[paul cotter (OP)]

I have been involved on the periphery of an argument on a different forum. A question has arisen as to whether a scalar can have a negative value, leaving out trivial examples such as temperatures below freezing point of water. What say ye?

[alancalverd]

Of course it can. Money is a scalar. The amount of stuff in my pocket is positive money, the number on my bank statement is negative money.

[Bored chemist]

Charge
Potential

[Eternal Student]

Hi,

    Good question, @paul cotter  and, I think, some good answers already.

    Basically, yes.
    You could consider whether there are some limits on every sensible physical quantity and hence whether we could start all scalar quantities from some reference 0 value instead of having negative values.

   Example:
     Let's consider  CHARGE  as suggested by @Bored chemist .
    Suppose that there is a fundamental neagtively charged particle, to keep it simple we might imagine this is something like an electron with a charge of -1.    It's more likely that the appropriate most fundamental charged particle is actually a quark but the details don't matter too much, we only need assume there is SOME most fundamental charged particle.
   Now, every charged particle should occupy some space and the universe may be finite in size.    Then there would be a maximum number of negative charges we can have.     Instead of labelling this as a charge of  negative  one  "Naked Scientist's Bazillion"  worth of charge,  we could choose to make this reference point 0 charge.   All other charges are then some positive number by comparison.    More interestingly, if the universe has an infinite amount of space then we could not do this and some genuine physically relevant scalar quantities must be allowed to have negative values.

    We've had past discussions about POTENTIAL,  such as discussing the Aharanov-Bohm effect   (see, for example, this post   https://www.thenakedscientists.com/forum/index.php?topic=86694.msg722832#msg722832  ).   There it was suggested that the choice of a reference 0 point for potentials may not be as arbitrary as we had thought.   Specifically, in most electrical circuit design it's only important what the differences in potential between various points on the circuit may be .... BUT the Aharanov-Bohm effect suggests that the actual value of the potential may really be important for some things.    Anyway, we can speculate that the refence zero point for potential may not be as arbitrary as we had thought and therefore it may also be possible to label the lowest possible potential you can get ever get as 0 potential and have all other potentials only as some positive number.

     Possibly, this sort of thing could be done for all the physically meanigful scalar quantities we know of....   I don't know.

Best Wishes.

[Bored chemist]

Let's assume that, on some other planet in the universe, when someone named "positive" and "negative" charges, they got it the right way round.
We can then follow the same argument and declare that all charges are negative.

[alancalverd]

Zero charge does have a a physical reality and is not arbitrary. Charged objects either repel or attract each other, but zero charge objects have no electrostatic field and therefore no mutual force. As long as there are protons, neutrons and electrons in the universe, I submit that positive and negative are significant concepts. 

AFAIK the definition of positive and negative is a historic oddity. If you rub ebonite with cat fur, the fur acquires a positive charge and the ebonite negative. The same applies with amber, ελεκτροη.

[paul cotter (OP)]

Thank you all for your replies. This was not a discussion I was directly involved in- intuitively I felt a scalar could have a negative value but did not have a rigorous proof that this was so.

[Bored chemist]

AFAIK the definition of positive and negative is a historic oddity.

As far as I am aware it was a reasonable, but incorrect, decision; they knew they could model electricity as some sort of "fluid" that moved from where there was too much of it ( a positive charge) to where there was a deficit.

But they didn't know which was which.
Benjamin Franklin who experimented with electricity in the middle 18th century made a choice:

When a.rubber rod that is rubbed with cat's fur the charge on the rod is called negative and when a glass rod is rubbed with silk the charge on the rod is called positive.   
It wasn't quite arbitrary
Franklin decided that sparks given off by an object charged by a glass rod (vitreous electricity) looked more like fluid leaking out than did the sparks from an object charged by a rubber rod (resinous electricity). Thus he decided that glass had an excess of electrical fluid.

[Eternal Student]

[LATE EDITING performed on 13/04/2025 just to fix some vocabulary and typing errors  etc.  It doesn't change the spirit of what was said.]

Hi again.

   It's been quiet and it seems like time to do something controversial and maybe @paul cotter  can use some of it to spark a more interesting discussion on this other forum he mentioned.   So I'm going to play devil's advocate here.


     Let's pick up on @alancalverd 's point:

Zero charge does have a a physical reality and is not arbitrary. Charged objects either repel or attract each other, but zero charge objects have no electrostatic field and therefore no mutual force. As long as there are protons, neutrons and electrons in the universe, I submit that positive and negative are significant concepts. 

     I appreciate the notion, I'm sure we all do - but the mathematics that appears just doesn't work properly.  It does NOT adequately explain how electrostatic attraction works in the universe.   I agree that some things do seem to repel or attract but the explanation based on having a charge and following the Coloumb force law is quite unsatisfactory when you examine it carefully.

       Basically, we are supposed to assume the fundamental electrostatic force is given by:

      F  =   Q₁  Q₂    /   (4πε r² )      (in direction   r₁  -  r₂ )

just the usual law for the force between   two charges of magnitude    Q₁    and   Q₂   at positions  r₁  and  r₂.

For convenience I'll call the direction of the force from a charge q acting on a charge Q as the vector d_q  rather than writing r₁  -  r₂.
Then the net force on any charged particle is found by summing all the individual forces from any other charges that are in the vicinity.

      F_{Net acting on  Q}   =   

∑    [  Q  q    /   (4πε r² )  ]    d_q
  Sum over all q       

Now the problem is that in a typical universe, we are then completely incapable of determining the overall net force on a given charge Q.

Why?  Because in a typical universe, we can assume a uniform distribution of other positive charges throughout all of space.   Sure there may be some more dense bits here and there, but overall it's pretty uniform.     So all we need to do is consider progressively larger spherical shells of space surrounding the one particle we are interested in.    The first 1 light year radius of space around our particle might be unusual but we can reasonably assume everything else is well approximated as having a uniform density  of ρ  positive charges per cubic metre.    The good news is that the actual magnitude of the force of attraction is falling off as  1/r²,  so we may hope that the contribution to our grand sum from  shells of space beyond  1 light year is going to get small and tend to some finite limit as   r → ∞.    The bad news is that the volume of space in a spherical shell of radius r is growing like r³.   Actually we're really only going to be interested in the components of force along narrow cones (directions), so we'd really be working with little patches of a sphere's surface that we need to consider - those surface areas grow only as the square of the radius,  r².   However, growth as r² is still precisely enough to compensate for the drop in magnitude of force as 1/r².

    Attempting to write more of the mathematics without support for LaTEX mark-up language is going to get messy but hopefully you can already see what I am leading to.....
    Trying to calculate the net force acting on the one chosen positive charge Q that we have decided to watch and study in our universe is just impossible.....  we would want to determine an indefinite improper integral  (integrating from radius= 0   to  radius → ∞)   BUT sadly,  that integral will be non-convergent.   To say that more simply, the integral is undeterminable, there will not be any one finite number and direction that we can declare as the net force acting on the particle.

LATE EDITING:   The correct term is an improper integral, this can describe a situation where the integration limits are written as infinity.  The term indefinite integral is reserved for integration without any limits specified at all, also known as finding anti-derivatives.  Sorry.

    This problem isn't limited to electrostatics,  a similar issue arises for Newtonian gravity - because it also has a 1/r² force law and the assumption of uniform density (mass density this time rather than charge density) is equally reasonable.    Overall we must note that declaring the following two things as the tenets of the theory actually fails to poduce any proper explanation of electrostatics (or gravity):
    (i)   That the fundamental force is given  by  something with form   Qq / r²     and,
    (ii)  The Net force is the sum over all other charges q that exist.

-  because that is tantamount to saying that in a simplified (but realistic) universe, we canot know and can never determine the net force acting on any given charge  (or a given mass if you wre using Newtonian gravity).

   On a minor note,  this problem of a non-convergent integral goes away if we can assume that the matter (or charged particles) in the universe actually thins out at large distances away from us (here on Earth).   If we can assume that, at some large radius, perhaps billions of light years from our chosen particle, the density of matter starts to fall as something like   1/r or just drops instantly to 0 (e.g. as if the universe does have an abrupt end or edge), rather than just holding constant,   then the integral becomes perfectly convergent.

Say it without the mathematics:
    That may have been too much mathematics and it's not to everyones liking.  So let's try to demonstrate the issue another way, we'll use the well known hollow-shell theoreom.    (Reference   https://en.wikipedia.org/wiki/Shell_theorem ).

     Here's the charged particle we're interested in:

⊕ 

It's  NOT   the only charged particle around,   all the white-space on the screen also has some positive charges in it but we can assume this is of uniform density across all of the space.   We are interested in just that one charge shown as a red circle with a + sign written in it that is in the centre of the screen.  We wish to know what the net force acting on it will be.

   So it's human to start drawing our shells around it and see what we can gather from the shell theorem.   Just to be clear then, it's natural and convenient to have the chosen positive charge as the origin (the centre) for all our shells.

   So here's a shell drawn with some radius, r around our charge of interest.

Shell-centre-origin.jpg (6.14 kB . 239x220 - viewed 736 times)

According to the shell theorem, the net force acting on our charge of interest from that shell is 0.    This will hold for any radius, r, of the shell.   We draw lots of thin shells with ever larger radius and we would cover all of space if we set the thickness of our shells at δr  and have them like Russian dolls just fitting snuggly inside each other and have  N of these nested shells with  N → ∞.   We can sum up all N of these zero contributions and get a total of zero.    So summing all the effects from every thin shell as  we allow the radius r → ∞    gives us just  zero.   We end up with  0  as the net force vector acting on our charged particle of interest.   
    That's probably exactly what we expect -  the situation looks perfectly symmetric around our charged particle of interest.   There are plenty of charged particles around it but they're all equally distributed and so there's no net force.    If we leave the particle alone then it should just sit still where it is.

   Next we do something else and I've probably given the game away a bit already by emphasising where we put our origin.   Nobody said we HAD to put the origin at our particle of interest and so let's just put it somewhere else.  The shell theorem doesn't demand that the origin is put in one particular place.  Indeed one of the beauties of the shell theorem is that our particle of interest could be off-centre inside some shell and that still won't matter.   Our particle can be nearer to one bit of the sehll than some other bit and it will still have net zero force acting on it.

Here's our (red) charged particle of interest and an interesting shell of space.


Shell-offcentre.jpg (22.06 kB . 476x374 - viewed 673 times)

I've shown the particle of interest and a shell centred just to left of it, with a radius (I'll call it the critical radius) which makes the shell precisely touch our particle of interest.   We are again going to use the notion of covering all of space by starting from this origin as the centre for a whole set of perfectly nested Russian-doll style thin shells.

Now the shell theorem allows us to ignore all of space that is ouside of that grey spherical region I have shown.   That is because our particle of interest will be INSIDE any shell with a radius bigger than this ciritical value.  Remember that the beauty of the shell theorm is that it doesn't matter that our particle is off-centred in those shells,  all that matters is that it is inside.  So, like before, most of the thin shells that will make up all of space contribute zero force to our particle of interest.  Summing up all those zeros, still gives us zero.   So, we can completely ignore MOST of the space in the universe.  We need only concern ourselves with that little bit of space which is contained within the spherical region shown in the above diagram (a spherical bit of space centred just to the left of our chosen particle and with that critical radius as shown / discussed).   Fortunately, the shell theorem does that for us as well.   For a particle OUTSIDE of a shell like that, the net force acting on it is equivalent to having all the charge contained in that sphere treated as a point particle of that charge located right at the centre of that shell.
    We only have the one thin shell which precisely touches our particle of interest to worry about,  since our particle is neither INSIDE nor OUTSIDE of that.   Obviously we deal with this by just insisting that we use progressively thinner shells.   Our particle of interest is either INSIDE or OUTSIDE of every shell except one which can be treated by a limiting process to be a shell of infinitessimal thickness and thus contribute no significant force to our particle of interest.
    Overall then, by putting the origin of all our thin shells just to left of our particle of interest, we will end up with a net force on our charge of size Q   that is given by   
   (4/3)πr³ ρ Q  /  (4πεr²)    =   
    rρQ  / 3ε         in the direction directly to the left   (if I've done my sums correctly).

[Where    r  is the critical radius = where we choose to put the origin of our shells away from our charged particle of interest;     ρ = the charge density of space which we are assuming is treated as some constant;    ε = the usual permittivity of free space;   Q = the size of the charge on our charged particle of interest. ]


    Now, the thing is that's not zero is it?   It's a different answer to what we obtain by placing the origin of all our shells right at the place where the charged particle of interest is.     Moreover we can choose to place our origin to the right, or above or below our charged particle of interest,  we can also move it further away.   We can have any size and also any direction of the net force acting on our particle of interest that we want.  Worse than that, they're all perfectly valid solutions somehow.

    Anyway, it's just of passing interest:    Netwon's law of gravity and similar notions for electrostatics do seem like they are reasonably complete and compelling theories that will allow us to compute the force(s) acting on any particle in the universe but actually they are a total failure in that task.....   OR ELSE    the universe is not as we might have imagined.   It may not be genuinely uniform, homogenous and isotropic (as described under the "Cosmological Principle") and/or it may be finite rather than infinite.   We can use some of these to alleviate the problem of having non-convergent integrals and/or invalidate the conditions required in the shell theorem so that we won't have contradicitions arising from applying the shell theorem.

   I don't know... but my opinion is that they (Newtonian gravity and Electrostatic force laws etc.) are just plain "wrong" in the sense of being only partially usefull theories.   For example, GR is a better theory of gravity than Newton and electrostatics is an equally crude approximation of some other underlying theory.   We should not be so quick to assume that Coulombs laws are suitable.  Then you must realise that without Coloumbs laws, a proper understanding of what a positive or negative charge may be or represent is lost.   We always imagine positive and negative charges and especially the meaning of their magnitude in the context of Coulombs laws.

    I hope that's something to think about.  I'll end my stint as the devils advocate now.

Best Wishes.

[Eternal Student]

Anticipated Counter-arguments:    There are some.

Typical counter argument 1:
   What if ρ is precisely 0 ?    Specifically the average cubic metre of space has just as many positive charges as negative and overall no net charge density.    Then the formula  rρQ / 3ε   for the net force predicted from applying the shell theorem just off-centred from the particle of interest always gives the value 0 anyway.   Then everything said above can be true but it's not contradictory:   The force in the south direction is 0,  in the North direction is 0,  it's all 0 in every direction  and  increasing r (the distance between where we centre our shells and the particle of interest) doesn't have any effect either,  10 times 0 is still just 0.   It's all just giving 0 net force acting on the particle of interest regardless of where you choose to centre the hollow shells of space.

Reply:   Well, I can't see anything to disagree with that.  It could be.

    However, there is no reason why the average charge density of space would HAVE to be 0.  Indeed it's quite an interesting coincidence if it HAS to be.
    We assume a net in-balance in the number of particles and anti-particles in the universe.   This is despite all the basic production methods for creating and annhilating particles suggesting that they always come and go in equal numbers.    The assumption that everything was of net neutral charge right at some sort of start for the univerese    OR    that it must remain in this sort of balance for ever after,   is no stronger.
    Even if, for some reason, the universe started out with a perfectly 0 or net neutral charge then there are some things which could dynamically change this balance as the universe evolves.   For example, we have models of charged black holes, indeed charge is one of the few properties they can have.   Assuming some small part of Coulombs law is a reasonable approximation, then we can see that these attract particles of the opposite charge into them faster than they would attract particles of the same charge.   The region of space around such a black hole therefore has a time-evolving charge density rather than having a constant charge densty at every time.  Some years ago we would have thought the overall charge is conserved,  i.e. that if some positive charge falls into a BH then the BH becomes more positively charged.   However the idea of Black Holes evaporating into just Haking radiation throws some mud in that water.   If a BH is near the end of its time then it may leave behind only photons (non-charged or net neutral particles) rather than leaving behind any of the charge it may once have had.


  But... I don't claim to know with any certainty.   Since this is in one of the main sections, it should be made clear that a lot of it is just reasonable speculation and was intended only to provide something to think about.
    However I wouldn't want anyone to think it's all without any main-stream scientific basis.
The essence of the ideas, such as recognising the problems (an outright inability) for the Newtonian theory of gravity to be a proper explanation of gravity on a universe-wide scale is not a new thing or something that hasn't been noticed already.   For example here's a link to an article written by John Norton around 1999, entitled   "The Cosmological Woes of Newtonian Gravitation Theory", that was published in the book entitled  "The expanding worlds of general relativity" which is still available for purchase today:
        https://sites.pitt.edu/~jdnorton/papers/cosmological-woes-HGR4.pdf
The pages numbered 272 through to 274 are sufficient to exhibit one simple version of the non-convergence problem.
The use of these same ideas for electrostatics rests on nothing more than recognising that it also involves a 1/r² law for the force of attraction.

Best Wishes.


   
   

[paul cotter (OP)]

Hi ES, I enjoy a bit of devil's advocacy. Just in case you feel I have lost interest in this subject I must explain that this is not the case. With the current good weather I am trying to catch up on very necessary garden work(I have recently purchased a chainsaw, to give some idea how dire the situation is) and I do not have time to do justice to your contribution.

[Bored chemist]

Now the problem is that in a typical universe, we are then completely incapable of determining the overall net force on a given charge Q.

Unless we ask for Mr Newton's help and multiply the particle's acceleration by its mass.

If it's not accelerating then we know that the rest of the universe's contribution sums to zero.

[alancalverd]

From ES:

in a typical universe, we can assume a uniform distribution of other positive charges

The theoretical physicist speaks! "Consider a uniform spherical cow in an infinite field.."

But we do live in a typical universe (indeed it is a perfect model of itself) which has, as far as we know, exactly equal numbers of positive and negative charges, and always will be overall electrically neutral. Experiment confirms this.

[Eternal Student]

Hi.

BC said:    ...If it's not accelerating then we know that the rest of the universe's contribution sums to zero...

    In other words, just observe and whatever happens, that is what happens.    If we're asked how potassium reacts with water we can just chuck some into water, observe and declare... "it reacts like that".    This evidently will work and will be perfectly correct but it lacks the scientific element of being able to predict anything.   Indeed there isn't much essence of being any sort of science:   It's just observing rather than modelling, building a predictive ability or gaining any usefull understanding.     What I'm saying about electrostatics, Newtonian gravitation or anything with a similar 1/r² law  is that it simply wouldn't be usefull as a method of predicting what will happen when we pick out one particle and just set it down somewhere in the universe and let it go.   We can just go and do it, observe it - and then we would know if there had been a net force on the particle  - but that's not much of a science and grants no insight into how it worked and why it did what it did.

   --------------------

  ..(The Universe).. has as far as we know, exactly equal numbers of positive and negative charges, and always will be overall electrically neutral. Experiment confirms this.

1.     What experiment?   I'm not an experimental scientist - what equipment exists to determine the net charge of the universe?

2.   Some things can have and do have a net charge or a net inbalance of charges.   As you said yourself a bit earlier... rub an ebonite rod on a furry cat and you have made two things which do have a charge in-balance.  Nature didn't explode or refuse to continue, it just goes on as usual.   How or why can you be so sure that the universe isn't a cosmic cat?

3.  As for the statement...   "and it always will"...  there are some things that could change the balance of charge in the universe as time progresses.   See an earlier post.... charged black holes may evaporate as Hawking radiation which leaves only a collection of un-charged or chargeless photons where there was once a charged BH.

Best Wishes.

[Petrochemicals]

To write off the extended universe at a distance as infinitesimal small is trying to write of the earth as non conductive, is this where the uniform universe stems from ?

I am also missing the point at how this is related to scalar ?

And with dank energy and matter being non uniform the problem is not so easy.

[Bored chemist]

" This evidently will work and will be perfectly correct but it lacks the scientific element of being able to predict anything."
What will happen if you drop potassium into water tomorrow?

[Eternal Student]

Hi.

....  What will happen if you drop potassium into water tomorrow? ....

   Without starting to build some model, I just don't know.   The easiest model to come to hand is that the properties of the potassium and of the water will be much the same tomorrow and all the laws of nature will be much the same.   But that's the essence of forming some model and scientific understanding.
     It fails when things aren't as we hoped.   Such as when the water has already been reacted with something else previously.
    A more concrete example is something like the double pendulum.  Here's an animation from Wikipedia of three such double-pendulums that were started with almost identical conditions but start to differ wildly in their behaviour after about 30 seconds.




  That's just a basic mechanical thing.... as a human being you would have thought that this ought to do the same basic thing tomorrow as it did today.   However... it's just "chaos" in the sense of being what chaos theory is all about.      However... it actually is a deterministic chaos,  provided we approach it with concepts like Newtonian mechanics and/or Lagrangian mechanics... we can start to understand it.    Meanwhile, watching one double pendulum do something today and just guessing it will trace the same path tomorrow will not help us.

Best Wishes.

[Eternal Student]

Hi,

I am also missing the point at how this is related to scalar ?

It has become a bit of a tangent to the original topic.   @paul cotter , the OP,  can call for a halt at any time.
I'm probably just about done here anyway  (although if @alancalved has knowledge of equipment or experiments that can determine the net charge of the universe I would genuinely like to hear about it).

Best Wishes.

[alancalverd]

All atoms have equal numbers of positively and negatively charged constituents.

When atoms disintegrate, charge is always conserved.

However we measure it, if we do so carefully and accurately, triboelectric charges and other induced static charges always match positive and negative charge.

However we separate charges, whether by electrostatics or electrodynamics, if we reconnect them with a conductor the eventual net charge is zero.

It is not clear to me how a black hole can acquire a charge other than by neutrons emitting negative pions in the early stages of gravitational collapse. This does not violate conservation of zero net charge - it's just another form of charge separation as occurs in beta decay and VanderGraaff generators.

 Yes, there will always be points of temporary positive or negative charge, but AFAIK there is no mechanism that allows the universe as a whole to possess a persistent overall charge.

[Eternal Student]

Hi.

   I'll try and keep this short, it may sound curt but that's not the intention - there's just no need to make anyone read a lot of stuff.

When atoms disintegrate, charge is always conserved.

    So what?   According to the most currently favoured cosmological models, there were no atoms or matter particles of any ordinary sort until a long time after the Big Bang  (hereafter abbreviated BB).
    There are two important issues:

1.     It isn't clear exactly how the matter particles (such as elementary charged particles) came about.   Just as we may think there should be an equal number of positive and negative charges created,  we also thought there should be an equal amount of ordinary and anti-matter.   However, it doesn't seem that there is an equal amount of anti-matter left around anymore.   More generally, the idea that CP (Charge Parity) violation does occur in nature seems to be fairly well recognised now   (reference:   Nobel prize given to Cronin and Fitch, 1980, for work on CP violation).

2.     Assuming charge is a fundamental conserved quantity, we do not know what that quantity is, in particular we do not know that the universe had 0 net charge to start with.
      Although conservation of charge requires that the total quantity of charge in the universe is constant, it leaves open the question of what that quantity is.
        [extract from Wikipedia:      https://en.wikipedia.org/wiki/Charge_conservation ].
 (I will admit it goes on to suggest it is probably net neutral but the important bit I need is that we do NOT know).

....  It is not clear to me how a black hole can acquire a charge other than by.......

    It is not to clear to us (us = the wider community with interest in Cosmology) that all Black Holes were created since the Big Bang.   Indeed it seems that there are too many massive BH around to have all formed in the 13.8 bn years since the BB.   Some of these BH may be primordial or might have just always been there.

None-the-less,  thank you for your time and comments @alancalverd.  It is natural to assume the universe has an overall neutral charge, I would do it.  However, it seems that we just do not know

Best Wishes.