Everything in the universe appears to be moving relative to something else, and the majority of that motion appears to be curved, as in a planet moving in an orbit about a central point. Some people, Johannes Kepler for example, thought that this demonstrated that natural motion is circular. Others, like René Descartes favoured the idea that natural motion is rectilinear, i.e. that matter is naturally inclined to move in a straight line, only deviating from this natural straight path if some other external force is acting on it (which is usually the case).

It could be argued that this difference is just a semantic issue, that the forces that cause deviation from the straight and narrow are part of the natural order so that the rectilinear idea is an unreal ideal state as far as matter is concerned. But Descartes’ idea got a lot of traction when Isaac Newton proposed that the observed curved motion could be attributed to the existence of two separate, very unequal and unrelated types of motion, firstly inertia or linear momentum which is a pre-existing state of motion and a second one which has its origin in mass itself and which acts on other matter causing the direction of the inertial component to constantly change. The earlier concept of angular motion or momentum was also recognized by Newton but only in the context of spinning bodies. He seems to have been the first to propose that it was conserved.

*“A top, whose parts, by their cohesion, are perpetually drawn aside from rectilinear motions, does not cease its rotation otherwise than it is retarded by the air. … greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time”*

– Axioms; or Laws of Motion, Law I. in The Mathematical Principles of Natural Philosophy. (See here)

This division of motion into two separate, unrelated, vastly unequal components was helped along by the new-at-the-time idea that curved motion could be represented by two motion components or vectors (h/t Robert Hooke et al.). Curved motion could be divided into two components, one tangential to the curved track and the other perpendicular to the curve.

Once two components have been identified then the way is open to assign two different origins to the motion. And that, via Newton, is what happened. He identified the perpendicular component as originating in the center of a mass and he went further with this line of thinking by proposing that separate masses are influenced by this component of motion, that they attract one other. The rectilinear component or vector was assumed to be inherent and lately it has been proposed that this inherent motion (it is called inertia) is a result of the event that set everything in motion, the proposed cosmological big bang. Note the if a curved motion has more than two components, as is the case for helical motion (which has three components) then the set of components is called a tensor, but still involves separating the complete motion into component parts.

To bolster his idea, Newton came up with a suitable expression that enabled the observed motion of a falling body, called g (little g), to be calculated from the mass of the earth and the distance between the center of the earth and the falling body. However, in order to obtain the correct result it is necessary to apply an adjustment factor (also known as a constant) to the mass/distance relationship. This adjustment factor is written as G (big G) and is called the universal gravitational constant. (See note below for details on how the value of G is determined)

Because this constant must possess dimensions , i.e. length, mass and time (the expression is not valid if the constant doesn’t have the dimensions of length^3 *, mass^-1, time^-2) it is called a dimensioned constant. This is in contrast with the arguably more fundamental constants (such as pi, which is just a number), which are called dimensionless constants. The other thing to note about G is that when everyday units are used to describe the masses and distance involved, it has an extremely small value. G’s value is hard to pin down precisely but depending on the units chosen for length, mass and time it is something like 0.00000000000676 m^3 kg^-1 t^-2.

Newton’s expression can be written as the force acting between two masses in the form,

F = GMm/r^2,

M and m are the gravitational masses involved and r is the distance between their centers. Note that, unlike length, where the product of two lengths can give an area, by itself, the product of two masses does not yield useful information. The more intuitive combinations of M + m or M – m do not work.

And note also that this form of the expression is similar to the expression for the force between two electrical charges (Coulomb’s Law), F = Kq1q2/r^2, with K being the constant and q1 and q2 the charges. (In contrast to the gravitation constant which is very small, Coulomb’s constant is large, 8.99×10^9 N m^2 C^−2 where N is a unit of force, the Newton and C is a unit of charge, the Coulomb. Note also that Coulomb’s constant has structure in that it is inversely proportional to another constant, *ε*_{0}, the permittivity of free space.)

From this expression the acceleration due to the force between the two bodies, called little g, can be derived. We do this by substituting mg for F (this relationship between force and the product of **inertial** mass and acceleration is another one of Newton’s insights and is known as Newton’s Second Law, which, incidentally, was contra Aristotle who proposed mass and velocity as the elements of force) so that the expression becomes,

mg = GMm/r^2, ** (see below)

and from there it is just a matter of using the simple mathematical expedient of removing the little m’s from both sides of the expression to obtain

g = GM/r^2.

Here we see that little m has been removed from the expression, hence the independence of little g from the (as we perceive it) falling mass.

And if we plug in the right values for big G, M and r we get the right (observed) value for little g, i.e. about ten meters/second/second.

Q.E.D.

No one knows exactly how Newton came up with this expression.

Note that Kepler himself was no slouch when it came to figuring things out. Wikipedia tells us that, …*His work led to the modern laws of planetary orbits, which he developed using his physical principles and the planetary observations made by Tycho Brahe. Kepler’s model greatly improved the accuracy of predictions of planetary motion, years before Isaac Newton developed his law of gravitation in 1686.*

About a century ago a new idea was proposed by Albert Einstein. This still involved the mass of the large body, but instead of the mass directly influencing motion it was here proposed that there was a more indirect effect with the mass now influencing spacetime causing it to curve so that any other masses would then follow the now curved shape of space (this curved track becomes the shortest distance between two points and is called the geodesic). The motion, both inertial and gravitational components, becomes a function of this curvature of spacetime rather than direct mutual attraction between the masses.

This necessitated the designation of gravity as a pseudo force (the motion itself can be “transformed away” by switching frames of reference ***) as opposed to a real force. In my opinion, this idea is significant because it includes spacetime into the mix as well as the masses themselves.

However recent observations of the cosmos have upset this apple-cart 🙂 and cast doubt on the relationship between gravity and mass. It has been found necessary to propose the existence of at least five times more mass than we can see to account for the observed motions of some galaxies. This affects both of the mass related causes discussed above. Because we cannot see this mass it is called dark matter. Unlike ordinary matter, dark matter, by definition, does not radiate energy. One could perhaps imagine dark matter to be like a very cold non-radioactive rock or gas that does not reflect or re-radiate incident (incoming) energy. Strange stuff, no?

Relativity at least finds a place for gravity in the contours of the the spacetime geodesic but it doesn’t fare so well when it comes to the other recent big thing in physical theory, Quantum Mechanics. This very successful model is strangely muted when it comes to gravity.

Quantum Mechanics deals with energy levels that give matter its structure and properties. These energy magnitudes overwhelm any effect that gravity might exercise and as a result it just doesn’t rate, everything works fine without it. But nevertheless it is dogged by demands that it be ‘unified’ with the gravitational geodesic assertions and transformations of relativity. Much time and effort has been devoted to resolving the issue and a number of add-ons such as Loop Quantum Gravity and String Theory have been developed in the attempt to include gravity in the picture. But all are problematic in some way, for example,

*…1) Loop quantum gravity is a way to quantise space time while keeping what General Relativity taught us. It is independent of a background gravitational field or metric. So it should be if we are dealing with gravity. Also, it is formulated in 4 dimensions. ***The main problem is that the other forces in nature, electromagnetic, strong and weak cannot be included in the formulation. Nor it is clear how loop quantum gravity is related to general relativity.**

*2) … String theory is a quantum theory where the fundamental objects are one dimensional strings and not point like particles. ***String theory is “large enough” to include the standard model and includes gravity as a must**. The **problems are three fold, first the theory is background dependant.** The theory is formulated with a background metric. **Secondly no-one knows what the physical vacuum in string theory is, so it has no predictive powers**. String theory **must be formulated in 11 dimensions, what happened to the other 7 we cannot see? …**

Quantum mechanics has a hard time ‘seeing’ gravity and doesn’t really need it to exercise its quantitative talents. It goes about its quantum scale business quite successfully without it. Perhaps Quantum mechanics is trying to tell us something about the nature of gravity, …that maybe it doesn’t exist as a discrete entity but is somehow an integral part of the geodesically guided inertial motion inherent in all matter, its motion groundstate. And whose origin is perhaps not based in matter itself. Maybe one needs to consider the geodesic energy state (if there is such a thing) rather than the matter energy state so that the raising of a finger changes the energy state of its surroundings rather than the energy state of the finger itself.

To recap, is it possible that the energy considerations of gravity are related to the alteration of the geodesic of spacetime, …where the energy variation is necessary to reconfigure the local geodesic, …because different locations require different inherent energy levels? That, together with a tendency for things to attempt to occupy the lowest motion groundstate in the local system, the center, may provide a more useful picture of this mysterious phenomenon. It may also lead to an understanding as to why gravity is only discernible at larger scales in that at quantum scale most matter does occupy its local groundstate where gravitation potential is a function of the state of the local geodesic rather than the mass itself.

It could also be that the current understanding of gravity may be returning anomalous results (via dodgy mass estimates) for the density of some of the components of the solar system. For example, although photographic evidence suggests a rocky composition, the latest estimate for the density of comet 67P is about 0.5 which is similar to the density of fluffy snow. Others think that the hydrogen sun model is obsolete and that iron is its most abundant element but the density is calculated to be about 1.5 which doesn’t allow for much iron content.

Given this state of affairs others have looked for a different origin for the cause of this strange phenomenon we identify as the force due to gravity. And given the similarity of the force expressions between two masses and two charges, some have proposed an alternative mechanism, that being the influence of the electron field. This idea still treats the perpendicular vector as a separate force but invokes the electron field rather than the proton (field?) which contains the majority of the thing we call mass. The neutron is a sort of hybrid of a proton and an electron so its mass is also proton related. Recall too that (at some scales?) the force associated with the electron field is about thirty-seven orders of magnitude greater than the gravity field (that is assuming that gravity does actually constitute a separate field) so their is plenty of scope for it to hide in the electron field.

It is also possible that gravity is mass related but operates differently at different scales, sometimes attracting, sometimes repelling. But nevertheless until we positively identify this extra hidden mass then the concept of mass related gravity has to be on shaky ground. 🙂

Apart from an aside, I haven’t mentioned spin in this polemic. That’s because gravity is not thought to play any role in this type of motion, it being due solely to primordial angular momentum and the conservation thereof which is balanced by self-adhesion of the mass. As I understand it, gravity only plays a part in the accretion of the body from the primordial disk.

———–

* Note that length cubed is the unit for volume. It may be worth looking into why G has a volume component in its dimensions.

** Note that with this substitution, Newton has equated gravitational little m with inertial little m. This equivalence is justified by the remarkable equivalence of earthly gravitational mass with inertial mass.

*… gravitational mass, is something that responds to the pull of gravity, tending to accelerate a body in a gravitational field. The other, inertial mass, is the property of a body that opposes any acceleration. That makes it rather odd that a large chunk of modern physics is precariously balanced on a whopping coincidence. **This coincidence is essential to the way we view and define mass. … *– New Scientist

But this equivalence of gravitational and inertial mass only applies on earth’s surface. Anywhere else like the Moon or Jupiter or wherever this remarkable feature does not apply, as even though inertial mass is the same, gravitational mass varies according to the mass of the body in question.

*… Because weight is directly dependent upon gravitational acceleration, things on the Moon will weigh only 16.6% of what they weigh on the Earth.*

*** … *Inertial forces can be ***“transformed away”**, as physicists say. Specifically, they can be made to vanish by simply making the transition from one observer to another (in our example, by making the transition from a rotating observer to a non-rotating one). The same is true for gravity, which can be made to vanish, at least locally and approximately, if you just let yourself fall freely. …

http://www.einstein-online.info/spotlights/scalar-tensor

This transformation process is informed by Einstein’s Equivalence principle which equates inertial mass with gravitational mass.

… *The starting point for general relativity is the equivalence principle, which equates free fall with inertial motion, and describes free-falling inertial objects as being accelerated relative to non-inertial observers on the ground. …*

**Determining the value of G.**

The value of this constant was determined about a century after Newton proposed it by Henry Cavendish using an apparatus designed and constructed by a geologist , Michel, who passed away before he had the opportunity to use it. The machine consisted of two small identical weights suspended at each end of a beam which was in turn suspended from a wire attached to the substantial housing intended to reduce temp fluctuations, vibration etc. The idea was that when large masses were moved into close proximity to the suspended masses the smaller object would move toward the large masses causing a small torque or twisting of the wire which would translate to movement of a suitably amplified motion indicator. The indicator, somewhat like a horizontal pendulum, oscillates about the point which indicates the attraction between the two bodies. (Presumably the oscillation pattern differs from the way the pointer would react were it under the influence of, say, a magnetic field where the oscillation would perhaps be more rapidly damped than the more elastic? gravitational attraction/field.) The amount of deflection of the pointer can be used to determine the force as the magnitude of the mass(es) are also accurately known.

The force acting to twist the suspending wire appears to exert a real force on the suspended masses, although it is not clear how one might distinguish between a physical force and the more recent idea of gravity as a pseudo force of distorted spacetime.

Below are two versions of how the results are used to calculate the value of G.

The next one is an abbreviated version of the full derivation outlined in the linked pdf.

There is an alternate opinion about what Cavendish actually determined from his observation,

*… Most physics books state that Cavendish performed the Cavendish experiment and measured the value of G, or the gravitational constant. However historical evidence suggests that Cavendish used the experiment to measure Earth’s density and did not actually calculate G – not until much later were Cavendish’s results used to calculate the value of G. …*

– Updated on 14/10/17 to include discussion of inertial vs. gravitational mass (a.k.a. weight)

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