Why is mass additive in the context of the Higgs boson?

[Avichal]

The picture of mass I have is as follows :- A particle, say an electron resists some force because of some intrinsic property it has called mass. Now this property is caused by the higgs boson particle (which I know nothing of).

When two such particles, in this case when two electrons are present the total mass will be the sum of the individual masses? Why is this?

 

[Simon Bridge]

You mean why do two particles resist twice as much as one ... (note: the property is called "inertia" - it happens to be the same as mass.)

If the inertia did not add, the it would take a different amount of work to move two masses separately as to move them together. How would the two masses know if they were "together" or not?

Maybe because they see each other's electromagnetic field?
Technically, two charged particles will have energy in their mass and in their electric field ... if the charges are bound (i.e. electron and proton) then there will be a mass discrepancy (mass of an atom is not the sum of the masses of it's components).

What adds up is energy and mass is a kind of energy.
It's the superposition principle all over again.

So the next is "why do the energies add up?"
Nobody knows - they just do. That's the rule.
If two identical objects are stacked one on the other, then the whole is twice as high as any one.
why is this?
why is 1+1=2?

Note: "why" questions like this are endless ...

 

[jedishrfu]

The best you can say here is that it is borne from experiment where adding the masses is consistent with getting the results that match the theory and so it is defined in the math of the theory.

Its similar to why do forces add vectorially in classical mechanics. Its consistent with CM experiments and so is defined as a legal operation in the math of the theory.

 

[Ethan0718]

The picture of mass I have is as follows :- A particle, say an electron resists some force because of some intrinsic property it has called mass. Now this property is caused by the higgs boson particle (which I know nothing of).

When two such particles, in this case when two electrons are present the total mass will be the sum of the individual masses? Why is this?

I think you need to know the origin of inertia mass. Inertia mass comes from Newton's law.

First, according to Newton's second law, ƩF=ma, you know mass is the ratio of net force and its acceleration. But what is net force? We can't say net force is its inertia mass times its acceleration because that's a circular argument. So, how to answer this problem? Or, I should ask "how to not to use the unknown "net force" to get the idea of inertia mass?".

Let's use Newton's third law, suppose that there are two ball colliding each other.

ƩF₁ = m₁a₁
ƩF₂ = m₂a₂

if we could neglect the frictional force, then the net force of each ball comes from each other.

ƩF₁ = F₂₁ = m₁a₁
ƩF₂ = F₁₂ = m₂a₂

Next, using Newton's third law,

F₂₁ = - F₁₂

m₁a₁ = - m₂a₂

m₁/m₂ = - a₂/a₁

Finally, let's define mass of the one ball is 1 kg, then we can know the other ball by the above equation. So the Newton's 2nd law is well-defined.

Second, we can consider two mass points system.


ƩF₁ = m₁a₁
ƩF₂ = m₂a₂

then sum them up

ƩF1 + ƩF2 = (F₂₁ + F_1,ext) + (F₁₂ + F_2,ext)

∴ƩF1 + ƩF2 = ƩF_sys,ext = m₁a₁ + m₂a₂ = (m₁ + m₂)a_c

a_c is the acceleration of center of the mass(system)

So, the reason why mass is additive is that's how we defined the center of mass.

If the acceleration of every particles in multiple-mass-points system is the same, then

you'll see the mass-additive property is guaranteed by Newton's second law.


PS. Newton's law is only suitable to single particle system. Euler's law of motion is advanced Newton's law and it can be used in multiple particles system.

 

[Avichal]

You mean why do two particles resist twice as much as one ... (note: the property is called "inertia" - it happens to be the same as mass.)

Yes, that's what I meant

If the inertia did not add, the it would take a different amount of work to move two masses separately as to move them together. How would the two masses know if they were "together" or not?

Maybe because they see each other's electromagnetic field?
Technically, two charged particles will have energy in their mass and in their electric field ... if the charges are bound (i.e. electron and proton) then there will be a mass discrepancy (mass of an atom is not the sum of the masses of it's components).

You say that "maybe" they see each other's electromagnetic field. So we are not sure about this?

Now that I asked this question many other similar questions are coming up like - Why is distance additive? Why is force additive? and more.
So the correct answer to all these questions is that because this is what we observe or is there a more deeper answer?

 

[Ethan0718]

Yes, that's what I meant


You say that "maybe" they see each other's electromagnetic field. So we are not sure about this?

Now that I asked this question many other similar questions are coming up like - Why is distance additive? Why is force additive? and more.
So the correct answer to all these questions is that because this is what we observe or is there a more deeper answer?

If you want to know the reason why forces is vectorially additive, then maybe you can check this:

law of equilibrium on the inclined plane

The story of Simon stevin

And The Science of Mechanics by Ernst Mach

 

[sophiecentaur]

When two such particles, in this case when two electrons are present the total mass will be the sum of the individual masses? Why is this?

We live in an essentially linear world. To within our limits of measurement, we usually find that we can apply simple arithmetic (Addition and Multiplication) when we combine quantities. Even when we impose extreme conditions, we nearly always find that the world is 'Monotonic', so twice as much of something will usually increase and effect, even when it's not actually proportional. There are notable exceptions to this, of course, particularly in complex systems (many biological and electronic examples).
To give us half a chance of getting a grip on Science, it's a good idea to start with the linear examples or where would we be? (The realms of Magic, or worse, I think).

 

[Dale]

When two such particles, in this case when two electrons are present the total mass will be the sum of the individual masses?

This isn't generally the case in relativity. In relativity the mass of a system is often greater than the sum of the masses of the individual particles.

 

[Avichal]

I never gave a thought before on why the physical quantities follow simple arithmetic when we combine them but now it's truly amazing to realize that they follow something so neat and precise.
I suppose there is no answer to my question, right? Mass is additive because it is, that's how nature is, right?

 

[Dale]

I suppose there is no answer to my question, right?

Did you miss Ethan0718's excellent answer in post 4?

 

[Nugatory]

I suppose there is no answer to my question, right? Mass is additive because it is, that's how nature is, right?

There are a couple of pretty good answers above, but if you want something less mathematical and more intuitive, you might think about what happens if you take a an object and break it into two pieces. Do you expect the two pieces together to weigh the same as the original object? Of course you do, and that's what mass being additive means.

 

[technician]

I never gave a thought before on why the physical quantities follow simple arithmetic when we combine them but now it's truly amazing to realize that they follow something so neat and precise.
I suppose there is no answer to my question, right? Mass is additive because it is, that's how nature is, right?

Nature is like that but it is also crafty !
If you ever find that 1 + 1 does not equal 2 you should hope that it equals 1.414 you have then found a crafty way that nature can behave...it is not difficult ! To get to grips with.

 

[Ethan0718]

When two such particles, in this case when two electrons are present the total mass will be the sum of the individual masses? Why is this?

Suppose here's a charged particle, for example, an still electron.

We know that the charged particle will emit radiation (photon) when it's accelerating.

So, let's push this charged particle and make its velocity be non-zero velocity from zero.

Assume its mass is m.

The change of its kinetic energy will be ΔE_k = $\frac{1}{2}$mv_final² - $\frac{1}{2}$m0²

--

However, it's not the energy you lose. Don't forget the emitting radiation. Let's be some energy,E_photon.

--

So the total energy you lose (or, the work done by you) during this process is

( $\frac{1}{2}$mv_final² - $\frac{1}{2}$m0² ) + E_photon = $\frac{1}{2}$mv_final² + E_photon

So, it's like you push a new-mass charged particle:

$\frac{1}{2}$mv_final² + E_photon = $\frac{1}{2}$m'v_final² - $\frac{1}{2}$m'0²

Obviously, m' > m

That's why you'll feel its mass greater than its still mass.
Same argument can explain the case you mentioned, the scenario that two electrons are present.


ps. actually, I prefer to think the electric field has some mass.

 

[Simon Bridge]

You say that "maybe" they see each other's electromagnetic field. So we are not sure about this?

No - we are sure that they see each others electric field ... I was proposing an answer to my question.

You should also see that if mass were not additive, then a balance would not work - a single 20g mass would not balance two 10g masses. You'd also be able to generate energy out of nothing ... but why conservation of energy? etc. etc. etc.

Now that I asked this question many other similar questions are coming up like - Why is distance additive? Why is force additive? and more.
So the correct answer to all these questions is that because this is what we observe or is there a more deeper answer?

It more that any answer anyone can possibly give for such a question just leads to the question being applied to the answer - you know, like when a child asks "why?" to everything, only more grown up?

Why is X like that?
Because Y is true.
Then why is Y like that?
Because of Z.
But why...

I could say it is because of the interconnectedness of physical laws - laws of locality and causality ... which are derived from the early Universe being quite small - but then you'd just ask: why was the early Universe so small? See?

This is the sort of question I am talking about when I say, 'Science does not do "why" questions - ask a philosopher.' (It seems to upset some people.) Empiricism can tell you what is, and how it comes to be, but does not tell you why it is like that.

The standard answer is to ask you to rephrase your question in terms of "how" or "what" to make it explicitly something that can be handled by science tools. Mind you, Ethan0178's earlier post is pretty good.

 

[Avichal]

No - we are sure that they see each others electric field ... I was proposing an answer to my question.

You should also see that if mass were not additive, then a balance would not work - a single 20g mass would not balance two 10g masses. You'd also be able to generate energy out of nothing ... but why conservation of energy? etc. etc. etc.

It more that any answer anyone can possibly give for such a question just leads to the question being applied to the answer - you know, like when a child asks "why?" to everything, only more grown up?

Why is X like that?
Because Y is true.
Then why is Y like that?
Because of Z.
But why...

I could say it is because of the interconnectedness of physical laws - laws of locality and causality ... which are derived from the early Universe being quite small - but then you'd just ask: why was the early Universe so small? See?

This is the sort of question I am talking about when I say, 'Science does not do "why" questions - ask a philosopher.' (It seems to upset some people.) Empiricism can tell you what is, and how it comes to be, but does not tell you why it is like that.

The standard answer is to ask you to rephrase your question in terms of "how" or "what" to make it explicitly something that can be handled by science tools. Mind you, Ethan0178's earlier post is pretty good.

Yes I am constantly reminded not to ask the "why" questions in physics here on PF but it's very tempting for me to know the "why" even though I might not get anywhere. Anyways I'll replace "why" with "how" now onward.

I think you need to know the origin of inertia mass. Inertia mass comes from Newton's law.

First, according to Newton's second law, ƩF=ma, you know mass is the ratio of net force and its acceleration. But what is net force? We can't say net force is its inertia mass times its acceleration because that's a circular argument. So, how to answer this problem? Or, I should ask "how to not to use the unknown "net force" to get the idea of inertia mass?".

Let's use Newton's third law, suppose that there are two ball colliding each other.

ƩF₁ = m₁a₁
ƩF₂ = m₂a₂

if we could neglect the frictional force, then the net force of each ball comes from each other.

ƩF₁ = F₂₁ = m₁a₁
ƩF₂ = F₁₂ = m₂a₂

Next, using Newton's third law,

F₂₁ = - F₁₂

m₁a₁ = - m₂a₂

m₁/m₂ = - a₂/a₁

Finally, let's define mass of the one ball is 1 kg, then we can know the other ball by the above equation. So the Newton's 2nd law is well-defined.

Second, we can consider two mass points system.ƩF₁ = m₁a₁
ƩF₂ = m₂a₂

then sum them up

ƩF1 + ƩF2 = (F₂₁ + F_1,ext) + (F₁₂ + F_2,ext)

∴ƩF1 + ƩF2 = ƩF_sys,ext = m₁a₁ + m₂a₂ = (m₁ + m₂)a_c

a_c is the acceleration of center of the mass(system)

So, the reason why mass is additive is that's how we defined the center of mass.

If the acceleration of every particles in multiple-mass-points system is the same, then

you'll see the mass-additive property is guaranteed by Newton's second law.PS. Newton's law is only suitable to single particle system. Euler's law of motion is advanced Newton's law and it can be used in multiple particles system.

Doesn't that assume that force is additive (I don't need to know why is that)? Anyways thanks for your answer.

 

[Ethan0718]

Yes I am constantly reminded not to ask the "why" questions in physics here on PF but it's very tempting for me to know the "why" even though I might not get anywhere. Anyways I'll replace "why" with "how" now onward.

Doesn't that assume that force is additive (I don't need to know why is that)? Anyways thanks for your answer.

Yes, it assumes that force is vectorially additive!

I've give you some suggestions about this question at post 6.

 

[epenguin]

Some historian can maybe be more precise but I think this argument goes way back. I think Galileo argued, and even people before him, that if all objects did not travel at the same rate under gravity, well they wouldn't, and so composite objects would separate out as they fell. And they realized all objects they knew of were composite.

 

[DrDu]

As DaleSpam already pointed out, mass is not additive in special relativity. Maybe the most relevant example is the mass defect in the fission of uranium nuclei where the sum of the masses of the fission products is less than the mass of the original uranium nucleus.
The point is that in special relativity, mass is the energy of a system in it's rest frame but for a composite system, the particles bound together can still have kinetic (and potential) energy due to their relative motion. This extra energy contributes to the mass in addition to the mass of the parts of the system, i.e. the energy they would have if they were at rest and widely separated.

 

[Dale]

Yes, it assumes that force is vectorially additive!

I wouldn't call that an assumption. It is a definition. Forces are defined to be additive by Newtons 2nd law.

 

[BruceW]

As DaleSpam already pointed out, mass is not additive in special relativity.

yes. although to be specific, this is the 'invariant mass' which is not additive. Conversely, the 'relativistic mass' is additive.

 

[Andrew Mason]

So, the reason why mass is additive is that's how we defined the center of mass.

If the acceleration of every particles in multiple-mass-points system is the same, then

you'll see the mass-additive property is guaranteed by Newton's second law.PS. Newton's law is only suitable to single particle system. Euler's law of motion is advanced Newton's law and it can be used in multiple particles system.

This seems a bit circular to me. This is just saying that the additive property of mass follows from the additive property of force and the additive property of force is implicitly assumed in Newton's laws. So you might as well say that the additive property of mass is assumed.

The additive property of mass is simply an observed property of nature. As has been pointed out, it applies only to mechanical interactions between bodies in which the energy of the bond between masses divided by c² is much less than the rest masses of the separate bodies.

AM

 

[jambaugh]

Here is a "micro-ephany" I had once upon a time. Firstly relating to this thread I re-emphasize that mass=rest energy. So we come to the question of conservation of energy and thence in the relativistic treatment to conservation of momentum-energy.

What is the meaning of "potential energy"? Consider starting with a definition of energy simply as the kinetic energy of a system. It is clearly not conserved in general. One could say we define potential energy to "fudge the books" i.e. the definition of potential energy is to create a conserved "total energy". Note that we cannot observe total energy since we cannot observe potentials, there's gauge degrees of freedom involved. We can however compare differences and thus observe (with the properly chosen definition of potential) a net change in total energy and hence balance the books on conservation of energy.

I think (at least for me) this sheds some light on the "why" question although not in the way one may have expected. The "why" part has as much to do with how we conceptualize nature as with the essence of that nature. We break it down into components that facilitate our understanding and that dictates e.g. resolving positions in space+time and idealization of cases to translationally invariant in these, which when we via Noether's work, connect those assumed symmetries to necessarily co-assumed conserved quantities.

The question then becomes, "why is (by suitable definition of a potential) total energy-momentum conservable?"

That I think relates in a very subtile way to the way we conceptualize physics in general, once we begin by thinking of (reversible) action on systems whose only meaning are their effect on the system, we, of necessity, have an associative algebraic structure with inverses when considering the composition of actions. We in short have a group, and with the continuum we get a Lie group which to be physically actualizable must be a "potential symmetry" and thence invokes a corresponding "potentially conserved quantity".

To bring this down to Earth... we see conservable energy because we seek to see energy in terms of dynamical laws with duration over time in some unchanging way. We implicitly choose to see conservable energy and thus may define a conserved energy. It is built into the assumption that "the laws of physics we see here-today we assume can be extrapolated to there-tomorrow".

I think that is as far as we can push the "why" question within science. You can of course step outside science and assert a purposeful creator, or assert accidental coincidence... just be aware that "Elvis has left the building" so to speak.

Understand, I am not talking solipsism here, quite. Nature does as nature does, and we seek to understand nature, we do not "invent" nature. But the how has two conditions. Our understanding must correspond to nature (which we achieve by insisting on empiricism) and it must be meaningful to us (which affects form and formulation to some extent). Both of these conditions affect the form our representations of nature take and I'm here asserting that at present our invocation of the concept of energy is dictated by both these conditions. That point must be understood when asking the question "Why".

 

[DrDu]

Conversely, the 'relativistic mass' is additive.

Which one, the longitudinal or transversal one?
Seriously, the concept of "relativistic mass" is outdated.

 

[BruceW]

I mean 'relativistic mass' as a synonym for energy. It's not really outdated as far as I am aware. I can see why you might not like the term though.

 

[Simon Bridge]

There are two main threads here -

- one ends up in a "by definition"... in a nutshell: perhaps we have defined our terms in such a way that the quantity we call "mass" (inertia, in the original question) is additive?
We can proceed from here by considering what would happen if we made another definition.
Basically the models get annoying quite fast... which suggests that we define mass to be additive because that makes the math easiest ... what makes the math easiest is what we come to view as a "better model". It is like there is more than one way to say something, and we tend to prefer the "plain language" approach and leave convoluted arty language to entertainment.

- the other ends up with conservation of energy.
This looks more promising - if energy were not conserved, what would happen?
It would be possible to get energy out of nothing ... situations would quickly build up so something starts generating lots of free energy sometime in the early universe ... any cosmologists want to weigh in on what that would mean?