an autodidact meets a dilettante…

‘Rise above yourself and grasp the world’ Archimedes – attribution

Archive for the ‘gravitation’ Category

reading matters 10

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New Scientist 3244 August 24 2019

Canto: Being dilettantes and autodidacts, we engage endlessly in educational reading, bootless or otherwise, so I thought we might take the effort to talk/write about, and expand on, what we’re learning from the texts we’ve perused, rather than providing ‘content hints’ as before.

Jacinta: Well of course science mags cover a wide range of topics at very various depths, so we’re going to limit ourselves to the ‘cover story’, if there is one. 

Canto: So today’s topic comes from a New Scientist that’s been hanging around for a while, from a year ago, but since quantum theory is more or less eternally incomprehensible, that shouldn’t matter too much. 

Jacinta: Yes I’ve heard of Lee Smolin, and in fact we can listen to many of his online interviews and lectures via youtube, and he’s described as a ‘realist’ in the field, which doesn’t mean much to me at present, but neither of us know much about quantum mechanics, in spite of having read numerous articles on the topic. 

Canto: You probably have to ‘do the math’, as the Yanks weirdly say.  

Jacinta: Well we won’t be doing much of that. The cover story is titled ‘Beyond weird’, and Smolin’s idea is that we need to move beyond quantum weirdness to something more coherent and unifying. He describes current quantum mechanical theory as comprised of two different laws:

The first… describes quantum objects as wave-like entities embodied in a mathematical construction known as a wave function. These objects evolve smoothly in time, exploring alternative realities in ‘superpositions’ in which they aren’t restricted to being in any one place at any one time. That, to any intuitive understanding of how the world works, is distinctly odd. The second law applies only under special circumstances called measurements, in which a quantum object interacts with a much larger, macroscopic system – you or me observing it, for example. This law says that a single measurement outcome manifests itself. The alternative realities that the wave function says existed up to that point suddenly dissolve.

Canto: So both of these laws – and of course I’m in no position to doubt or to verify their mathematical exactitude or explanatory power – make little sense from a ‘common-sense’ or ‘realist’ perspective, in which objects must always be objects and waves waves, and, if objects, they must be in a particular place at a particular time, regardless of anything observed. So it seems perfectly cromulent to me that Einstein and no doubt many others found something incomplete about quantum theory, in spite, again, of its apparently vast explanatory power. Like it was an intellectual placeholder for something more real or coherent.

Jacinta: Well Smolin seems to be one of those dissatisfied physicists, – he mentions de Broglie and Schrödinger as others – pointing out that the two laws are in apparent contradiction, with the second law unable to be derived from the first. The theory also ‘seems to’ violate the principle of locality, in which forces are dependent on distance. Quantum entanglement does away with that principle. So Smolin sees a way out by trying to incorporate gravity into the quantum world, or at least trying to connect the general theory of relativity and quantum theory into a seamless whole, as their current incompatibility constitutes a major problem. General relativity presents ‘a smooth, malleable space-time’, while quantum theory suggests ‘discrete chunks, or quanta, of space or space-time’.  String theory and loop quantum gravity are some of the attempts to bridge this divide, but these are currently untestable theories. Also, apparently general relativity is compatible with our perception of the flow of time, whereas quantum theory is more problematic, an issue which, I think, Gerard ‘t Hooft attempts to address in his essay ‘Time, the Arrow of Time, and Quantum Mechanics‘ . 

Canto: Yes, he feels that time, with its arrow pointing eternally forward, with no need for or possibility of reversibility, must be an essential element of a grand physical theory.

Jacinta: Maybe. He’s saying I think, that any explanation of our world, any theory, is arrow-of-time dependent, as it necessarily involves preceding causes and antecedent consequences. But let’s just stick to Smolin’s article. He argues that both relativity and quantum theory have issues with the conceptualisation of time. And there are problems, such as dark matter and dark energy, which don’t easily fit within the standard model. So he feels we need to go back to first principles, ‘in terms of events and the relationships between them’. So, according to these principles, space is an emergent property of a network of causal relationships through time.

Canto: Well to keep more strictly to Smolin’s description, he has five hypotheses. One – the history of the universe consists of events and relations between them. Two – that time, as a process of present causes and future consequences, is fundamental. Three – that time is irreversible, cause can’t go backwards and ‘happened’ events can’t unhappen. Four – that space emerges from this cause-consequence chain. Five – that energy and momentum are fundamental, and conserved in causal processes. 

Jacinta: Good, and this is an ‘energetic causal set model’ of the universe, as he and others describe it, to which he’s added a sixth hypothesis, derived from ‘t Hooft, which says that ‘when two-dimensional surfaces are defined in the emerging geometry of space-time, their area gives the maximum rate by which information can flow through them’.

Canto: Now that sounds horribly mathematical. I do note that area = space and rate = time, and so this hypothesis somehow marries space-time with information flow?

Jacinta: Yes, it’s all threatening to move beyond our brains’ event horizon here. Smolin says that ‘in this picture’, and I’m not sure if he’s talking about the ‘picture’ derived from the sixth hypothesis or by all six taken together, but ‘in this picture, every event is distinguishable by the information available to it about its causal past’. This he calls the event’s sky, because the sky, or what we see (speaking about horizons) at any one instant, is what he calls ‘a view of its own causal past’. This has to do with the speed of light – we can’t see what we can’t see. And this sixth hypothesis, combined with the first law of thermodynamics, can apparently be used to derive the equations of general relativity, bringing gravity into the picture. 

Canto: I don’t get the laws of thermodynamics.

Jacinta: The first law is about energy used in a closed-system process, which can be transformed in that process but is always conserved. Anyway, we’ll try to quit before we get in too much deeper. We know that there’s a ‘measurement problem’, a problem of causality in quantum mechanics, in which it is said that a measurement, or observation, ‘collapses the wave function’ to define a particle’s specific place at a specific time. This is counter-intuitive, to put it mildly, and highly unsatisfactory to many physicists, because it seems to make a mockery of how we understand causality. It seems to be a long-standing impasse to the unification of the two major theories. So we’ve only described a fraction of what Smolin has to say here, and there’s also the problem of entanglement. In ‘classical physics’ proximity matters in a way that it doesn’t in quantum theory. Smolin describes, or mentions, a lot of work being done on ‘ensembles’ in an attempt to solve this measurement problem.

Canto: I think one of the issues that the ‘realists’ are concerned with, but perhaps deliberately not mentioned in Smolin’s piece, is the many worlds hypothesis, or the multiverse, embraced for example by Max Tegmark in Our mathematical universe. Neil Turok is another skeptic of this apparent solution to the causality impasse. 

Jacinta: Yes, I don’t think Smolin is an embracer of the multiverse, tantalising though it is in a sci-fi sort of way. Of course we don’t have the mathematical wherewithal to give an informed view one way or another, or to know whether mathematical wherewithal is what’s really needed. I’ve heard it said – possibly by Tegmark – that a multiverse fits so neatly with the mathematical equations that we need to accept it against our intuitions, which have been wrong in so much else. I don’t know… we’ll just have to watch with interest this intellectual battleground, and see if anything decisive crops up in what remains of our lifetimes.

Canto: Singular or plural…

Other references

The universe within, by Neil Turok, 2012

Our mathematical universe, by Max Tegmark


Written by stewart henderson

September 12, 2020 at 1:08 pm

towards James Clerk Maxwell 6: Newton’s universal law of gravitation and G

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Newton’s law of gravity goes like this:

{\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}},}

where F is the force of gravitational attraction, G is the constant of proportionality or gravitational constant, m is an entity, particle or object with a particular mass, and r is the distance between the centres of mass of the two entities, particles or objects.

What’s the relation between all this and Maxwell’s electromagnetic work? Good question – to me, it’s about putting physics on a mathematical footing. Newton set us on this path more than anyone. The task I’ve set myself is to understand all this from the beginning, with little or no mathematical expertise.

The law of gravity, in its un-mathematical form, says that every object of mass attracts every other massive object with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

It seems often to be put about that Newton was revolutionary because he was the first to wonder why objects fell to the ground. This is unlikely, and Newton wasn’t the first to infer an inverse square law in relation to such falling. Two Italian experimenters, Francesco Grimaldi and Giovanni Riccioli, investigated the free fall (where no force acts besides gravity) of objects between 1640 and 1650, and noted that the distance of the fall was proportional to the time taken. Galileo had previously conducted free fall experiments and found that objects fell with uniform acceleration – an acceleration that is proportional to the square of the elapsed time. Nor was he the first to find a time-squared relationship. The point of all this is that science doesn’t proceed via revolutions proceeding from one brilliant person (which shouldn’t diminish Galileo or Newton’s genius). The more you find out about it, the more incremental and fascinatingly collaborative and confirmative over time it is.

Galileo used the geometry of his time to present his time squared law, but algebraic notation, invented principally by Descartes, superseded this approach in the seventeenth century.

What about the gravitational constant? This appears to be a long and complicated mathematical story. I think it tries to answer the question – why do objects fall to Earth at such and such a rate of acceleration? But I’m not sure. The rate of acceleration would have been easy enough to measure – it’s approximately 9.8 m/sec2. This rate would appear to be caused by the mass of the Earth. The Moon has a fraction of Earth’s mass, and I believe the gravitational force it exerts is approximately one sixth that of Earth. It has been measured as 1.62 m/sec² (for Mars it’s 3.71).

It’s frustratingly difficult to get an explanation online of what the gravitational constant (G) is or really means – without very quickly getting into complex (for me) mathematics. Tantalisingly, Wikipedia tells us that the aforementioned Grimaldi and Riccioli ‘made [an attempted] calculation of the gravitational constant by recording the oscillations of a pendulum’, which means nothing to me. Clearly though, there must be some relationship between G and the mass of the Earth, though how this can be ascertained via pendulums is beyond me. Anyway, on with the struggle.

We do have a number for G, or ‘Big G’, as it’s called (explanation to come), and it’s a very very small number, indicating that, considering that the multiplied masses divided by the square of the distance between them then get multiplied by G, gravitation is mostly a very small force, and only comes into play when we’re talking about Big Stuff, like stars and planets, and presumably whole galaxies. Anyway here’s the actual number:

G = 0.0000000000667408, or 6.67408 × 10-11

I got the number from this useful video, though of course it’s easily available on the net. Now, my guess is that this ‘Big G’ is specific to the mass of the Earth, whereas small g is variable depending on which mass you’re referring to. In other words, G is one of the set of numbers in g. We’ll see if that’s true.

Now, looking again at the original equation, F stands for force, measured in newtons, m for mass, measured in kilograms, and and r for distance in metres (these are the SI units for mass and distance). The above-mentioned video ‘explains’ that the newtons on one side of the equation are not equivalent to the metres and kilograms squared on the other side, and G is introduced to somehow get newtons onto both sides of the equation. This has thrown me into confusion again. The video goes on to explain how G was used by Einstein in relativity and by Max Planck to calculate the Planck length (the smallest possible measure of length). Eek, I’m hoping I’m just experiencing the storm before the calm of comprehension.

So, to persist. This G value above isn’t, and apparently cannot be, precise. That number is ‘the average of the upper and lower limit’, so it has an uncertainty of plus or minus 0.00031 x 10-11, which is apparently a seriously high level of uncertainty for physicists. The reason for this uncertainty, apparently, is that gravitational attraction is everywhere, existing between every particle of mass, so there’s a signal/noise problem in trying to isolate any two particles from all the others. It also can’t be calculated precisely through indirect relation to the other forces (electromagnetism, the strong nuclear force and the weak nuclear force), because no relationship, or compatibility, has been found between gravity and those other three forces.

The video ends frustratingly, but providing me with a touch of enlightenment. G is described as a ‘fundamental value’, which means we don’t know why it has the value it does. It is just a value ‘found experimentally’. This at least tells me it has nothing to do with the mass of the Earth, and I was quite wrong about Big G and small g – it’s the other way round, which makes sense, Big G being the universal gravitational constant, small g pertaining to the Earth’s gravitational force-field.

Newton himself didn’t try to measure G, but this quote from Wikipedia is sort of informative:

In the Principia, Newton considered the possibility of measuring gravity’s strength by measuring the deflection of a pendulum in the vicinity of a large hill, but thought that the effect would be too small to be measurable. Nevertheless, he estimated the order of magnitude of the constant when he surmised that “the mean density of the earth might be five or six times as great as the density of water”

Pendulums again. I don’t quite get it, but the reference to the density of the Earth, which of course relates to its mass, means that the mass of the Earth comes back into question when considering this constant. The struggle continues.

I’ll finish by considering a famous experiment conducted in 1798 by arguably the most eccentric scientist in history, the brilliant Henry Cavendish (hugely admired, by the way, by Maxwell). I’m hoping it will further enlighten me. For Cavendish’s eccentricities, go to any online biography, but I’ll just focus here on the experiment. First, here’s a simplification of Newton’s law: F = GMm/R2, in which M is the larger mass (e.g. the Earth), and m the smaller mass, e.g a person. What Cavendish was trying to ascertain was nothing less than the mass and density of the Earth. In doing so, he came very close – within 1% – of the value for G. Essentially, all that has followed are minor adjustments to that value.

The essential item in Cavendish’s experiment was a torsion balance, a wooden bar suspended horizontally at its centre by a wire or length of fibre. The experimental design was that of a colleague, John Michell, who died before carrying out the experiment. Two small lead balls were suspended, one from each end of the bar. Two larger lead balls were suspended separately at a specific distance – about 23cms – from the smaller balls. The idea was to measure the faint gravitational attraction between the smaller balls and the larger ones.

the ‘simple’ Michell/Cavendish device for measuring the mass/density of the Earth – Science!

Wikipedia does a far better job than I could in explaining the process:

The two large balls were positioned on alternate sides of the horizontal wooden arm of the balance. Their mutual attraction to the small balls caused the arm to rotate, twisting the wire supporting the arm. The arm stopped rotating when it reached an angle where the twisting force of the wire balanced the combined gravitational force of attraction between the large and small lead spheres. By measuring the angle of the rod and knowing the twisting force (torque) of the wire for a given angle, Cavendish was able to determine the force between the pairs of masses. Since the gravitational force of the Earth on the small ball could be measured directly by weighing it, the ratio of the two forces allowed the density of the Earth to be calculated, using Newton’s law of gravitation.

To fully understand this, I’d have to understand more about torque, and how it’s measured. Clearly this weak interaction is too small to be measured directly – the key is in the torque. Unfortunately I’m still a way from fully comprehending this experiment, and so much else, but I will persist.


Go to youtube for a number of useful videos on the gravitational constant

Written by stewart henderson

July 14, 2019 at 4:51 pm