1

General Science / Re: Does All Of Mathematics Come Down To Simple Arithmetic?

« on: Yesterday at 22:52:38 »

Hi.

(I started this several times and deleted it.   I finally decided I might as well post something, convsersation has been very light at home lately.   It was supposed to be short but then it got late and I lost my way, sorry).

   One odd thing is that @Jimbee  seems to make an opening post and then never makes another post on the same thread.    (I've checked the most recent 10 posts listed on your profile, @Jimbee ).     You are most welcome to join in the discussion at any time, it's a forum.    Otherwise, a few of us will just use the space to discuss whatever topics arise no matter how tangential to the OP they have become.   That's also OK but if you want something specific discussed or more info on anything that has been raised then you can say so.

     I'm not going to start anything new.   I will just comment on some bits that have already appeared.

Someone on another message board once said, that all of science is reducible down to math.

      In the UK we condense the word Mathematics to Maths instead of Math.   It may look silly when you analyse it, since it would seem that Maths is a plural form and implies there are many different Maths you could use.   Well there probably is and at the same time it's all something with a lot in common, as implied by others ( @varsigma 's post is most closely related).   
      A lot of people who have only studied the compulsory bit of Maths at school may think that Maths is just arithmetic and so more advanced Mathematics must just be arithmetic but the sums are longer.   Somewhere at the top end of school or early University you may learn and accept that "the Mathematics" wasn't about how numbers combine.   They were just an example of something with a suitable algebraic structure.   A bit further along in University you may study set theory and learn or accept that everything you've been doing in Mathematics is just set theory and all mathematical objects are just sets.    ( @Bored chemist  was the first one to imply set theory is important by mentioning the 1+1=2 proof  and then  @evan_au mentioned Peano's axioms directly in his post soon afterwards,  which you can Google if you wanted more information).   You will probably be both surprised (or disappointed?) to realise that everything you've done was based on something so simple but also quite pleased that at least now you know what it is you've been doing and working with.   You'll be fairly happy knowing that in Mathematics "sets" are the equivalent of "atoms" in Physics,  they are the most fundamental object and everything else is just built up from them.
      Near the top end of University you may learn something else.   The modern thinking is that sets and set theory aren't the only place you could start from.   For example, we can obtain all the results of an advanced topic in Mathematics like "representation theory" starting from "set theory".   However we can also obtain set theory from representation theory.   Therefore, it is quite arbitrary whether you consider sets (and set theory) or representations (and representation theory) as your most fundamental or atomic objects from which all the rest of Mathematics can be built.
     Indeed, there may be many places that you could start from and consider as your most fundamental objects.  Some popular choices inlcude starting from geometry,   or from the notion that mathematics can be built up from a set of fudamental or atomic algorithms.
     Anyway, why does this matter?   Well, it means that you probably could start from the Natural numbers and arithmetic.   On the face of it that seems to offer some good algebraic structure to begin with and you probably CAN build all the rest of Mathematics from this.    We seem to have turned a full circle, the notion people may have had at school - that all Mathematics is just arithmetic - may actually be right.

    It's probably time to address some of the other posts that have been made:

Not really, once you get into calculus you need to understand the concepts of limits and functions.

    It would take a lot of lines to define or construct limits in terms of sets but functions we can do very quickly.

You may think:        f  is a function.    This means it's a rule taking a number to another number.   For example,  the rule might be to double the number and we can write   f(1) =2,     f(2) = 4,      ... f(6)=12,....   and in general   f(x) = 2x.
It's important to note that f isn't the image you obtain,  f is "the rule" that maps a number to its image,  we have some notation that really attempts to make this clear and instead of defining the function as f(x)=2x we can write
   f :   x  --> 2x

So how is this mathematical object, a function, actually a set?
    f is this set,   
f =  {  (1,2),   (2,4),  ...... ,  (6,12), ...... }
in words it is the set of ordered pairs of numbers  like   (1,2)   and  (2,4)   where the first ordinate is a number in the domain of f and the second ordinate is the corresponding image under f.

Where we write    f(1)=2      that has just been short-hand for writing     (1,2) ∈ f         
  (in words,   (1,2) is a member of f ).

Note that an ordered pair  like  (a,b)  can be written as a simple set.
(for brevity, I'll just provide a reference:    https://en.wikipedia.org/wiki/Ordered_pair#Kuratowski's_definition   - there are actually many ways to define an ordered pair but Kuratowski's definition is widely used).

Anyway, the point is that f is a set and it's elements are just sets.   It's all sets.   Notation like f(x) = 2x has always just been shorthand for the set theoretic expression    ∀x( (x,2x) ∈ f)   although it's easier to read if we just write out the set f this way,
   f =   {  (x,2x)   :  x ∈ Z }

    The keen eyed may notice that there is still something in these sets, e.g. the ordered pair (a,b) has the element {a} in it.  This means that    a    must be something,  perhaps we still need the natural numbers to start from.   One of the things that Peano's axioms shows us is that we can construct the Natural numbers just from the empty set,  ∅.   So, it turns that everything can be a set and we need only the most basic of sets, the empty set, as the thing we start from.

    I've taken the time to set out clearly that a function is a set because it helps to make the jump to seeing that every, absolutely every, mathematical object is a set.    Numbers, functions, vector spaces, modules,  whatever mathematical object you can think of....   they're ALL just sets.
     (.....  well, anyway....   that's some hint of how it feels when you finally study set theory at University...  in modern parlance it's an  " o.m.ds. - we've actually found our atoms" moment).

    I'm not going try and get from Natural numbers and arithmetic back to sets and set theory, this has probably been long and boring enough.   I'll just repeat what was said above - you probably could start from Natural numbers and arithmetic and take those as your atoms.

    Of course what @paul cotter  actually said was the following....

 

....  you need to understand the concepts....   

      This is regarding Mathematics as something a human being does, treating "Mathematics" as a verb rather than as a noun.  If we treat it as a noun, then Limits and functions will exist irrespective of whether some human being understands the concepts.  There is a human need to understand some concepts that don't seem to be like arithmetic because we are not computers and any attempt to set this out as arithmetic could take years.
    By the way, considering Mathematics as a verb is very good way to consider what Mathematics is.   Unlike a real scientist, mathematicians do very little.  There is nothing new they can find, like a new trans-uranic element, all of Mathematics is already there and could have been seen.  A mathematician is just someone who can demonstrate that one thing is the same as another thing in very small steps that another human being can follow.  A smarter being would have just seen it and made the jump in one go.   The common phrase is that God would be a poor Mathematician because he has no need for it.

... and the square root function doesn't break down into addition and subtraction....

Taylor series or Power Series.   The square root function is just a sum of terms,  each term involves only integer powers - which are clearly just based on (an integer number of) multiplications ....  multiplication being based on addition.   It's all just addition and subtraction.

Similarly, you could write code for square roots,.....

    Indeed, for any analytical function because even if nothing easier presents itself, you'd still always have power series.

The familiar "Commutative" property of addition, subtraction and multiplication is a result of the integers forming a mathematical "Field".

    Was it late when you wrote that?   
(i)   Subtraction is not commutative.   5 -1   ≠   1 - 5.
(ii)   The integers do not form a field.   A requirement for a field is that multiplicative inverses exist for every non-zero element.    An integer like 5 has the fraction 1/5 as a multiplicative inverse but sadly this fraction  isn't in the integers.  The Rationals would form a field, the integers won't.   This is important because the integers should be recognised as being "incomplete".   It's an example of how human beings did manage to develop more advanced and idealised algebraic structures even if they started from a place that was littered with imperfect and incomplete structure.
(iii)  You imply cause and effect where none exists.  "A field" is just the name we give to this sort of structure not any explanation for it.   "Light travels at the speed of light because it's a luxon" offers no explanation for why light travels at this speed, the term "luxon" is just the name we give to a particle travelling at that speed.
    Seriously, @evan_au, you write very well and spend the time to put in references and hyperlinks.  Most of the time they're the best and most easily read and understood things in every thread.   People may read your posts, so in the future you just can't post if it gets too late.
     

The proof that 1+1=2 can not be simple arithmetic, because that would be circular reasoning.

    Circular isn't necessarily wrong.   A proof can be one line if the statement you want to prove is an axiom.

    As breifly hinted at above,  Mathematics may be very circular.  There may be many places you could start from to build up all the rest of Mathematics.

Is it meaningless?
    Now, I wouldn't want anyone to think that Mathematics must be junk because anything based on a circular argument is usually junk.   Quite the contrary, it may be one of the strengths of Mathematics and provide some assurance that it is something worth studying.
     No matter what you start from and what you consider as your fundamental objects, you'll probably end up being able to re-create the entirity of Mathematics.   So, for example, aliens on planet A who are more interested in some geometry that appears in their environment instead of sets should still end up with the same complete works of Mathematics that we have.   Other aliens on planet B may be completely blind, they can't see any shapes, their main sense might be feeling vibrations (like a spider on a web).  Studying the harmonics that can exist on a string will probably also allow them to develop the entirity of Mathematics, even though the natural number system they learn in school may seem very alien to us.  For example a string vibrating with the first harmonic could be driven again at the same frequency and all that happens is that the amplitude gets bigger, the frequency doesn't change.   So, for them, their logical number-like system is    1 + 1 =   1    If we said to them 1 + 1 = 2 they might think we were idiots who never went to school.   However, their Mathematicians will realise that we are just describing a slightly different structure:   If you join two strings together and each was vibrating with it's 1st harmonic, then you can end up with a new string (of twice the length) and that would be vibrating with its 2nd harmonic frequency,  so they will understand that we are just using a different algebraic structure.   Anyway, hopefully you get the point...  all algebraic structures and all of mathematics could be re-created by studying the vibrations on a string.

    Of course, this tacitly assumes that several things in nature do follow or have some structure which we may call an algebraic structure.   Well.... it seems that it does but that's a discussion the scientists can have in another thread if they wish.  Historically, mathematics was rooted in some stuff that seemed to appear in nature.   Later on it's possible to just propose some new axiom system that may have no basis in nature or reason to assume its truth and still declare that you are perfoming or doing some Mathematics.   However, I would still argue that this is justified by some structure existing or being exhibited in Nature.  Specifically, we can study an arbitrary structure and axiom system because we can argue that there is some Mathematical system that does seem to hold in Nature, so it's sensible to conisder "what if" scenarios..... what if there were other axioms and other structures? 

It's too late and too long already.  I'm signing off.

Best Wishes.

2

General Science / Re: Is There A Pattern In The Number Pi?

« on: 04/10/2024 15:12:17 »

Hi.

      I quite like everything that's been said already.   However, it's just faster and more natural to start with some minor arguments before going on with anything else.

     

I understand that there is a proof of no possible pattern.

      Possibly - but a lot of texts on the subject would just consider "a pattern" as a simple repeating pattern of digits in the decimal expansion.    There certainly isn't one of those.      @Jimbee 's question is harder because they allow more complicated things to still be considered as "some pattern".
    This leads to the comments made by @evan_au   --->   There certainly are some rules or algorithms (e.g. those based on summing an infinite series) that you could use to generate as many decimal digits for Pi as you require.   The most generalised notion of there being some pattern or rule for the digits would just be that a suitable algorithm to generate the digits does exist.

   @evan_au ,  you make this point well but then switch back to discussing "patterns" as if they could only mean a simple pattern of repeating digits.   

- 1/1000001 is zero in the first 6 decimal places
- So any pattern that you might detect in the first 6 decimal places will be destroyed by adding in this factor which does not affect the first 6 decimal places.

     An algorithm exists to determine an arbitrary number of digits.   So "some rule" exists to determine the next N digits given the first N digits,  even if all I do with the first N digits is just count them so that I know I will need the N+1 through to 2N  digits calculated.

- - - - - - - - - -

I think it's time to return to the first line of the OP and go along with the spirit of @alancalverd 's first reply.

Why does it seem so hard to calculate the value Pi?

       It's only because the language and development of mathematics has brought your attention to it and given some cause to think that we may want to express these sorts of ratios in a certain way.
       As suggested by @alancalverd , we could use a system where Pi seemed perfectly natural and could even be considered as just one unit and written as  1.    However, that would make some other numbers seem strange or "irrational", so we'd just be moving the problem somewhere else.
       The only thing we can't do is make ALL numbers seem perfectly rational.   The ancient Greek mathematicians and philosophers were very worried about this, so you're in good company if you feel that all numbers should be rational or well behaved.   Indeed it turns out that, regardless of what you take as your number 1, when you develop this into a complete number system there will always end up being vastly more numbers that are "irrational", transcendental or badly behaved.
        This is worth making clear:   There are vastly more irratonal numbers than rational ones.   It is a miracle that we can do so much while confining ourselves to just some numbers that seem natural - but we can.     If you dropped a pin into the Real number line and picked a number "at random" the chance of it being a well behaved rational number is zero (or "infinitessimally small" but it would need another whole forum post to discuss whether an infinitessimal is precisely equal to zero).   The truth is that most real numbers will be un-natural slippery little scallywags.   Almost everything you see - some length, some weight (etc.) will be an irrational number of units in some system that anybody else may choose to measure those things in   (e.g.  metres, inches or "hands" etc.).  So these things would always require an infinite and non-repeating set of digits to write it in a decimal form for that other person.
     The good news is that, regardless of what quantity you start from and consider as one unit and write as just 1,  an identical structure emerges and behaves as the number system we are familiar with and were taught about in school.   Thus, we have no difficulty in recognising that when we add 1 sheep to a field that had 2 sheep in it, we have a field with 3 sheep in it.    Similarly, if we add 1 cow to a field of 2 cows, we have 3 cows in the field.   This holds perfectly well even though     1 sheep     is clearly not the same as     1 cow.    It doesn't matter, the mathematical structure is the same and will  behave the same whether your unit is the cow or the sheep.   However, it does matter if we need to make an exchange, attempt to make some comparison or find a ratio between these two systems.   If we went to market and wished to barter and swap our cows for someone else's sheep then the chances are that 1 cow is actually worth an irrational number of sheep.   I'm going to guess a cow would provide a farmer with 3 times as much meat as a sheep but it's a unlikely to be exactly 3 times.   As discussed earlier (dropping a pin into the real number line) the cow-to-sheep ratio is never going to be a perfectly rational number.  So, in principle, we could never make a fair exchange of some sheep for any number of cows - beacuse no natural multiple of cows would ever be exactly equal to some other natural multiple of sheep.   Problems with the ratio of two quantities being badly behaved (irrational) are to be found everywhere and we do encounter them ALL the time.   Human beings are just very good at making approximations and getting a working solution.   In the history of man, we have made an exchange of some sheep for some cows.   The absolute impossibility of any fair exchange existing just went un-noticed.     

Best Wishes.

3

Technology / Re: After reading this, would you let AI drive your car?

« on: 26/09/2024 13:10:23 »

I?ve always been a bit skeptical about AI taking over driving, especially with cars that I actually care about.

       That's quite an amusing one-liner, thank you.   The other road users and pedestrians don't matter as much.

4

New Theories / Re: Could quantum mechanics be wrong?

« on: 08/09/2024 20:13:46 »

Hi.

[ES said]  So the problem starts with the way the question was phrased.
[Hamdani replied]  Can you suggest a better way?

     Obviously it's up to you to create your own question.   You can ask whatever you want.

      However, Gemini will be lead by the question you asked.  It rarely considers that the user has some faults or misconceptions (other than just spelling errors) in the question they ask.

     I am also going to be lead by the question but I am able to recognise that the question involves a classical phenomena while you seek an explanation involving quantum or discrete phenomena.
So the question is a special case of a more general question:
    Can Quantum Mechanics recreate or explain ALL classical phenomena?
    --->  I don't know and possibly we (physicists) haven't tested them all and nobody knows  BUT  it seems that it does recreate a great many of them very sucessfully.   Since the early development of QM one standard "test" or criteria for establishing the reasonableness of any new piece of quantum theory has been that it will reduce to a well established classical theory under suitable conditions  (for example, in the limit  ħ --> 0 ).
[Reference:  See https://en.wikipedia.org/wiki/Correspondence_principle  for some more discussion ].   It (Quantum Mechanics) also models or predicts some things which classical physics would not adequately model or explain.

How does it explain electrostatic and magnetostatic fields?

         With various pieces of mathematics which I would struggle to explain here in a short amount of time.    Typically you build (or propose) a Lagrangian that you believe describes the system.  There are some commonly used models that fit with observations well, so you don't just have to dream up a new Lagrangian every time.   It is then common to apply what is often called "Propagator Theory" and identify a suitable "propagator".

- - - - - - - -
An extract of some notes from the Physics module "Theoretical Elementary Particle Physics" by Nottingham University, United Kingdom.

Propagator theory.jpg (118.58 kB . 868x539 - viewed 42 times)

- - - - - - - - -
    It is possible to construct Feynman rules and suitable Feynman diagrams from these expressions.   With these sorts of diagrams, it is common to imagine that a pair of interacting particles (say two electrons with the usual negative charge) would be seen to move along for a short while and then exchange a virtual photon and end up moving slightly differently.   That is understood as the electrostatic repulsion between the electrons.    However, these Feynman diagrams are really just for convenience and provide a simple way to "visualise" and describe what is happening.   It is just the mathematics, the evolution of the wave function(s) we would obtain as an approximate solution using propagator theory that is really being shown in those diagrams.   The "virtual particles" are not something concrete that we could actually find or detect.

Let's paraphrase or summarise this again:  So how does this QM theory explain electrostatic interactions?   Well, it doesn't even attempt to "explain" these interactions.  It just provides an abiltiy to calculate the results.   It (the theory) just shows how the wave functions should evolve.   We (human beings) add a layer of interpretation on the top of this, we use terminology like "virtual particles" and build a useable and more intuitive way of thinking about or "explaining" the interaction to our satisfaction.

Concerning the double slit experiment with a single photon,  Hamdani said:  You seem to interpret one glowing spot on the screen as an event of a single photon being detected. Have you considered some alternative interpretations or explanations?

     Yes and it is very much as you described in a later post.

In our previous paper1 we pointed out that, strictly speak-
ing, we are not detecting single photons of light but rather
single photoelectrons liberated by the light impinging on the
detector; this is still true in the present experiment.
Furthermore, the detection of a photoelectron does not neces-
sarily imply that a single photon arrived.

    I don't know if we can ever be sure that we will detect a single photon in the sense of a small packet of light hitting only one spot.   No matter how much we think that only a single particle of light has been created and sent into the apparatus, it could still be a spread out wave of stuff that actually arrives at the detector screen.   Due to some properties of the screen all the energy is collected and deposited into just one small bit of the screen so that only that bit of that screen will glow.   
     It's interesting.  Is there a particle of light to be found and detected or is that just an object of convenience?  I don't know.  It's not necessarily important.   Simple Quantum Mechanics would quite like to have a particle of light, a photon, but provided the end result is an interaction that looks like a single particle interaction then "a photon" could just be an object of mathematical convenience.      With just a minor change in semantics, the popular phrase about light would still apply and the same Mechanics can be used to predict results:  "it travels as a wave and interacts as a particle".

...By the name only,...

     I didn't really understand what you meant here.    QED  ,  QCD,  QFT   and what I have tended to call simple QM differ by more than just their names.    For example, Simple QM is based on the original Schrodinger Equation which is non-relativistic in nature.   QFT is a relativistic quantum theory and was motivated by alternatives to the Schrodinger equation such as the Klein-Gordon and Dirac equations.

Best Wishes.

5

New Theories / Re: Could quantum mechanics be wrong?

« on: 08/09/2024 02:49:37 »

Hi.

Can quantum mechanics explain the generation, propagation, and reception of radio waves?

Short Answer:
   Probably not in an easy way for human beings to understand. 
   It probably is possible to use QM and apply a suitable interpretation of probabilities or the frequency of detection of photons to re-create the usual gross-scale classical radio wave propagation results - but this would be complicated.
   At the fundamental level, in most QM models there aren't any "radio waves".   So the problem starts with the way the question was phrased.

Medium detail / further discussion:
     The name "radio wave" that you are using in the question describes a phenomena that we (human beings) have come to ascribe some meaning to.   This thing,  "a radio wave", is an essentially classical physical phenomena.
    We imagine a radio wave as a ...  (you know) .....  spread out and continuous wave thing.  We expect to be able to measure an Electric and a Magnetic field and a radio wave will be a suitable oscillation in these fields.  There are many versions of Quantum Mechanics but most of them don't have that sort of thing as their subjects or objects of interest.   The simplest QM models consider all objects as if they are particles.   For example, we might consider a particle of light, a photon, as a suitable QM object but we do not consider an e-m wave as a suitable QM object.    QM objects just happen to have a wave function associated to them BUT they are essentially discrete particles and would always be detected as a discrete particle.
      More advanced QM theories, such as the mainstream versions of QFT, will consider the most fundamental objects to be fields.  For example, there will be a photon field that exists everywhere in space.   The thing we call a photon would not be treated as a fundamental QM object, it is just how we may interpret a certain oscillation in that underlying photon field at some place.    So, in QFT, there is a photon field but there is NOT any fundamental Electric field or Magnetic field.   These things, the electric and magnetic fields are just gross-scale fields that can emerge from the more fundamental fields of the particles in the standard model of particle physics.   QFT doesn't aim to deal with oscillations in the Electric and Magnetic fields, it only deals with the fundamental particle fields.   You need to add a layer of interpretation on to the top to translate what happens in these fundamental particle fields into something that can be observed in the gross-scale Electric and Magnetic fields.   That can be done in many versions of QFT but is rarely simple.
    When you are asking about "a radio wave", you have moved attention to something in terms of the ordinary (or gross-scale) Electric and Magnetic fields and not on any fundamental object (a photon in simple QM, or the photon field in QFT).   So the question cannot be answered without translating the results of QM back into something classical involving the Electric and Magnetic fields.   One of the simplest or most naive ways of making this translation is to just analyse the probablities or frequencies with which a QM object would be found at some place and time.   So, we might imagine that an e-m wave is a collection of many photons or that an experiment involving one photon has been done many times and the e-m wave we observe is the combined average of all of these.

Example:    If you could fire just one photon through the usual double-slit experiment apparatus,   then you probably won't get an interference pattern on the screen at the end.   Instead we think you'll get just one spot to glow on the screen.   However, if we repeat the experiment many, many times then we expect to see that the places where the screen glows and how often that bit of the screen glows will correspond to the usual interference pattern for a classical wave being put through that apparatus.    So, one individual photon does NOT behave like an e-m wave - but a thousand or a million of them would seem to reproduce classical radio wave-like patterns and behaviour.

NOTE:   I'm really not convinced anyone has actually done a genuine version of Young's double slit experiment where only a single photon is fired with a clear interval of time before the next is fired.   However, this demonstration gets close and claims to be using such a thin and dim stream of light that only "a few" photons per second will be produced.
https://sciencedemonstrations.fas.harvard.edu/presentations/single-photon-interference


   So, the important point is simply that "a radio wave" isn't the normal subject for any QM model.  It demands some form of interpretation to put the results predicted by any QM model back into terms which would make sense when you consider "a radio wave".   
   If we wished to be controversial we would say "there are NO radio waves... just QM objects that get detected at certain times and places.  On a gross scale, that looks like a radio wave is there".    ... However... I'm not that controversial or convinced that a simple QM model is some ultimate truth....  there are many versions of QM, all have some use and classical e-m theory is also extremely useful.   

Another minor note:    Your (Hamdani) extracts produced by Gemini seem to be blending many different versions of Quantum Mechanics.    For example,  QFT is a specialised theory and there are still about 3 versions or flavours of QFT that are in current use.   I don't think Geminii is reliably identifying where things can and cannot be combined.   For example, the simplest QM models are not like QFT.   They will involve the original Schrodinger equation, which is shown here:



That has a mass term, m, in the Hamiltonian for the QM object under consideration    (it's in the square bracket, h squared over 2 m).   This means it's actually useless for photons which should have 0 rest mass.   None the less, we (physicists) do generalise a lot of results from such a QM theory for massive particles to massless particles like photons.  For example, the Youngs double slit experiment can be modelled for electrons or even bigger particles like carbon bucky-balls.  We can (and have) then done experiments with these things and obtained a good match with the predictions from the simple QM model.   It is natural to assume a similar thing for photons, even though the zero mass term would technically make the simple QM model invalid.   If you read what I've said (above) about firing single photons then you'll hopefully notice that I've got phrases like "we think".   
     I believe that how you reconcile a classical radio wave with a suitable QM object ("a photon") is subject to a lot of interpretation. 

[Disclaimer:     I'M NOT AN EXPERT.  ]

Best Wishes.

6

General Science / Re: what is Nothing or ?energy landscape.? or waves

« on: 11/07/2024 20:17:47 »

Hi.

    I wasn't at the same Solid State course, so I don't really know what @varsigma was suggesting.
However, I'm suspicious that you may be wrong @alancavlerd but in a nice way and you have tried to make a reply that is still reasonably sensible.

      When considering light travelling into some media from some other media,   it is common to model the e-m wave by considering the E field component only.    By Maxwells equations, the B field would just be a scalar multiple (with orthogonal direction) anyway.   So we need only consider the E field.

    So we  have,   E field at the position z (a 1-dimensional position co-ordinate measured along the direction of travel) and the time t   =

     
      E (z,t)  =   A.e^{i(kz - ωt)}                                                      

[EQUATION 1]

     

Where, as usual, k can be identified as the wave number for a component of the e-m radiation that we're interested in.    The whole collection of e-m radiation will be expressable as a sum of several different rays with different k numbers.

     The wave equations are linear, as @varsigma seemed to be saying, so that multiples of solutions are solutions and sums of solutions are also still solutions.    Hence, rather than considering oscillations of the form  Cos(ωt) or  Sin(ωt)  we can consider solutions of the form   Cos(ωt) + i.Sin(ωt).       With Euler's formula  that means we need only consider solutions of the form  A.e^{i(kz - ωt)}   and the complete solution will just be a sum of a few of these rays with some different k wave numbers     (that is to say,   W.L.O.G. all fundamental solutions can be treated and described in the form of [EQUATION 1] as appears above ).

     Since we can have the A (amplitude) term arbitrary,   there's nothing very special that would happen if that was allowed to be a complex number, it always could be.
 
Alan said:

giving A an imaginary component allows for, say, reflection (P positive) or transmission (P negative)

    Which may be how things are done in some text you read but in the texts I've seen, a reflected ray wouldn't be marked out just by having an imaginary amplitude only.   Instead a reflected ray will have a wave number, k, that is precisely the negative of the wave number k for a ray travelling in the forward direction.   That, I hope you will see, makes sense,  the wave number k always does tell you about the direction of travel of the ray.   
    The amplitude being real or imaginary need not relate to anything physically observable,  since as you pointed out it is generally only the square of this Amplitude  (the Intensity,  I = |A|² )  that we could see anyway.

  However, if we allow the wave number, k, to be imaginary then we do get something useful.

    Assume k is complex and write it as the sum of real and imaginary parts as usual,      k =  K_real  +  i. K_imag.   Then we see that the net effect in  [EQUATION 1] above from a non-zero imaginary part for k,    K_imag >0,   is to convert the amplitude  A  into something of the form   A.e^-k_im.z      ~    exponential fall off in Amplitude with distance z along the direction of travel.

    This is a fair model of how light may penetrate another media.   For example, if the new media is perfectly transmissive then k_im = 0 and there is no exponential fall off in Amplitude as the wave travels into it.     Otherwise for media that is (on a macroscopic scale) opaque to this wave number k of light,  this sort of exponential decay of Amplitude is a better approximation to what actually happens and/or what QM predicts.    For example,  the solution of the Quantum Mechanical "particle in a square well potential" shows that the wave function does penetrate the region outside the well, a classically forbidden region, but this wave function falls off exponentially.

      Permitting a complex valued wave number k, allows various situations to be modelled.    Since we can express the wave number k in terms of a refractive index, N, for the media,  this is equivalent to using or permitting a complex valued index of refraction.

    It is thus, the use of complex numbers for the wave numbers { k } representing the collection of e-m rays that I suspect @varsigma was talking about and not so much the use of complex numbers just for the Amplitude.

 ------
The rest of what you ( @alancalverd) said looks OK.

------

NOTE:    The above was fairly well established scientific information and not just something out of my head.  It may not match exactly what @varsigma or @alancalverd was talking about but the information is still factually accurate to the best of my knowledge and belief.    However, the comments below are speculative.  It is just an attempt to engage in discussion on the topic raised by @varsigma and others.

Is a light wave a complex number?

     For most purposes, we could model the situations using only real valued everythings.....  real  K wave numbers, real index of refraction, real Amplitudes etc.      and just manually or directly add in the exponential damping for the Amplitude where a given k wave number cannot ordinarily continue travelling through some media or through some place.    It is just easier to use complex valued wave numbers k so that the exponential damping effect is automatically taken account of.
     However, the whole thing is a bit of a fudge.    There isn't a lot of direct reason to assume a light ray entering a forbidden region (e.g. some media that is opaque to it) would fall off exponentially instead of just abruptly stopping and being completely reflected at the point or place of entry to that forbidden region.   As mentioned, the main motivation is from Quantum Mechanics and the observation that the QM wave function does often enter classically forbidden regions and just falls off exponentially.    However, since that is our main motivation AND then the results we obtain from the mathematical model of light as oscillations in the E and B fields when we do use complex valued wave numbers really do match experimental observation so well,  it is very tempting to think that there is some more fundamental link here.
    For example, maybe all of Maxwells equations and what we describe as the E and B fields  are just a simplified or limiting case of Quantum Mechanics.  So that light is just some QM wave phenomena instead of an oscillation in the classical E and B field.    There are many discussions available which suggests that the use of complex numbers is fundamental or essential in Quantum Mechanics,  wave functions must be complex valued.   As such maybe light actually REALLY IS best considered as some wave phenomena that must be assigned a complex value.   

Best Wishes.

7

General Science / Re: what is Nothing or ?energy landscape.? or waves

« on: 06/07/2024 20:56:32 »

Hi.

You've had some good replies already.   Here's another summary.

Do particles pop in and out of existence?

    Maybe.    However that is the main visual or sensible interpretation for what happens.    It is often easier to imagine or conceptualise what is happening as if particles were poppoing in and out of existance.     You went on to ask -

Very confused now... any answers that can be put into visual terms?

Virtual particles aren't actually particles, apparently.

    That's about right.   They are generally regarded as a useful way of visualising something or providing a simple explanation for some phenomena that do seem to happen.
    Minor Note:   Virtual particles can be used as a mathematical tool for various situations but the easiest one to understand is their use to represent the exchange particles that mediate some of the known forces like the electric repulsion between like charged objects.   That's what I will restrict my dicsussion to:  Virtual particles as force carrying particles.

Example:      Two electrons would repel each other.   In simple terms we recognise that an electrostatic repulsion exists between them because they are particles with the same charge.   We could just leave it there and say - there is an electrostatic repulsion - it is caused or manifested by something that may or may not be anything like some particles being exchanged between those electrons.    However, in most versions of Quantum Field Theory, we are able to model that interaction as if virtual photons were being exchanged bethween them.   These virtual photons are the force carrier particle for electrostatic repulsion.   
     In simple terms, one electron throws a virtual photon at the other electron.
(i)  The throwing process imparts some momentum on the electron it left from.   You may have noticed that if you throw an object away from yourself, e.g. let off a fire extinguisher so that plenty of foam or Carbon Dioxide gas rushes away from you in one direction, then you will experience a push in the other direction.
(ii)  When that virtual photon hits the other electron, that will impart some momentum to that receiving electron.   You may have noticed that if something hits you then you are pushed in the direction of that thing which hit you.

They cannot be detected by experiments, right?

     Correct... although since "virtual particles" can be the force carrying particles for physical forces we certainly can observe, we must infer one of the following:
     (i)   Either the virtual particles were there (because the force has been experienced),
     (ii)  or else, our model of representing certain forces as if virtual particles were being exchanged is just inappropriate,  we should think of how the force is produced in some different way that didn't involve exchanging virtual particles.

So, how do we know that it's correct?

     Within reason we don't.   "Virtual particles" can be just a name we give something.    For example, electrostatic forces do seem exist between charged objects and we had to call the mechanism for that force exchange something.

    They do not HAVE to be there.   There is at least one formulation of QFT (Quantum Field theory) that does NOT require the use of virtual particles:
Although widely used, they (virtual particles) are by no means a necessary feature of QFT, but rather are mathematical conveniences - as demonstrated by lattice field theory, which avoids using the concept altogether.            

[Quote from  Wikpedia:   https://en.wikipedia.org/wiki/Virtual_particle ]

 

If partials do appear and vanish back to wherever, what replaces the void they occupied?

    I guess you use the term "partials" to mean "virtual particles" or something similar.    This might even be the wrong way of asking this question.
    You seem to think that these virtual paricles did exist and so that when they pop out of existance there should still be something left there, some energy or something which is roughly equivalent to that virtual particle.    Hence the total energy content of some space is always greater or equal to 0 and whatever value it may have been, that is protected and must be preserved at all times.
     Another common way of thinking about virtual particles is that energy has been "borrowed" for a short while.    There was nothing like some mass in a region of space but a virtual paricle can pop into existance by "borrowing" some energy or running up a slight energy deficit somehow.   Hence nothing would need to be left behind when the virtual particle pops out of existance:   It is just the (brief) appearance of the virtual particle that is strange and may seem to violate some conservation of energy laws,   its disappearance is not strange, that is merely restoring things to how they should be.
    Minor Note:     It wasn't clear to me what @Halc was saying about "there needs to be positive energy from somewehre or else they're just transient"....    but overall I think we are saying much the same.     I think the key is that Virtual Particles are transient (short lived) in most applications to QFT, the energy can be thought of as just having been borrowed from "somewhere" or "nowhere" for a short while.

If the universe is just made up of waves, and it's these waves that create the partials, then it's just an oscillation of a wave?

    Yes, that is one possible way of interpreting  QFT  (Quantum Field Theory).

So what's the wave?

      No idea.   As human beings we expect all waves to be waves IN SOMETHING.   They need to be waves in some media,   for example water waves are waves we can create in the medium of water.    Historically, physicists thought that electromagnetic waves were also waves in something and that medium was often called "the ether".   (Just in case you didn't know - "the ether" idea is not current or popular any longer).    It is possible that waves can exist without needing to be waves in some medium.   So there may be nothing tangible or like any sort of substance that we could say is associated with these waves.
     We (human beings) may not have the ability to feel these waves and so it is only of limited value to visualise them as if they are waves in some sheet of fabric.   It's a fabric that may have no substancewe we can feel.
     For comparison, a computer cannot feel a human finger hitting the keyboard.   However the computer is able to detect that somehow a key has been activated.   Indeed if, like me, you sometimes don't press the key hard enough then the computer is unaware that anything like a finger existed or ever touched that key.  We (human beings) may not be able to feel what these Quantum Waves are doing, however when things come together in sufficent magnitude to form a virtual paricle, then we may experience the final effect as an attractive or repulsive force acting between two objects.


I hope that helps.   Best Wishes.

8

Just Chat! / Re: Fed up with the lack of reasoned discourse

« on: 29/06/2024 02:41:15 »

Hi.

         A spinning magnet - that can be interesting.   Indeed we had some discussion here (TNS forum) about a year ago.   You ( @paul cotter ) contributed to that old discussion but it was a long time ago.   I'm going to recap some of it.

       In general, a thing that is rotating should just keep rotating at the same rate, that's basic angular momentum conservation.   However, in outer space that doesn't seem to be what happens....    well... at least not if the spinning thing is actually a magnet.    In that situation it does seem that the object emits e-m radiation and MUST eventually stop spinning.

      Let's put conservation of momentum and angular momentum to one side for a moment.    If a spinning magnet emits e-m radiation then it is losing energy.   Energy conservation MUST apply.    If energy is being emitted or lost from the spinning object then it came from somewhere.   Either the spinning object is losing some mass over time, or else it is losing kinetic energy by slowing down its rate of spin.    It doesn't seem that a lot of mass is being lost, so the object must be slowing down its rate of rotation even though there is no obvious source of external torque being applied to the object.

     A long time ago, I started writing a new forum post based on this - but it became too long and had to be split into many parts.    In the end, I decided it was a monologue rather than a discussion so I never actually posted it.

   (Part II) had much of the mathematics in it.   Not a lot of that displays well now that LaTEX markup isn't available.     In words, it showed that the far field produced by a spinning magnetic dipole has the right characteristics to be considered as e-m radiation.        "Does a spinning magnet in space emit e-m radiation" --->  yes, it does.

    (Part III)  was based on an objection that @alancalverd raised in the (very old) original forum thread concerning the notion that the spinning object would ever slow down.   Where is the torque coming from to slow the magnet down if it's in outer space?     How is angular momentum conserved?   etc.
    The answer mainly seems to be that the e-m radiation that is emitted will be circularly polarised.   This allows the e-m radiation itself to have or "carry away" some angular momentum.   Well.... that explains "where" the angular momentum has gone but it still doesn't explain the source or cause of any actual torque acting on the spinning object.
      The torque acting on the spinning magnet seems to arise by consideration of the near field (near to the dipole object) as opposed to the far field approximation.   The mathematics here is just too complicated (for me anyway) and the other references I chased up seem to use only numerical (computational) approximations instead of finding exact analytical solutions.   In spirit, it seems that the near field changes at a rate slightly slower than the dipole is rotating,  so that the B field in the local space differs from the magnetisation, M vector, within the object itself.   As such the B field pulls on the magnetised object slightly and effectively puts a drag or retarding torque on it.   So, let's re-phrase this because it's quite weird when you think about it.......   What exerts a torque on the spinning magnet in free space?  Well... it seems that space does or whatever it is that we recognise as the B field in space.  Space isn't exactly "nothing", it has B fields in it.    The spinning object will spin at the speed it wants to spin, of course, and the Magnetisation vector M of that object should change at that rate - however space a few centimetres away cannot instantly change its B field when the magnetised object has moved.   The maximum propagation speed of a new B field is c,  so the B field just a small distance away is still the old B field.    This is a very simplified and wordy rather than mathematical version of what appears to happen but I hope it makes some sense and provides the gist of what seems to be going on.    It raises a number of other questions... space doesn't seem to be just empty space, it's almost as if the B fields in it are something tangible - but that's probably a whole other thread and not something we need to consider here.

You mentioned the following (which is why this example is relevant):
 

The discussion was about possible exemptions to the conservation of momentum because of the finite speed of em propogation.

     The spinning magnet in free space seems to be an example where instead of being a problem that may break the principle of conservation of (angular) momentum,  the finite speed of waves propagating through the E and B fields is actually NECESSARY, indeed it is ABSOLUTELY ESSENTIAL to maintain the principle of conservation of angular momentum.   It is the (tiny) delay between the magnetisation vector of the magnet changing its orientation and the changing of the B fields in nearby space that puts the retarding torque on the spinning magnet.   So, a magnet spinning in free space will slow down and eventually stop spinning and no laws or principles of conservation are violated.

Best Wishes.

9

Question of the Week / Re: QotW - 24.06.28 - Are there stars which aren't part of a galaxy?

« on: 21/06/2024 21:02:53 »

Hi.

Possibly.
     The definition of what is a galaxy, or exactly where one ends is vague.   Hence, under one definition of a galaxy, a given star may be in a given galaxy but under another definition it is oustide any galaxy.

One reasonable definition of a galaxy:
    Galaxies consist of stars, planets, and vast clouds of gas and dust, all bound together by gravity.

[Taken from NASA website.     https://science.nasa.gov/universe/galaxies/ ]

    The key here is that we have some understanding of what it means for gravity to be generally keeping a collecton of things (stars, gas, plantets etc.) together.    Gravity follows an inverse square law with distance.   This means that at some distance from the centre of what we would define as the galaxy, gravity will be too weak and the star(s) this far out are not considered to be part of this galaxy any longer.

The definition of a galaxy is then mired by the existance of dwarf galxies apparently inside or just on the periphery of a larger galaxy.   For example, our Galaxy (the Milky Way), has the (Large and Small) Magellanic Clouds as dwarf galaxies.   They can be though of as small galaxies in their own right but they have become trapped in the gravitational well of the Milky Way and now orbit around the periphery of our galaxy.    So, that they are gravitational bound to our galaxy and could reasonably be considered as just a part of one big galaxy instead of being called galaxies on their own.      We can consider ever larger structures (regions of space) until we identify regions we call Clusters or Super-Clusters.   All the mater in these regions is gravitationally bound together, they do not tend to drift apart, but they appear much as many seperate galaxies orbiting around each other in some complicated way (literally "a cluster of  galaxies").    However, since everything in this cluster is gravitationally bound together, it forces us to question whether we should just define the whole cluster as one really big galaxy.

-------
   So, there probably isn't going to be a definitive answer to the original question.   If there's a star in space somewhere then it is probably inside someones definition of some galaxy.
   None the less we can still try to address some of the spirit of the question:    Stars don't tend to form in complete isolation from other stellar bodies.    Based on our simulations of cosmology and the evolution of the universe,  matter tends to distribute itself into filaments spaning through space.   The overall structure is often referred to as a cosmic web   (you can Google the term Cosmic web to find more info if you wish).    Stars will form only where there is matter for them to be formed from.   So stars tend to form along these filaments rather than in the voids in between them.   Once stars have formed, gravity naturally tends to start attracting other material and vice versa (other material will attract the star), so that stars will usually end up with other stars orbiting around them and/or they are pulled into orbit around some existing cluster of stars.

Exceptions
   Can there be a star that is NOT gravitationally bound to any galaxy?   Yes.    Over enough time, it is likely that every star would eventually be attracted to some some galaxy and orbit in some stable fashion.    However, for a short while (in Astronomical terms that can be a million years), some stars can be "flung out" of a galaxy and fly through the voids for a while.
     Although stars cannot form in the voids between galaxies (since the density of matter is far too low), there are in fact large numbers of intergalactic stars. It has been estimated, for example, that 10 per cent of the mass of the Virgo galaxy cluster is in the form of these stellar interlopers........
     ..... stars can be expelled from their parent galaxy if it collides, merges or passes close to another galaxy.

[Taken from:   BBC Science Focus.    https://www.sciencefocus.com/space/are-there-any-stars-between-galaxies]

Best Wishes.

10

Physics, Astronomy & Cosmology / Re: why do a lot of people confuse between interference and diffraction?

« on: 28/04/2024 02:37:08 »

Hi.

If left uncorrected, the confusion will spread among students. To make it worse, they will get positive feedbacks in the exams by giving wrong answers.

1.   It's not in my power to make all students and teaching staff use the terminlogy only a certain way, sorry.

2.    Some of the important ideas for diffraction and interference are the same.   Blending the terms may not be as bad as you think.   Scientific terminology evolves just like all spoken languages evolve and develop.   If it became essential to point out some differences between one thing and another, there are other ways and other terminology we can use.

Evolving terminology isn't always bad:
       In Britain, the word   "inflammable" meant that something could easily be set on fire or was some kind of fire hazard.   The etymology of the word is from the Latin inflammabilis.    However, English started to evolve in a way where adding "in" as a prefix in front of a word indicated the negation or opposite.   
Examples:   Hospitable  and  Inhospitable;      Tolerable   and   Intolerable;   Edible and Inedible......  plus many others.

As a consequence some people thought that "inflammable" meant that something would NOT burn and indeed, when surveys were done in some workplaces, a few people thought they might be able to use an inflammable material to put a fire out if one started somewhere.   Obviously this is a serious problem.   So the resolution was to change the terminology assocated with something that will burn or ignite easily easily.   HazChem warnings now say  "Flammable"  instead of "Inflammable".     This was probably a senisble decision and may have saved lives.    Technically, it's wrong on many levels, the Latin roots have been lost and the word "Flammable" was just nonsense that wasn't in any dictionary when the decision was made.   However, it was still probably the right decision.

Best Wishes.

11

Technology / Re: Does regenerative braking have a hazardous complication?

« on: 27/04/2024 14:33:10 »

Hi.

In Ireland when changing down in a manual the recommended method is to lightly touch the brakes

    As far as I know that's not written in the "highway code" booklet (for the mainland UK) or generally expected - but I have once had such advice from my driving instructor.

    There are also alternatives, like double de-clutching   (double clutching in the U.S.):   The strongest slowing down effect is caused within moments of lifting your foot off the clutch pedal when you change to a lower gear.   This is when the engine, drive shafts on the engine side of the gear box and gear wheels in the gear box which were only spinning at (for example) 2000 rpm has to increase to 3000 rpm since you are initially at the same road speed but will now be in a lower gear.    An option then is to shift gear to neutral, ensure the clutch pedal is then released, briefly touch the accelerator pedal to spin everything (engine and the gears in the gear box) and then clutch and shift again to the new lower gear.   This isn't in the highway code either but it can be done and you will soon get quite good at it.
https://en.wikipedia.org/wiki/Double_clutching_(technique)       (that article talks about reducing wear-and-tear rather than application in racing).

    Another level up again is to just change gear without even using the clutch pedal.  You ease your foot slightly on the accelerator to reduce the pressure or contact force between gear teeth, then move the gear stick to neutral, then increase pressure on the accelator again (but it won't be any seconds, just milli-seconds because without any load the engine will pick up rpm almost instantly), then when the new gear is spinning at about the right rpm you'll push the gear stick again and engage the new gear while everything is still spinning.   With a lot of practice (which involves a lot of terrible noises from the gearbox while you fail) you can perfect this.   To practise this, the usual requirement is to use your dads car and be about 17 - 18 years old.  Your dad will have enough money to get their gearbox repaired and re-built after all the times you failed and have broken gear wheel teeth while trying to enage a gear wheel spinning for 4000 rpm to another gear wheel that was only spinning for 2000 rpm  etc.  (and after-all, new drivers do make mistakes even while making less extreme racing gear changes).   Once you own your own car, you won't try it again because the technique of double de-clutching is actually good enough and considerably cheaper.

    Anyway, all of these driving techniques are fast becoming obselete because most new cars will be fully automatic.   These days, they (automatics) are better and don't result in a loss of performance (e.g. fuel economy). 

     I suppose we should address the original question:
Does regenerative braking have a hazardous complication?
   Possibly but the driver behind shouldn't be following that close.   If they are then according to the highway code you must now adjust your own driving appropriately.  For example, reduce your speed so that they can stop in their braking distance.   It's not exactly what they deserve but it's what is safe and recommended.   Her Majesty's Secret Service takes a dim view of it if we keep deploying the special counter measures on ordinary civillian drivers who were just too close.

Best Wishes.
[ These days     Her = His       but all the films are based on Her Majesty's Secret Service ]

12

Technology / Re: How does my car calculate fuel economy?

« on: 20/04/2024 18:23:38 »

Hi.

   I don't really know a lot about cars.

   I have a red one with 7 seats but I don't usually put all the seats down.  So I reckon it has the fuel economy of a 5 seater on average.

    I've also noticed the trip computer displays an economy of 0.1 mpg while I'm waiting at the traffic lights.  The hand-brake is holding, so I reckon that's because the earth is moving through space all the time.

    It runs on tyres.   This car has a diesel injection engine while my old one had a petrol injection engine.  I can use the same tyres, the engine just injects them differently.

Not sure where ES is based

   It's a fairly ordinary house with a B rating for energy efficiency but I can't always park the car close to my house and I don't know what efficiency the other houses offer.

Best Wishes.
(perhaps I should keep my day job and not consider a career as a comedian).

13

New Theories / Re: How does light propagate? By what means does light get from Point A To Point B?

« on: 20/04/2024 14:54:17 »

Hi.

What causes the asymmetrical difference? It speeds up in one case, while slows down in the other case?

    The light was trying to travel towards the place you measure distance from, or away from that place.  For an expanding space, this matters.

    The terms "speed" or "velocity" were used informally.   Ratios of change in physical distance with time were being described.   In a flat Euclidean or Minkowskian space this is exactly what we can call a speed or velocity.

Best Wishes.

14

Technology / Re: How does my car calculate fuel economy?

« on: 19/04/2024 02:18:57 »

Hi.

   I don't really know where your figures came from.  My car hasn't improved its fuel economy this week compared to last week.  52 mpg sounds unrealistically high for the average motor car, what country did the figures come from and is it all downhill there?

Best Wishes.

15

New Theories / Re: How does light propagate? By what means does light get from Point A To Point B?

« on: 16/04/2024 01:38:50 »

Hi.

I think I can compress ES's post to a short line.

   I do like it and it is a bit shorter than my post.   I did have to read the line twice, make inertial frame a bit more obvious and I wouldn't have had to.

Best Wishes.   

16

New Theories / Re: How does light propagate? By what means does light get from Point A To Point B?

« on: 15/04/2024 19:45:24 »

Hi.

   First of all, I think 3 people in succession have now said something slightly opposing @Kryptid .   So, just to be clear, we don't dislike you @Kryptid .    Quite the contrary, people probably wouldn't have bothered making comments if they didn't think you and your ideas were sensible and could be amenable to discussion.

    So, let's challenge the comment made by @alancalverd slightly instead.   It's not his fault, it's mainly just the "SI council" of people who make changes to the measurement system are often about 10 years behind the state of the art in science.  (I don't know if they are called the 'SI council' but that's what I will refer to them as).   I suppose they have their reasons, for example why change a system that works well enough until there's a good reason for it, especially if it could be expensive for all instititutions to re-callibrate all their equipment etc.   They may also feel that certain things are simply beyond their jurisdiction and control.   For example they probably concentrate on providing a system that works here on Earh and over a few years that we can call "now" rather than suggesting all precise callibrations and definitions refer only to a region of outer space that is well behaved.

    Anyway, this is how it is:   The SI council changed the definition of a metre much as described by @alancalverd .   That happened in 1983 and was extremely useful and sensible because it puts things on a footing that agrees nicely with the theory of special relativity.   Sadly, it did NOT pay a lot of attention to the later theory of General Relativity and has left a lot of "mud in the water".
    So that's a fairly bold claim and I'd better have some evidence for this instead of just walking away.

1.  It's not just me who thinks the current SI definition of the metre fails to recognise the impact or role of General Relativity.   The CIPM (International committee for weights and measures) are aware of these limitations.

The CIPM issued a clarification in 2002:

    Its (..the speed of light..) definition, therefore, applies only within a spatial extent sufficiently small that the effects of the non-uniformity of the gravitational field can be ignored (note that, at the surface of the Earth, this effect in the vertical direction is about 1 part in 1016 per metre). In this case, the effects to be taken into account are those of special relativity only.

   (The CIPM)   ....Considers the metre to be a unit of proper length and thus recommends this definition be restricted to "lengths ℓ which are sufficiently short for the effects predicted by general relativity to be negligible with respect to the uncertainties of realisation"....

Various extracts from  https://en.wikipedia.org/wiki/Metre#Speed_of_light_definition

2.     Credible physicists have also written about this:
    This is an article by various contributors and held on the University of California website:
https://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/speed_of_light.html
     
   We'll only need a few sentences from it:
.....in the presence of more complicated frames and/or gravity, relativity generally relinquishes the whole concept of a distant object having a well-defined speed.  As a result, it's often said in relativity that light always has speed c, because only when light is right next to an observer can he measure its speed? which will then be c.  When light is far away, its speed becomes ill-defined.  But it's not a great idea to say that in this situation "light everywhere has speed c", because that phrase can give the impression that we can always make measurements of distant speeds, with those measurements yielding a value of c.  But no, we generally can't make those measurements.  And the stronger gravity is, the more ill-defined a continuum of observers becomes, and so the more ill-defined it becomes to have any good definition of speed.

     I'll need to elaborate on this a bit:   If we are considering a solution to the EFE with a scale factor that changes significantly every second,  then even 1 light second, is still too far away to be able to treat this as the local vicinity of the light now.   We can't just measure the physical distance between the light and the point of emission and divide by 1 second.    Where we seek precision over the definition of the speed of light, or the metre, then even 1/300 000 000 th of a second is too long and has allowed the light to move too far away.   Specifically, the SI definition of the metre works here on Earth and right now only because the scale factor won't have changed very much over  1/300 000 000  of a second in this region of space.


3.     Let's just see how the physical distance from the point of emission and the particle of light evolves from first principles.
        We'll have a simple  FRW metric for a universe with 0 spatial curvature (which is what we think we do have in our universe). 
        dS²   =   a(t)² ( dr² + r² dΩ² )  - c²dt²

[Reference:  https://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric#General_metric ]

   a(t) = scale factor;       r = reduced circumference co-ordinate   ---> it's a co-moving co-ordinate based on a radius as used in spherical polar co-ordinates upto some constants or choice of units;     dΩ² is based on the polar and azimuthal angles for the co-moving co-ordinate system,  dΩ² = dθ² + Sin² θ dφ².
    We'll centre the co-moving spherical co-ordinate system at the place of emission of the light at t=0 and will only be considering light that progresses along the radial direction, so dθ = dφ = 0 and the metric simplifies to:
     dS²   =   a(t)² dr²    - c²dt²

  Light is a null ray,   dS² = 0,    so  a(t)² dr²   =  c²dt².   taking the positive square root is appropriate for an outward travelling ray which is what we will be considering  (the light was emitted at r=0 and t=0 and travels outward).   So we have, 

       dr   =    c dt / a(t)         

   That's the relationship between a differential change in time, dt, and a differential change in radial co-ordinate, dr.   This holds at all times and at any place along the path taken by the light.   Note that this t and r are the co-moving times and r co-ordinate.

    So from time t=0 to t = an arbitary t,  the  total change in the co-moving r co-ordinate for the particle of light is given by a simple integration.....

(... LaTeX mathematics support is still not working....so it's not so simple....  I'll just insert an image....)


Delta r.jpg (8.76 kB . 438x126 - viewed 255 times)

The initial radial co-ordinate of the light  r(at t=0) = 0,   the radial co-ordinate of the light at time = t  is r(t) = R_H    (I've got to call it that, that's the image I could find - they were calculating a cosmological horizon so a letter H made sense for them).

   Recall that this, r(t) is a co-moving co-ordinate,   we multiply by  a(t)  = the scale factor at time t,  to obtain the physical distance  R_H(t)    between the  particle of light and the place of emission at time=t.


Delta R.png (14.25 kB . 528x139 - viewed 260 times)

In the situation where a(t) is strictly increasing with time,  i.e.  we say the universe is expanding,   then the integrand above is strictly greater than  dT / a(t)  while  T<t.       ( I can't get the tilda above the t and have used a capital T,   just evaluate the integral knowing that  a(T) < a(t)  while  T varies between 0 and t  in the integral ).   
    We have (omitting limits on the integral because LaTeX isn't avalaible),     a(t) . (1/a(t)) . c . ∫ dT    =   c ∫dT    =    ct .
   So we have  R_H  =  the physical distance from the point of emission to the particle of light  >  ct.   Hence the speed of the light  (if we define it as physical distance between point of emission and the particle of light after 1 second)    >  c.
    We get equality if the scale factor a(T) in the aforementioned integrals remained effectively constant throughout the whole time between t=0 and t.   If the universe expanded during this time interval, then we don't.    Hence, the SI definition of the metre doesn't actually work when you consider General Relativity and an expanding universe.  I mean it is what it is - that's how they defined it and what we have to use - but it just doesn't quite define what we would really want it to define.   It gives a satisfactory definition of a metre and an appropriate speed of light follows from this ONLY in a region of spacetime (which means over a small time and over a small region of space) where the scale factor can be assumed to hold constant.
    It turns out that having an expanding universe is only one way in which GR challenges the SI definition of a metre.   Other spatial curvatures (which are often described as gravitational fields) can also present problems.

    The safest way to talk about "the speed of light" is to refer only to what someone could measure as a differential rate of change while they are right at the same location as the light.   The SI definition of the metre is just some  "mud in the water".    We shouldn't measure the distance from an initial position to a final position, not even over a tiny time interval like 1/300 000 000  th of a second   because GR suggests that various things about the metric or gravity can still mess up that distance.
    Maybe the SI council will adapt the way a metre or the speed of light is defined in the future but provided the Universe doesn't suddenly return to expansion rates like the epoch of inflation or become very close to an extreme source like a gravitation like a black hole, then what we have as the definition of the metre will, I suppose, just have to do.
    Anyway, that's why it might seem like there are at least two slightly different sorts of replies that may seem slightly contradictory:
     The General Relativists   (e.g.  @Halc )   suggesting the speed of light is only a local thing.   Also me who will stand by the statement that in 1/300 000 000 of a second light may not have travelled 1m  of physical distance from where it was, we need to know about gravity or the evolution of the metric before we could determine that physical distance (e.g. assume the scale factor ≈ constant).   
     The Practical-ists   (e.g. @alancalverd )  suggesting that the metre is the metre as set by the SI council and that's the deal.   The speed of light follows.   So, regardless of where you are, how fast the universe is expanding, or the existence of unsual gravitational fields in the region, light will have travelled 1 m  after 1/300 000 000   of a second.
   
Best Wishes.

17

New Theories / Re: How does light propagate? By what means does light get from Point A To Point B?

« on: 15/04/2024 01:08:00 »

Hi.

It takes light one second to travel 299,792,458 meters. Of course, over the course of one second, a portion of space of that distance will expand to be ever-so-slightly longer than 299,792,458 meters. As such, it will take ever-so-slightly longer than one second for light to travel that distance because it can only travel 299,792,458 meters in one second.

      I'm not sure if you meant to do it but this is actually a perfect way of suggesting that light will change its speed. 

    Let's keep it simple and have light travel at 3 x 10⁸ m/s precisely.    Let's assume a constant rate of expansion for space set at 10% per second.
    Let's have some light emitted out to space by a man with a flashlight on a planet.   We'll assume the planet is co-moving with the expansion of space.   We also ought to make it clear whose time we're using,  we'll use co-moving time, so that will be the time recorded by a stopwatch that stays with the man on the planet.
    Now, you ( @Kryptid ) claim light will travel 3 x 10⁸ m  after 1 second, that's the distance you can measure between the planet and the light after 1 second.    The problem is, the space which it has covered will now also start to expand.   So after 2 seconds,  the light has travelled on a bit further.   However the total distance between the planet and the light will be a bit more than  6  x 10^{8 m/s}   (2 lots of c).   Specifically, it will now be this far away from the planet:       
           3 x 10⁸             +          (1.1  x  3 x10⁸)                      =           6.3 x 10⁸ m
[Distance moved t=1 to t=2]   +   [110% of the distance travelled t=0 and t=1]
   
    Between t=1 and t=2, which is just 1 second for the man on the planet,  the light became  3.3 x 10⁸ m  further away from the planet,   so it's average speed over that second was   110% of c.

    A similar calculation shows that the average speed of the light would be 3.63 x 10⁸ m/s between t=2 and t=3.   In general the speed of light seems to be increasing by 10% every second.

    Anyway, it can be some fun to think about it for a while.    It's fixable but usually leaves us forced to talk about the movement of light through that bit of space which is local to it.   Specifically we cannot consider the rate of change of physical distance from any point that is even just slightly remote from the particle  - which can take a page to fully explain and develop the idea.  [ Sean Carroll in "Spacetime and geometry" takes most of a chapter].
    When we say light has the speed 3 x 10⁸ m/s we don't really mean that it will be 3x10⁸ metres away from where it was emitted after 1 second.   We wouldn't be able to say anything about the physical distance between the light and the place of emission unless we also knew something about the curvature and especially the evolution of the scale factor in this region of space and over this time.
    So, that can be used as an alternative way to mitigate the earlier comment from @hamdani yusuf

If we insist that speed of light in vacuum is constant while also accepting that space is stretching, it implies that we also need to stretch the time by the same amount. So far, I haven't found any source for the latter.

   No, we don't.   Mostly because the speed of light isn't as restrictive as you might imagine.   It's a description of the movement of the light through space that is absolutely local to it (which is a different bit of space at every instant of time).   It certainly isn't enough on its own to tell us about the physical distance between the light and any place that is remote from that light.   If space is expanding rapidly with time, this distance (from the light to the place of emission) can grow much faster than 3 x 10⁸ m/s, that isn't a problem and that does happen.  We do not need to scale time in any way because we are under no obligation to keep that rate of change at 3 x 10⁸ m/s,  the speed of light never meant that, it only meant that light has a local speed of 3 x 10⁸ m/s.

Best Wishes.

[Overlap with @Halc who posted just before I did.   At a glance, we have said similar things, just in different ways and it won't hurt to have two different styles of explanation].

18

Just Chat! / The fair use of LLM and AI in education.

« on: 14/04/2024 02:45:59 »

Hi.

      Did you know that a lot of Universities run quite a lot of submitted material, like essays, through a computer to check for plagiarism?   You probably did know that.   
      Did you know they now also include a quick assessment of the probability that this content was substantially created by some LLM  (Large language Model - like Gemini, ChatGPT etc.)?   You may have known that but LLM's are quite new so they've only recently started it.
     If you submit an essay with a high chance of having been generated by a LLM then it can be rejected or the person could be called in for a viva examination.
     However, it seems to be bit one sided.   There's one student I know who felt strongly that the feedback and comments on their essays were very generic and not especially useful.  They ran the feedback through a LLM checker and - well, you guessed it.   Perhaps they should reject the feedback and insist the marker comes in for a face-to-face meeting where the essay is properly discussed for 10 minutes?

    On the other hand, perhaps it's not all bad.   Eduaction may end up becoming a game or a test to see who can game the system better - but we will have grauates who can play the game very, very well.  Students will be working not just to prevent triggering an LLM detection but to also get future iterations of their LLM assisted essays adjusted in line with output generated by the markers LLM system.  This will help the markers AI system pick up key elements and hopefully grade it more highly.   We'll have future graduates who are highly skilled with this new LLM technology.
    That will be important because LLM and AI will only continue to develop, so our new graduates will need to be good with this technology.  There's got to be a bright side to everything.

Best Wishes.

19

Question of the Week / Re: QotW - 24.04.19 - What language do the profoundly deaf think in?

« on: 13/04/2024 01:12:52 »

Hi.

  INTERNAL  MONOLOGUE.   That seems to be the most prevalent official terminology to describe the common way of thinking where your thoughts seem to arise as if they were articulated by some voice inside your head.

Other terminology also exists:   Autocommunication;   Inner Speech;  Intra-personal communication.
With search terms like this, there's a wealth of information (including YT videos, if you don't feel like reading) already on the internet.

However, it's not universal: Some people do not have any such internal monologue.   Some people report organising or experiencing their thoughts as something more like pictures or images and sometimes in other ways.

This website seems to claim some authority and has created a short single web-page discussing the internal monologue of deaf people.    Warning:   That's an external site and it is apparent that they are a commercial enterprise that sell courses and training material in sign language.   However, they don't sell mind training and their comments about internal monologue for the deaf shouldn't be too biased.
https://www.lingvano.com/asl/blog/how-deaf-people-think/

Their key phrase is as follows:
If a person was born Deaf and is primarily using sign language as their way to communicate, it?s very likely that this person will also think in sign language.

Best Wishes.

20

New Theories / Re: Stationary model of the solar system

« on: 12/04/2024 16:40:14 »

Hi.

This is something I started writing a day ago but @Kryptid answered most of the questions.   You ( @Yusup Hizirov ) seem to want some other explanation, so I might as well post this bit of what I was writing.

1. Due to what forces does the Universe expand and contract?

     The short answer would be "gravity" but the longer answer would require explaining that gravity is not a "force" under the model used.   In GR gravity is just an effect from the curvature of spacetime.   Under GR, space is not a static background that we can ignore.  Space is a dynamic thing, it changes its behaviour with time.
    Most noteably, the nature of space (I'd say "metric" but just read "nature of space") responds to the content of that space including how that content is distributed in space.   We then have something you could naively consider like a feedback situation:   
    Some content of space changes the nature of space, which changes the movement of (or even just the volume of space occupied by) the content - so the distribution of content is changed, which in turn changes the nature of space, which in turn changes the movement and distribution of the content, which.....
    Overall, starting from the EFE (Einstein Field Equations) there actually aren't many static solutions that we can find.   As soon as we put any content into the universe we end up with space that must change in some dynamic manner.
    It's difficult to say what caused what, it's the chicken and egg dilemma.   All we need to care about is that at some early time, there was some content of the universe and some space in which this content existed,  after that we have a dynamic inter-play between the content and the curvature of space.   
    This is perhaps one of the key differences between Newtonian Mechanics and (General) Relativity.   In Newtonian Mechanics, a moving particle didn't change the nature of space in any way.  Space is just a static background in which particles will move.  As such we can assume that an ordinary sort of Newtonian force must be there and is required to change the motion of any particle.    In GR, space will respond and change as a particle moves through it.

Do we really consider how space responds to the movement of a particle when using GR?
     Not always.   For a small particle, let's say a starship, the change in space caused by this particles movement through it is negligble in comparison to all the other sources of gravitation that exist in the space around it (like planets and stars) and we can reasonably assume it just moves through space without seriously affecting it.   In technical terms, we'd say the starship can be treated as a test particle that is just moving through a background of space whose properties were determined and set by all of those other big things in the region and not by the starship.
    However, when you're considering the whole contents of the universe (as we do in Cosmology) it's clear that you certainly cannot ignore the response of space to the movement of this content.  The content is the whole content of the universe, there is nothing else bigger or more significant as a source of gravitation in the region.   We must seek solutions where the response of space is completely determined by the evolution (e.g. movement and distribution) of the content.    We are precisely in the situation described above,   something we can view as a feed-back situation  and space has to be a dynamic thing.   

Where is dark matter located in the Solar System?

    It's hard to say.    There could be some right next to me and you at the moment.   It doesn't interact much with anything.   There are plenty of experiments underway to try and directly detect some dark matter, i.e. some that is right here on planet earth with us.
    We have some models and theories for how dark matter may be distributed in a galaxy but a galaxy is a much bigger thing, a huge scale up from a single solar system.   There are no models for the distribution of dark matter within an individual solar system that I've seen.

Is there an equation for dark energy, and in what units is dark energy measured?

     There are several equations where a quantity you can call "dark energy" appears. 
     We still can't assume that "dark energy" must be a substance or some content of the universe.   It's just a name we associate with some effect which accounts for the apparent expansion properties of the universe.  In particular, the universe is expanding and more importantly the rate of that expanson is accelerating.
      One way in which we could explain this is to have a strange substance as some content of the universe.  Another way involves adjusting the rules for the way that spacetime curves in response to the content of the universe.  Just using General Relativity we have the ability to have a cosmological constant on one side of an equation, or else to move this to the other side of the equation and consider it as some additional content in the stress-energy tensor.    This corresponds exactly to what has just been discussed  (keep the content and adjust the rules    vs.    adjust the content and keep the original rules).  Technically,  the cosmological constant is only associated with what we should call "vaccum energy" rather than "dark energy" but the distinction is frequently blurred.   For example, "Dark energy" may have an energy density that changes with time, whereas "vaccum energy" is considerd to have a fixed energy density at all times - but this is a minor detail:   The current version of the big bang theory known as the ΛCDM model works quite well and assumes we have a vaccum energy which will only ever have a constant density at all times.
     There are also other theories where the rules of movement for objects are changed in other ways, for example MOND  (Modified Newtonian Dynamics) and doubtlessly many other theories that I'm hardly aware of.   The effect which we, and especially the Pop Sci media, have called "dark energy" is quite possibly a combination of many of these things.   So, despite the name suggesting it is some sort of energy in space, you have to step back and realise that it may not be anything like that.   It's just some effect for which one explanation includes the existence of some strange substance or content of the universe.

     Let's confine our attention to "vaccum energy" which is sufficient to explain most of what we would want "dark energy" to be.   One obvious choice of an equation to look at is the Einstein Field Equations with the cosmological constant and declare that the value of the cosmological constant is an appropriate measurement of the combined effect we could call "dark energy".
    According to Wikipedia, this is a fairly reasonable value that agrees with current results:      Λ = Cosmological constant ≈   1.1 x 10^-52  m^-2.     It would have SI units  of  "per square metre".

     As I mentioned earlier, this is considering "dark energy" as just a change in the rules for how space responds to content.   Let's move the cosmological constant to the other side of the equation and consider what it would represent if it was some content, some sort of substance or energy, in the universe.

    If you consider dark energy as some content existing in a FRW universe, then it is treated as if it is an ideal fluid.   
The word "ideal" means that it doesn't support or exhibit any shear stresses  (so it has 0 viscosity).   The word "fluid" is quite nominal and we cannot run too far with the notion that it is like an ordinary fluid we have experience of in our normal life.  In particular, this thing has a positive energy content (e.g. it has some mass or energy in it) but it has a negative pressure.   There's no ordinary fluid in our normal lives that could have negative pressure.   

    The values for the sort of energy content and pressure it would have you can find in various places.    Wikipedia report an energy density approximately 5 x10⁻¹⁰ J/m³ .   I haven't spent the time finding the pressure in SI units.   The pressure would be precisely -1 times that energy if we work in natural units where c = speed of light=1 etc.   You could find (or calculate) that pressure in some other units like Pascal if you wish.

Best Wishes.