# Invariant Mass vs Proper Mass

pmb
"Invariant Mass" vs "Proper Mass"

I see that there are many people here who prefer the idea that the mass of a particle is the magnitude of the the particle's 4-momentum.

However that is known as "Proper Mass" and some simply say "mass." However that idea is limited in use. It can't be applied in general.

Refer to this diagram in what follows --
http://www.geocities.com/physics_world/4-momentum.gif

When people measure things in Newtonian Mechanics one tends to measure them "simultaneously." E.g. It would be incorrectm, Newtonian Mechanics to measure the momentum of particle #1 at t = t_1 and then measure the momentum of particle #2 at t = t_2 and then add them - that wouldn't be a very meaningful quantity.

In the diagram --
C_1 is the world line of a particle #1 which has a non-constant velocity.

C_2 is the world line of a particle #2 which has a non-constant velocity.

The 4-momentum of each particle is well defined along its own world line. But when you try to add them - trouble! The magnitude of the sum of the two 4-momenta will be invariant. But not meaningful.

In reference to the diagram ---

Observer in frame S - This observer could add the 4-momenta of the two particles but he'll be tempted to use the values he measures at the "same time" as measured in his frame. In this case that corresponds to Events A and B.

Observer in frame S' - This observer could add the 4-momenta of the two particles but he'll be tempted to use the values he measures at the "same time" as measured in his frame. In this case that corresponds to Events A and C.

Each observer will measure 4-momenta and each will get a 4-vector with an invariant magnitude. However they will disagree on the magnitude.

The problem vanishes when all the particles move force free.

This problem does not exist with relativistic mass.

Pete

marcus
Gold Member
Dearly Missed
The invariant mass is just another name for the rest mass.

John Baez (moderator of Usenet sci.physics.research) has a usenet physics FAQ and some physics tutorials at his website which covers this. You may have misunderstood. Your post suggests that the invariant masses of the particles change
thru time as they go thru their motions (as in your diagram) but they dont.

The rest masses of the particles stay the same all the way along.
A particles rest mass or invariant mass is just the inertia of the particle measured when it is at rest.

http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html

I will paste in an exerpt, so you dont have to surf to it if you dont want. He says here that:

"Of the two, the definition of invariant mass is much preferred over the definition of relativistic mass. These days, when physicists talk about mass in their research, they always mean invariant mass. "

He says:

"At zero speed, the relativistic mass is equal to the invariant mass. The invariant mass is therefore often called the "rest mass."

He gives a formula for calculating the invariant (i.e. rest) mass of a moving particle from the components of the 4-momentum:

m_0 = sqrt(E^2/c^4 - p^2/c^2)

And also a formula for computing the "relativistic mass" from the invariant mass:

m_r = m_0 /sqrt(1 - v^2/c^2)

It is interesting that the magnitude of the 4-momentum does not change as the particle in your picture speeds up and slows down and goes thru its paces. The minus sign in the formula for the magnitude takes care of that. this may be all familiar to you but
there are some differences in detail as regards your post.

[[Does mass change with velocity?
There is sometimes confusion surrounding the subject of mass in relativity. This is because there are two separate uses of the term. Sometimes people say "mass" when they mean "relativistic mass", m_r but at other times they say "mass" when they mean "invariant mass", m_0. These two meanings are not the same. The invariant mass of a particle is independent of its velocity v, whereas relativistic mass increases with velocity and tends to infinity as the velocity approaches the speed of light c. They can be defined as follows:

m_r = E/c^2
m_0 = sqrt(E^2/c^4 - p^2/c^2)

where E is energy, p is momentum and c is the speed of light in a vacuum.

The velocity dependent relation between the two is

m_r = m_0 /sqrt(1 - v^2/c^2)

Of the two, the definition of invariant mass is much preferred over the definition of relativistic mass. These days, when physicists talk about mass in their research, they always mean invariant mass. The symbol m for invariant mass is used without the subscript 0. Although the idea of relativistic mass is not wrong, it often leads to confusion, and is less useful in advanced applications such as quantum field theory and general relativity. Using the word "mass" unqualified to mean relativistic mass is wrong because the word on its own will usually be taken to mean invariant mass. For example, when physicists quote a value for "the mass of the electron" they mean its invariant mass.

At zero speed, the relativistic mass is equal to the invariant mass. The invariant mass is therefore often called the "rest mass"... ]]

• Batuhan Unal
pmb
marcus wrote -

re - "The invariant mass is just another name for the rest mass."

Invariant mass is the more general name which applies to systems of particles. And that, of course, means it applies to single particles as a special case. The name "invariant" is used over "rest" since there's nothing which is at rest. Only thing they have in common in this respect is a zero-momentum frame,

re - "Your post suggests that the invariant masses of the particles change thru time as they go thru their motions (as in your diagram) but they dont."

That is not what the diagram suggests at all. Any single point particle has in invariant mass. Even the system of partiles has an invariant mass. However with the system it's only a meaningful quantity when a well defined meaning can be given to it. For example: In the diagram there are two particles whose velocities are not constant. Therefore each particle's 4-momentum is not constant. The value of the 4-momentum of each particles is different at each point along its world line. Each individual particle still has a well defined 4-momentum. However if you want to add the two 4-momenta together then you have to choose which events to evaluate them at. Different events -> different 4-momenta -> different sum -> different magnitude of sum. If the 4-momenta of *all* the particles is constant then this is not a problem since the 4-momenta along each worldline in that case varies. But in the case where any of the 4-momenta are not constant, then there is no unique way to add them (if they change velocity only upon contact forces then its not a problem either). Each observer, had he measured the values of the components of the 4-momenta "at the same time" will then get a different 4-momentum since he is evaluating the 4-momenta at different events.

So the meaning of "invariant mass" when the system has external forces acting on it is ill-defined and thus not physically meaningful.

This is similat to having a non-constant electric field. The electric field is an example of a Cartesian vector. The magnitude of a Cartesian vector is independant of the system of coordinates. I can take the value of the field at different points and add them. The result will still be a Cartesian vector but its not physically meaningful. The sum of Vectors has a unique meaning when all vectors are evaluated at the same point. Or if the field is constant.

re - "The rest masses of the particles stay the same all the way along."

Each particle's rest mass does yes.

re - "Of the two, the definition of invariant mass is much preferred over the definition of relativistic mass. These days, when physicists talk about mass in their research, they always mean invariant mass."

One of the reason's I posted this was because wide sweeping comments like that are simply incorrect.

re - "It is interesting that the magnitude of the 4-momentum does not change as the particle in your picture speeds up and slows down and goes thru its paces."

No individual 4-momenta changes that is true.

re - " The minus sign in the formula for the magnitude takes care of that. this may be all familiar to you but there are some differences in detail as regards your post."

I'm very familar with it yes. But you have to use caution when using it because it's not always meaningful to use. But relativistic mass is.

It appears that you didn't understand the way I phrased my post. Here is the way it's described in "Gravitation and Spacetime - Second Edition" Ohanian, and Ruffini, page 114 - footnote

(see http://www.geocities.com/physics_world/ohanian.htm for meaning of symbols)
* Since P^m is the sum of two four-vectors (see Eq. ), at first sight it seems obvious that it should be a four-vector. However there is a catch: in Eq.  it is implicitly assumed that two four-velocities are evaluated 'at the same time'. Since Lorentz transformations do not preserve simultaniety, the meaning of 'at the same time' is different in two frames, and this difference could complicate the transformation law of the sum . However, for the special case of free particles, Eq.  is obviously correct since in this case the four velocities are constant, and it is irrelavant whether the velocities are evaluated simultaneously or not. It then follows that Eq.  must be correct for particles that were free (did not impact) at some time in their past: the momenta P^m and P'^m are constant by hypothesis, and if they had the transformation law  at one time they must keep it forever.

It thought that needed explaining since it sounded like the two 4-momenta wasn't a 4-vector. It is. There is jut no unique way to add them.

Pete

marcus
Gold Member
Dearly Missed
Pete, thanks for the detailed answer and for
transcribing the passage from Ohanian+Ruffini.
I understand what you are driving at much better now.

I must admit that I still do not fully understand what
appeals to you about the concept of "relativistic mass"
but the concepts one uses are to some extent matters of taste
and to that extent not subject to dispute.

pmb
Originally posted by marcus
Pete, thanks for the detailed answer and for
transcribing the passage from Ohanian+Ruffini.
I understand what you are driving at much better now.

I must admit that I still do not fully understand what
appeals to you about the concept of "relativistic mass"
but the concepts one uses are to some extent matters of taste
and to that extent not subject to dispute.

It appeals to me because of relativity. In every sense of the word "mass" its "relativistic mass" that applies. I've seen people being drawn to wrong conclusions because they thing in terms of "rest mass" and not students but relativists.

If you'd like - e-mail me at [email protected]. I'll forward a copy of the paper I wrote on the subject. It explains it all in detail.

Meanwhile see

http://www.geocities.com/physics_world/photon_in_box.htm

Pete

pmb
Actually now that you mention it I too must admit that I still do not fully understand what appeals to you about the concept of "rest mass"???

If you were to define "rest mass" how would you define it? If you define it in terms of E and P then how do you define P?

Pete

Alexander
The confusion is because it is the relativistic mass which increases body's inertia and gravity. (And we got used to associate gravity and inertia with something called "mass" rather than "energy", that is why we consider increase in inertia/gravity of body (due to change of reference system, say from stationary to moving versus body) as being increase in relative "mass" rather than in relative "energy").

pmb
Alexander wrote
The confusion is because it is the relativistic mass which increases body's inertia and gravity. (And we got used to associate gravity and inertia with something called "mass" rather than "energy", ..

IMHO - They aren't the same thing. They're related but they aren't identically the same. The relation E = mc^2 does not mean that mass and energy are the same thing but that when there is an increase in one there is a decrease in the other. I.e. If a body radiates energy of an amount L then its rest mass decreases by any amount L/c^2.

Saying "E=mc^2 means mass and energy are the same thing" is like saying that "E=hf means frequency and energy are the same thing" which sounds just plain silly to me.

The very meaning of the terms are different. One corresponds to a constant of motion (i.e. Energy) and one corresponds to inertia nad gravity.

re - ".. we consider increase in inertia/gravity of body (due to change of reference system, say from stationary to moving versus body) as being increase in relative "mass" rather than in relative "energy")."

I fully agree! :-)

Pete

marcus
Gold Member
Dearly Missed
Originally posted by pmb
Actually now that you mention it I too must admit that I still do not fully understand what appeals to you about the concept of "rest mass"???

If you were to define "rest mass" how would you define it? If you define it in terms of E and P then how do you define P?

Pete

As you may have guessed, I do not care what concepts you prefer to use----suit your own taste. But you ask what appeals to me about my concept of mass, which i think is the contemporary mainstream one.

And you ask how mass (it is redundant and confusing to say "rest" mass because it suggests there is some other sort) is defined.

So I will try to reply. Traditionally mass is inertia, and a body's inertia is independent of the direction you push it.

Around 1904 and 1905 Lorentz and maybe Einstein independently realized that a moving body does not have an unambiguous inertia because the acceleration response depends on the direction the force is applied---in the line of motion or crossways. Lorentz proposed the terms "longitudinal mass" and "transverse mass". I am not too sure about the history but I think it was around then and it wasnt just Einstein---Lorentz was in on it.

Anyway once you realize that then the rest inertia is the only kind of inertia there is. Mass has no meaning except as the inertia of a body at rest.

Also Einstein never claimed E=mc^2 except for a particle at rest.
If the particle is moving the formula is
E^2 = m^2c^4 + p^2c^2.
This reduces to the familiar E=mc^2 in the special case where the thing is at rest.

One of the PF mentors Tom suggested an introductory special relativity text for review, in case you want to bring your definitions up to date. This stuff about mass is in section 6.4.
The book is by a guy at Princeton Inst for Adv. Studies, David Hogg, and the date is 1997. He says "relativistic mass" is "archaic and ugly" and says "if you are old enough you may have heard" of it and if so "forget all you ever heard."
Einstein deplored the concept of "relativistic mass" and urged that not be taught. A letter he wrote in the 1940s about this is quoted sometimes. There seems to be a growing consensus that we need to get rid of the concept because of the confusion it causes.

But I personally do not care what you mean by mass---ultimately taste in language is an individual matter. What I mean by it is inertia so I have no other choice.

marcus
Gold Member
Dearly Missed
Einstein's disapproval of "relativistic mass"

In a 1948 letter to Lincoln Barnett, Einstein wrote

"It is not good to introduce the concept of the mass M = m/(1-v^2/c^2)^1/2 of a body for which no clear definition can be given. It is better to introduce no other mass than `the rest mass' m. Instead of introducing M, it is better to mention the expression for the momentum and energy of a body in motion."

Lincoln Barnett was the author of "The Universe and Dr. Einstein" (Norton, 1948)

A photo of this letter was included in an interesting article on mass by Lev Okun in the June 1989 issue of Physics Today.

pmb
marcus - I'm a bit confused here!? You specifically stated
I must admit that I still do not fully understand what
appeals to you about the concept of "relativistic mass" ...

Then you tell me
I do not care what concepts you prefer to use.
For future referance - When you say "I don't understand" then I should not assume that you *want* to understand?

re - "And you ask how mass (it is redundant and confusing to say "rest" mass because it suggests there is some other sort) is defined."

Actually "rest" mass is *not* redundant if both "relativistic mass" and "rest mass" are being discussed. And I didn't ask how mass is defined. I asked

"If *you* were to define "rest mass" how would you define it?"

re - "So I will try to reply. Traditionally mass is inertia, and a body's inertia is independent of the direction you push it."

That's only because you're thinking in terms of "F=ma" which is incorrect as a defintion. F = dp/dt where p=mv. Inertia represents a resitance of a body to change its momentum

And in 1906 Max Plank showed that the force on a particle is given by

F = dP/dt

where P = Mv where
F = force vector
P = momentum vector
v = velocity vector
M = relativistic mass

This then became the proper definiton of "mass" since its the one that fully fits into general relativity.

re - "Anyway once you realize that then the rest inertia is the only kind of inertia there is. Mass has no meaning except as the inertia of a body at rest."

That's because you're using an incorrect definition of "inertia".

Also Einstein never claimed E=mc^2 except for a particle at rest. Not true at all! In fact Einstein specifically stated that light has mass. In "The Evolution of Physics" Einstein was commenting on the observation made by an observer inside an accelerating elevator that light is ‘weightless’ he writes -

But there is, fortunately, a grave fault in the reasoning of the inside observer, which saves our previous conclusion. He said: “A beam of light is weightless and, therefore, it will not be affected by the gravitational field.” This cannot be right! A beam of light carries energy and energy has mass.

More appropriately he spoke on this in a paper he published in 1906 regarding his famnous (what has become to be known as) "photon in a box" thought experiment From "The Principle of Conservation of Motion of the Center of Gravity and The Inertia of Energy," Albert Einstein, Annalen der Physik (1906) - Speaking of a pulse of light with an energy "E" Einstein writes

However, if one assumes that any energy E possesses the inertia E/c^2, then the contradiction with the principle of mechanics disappears.

re - "If the particle is moving the formula is E^2 = m^2c^4 + p^2c^2.
This reduces to the familiar E=mc^2 in the special case where the thing is at rest."

But what is "m" to begin with? There are 3 quantities which are in that equation. E we know we can measure. That leaves "p" and "m". But "p" is defined as p = gamma*m*v so you're have to define "p" in a way that does not require this relation otherwise there is a cirluclar arguement.

re - "One of the PF mentors Tom suggested an introductory special relativity text for review, in case you want to bring your definitions up to date."

My defintions are "up to date" thank you. Seems that you don't have a better understanding of how mass is defined - not everyone defines it that way. But you seem to believe that! That assumption is an error.

re - " This stuff about mass is in section 6.4. The book is by a guy at Princeton Inst for Adv. Studies, David Hogg, and the date is 1997. He says "relativistic mass" is "archaic and ugly" and says "if you are old enough you may have heard" of it and if so "forget all you ever heard."

So? That's his "personal" opinion and not one universally accepted. There is even recent GR text are out which is written by one of the more well known and well respected relativist who defines it differently. It's rather condescending to tell people that what they use is "archaic" when there's nothing wrong with the idea.

re - "Einstein deplored the concept of "relativistic mass" and urged that not be taught."

Nope. He suggested not to use the relation "M = gamma*m" and that does not apply to light. As I've shown you, when it comes to light - its a different story. In Einstein's ussage - If there is light with energy E then it has a mass of E/c^2.

Pmb

Alexander
Guys, consider the following example: electron is slowly moving back and forth a big massless black box (fine, very light box if you don't like the word "massless"). Let's say electron velocity is much less than the speed of light c (it indeed is if the box is big).

You don't know what is inside the box - mass of the box for you is then simply m, where m is rest mass of electron.

Now let's say you squized box to much smaller size. Due to uncertainty principle dxdp~h elctron now is moving inside like crazy (with large p). Let's say, you squized box to such size that electron is moving with ultrarelativistic momentum (p^2/2m >> mc^2).

Now what is the mass of the box (again, assuming that you don't know what is inside)?

Well, now mass has to be way more than original mass of electron m. In fact, it is now so huge that you can even neglect original mass of electron (thus it does not matter if you take an electron, or, say, a photon, or neutrino, etc). Weighing (or pushing) the box will give you the ONLY information about its inertia/gravity.

Now, let's say, the box already comes in small size only. And again, you don't know what is inside (say, box is so small that nothing in your posession is smaller than box, thus nothing can be used to investigate box content/structure). May be, there is just a massless but moving like crazy photon trapped in.

What would you conclude about MASS of this tiny box? Would you just say that it is zero while you scale or spring says "1 kg" ?

(By the way, all "elementary" particles are such "boxes" - we simply don't have fine enough scope at present to "see" what is inside, and thus label them "elementary" or "fundamental". And we assign them "rest masses" as we measure these masses from outside, while in fact there may be no "rest" mass inside whatsoever - only energy).

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