Did Einstein Shag Marilyn Monroe: "E=mc²" by David Bodanis

This is not really a book about E=mc² equation per se, i.e., it's not a physics book. It's more a book about the atomic bomb which was something I was not expecting. For those of you expecting something more Physics-oriented, here's a quick rundown of the equation.

There's a lot of confusion surrounding this equation caused by oversimplification. As it stands, the equation gives the energy equivalence of the mass of an object, and as this post goes on to say, there's a more complicated expression connecting energy and momentum in a reference frame in which the momentum is non-zero: E^2 = p^2c^2+m^2c^4. So yes, mc^2 gives the total energy only in the rest frame. Einstein did initially introduce two sorts of mass, the "rest mass" and the "relativistic mass", and if you interpret m as the relativistic mass then E=mc² is valid in all inertial frames. But Einstein distanced himself from the concept of relativistic mass late in life, and it is no longer taught in physics courses and not used by physicists, at least not by particle physicists. But its legacy lingers on, unfortunately, particularly in popular science.

One of the sillier uses of the equation I've ever read some years back when someone tried to argue that if you load files into an electronic device such as a Kindle, it gains energy, and therefore gets heavier. A slightly less daft version is actually taught in some SR courses, namely that objects get heavier if you heat them up - despite the fact that Einstein's "rigid bodies" lack any sort of internal structure, and hence are physically incapable of heating up!

People also confuse E=mc² with F=ma. The latter (Newton's second law) relates force to acceleration. When the car is cruising its engine is exerting just enough force on the wheels to overcome friction (including air resistance), so there is no net force on the car and the speed stays constant. It accelerates when there is a net forward force, although the backward force you feel inside the car as a result is sometimes called a "fictitious force" which arises because Newton's laws don't hold in an accelerating reference frame.

Special relativity is quite distinct from all this. The rather surprising relationship between energy, mass and the speed of light arises from deductions made from the two basic postulates of the theory - the principle of special relativity, and the principle of the constancy of the speed of light. But you have to be travelling at speeds close to c to notice any effect.

Although no one did it at the time, if you plug the numbers into Maxwell's equations, they work fine for moving charges up to a speed of c, then they generate an inconsistency for faster speeds. So you could say that they indicate that c is the greatest speed at which charged particles can move. This might have led someone to wonder why it was impossible for charges to move faster than c - if someone had done so, the idea that the speed of light was some kind of universal constant could have been discovered earlier. But no one did this until after Einstein had put forward his ideas - perhaps because Maxwell's equations are hard to get your heads round, so few people would have understood them well enough to really grasp the inconsistency.

The nitty-gritty of the equation is as follows. In the derivation of relativistic kinetic energy:

KE = mc^2/(1-v^2/c^2)^(1/2) - mc^2 where m is the rest mass of the object.

OK, so an object is moving at v relative to me and this is its KE. This is an exact equation.

At low v the first RHS expression expands to [an approx. -(*)]:

mc^2(1+(1/2)v^2/c^2)

After multiplying through by mc^2 and subtracting mc^2 gets (1/2)mv^2, the classical kinetic energy. So the origin of the classical KE is in the bottom 1-v^2/c^2 term x. Of course classical KE can be simply found by calculation. the energy needed to get a mass "m" to a velocity "v", but it's so satisfying to see it nicely pop out of the relativistic mass equation (as it should!)

It's interesting to think like this too...when an object (relativity) is coming towards me I see its length contract (space-time is different), so in a way, an object that has just been sitting there (doing nothing) and then gets imparted an energy from a force, is suddenly behaving according to relativity (which has at its base the in-variance of laws at different speeds). So one would kind of expect, intuitively, to see its mass/energy vary with speed (and I guess one could do some hand-waving arguments to show this must increase) - just as it's clock sitting on it slows down (from my perspective).

Fundamentals to do with the object change, so I guess even here in special relativity, there's the hint that mass is linked into space-time etc. etc. and a clue to general relativity - where mass/energy actually distorts space time. I think it's really good to think of fundamentals like this because you can just gently see where all these things came from.

If it's a Newtonian object its rest mass is zero and mass is undefined if it's just sitting there in space staring at me, being only defined as m = F/a. When I kick it, it magically "appears"! Alternatively if it's going past me at v, m = 2(kinetic energy)/v^2, so now “m” is defined, but this has relied on the object being given a force anyway. However I can make “m” go away by moving at the speed of the object - I measure a KE of zero. Such is the appearance/disappearance of inertial mass, only existing in relation to forces.

A completely different mass is Newtonian gravitational mass from:

F = GmM/r^2.

Here, F is only defined when “m” and M exist in space. Only one, force is undefined. But if F is undefined mass is undefined...same issue above...mass/forces defined together.

If we put ma = F = GmM/r^2 then:

a = GM/r^2 but we are doing something naughty here. Mixing inertial mass into the gravitational mass eqn. What results is an object M in space, just sitting there, but it is producing an instant effect over space (not limited by c speeds), and “a” is the gravitational field strength.

But from Einstein, an object sitting in space does have “m” defined! m = E/c^2. And you cannot magic it away like above by going to another reference frame. So where does this m come from? Space itself? Marilyn Monroe? For Newton, “m” means something when changing motion happens or, for a different phenomenon, its gravity. With Einstein, you just require the laws of physics to look the same in all reference frames, this implies c is constant...then m = E/c^2. So mass and energy intimately tied to space-time, clues for general relativity, quantum theory. Newton collapses under conceptual contradictions, Einstein opens up much more stuff.

  

There are people writing here who think that such equations are examples of "mathematical idealism" and also seem to think that they have never been empirically corroborated. The same people seem to think that philosophy stopped with Hegel in the same way that some Catholics think that it ended with Aquinas. And in the same way that such Catholics interpret everything in terms of Aquinas those who follow Hegel insist on everything being interpreted in terms of his ideology. As a friend of mine likes say to debunk Einstein every chance he gets, the real equation is: E = MC^2 + 0.5 and it's been covered up by the New World Order Tiberians.  I always tell him I don't care about stuff like that. What I really want to know is whether he or did not shag Marilyn Monroe. 

Never mind all this scientific mumbo-jumbo.

That’s what Bodanis should have written (I know I sound smug but I hate books that don’t address what’s in the title ffs!!! If I had wanted a book on the atomic book I’d have bought one!).

NB: (*)

KE = mc^2/(1-v^2/c^2)^(1/2) - mc^2 (m here is the rest mass) - which is really what we are dealing with.

or KE = m(r)c^2 - mc^2 where m(r) is the relativistic mass.

When v << c but not zero, sure when you expand the first eqn. you will get higher powers adding to the classical KE (1/2)mv^2 but these are not important measurement wise. So you then get v^4/c^4 terms and so on meaning you never get away from rel. effects to the KE at low v.

But...exactly the RHS = the total rel. KE but approx. = the classical KE. So that's OK.

Finally...you'll see that even when the KE is zero, all terms drop out to zero and the mc^2 for rest mass cannot be affected. It sits by itself.

The mc^2 actually contribute to cancelling out all the KE. But always, Total KE = rel. energy - rest energy

Cantorian Sets: "Beyond Infinity - An expedition to the outer limits of the mathematical universe" by Eugenia Cheng

“If this be not that you look for, I have no more to say, but bid Bianca farewell for ever and a day.”

In “The Taming of the Shrew” by William Shakespeare (quoted by Cheng in the book)

Eugenia Cheng starts by saying right at the beginning of the book, "Infinity is not a number," and I think it really helps to get that misconception out of the way at the start. As soon as we one gets past that hurdle the rest is just a piece of cake.

As pointed out numerous times by Cheng, Cantor is the accepted authority on this, but are there alternatives?

Cantor Infinities

Key Idea: You can put the even numbers in one-to-one correspondence with the whole numbers and say that this demonstrates they have the same cardinality.

1->2

2->4

3->6

...

This shows that the set of whole numbers is the same size as the set of even numbers.

This seems counter-intuitive - and it's usually a real challenge to anyone encountering it for the first time, but if you do accept this then all sorts of deep and interesting mathematics follow. The way I think about this is, it's not a natural property, it's not a statement about the world*; it's Cantor's definition of infinity, let's go along with it and see what happens.

Non-Cantor Infinities?

But what about the intuitive (naive, even) point that that in between the even numbers you have the odd numbers, therefore the set of even numbers is smaller than the set of whole numbers?

2, 4, 6, 8, 10, …

is obviously smaller than

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...

Therefore, the set of whole numbers is obviously bigger than the set of even numbers.

There's a hidden assumption that if I add an element to a set then the set gets bigger. That seems like a reasonable assumption to me. Are there "non-Cantor infinities"? To understand current thinking on infinities one must understand Cantor, but maybe one should also consider alternatives.

What Cheng proposed is nothing new in the field of number theory dealing with Infinity: “The general question is: How should we compare the size of infinities? So far, we have decided that the real numbers are unaccountable, that is, that there is no possible way to match them up precisely with the natural numbers because some real numbers are doomed to be omitted. Intuitively this means that there must be ‘more’ real numbers than natural numbers, but what could this possibly mean if they’re both infinite? Some infinities are bigger than others – how is that possible, seeing as infinity is already infinitely big? Isn’t it the biggest thing that there is? How can anything be bigger that it? Just like questions of the soul, everlasting life, and whether or not I’m fat, this comes down to definitions.

The way to dealt with infinities is the following: Start with a simpler question. How do you determine if two finite sets are equal in size? The basic process is to place the two sets in one to one correspondence.  This is key to answering the question about infinities. I know that I have the same number of fingers as toes because I can pair them up with none left over. I know that I have more ears than noses because when I attempt to pair them up I find I make one pair, say nose paired with left ear, and have one ear left over. Now this "one to one correspondence" sounds a bit of an odd process - after all why not simply count each set? The reason is because for infinite sets you cannot count them. But even for infinite sets you can still apply the very basic process of placing it in one to one correspondence.

And as Cheng so ably demonstrates, this can be done for some infinities such as the integers, the prime numbers, the rational numbers so they are equal in size, but fails for others like the integers and the reals. The mathematical proof is quite far more rigorous than that, but the key idea is that simple.

Take the Hilbert Hotel Paradox which Cheng uses throughout the book (bear with me; I’m going to rephrase Cheng here). You have an infinite amount of rooms in your hotel. An infinite amount of people turn up. You fit them one per room. Fine. Then a new person turns up demanding a solo room. How to fit them in? Easy. Move the person from room 1 into room 2, the one who was in 2 into 3, and so on. Infinite amount of people, infinite amount of rooms, all have a room, but now room 1 is free. Infinity + 1 = infinity. Then an infinite amount of new people turn up. They also demand solo rooms. How to fit them in? Easy. Move the person from room 1 into room 2. From room 2 into room 4. From room 3 into room 6. Now you have an infinite amount of odd numbered rooms free. Infinity x 2 = infinity.

So far so good.

But imagine trying to count up all the possible fractions. Someone gives you an infinite list of numbers where the decimal points go off to infinity. You start numbering them; the first one is no. 1, the second is no. 2, and so on. And you do it. You number them all, 1 to infinity.

Then that someone plays a trick. The first set of decimals are like this:

0.11111...

0.2222...

0.3333...

0.4444...

0.6745969...

What they do is change one number in the decimal expansion. They change the first digit of no. 1, the second digit of no. 2, the third digit of no. 3. And so on to infinity. You've now got a number that starts 0.23451..., which is different in at least one digit to every single one that you've numbered to infinity.

You can't add it to the list. Your list of fractions is therefore bigger than your infinity.

This is therefore a second, bigger, infinity than the one of all counting numbers. The infinity of counting numbers is called aleph-null; that of all fractions as well (real numbers) is called R. And you can take this further into ever more mind-boggling and inexplicable higher levels of infinity. If you can prove there are any infinities in between these two, or even that there are no infinities between these two, you would get lots of buns from the mathematical community.

What about the primes which Cheng barely mentions? First, the number of primes is the same as the number of normal (natural) numbers (which is also the same as the number of perfect squares). This is because they can be put into a one-to-one correspondence like this:

1.............. 1st prime number.............. 1 squared (1)

2.............. 2nd prime number............ 2 squared (4)

3.............. 3rd prime number............. 3 squared (9)

4.............. 4th prime number............. 4 squared (16)

...

1000........ 1000th prime number....... 1000 squared (1,000,000)

...

Billion....... Billionth prime number.... 1 billion squared

...

So - and this is counter-intuitive - there are not "more natural numbers" than there are primes or perfect squares because for any natural number, you can match it to a prime or perfect square. This is a peculiar property of infinite series like this - that you can put them on a one-to-one correspondence with a subset [It's also possible to order the rational numbers (fractions) like this. So, the number of factions - 1/2, 3/4, 7/8, etc. - is the same as the number of natural numbers. Again, counter-intuitive.]

However, there are still different sizes of infinity. In fact, there are an infinite number of sizes of infinity. These are called Aleph-Null, Aleph-One, Aleph-Two, etc, (and there are also sizes of infinity that fall outside this series). There are Aleph-Null natural numbers, Aleph-Null primes, Aleph-Null perfect squares, Aleph-Null fractions/rational numbers. There are more than Aleph-Null real numbers though (real numbers include irrational numbers like pi or the square root of two that cannot be expressed as a fraction).

In the 19th century, Georg Cantor showed how to compare the sizes of sets, including infinite sets, by putting the elements of the sets into one-to-one correspondence. Here you can think of a set as just a collection of objects, e.g. numbers. For example, in the finite case, you can put the elements of the sets {1,2,3} and {4,5,6} into 1-to-1 correspondence, or pair them off, showing they are the same size: 1<->4, 2<->5, 3<->6. You might say you could just count the elements in both sets and see that they are the same size, but the action of counting is in fact putting the objects being counted into one-to-one correspondence with the counting numbers, if you think about it.

One can think of finite examples where one can prove two large sets are the same size without having to count them. E.g., at a mass wedding of Moonie couples, if there are no men and women left over in the stadium then we know there are an equal number of men and women i.e. the men and women have been put into a (romantic) one-to-one correspondence, so the set of men is the same size as the set of women, but we don't know the size.

In the infinite case, we can't count the elements of the sets but, like with the Moonies, if an infinite set can be put into one-to-one correspondence with the infinite set of counting numbers or natural numbers, namely {1,2,3,4,5,6,7, etc.} then we consider it the same "size" as the set of natural numbers and we say that infinite set is "countable". Note that strict subsets of the natural numbers can be as "big" as the set of natural numbers. e.g. the set of even numbers {2,4,6,8,10, etc.} is missing half the natural numbers but is countable. To see this, you need to put the even numbers in one-to-one correspondence with all the natural numbers. This is easily done: 1<->2, 2<->4, 3<->6, 4<->8, 5<->10 etc. It can be proved that every even number will be in this list (just divide it by 2 to find its partner). Therefore, the set of even numbers is the same size as the set of natural numbers and so is countable. 

You provide two infinite sets of numbers in your question. The set of primes was proved infinite by Euclid, and the normal numbers or composite numbers are simply made by multiplying the primes together and so the set of composite numbers is also infinite. The set of primes and the set of composites are subsets of the natural numbers. The natural numbers have a natural ordering, namely, 1<2 1-to-1="" 1="" 2="" a="" also="" and="" any="" are="" argument="" as="" be="" biggest="" both="" by="" can="" composite="" composites="" correspondence="" countable.="" countable="" each="" etc.="" for="" idea="" in="" into="" is="" it="" makes="" match="" matching="" natural="" nbsp="" next="" number="" numbers:="" numbers="" of="" other.="" p="" prime="" primes="" put="" same="" sense="" set="" size="" smallest="" so="" subset.="" subset="" taking="" the="" then="" therefore="" used="" we="" with="">

Cantor also proved that the set of real numbers (all the finite and infinite decimal numbers e.g. 2, 4.5, PI, sqrt(2), etc.) cannot be put into one-to-one correspondence with the natural numbers and is therefore uncountable. (see: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument). So, in some sense the infinity of real numbers is "bigger" than the infinity of natural numbers. This answers one’s other question: there are many degrees of infinity.

The confusing part is this. If I have a bag of all the natural numbers and take natural numbers out one at a time in the order 1,2,3,4,5, etc., then whatever natural number you may come up with, however big, I will eventually draw it from the bag given enough time. The point is that the same is not true of the real numbers. If you have a bag of all the real numbers and take real numbers out one at a time, in whatever order, I can build a real number (in fact many real numbers) as you do this which you will probably never draw out of the bag. This is shown by Cantor's diagonal argument also used by extensible by Cheng.

The question then is: who cares if there are different size infinities? Well, the set of real numbers, which Cantor showed to be uncountable, is required for calculus which is crucial to physics, engineering, applied maths, etc. And in fact, some mathematicians are sceptical of these ideas about infinity and try to prove calculus theories using only finite maths.

Also, Cantor considered the obvious question: is there an intermediate infinity strictly between the infinity of the natural numbers and the infinity of the real numbers in size? Cantor conjectured that there was no such intermediate infinity and this is known as the Continuum Hypothesis or CH. Cantor apparently went insane trying but failing to prove it. In the 20th century, Gödel and Cohen showed that CH can neither be proved nor disproved from the accepted foundations of mathematics, which explains why Cantor couldn't prove it - it's not provable nor is it disprovable! So, CH is an example of a mathematical statement which is undecidable, and is part of the crisis in the foundations of mathematics. 

This is one of the top experts in the field, Hugh Woodin, explaining the current state of play in this crisis:

In an ironic faint echo of Cantor's descent into insanity over a century ago, in this talk Woodin says that 15 years earlier he believed CH was false, and now he believes it true!

How bad is this mess in Maths? Well, to prove that arithmetic (+,-,x,/) is consistent i.e. that you won't end up with contradictions, you need to first build a bigger system than arithmetic in which you can prove the consistency of arithmetic. But you then can't prove the consistency of the bigger system without building an even bigger system, and so on. So, it's theoretically possible that a primary school child doing their sums could end up proving 0=1, which is clearly a contradiction. No mathematician believes this will happen, but no-one can prove it won't currently. The video shows the lengths mathematicians have gone to try to resolve this issue.

Cheng's book is what you need to study first before you can get an idea of what the video is about, although the talk is considered as aimed at the layperson.

It's when Cheng started tackling Calculus and the infinitesimal small that things started going downhill.  What happened to Newton and Leibniz???

Newton Made Easy: "Newton's Principia for the Common Reader" by Subrahmanijan Chandrasekhar

As a math and physics graduate back in the day, I applaud some of the Physics Professors choices when it comes to choosing the best books in Physics, and I also decry a lot of the works on that supposed imaginary list as being, in the grand scheme of things, quite trivial. I too would have assumed that importance and even profundity - if I dare use such a potent word - would carry some merit for non-fiction works, but, alas, i was quite mistaken it seems. To try to be fair though, as I said elsewhere on this blog, I think that the main problem for the arts and humanities mob is maths. As in their cluelessness about it. It completely underpins the natural sciences, and has to be mastered to at least some extent. Newton's Principia is virtually gibberish to even highly trained modern readers, even when it's not in Latin, which is why I would recommend Newton's Principia: For the Common Reader. This is an abridged and tastefully modernised version by the great Indian astrophysicist Subrahmanijan Chandrasekhar. As a tasteful modernisation should do, it preserves the spirit of the original almost entirely. Unfortunately, in general terms though, to understand 'decent' books on physics and/or mathematics, you have to learn a fair amount of, well....physics and mathematics.

Think of Darwin for instance: since his time, biology has become almost as mathematical as physics, which incidentally gives the lie to a remark once made by Kant (*) that 'there will never be a Newton of the grass blade.' In all probability, Darwin would not be able to understand many mathematical biology books written in the last 50 years, whereas i suspect that Newton, with some effort and a few rapid shifts in certain aspects of his world view, would probably be able to cope with everything in physics up to the early 1900's, before slowing down to digest relativity and quantum theory. Only slowing down though, and that mostly to learn the relevant 18th, 19th and 20th Century maths that postdated him:)

This then, or so it seems, is the peculiar difficulty with mathematics: I know nothing technical about music whatsoever, but I can appreciate the glory that is, for example, Bach without knowing how to read a note of music. But to appreciate calculus, and all that flows from it (**), I must learn calculus, and other mathematics besides. Which, I am always being told, is beyond most people apparently.   I still think that people should make more of an effort though. If I can read, for instance, Milton's  “Paradise Lost” (several times over the years), then the arts mob can be expected to cope with C. P. Snow, or Koestler's “The Sleepwalkers” can't they?

NB:

(*) Though i certainly think that the Critique of Pure Reason should be on the list somewhere.

(**) Rutherford, the discoverer of the structure of the atom, was fond of saying that 'there are two kinds of science: physics and stamp-collecting', but he was put in his place by the world's then leading mathematician, David Hilbert, who, on hearing of this, replied: 'physics is too difficult to leave to physicists; it can only be done properly by mathematicians.'

Physics and Computer Science for Laymen: "The Emperor's New Mind" by Roger Penrose

(My own copy)

Penrose certainly has a generous idea of his readers' mathematical ability. It's a kind of running joke among Penrose-fans: he always starts his books by saying you'll find it tough going if you haven't got a 12th Year (in Portugal)/GCSE (in the UK) in math, but that he'll explain it as he goes if you haven't. Twenty pages later you're on Gödel and conformable geometry. He doesn't do it deliberately; he really does believe his books are popular science. How can you not love him? I purchased an on-line kindle edition of this book back last year via Amazon and it was more about bringing myself up to date (I read it for the first time in 1991 when the book came out), although such things are never truly current due to Theories being debated and tested for very many years within Scientific Realms. Roger Penrose's books are as stated often inclusive of more mathematical devises than many books aimed at more laymen realms, so I often regard them as perhaps Bridging that gap between Solid Science Headaches and Laymen 'I read an article and am a common law know-all expert'. What to say about this re-read? What I liked within this particular tomb over multiple other works and writers from his genre was that Penrose took the opportunity to use more analogy & metaphor in his descriptions of the function of the ideas. So, you read the math and then you might see an image or descriptive wording of the comparisons. Overall as I have stated above, this is potentially overwhelming for your Mr/s-Average-Layman and Not-Hard-Core-Enough-for-Scientists. Though for anyone looking for a kind of Half-Way-House bridging zone, this book may well serve and be fit for the purpose.

Anyone read Raymond Kurzweil's "The Singularity is Near"? It discusses this exact same stuff in more detail. Personally, I think it's a little more optimistic about strong AI than reality warrants. Penrose has interesting ideas about quantum consciousness, but I don't see humanity making any major gains on "the hard problem of consciousness" any time soon. It's like the holy grail of several disciplines. In reading about an electromagnetic theory of consciousness, a good point was brought up - if the brain develops its own field, then contemporary traditional computer engineering will never achieve the goal of true AI, simply because now actively creating shielding against short-circuits won't allow any field to develop. The closest I've seen to understanding consciousness seems to be the promising ideas about quantum theories of consciousness. We have yet to develop computers that can grow their own neural connections, much less create consciousness. It's been proven that memory doesn't reside in any one neuron - you can't create a lesion which excises a given memory, for instance. There's got to be some kind of unknown field. Until we understand the toughest riddle of all - consciousness - we will never develop true AI. Nor, I might add, will we be able to do what Kurzweil suggests - "uploading the mind". Bullshit. The day we can achieve either of those things we will also be able to create the ability to travel at the speed of light, but transmitting our consciousness, with perhaps genetic meta-data to reconstruct the body nanoscopically at the other end. It's the stuff of SF, to be sure. 

(Bought in 1991)

One only has to refer to Alan Turing's famous "Can Machine's Think?" to understand the singularity is total nonsense.  Machines, hardware are nothing w/o a program to function it, which is nothing more than a theory.  When one compares the structure of our ability to speak (our language) with that of programming, it's not even close to being in the same field of discussion.  Programming languages follow a somewhat commonsensical pattern, because we're the ones manipulating it. When one looks at the core structure of our language, it's so remote from common sense notions that it is not for the ear, but more soothing for the mind.  It's striking the deeper we look into out cognitive faculties, the more it diverts from our presumptions.  Man's fascination with machines and their impact began to fully realize itself with Galileo and Descartes, only to get smothered rather abruptly forever (apparently not) by Newton not too long after.  We should pay much more attention to history, most certainly politically but scientifically as well. 

NB: It's worth pointing out that towards the end of his life Hawking gained an interest in what is known as model dependent reality...which actually totally negates any notion of a "theory of everything". So, Hawking's book "The Grand Design" actually contradicts the very idea that a theory of everything is even possible. Hawking also worked with Professor Paul Davies on ideas about reverse causality....a very fringe and not widely accepted area of speculation. All of which shows he was open to new ideas.