so on. They would simply have negative mass-energy, instead of the ordinary mass-energy that we ordinarily deal with. (Anti-matter has ordinary mass-energy, because it has always been associated with the famous equation E=mc², and negative mass-energy is not normally part of that equation. For negative matter and negative mass-energy, the equation would be written (-E)=(-m)(c²).) A link:

Next, it is known that in an ordinary “perfect” vacuum, virtual particles of all possible types are constantly popping into existence out of Nothing, persisting for tiny periods of time, and then vanishing again. (A really perfect vacuum would stay empty.) This means that if particles of negative matter can exist, they should be popping up in the vacuum, also. There is a very large discrepancy between General Relativity and Quantum Mechanics, regarding properties of the vacuum, that might be resolved if negative-matter particles were included. If it can be resolved that way, then that gives us more reason to keep speculating about particles of negative matter.

There is also another and completely different reason for physicists to like the possibility that negative mass-energy might exist. It has to do with Symmetry; the “T-Symmetry” mentioned earlier is but one of three very general observations in Physics. Here are the other two:

There are ways of combining the various Symmetry rules, and the main reason this has been done is because the Weak Nuclear Force is able to violate those rules. In general, physicists like Symmetry, and don’t like imbalances in it. Here:

Despite that grand “CPT Symmetry” rule, there are suspicions that the Weak Force may be able to violate it, also, leaving Physics in an unbalanced and unpretty state. So, enter the notion of negative mass-energy! This actually is so different a thing that it would be associated with a completely new Symmetry Principle. For proof, just start with the equation for the energy of a photon, and ask what form that equation should take, to describe a photon of negative energy. Here’s the equation:

The most logical thing to do is to note that the “dimensional units” of Planck’s Constant include Energy, so if it is used to talk about photons of negative energy, then the Constant should be negative in those cases.

The net result is that we can now imagine a really Grand Symmetry (combined with CPT Symmetry) between a physical realm in which Planck’s Constant is ordinary, and an opposite realm in which the Constant is negative. There doesn’t seem to be any possible way for the Weak Nuclear Force to violate that Grand Symmetry, and so that is why physicists might like for negative mass-energy to be more than just a speculation.

It’s important to keep in mind that the Uncertainty Principle, especially the Energy-Time formulation of it, is responsible for allowing ordinary virtual particles to appear out of Nothing. That is, even if a volume of Space was utterly empty of ordinary mass or energy, the total energy content of that volume must still be Uncertain (even varyingly Uncertain). The Law of Conservation of Mass-Energy has a temporary loophole in it, courtesy of the Uncertainty Principle, and therefore virtual particles can and do temporarily exist. Furthermore, it should be obvious that the preceding could work negatively, too. There is no reason to think that the average zero energy of an empty volume of Space should .only. fluctuate per Uncertainty on the positive side of zero, and so, by fluctuating on the negative side, virtual particles of negative matter should be able to appear.

Next, due to the existence of other Conservation Laws, virtual particles always appear in pairs, the members of which have various opposite properties. For example, if one has an electric charge, the other will (and must) have the opposite electric charge, so that the net affect is that the Conservation Law (for Electric Charge in this case) is not violated. This leads us quickly to the Question, “Can a pair of virtual particles appear such that one has ordinary mass and the other has negative mass?” They are obviously opposites with respect to the Energy Conservation Law! In fact a loophole in that Law is not even required, in order for them to appear! However, there is more than one Conservation Law, and in this case another one applies, the Law of Conservation of Momentum. It is easy to show that if an ordinary virtual particle pops into existence moving one way, and a negative-mass particle appears moving the other way, then Momentum would not be Conserved. Does this mean such an event cannot happen?

Well, remember that the Uncertainty Principle has more than one formulation, and in this case the original Momentum-Position formulation could apply. It creates, after all, a locale-associated (limited space) loophole in the Law of Conservation of Momentum, in exactly the same way that the Energy-Time formulation creates a time-associated (temporary) loophole in the Energy Conservation Law. We therefore speculate that such appearances of ordinary- and negative-matter virtual-particle-pairs are indeed possible.

Next, we note that ordinary pairs of virtual particles are able to interact with real energy in a special way: If the right amount of real energy comes along, the virtual particles can absorb it and as a result become genuine real detectable particles. (This is the process by which all observed anti-particles are created.) Does this mean that if we have a pair of virtual particles such that one has ordinary mass and the other has negative mass, and a “right amount of real momentum” comes along, then they can absorb it and as a result become genuine real detectable particles? Why not? The most obvious answer to that is, “We have no idea what form momentum can take, such that it might be able to take part in that interaction and be absorbed.”

Is there any way to obtain an idea about that? What if we pretended we had some real negative matter, and studied its interaction with ordinary matter? If they wiped each other out of existence (a process called “nullification”, not to be confused with the “annihilation” that occurs when ordinary matter meets anti-matter), then Momentum would be left over, of exactly the same type/form as the momentum described as needed in the previous paragraph. The advantage here is that we can mathematically study nullification events in different Reference Frames, to acquire more information about that leftover momentum. So, if in some “original” or “O” Reference Frame, 1 unit of mass is moving at 2 units of velocity, and 1 unit of negative mass is moving at -2 units of velocity, and then they nullify, leaving no mass or energy behind, then we can examine the event from a number of viewpoints:
Reference Frame:  I   J   K   L   M   N   O   P   Q   R   S   T   U
     Mass1 Mass: +1  +1  +1  +1  +1  +1  +1  +1  +1  +1  +1  +1  +1
 Mass1 Velocity: -4  -3  -2  -1   0  +1  +2  +3  +4  +5  +6  +7  +8
 Mass1 Momentum: -4  -3  -2  -1   0  +1  +2  +3  +4  +5  +6  +7  +8
 Mass1 KineticE: +8  +4½ +2  +½   0  +½  +2  +4½ +8 +12½ +18 +24½ +32
     Mass2 Mass: -1  -1  -1  -1  -1  -1  -1  -1  -1  -1  -1  -1  -1
 Mass2 Velocity: -8  -7  -6  -5  -4  -3  -2  -1   0  +1  +2  +3  +4
 Mass2 Momentum: +8  +7  +6  +5  +4  +3  +2  +1   0  -1  -2  -3  -4
 Mass2 KineticE:-32 -24½ -18 -12½ -8 -4½ -2  -½   0  -½  -2  -4½ -8
     Mass After:  0   0   0   0   0   0   0   0   0   0   0   0   0
 Momentum After: +4  +4  +4  +4  +4  +4  +4  +4  +4  +4  +4  +4  +4
 KineticE After:-24 -20 -16 -12  -8  -4   0  +4  +8 +12 +16 +20 +24

The most obvious results are that in every Reference Frame, no mass is left over, but the same quantity of momentum is left over. Also, while in the original Reference Frame no kinetic energy is left over, some is left over in all the other Reference Frames. So, if we have some weird massless momentum-possessing .thing, it might be considered less weird since it can be associated with energy, if not mass. Indeed, the simplest way to describe it is to say it is a “quantity of pure momentum moving at some velocity”, and we can do some divisions to compute that velocity in each of the above Reference Frames:
 Velocity After: -6  -5  -4  -3  -2  -1   0  +1  +2  +3  +4  +5  +6

Could we get away with calling that thing a “quantum of momentum”? The fact that it has the same magnitude in all those different Reference Frames indicates that such is not an utter impossibility, and in Physics these days everything .else.seems to be quantized: Mass, Energy, matter, charge, …, even Space and Time have been examined in terms of quanta, and the goal here is to quantize gravity. So why not add momentum to the list, too, while we’re at it? In this speculation, we will do exactly that.

Now let’s consider a well-accepted belief of most physicists, that any sort of imaginable elementary thing, if it can actually exist, is also able to exist (already exists!) as a virtual particle. In the present cases, if particles of negative mass and quanta of momentum can actually exist, then they should be present among all the other virtual particles in a vacuum. We’ve already covered appearances of virtual negative-mass particles, and if they can really exist perhaps quanta of momentum have to also exist as a logical consequence. Obviously such a quantum (a real one) would be “just the thing”, if absorbed by a pair of virtual particles having opposite masses, to cause them to begin existing as real and non-virtual particles. This would be quite equivalent to a pair of virtual particles, one matter and one anti-matter, absorbing a quantum of real energy, and thereby becoming real and non-virtual particles.

But what about those virtual quanta of momentum? Besides the possibility that they might be their own antiparticles (like photons), and therefore could spontaneously/temporarily appear, is there some other quantum mechanism that can cause them to exist? Possibly! The mechanism could be called “virtual nullifications”. See, it is well known that ordinary real particles can interact with virtual particles; this is fundamental to how forces such as ElectroMagnetism work. Also, at the subatomic level of quantum events, real particles are literally surrounded by virtual particles that are constantly popping into temporary existence and vanishing again. Does anyone dare say that no interactions between them are possible?

As a specific example, consider an electron/anti-electron pair. If this pops into virtual existence right next to an ordinary real electron, why can’t the virtual anti-electron annihilate the real electron, causing the virtual electron to .become.the real electron?

Now let’s repeat that example, involving a pair of virtual particles that includes an electron and its negative-mass opposite. If these appear next to an ordinary real electron, the virtual negative-mass-electron would nullify out of existence the real electron, and the virtual electron would become the real electron. And what happens when nullification occurs? A quantum of momentum appears –only in this case, since the whole event was virtual, the quantum of momentum must itself be virtual. (Note that such a description can apply for any real particle, even for something that has no mass but only energy, like a photon.)

Events of the preceding sorts can be very closely connected to the wave-particle duality. Real particles that have more mass-energy can interact with virtual particles at a greater rate than real particles that have less mass-energy. This means that virtual quanta of momentum can be generated by some overall quantity of real matter at a rate that is directly proportional to its total mass-energy. It means that quanta of momentum might also be the solitons/gravitons that were previously described here! That could be a Good Thing, since physicists prefer fewer particles to many, and here we have an opportunity to say, despite all the talk about quanta of momentum, we haven’t really been talking about some new kind of particle!

And now we reach the main point of that rather significant digression. If a graviton is also a quantum of momentum, which we have analytically described as a quantity of momentum moving at some velocity (and notably not at light-speed, either!!!), then now we can manipulate that description in more mathematical terms, seeking a specific goal. Remember the question about how absorbable are gravitons? If Energy is (E) and if momentum is (p) and if velocity is (v), then for a quantum of momentum, (E=vp). Suppose one of those was moving along and happened to encounter a rest-mass (m); be absorbed? Remember that the very definition of a “quantum” includes the specification that it cannot be subdivided; this means that if a quantum of momentum is to be absorbed, the mass must acquire exactly the kinetic energy and the momentum that it possesses, no more and no less. Otherwise it simply cannot be absorbed!

We have no reason to think that the momentum and the velocity of a quantum of momentum must exist in some particular combination, and as a result the relevant algebra indicates that the equation (p/2v=m) must be satisfied for it to be absorbed. More, this relation must be exact to the limits of Certainty, which implies that only a very very very very tiny percentage of all interactions between masses and gravitons will result in absorption. .That’ gravitation becomes the weakest force in the Universe!

We close these descriptions with two final observations. First, when accuracy to the limits of Uncertainty are required, so also must the math be done in the more precise terms of Relativity (compared to Newtonian math). Here: (p/2v)[1-(v/c)²]=(m) –and this page shows how that result was obtained: Second, Isaac Newton first asked a Question, “Why does inertial mass seem to be identical to gravitational mass?” Unanswered and postulatized for centuries, it appears we now can say, “Because the quantum of momentum is the same thing as a graviton.”

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