# Speed Increases Mass?

#### mijoon

pmb_phy said:
Actually Einstein held that the mass of a particle will be altered in a gravitational field. That notion of mass is identical to the notion of relativistic mass in definition..

A particle is always at rest in its own reference frame so as such I don't understand what you mean by it seeing spacetime differently. Can you clarify please?.
You have answered your own question . The particle is at rest in it's own reference frame, but moving in others . It's (hypothetical) clocks and measuring rods must measure time and space differently than those in other reference frames.

Particle's don't make measurements. Observers do. Hence the observer dependant nature of mass etc.
Beauty may be in the (subjective) eye of the beholder, but mass is an objective property of the particle itself.

Why? Its quite possible to do all calculations in special relativitywithout even knowing what proper velocity is. If you want to stick to 4-vectors and Lorentz invariants then you really shouldn't use the term "speed" or "velocity" etc. In such a case the magnitude of a particle's speed is the magnitude of the particle's 4-velocity and that is always c.
Arithmeticly you may get the right answer, but that does not answer the original question, "where does the extra mass come from?" .
As you, yourself said, the particle is always at rest in its own frame. Therefore, it always has the same mass in that system.

#### mijoon

DanielK said:
OK, I just wanted to emphasize that the concept of mass is clear in relativity and there is nothing to worry about.
Not quite as clear as we'd like to think.....But it's all good , clean fun.

#### Chronos

Gold Member
Hmm, relativistic mass increase only has meaning when you are talking about inertial resistance. Otherwise, highly accelerated objects would acquire infinite gravitational attraction and a single proton could gravitationally collapse the entire universe. Obviously, that has not happened.

#### pmb_phy

According to relativity mass is invariant because it is the "length" of the energy-momentum four vector (mass-shell condition).
[/quote]
That's the invariant mass, aka proper mass, aka rest mass.
That is rest mass is what relativity calls "mass".
A theory doesn't call something mass. People do. Some relativists call proper mass "mass" and some relativists call relativistic mass "mass". To understand what I mean by this think of the reason people call the magnitude of a timelike spacetime displacement the proper time and the time component of that same 4-vector time. Would you prefer to get rid of the term "proper time" and call that magnitude "time" and then forget about naming the time component? Same thing with a spacelike spacetime displacement. Now consider 4-momentum. The magnitude is proper mass and the time component is mass. The magnitude of 4-momentum equals the time component as measured in the rest frame. Why would you want to call the time component something other than mass and then call that same time component "mass" when the frame of reference is just right?

By the way, it is not true that proper mass is defined that way. It would be a circular defintion.
"Mass increasing", "longitudinal and transverse mass" are concepts that come into consideration when one wants to insist on the usual, newtonian sense of mass, as in the equation F=m a, or one misinterprets the mass-energy equivalence by thinking that it holds for a moving mass, too, thereby attributing speed dependency to mass.
I'm sorry to inform you but that is a common misconception. In the first place it is very far from being a misconception. Especially if one knows how each term is defined. E = mc2 does not define relativistic mass. Far from it. That is a relation which is derived from the definition of relativistic mass as the m in p = mv and the conservation of energy. I.e. m is defined such that p = mv is a conserved quantity. For a tardyon m is a function of speed, m = m(v). Proper mass is then defined as m0 = m(0).

Many relativists (e.g. J.D. Jackson, Hans C. Ohanian, etc) do this in a very similar way. They assume that p = Mv is a conserved quantity where p is 3-momentum and v is 3-velocity. M is assumed to be a function of speed, i.e. M = M(v) where v is the speed of the particle. They then define m = M(0) as the "mass" of the particle. Different strokes for different folks.

Inertial energy E is then defined as the sum of kinetic energy and rest energy. From those two definitions and the derivation of m = gamma*m0 for tardyon's then the relation E = mc2 is then derived. It would be a serious error to think that relativistic mass is defined as m = E/c2. It has never been defined that way in any serious treatment of special relativity. When Tolman and Lewis first proposed this definition (and its a definition - note that this is the topic of this thread so far - definition). I.e. Tolman and Lewis defined the mass of a particle to be the m such that p = mv is conserved. All derivations of the relationship for relativistic mass I've ever seen employ this notion.

For the derivation of E = mc2 for a tardyon see
http://www.geocities.com/physics_world/sr/work_energy.htm
(I use T for E in that page. I like to use E for total energy = kinetic energy + potential energy + rest energy)

This derives E = mc2 in a manner similar (not identical) to Einstein's 1905 derivation (mine is clearer).
http://www.geocities.com/physics_world/sr/mass_energy_equiv.htm
Einstein's derivation made no distinction between rest mass and relativistic mass since he used a low speed scenario in which the rest mass = relativistic mass. I do not.

This is a description of Einstein's 1906 derivation
http://www.geocities.com/physics_world/sr/einsteins_box.htm
In the second section of that paper Einstein basically stated that light has mass. This was the first, but not the last, place he did that.

There is nothing Newtonian about F = dp/dt. It's as valid in special relativity as it is in Newtonian theory. Many modern texts use it in their relativity sections. It was not Newton who wrote F=m a. That was Euler who did that. Newton wrote F = dp/dt. The increase in mass (relativistic mass) is not related to any force equation. It is defined and derived soley in terms of momentum. Nothing else whatsoever.

re - "DW answered this." - I don't know what dw wrote. I blocked all his posts due to his past posting habits. i.e. He has a tendacy to ignore all facts given to him and all corrections made against his claims. So I don't bother anymore.

The topic of this thread is the speed depenance of mass. That means that one is speaking of observing/measuring the mass of a particle in a frame of reference in which the particle is moving, not at rest. Rest mass is the mass as measured in the frame of reference in which the body is at rest. Don't forget, relativity has a lot to do with what different inertial observers measure from different frames of reference. In fact all measurements in the lab are such measurements. E.g. one can measure the relativistic mass of a moving charged particle, of a known charge, by measuring the radius of curvature of its path in a cyclotron. One then uses

p = qBr = mv

or

m = qBr/v

re - "Hmm, relativistic mass increase only has meaning when you are talking about inertial resistance."

That' incorrect. If you've ever read or followed the derivation of the speed depedance of relativistic mass then you'd know that the derivation has nothing to do with force or acceleration. It all has to do with the conservation of momentum. This is what is known as Weyl's definition of mass. Planck did what you're thinking of in 1906 after Einstein's paper was published. Planck showed that the Lorentz force can be expressed as

F = dp/dt = q(E + vxB)

where p = gamma*m0. That was the first sign in relativity of relativistic mass and, as I recall, Einstein liked it better than his notions in his 1905 paper (but don't quote me on that). However this definition was still connected to electrodynamics. It wasn't until later that the relation was shown to be true apart from electrodyamics. That was done by Tolman and Lewis and it was in a landmark paper. After that paper relativity was brought into the domain of mechanics and out of its restricted domain of electrodynamics (i.e. relativity articles started to be published in mechanics journals etc).

Regarding m is defined such that p = mv is conserved - Pick up any text in which the expression for relativistic mass (or relativistic momentum) is derived and that is what you'll see (E.g. see Jammer's new book on mass). If you don't have a text handy then see

http://www.geocities.com/physics_world/sr/inertial_mass.htm

re - "Otherwise, highly accelerated objects would acquire infinite gravitational attraction and a single proton could gravitationally collapse the entire universe. Obviously, that has not happened."

Do you mean objects with high speed? If so then the faster an object moves the greater is gravitational attraction. See Measuring the active gravitational mass of a moving object, D. W. Olson and R. C. Guarino, Am. J. Phys. 53, 661 (1985). The faster a body moves the greater its weight too (due to increase in passive gravitational mass). Also, its quite common to see modern GR texts speak of the mass of radiation. Ohanian's text Gravitation and Spacetime - Second Ed. is one such example.

Note: Sometimes people tend to confuse "invariant" with "independant of the observer." These are not the same concepts. Invariant means "independant of the coordinate system used to evaluate the quantity" whereas "observer independant" means "makes no reference to an observer". E.g. if you take the scalar product the four momentum of a particle with the four velocity of a particular observer, then divide by c2, you'll get the relativistic mass as measured by that particular observer.

For calculation see bottom of
http://www.geocities.com/physics_world/ma/invariant.htm

The scalar product is an invariant, i.e. it does not matter which set of coordinates you use to evaluate it. A similar thing happens with the electric field. If you take the scalar product of an observers 4-velocity with the EM tensor then you'll get another 4-vector. That 4-vector is called the electric field 4-vector. For this definition see Wald page 64, Eq. 4.2.21 or Thorne and Blanchard's new text (somewhere online - I forget where).

You seem to think I haven't considered all these objections. Not only have considered all these objections but I covered them, and more, in that paper I wrote on the topic of mass in relativity which I posted a link here to several times. However I found many typos an gramatical errors in it and have taken it off line. It will be back on the internet in late fall when I'm better (herniated disk).

Later folks! Have a great summer!

Pete

ps - For those who are interested in this topic see the section on relativistic mass in Concepts of Mass in Contemporary Physics and Philosophy, Max Jammer. Jammer explains why P0 = mc where m = relativistic mass. I.e. he argues why that should be the definition of the time component.

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#### DanielK

pmb_phy said:
A theory doesn't call something mass. People do. Some relativists call proper mass "mass" and some relativists call relativistic mass "mass". To understand what I mean by this think of the reason people call the magnitude of a timelike spacetime displacement the proper time and the time component of that same 4-vector time. Would you prefer to get rid of the term "proper time" and call that magnitude "time" and then forget about naming the time component? Same thing with a spacelike spacetime displacement. Now consider 4-momentum. The magnitude is proper mass and the time component is mass. The magnitude of 4-momentum equals the time component as measured in the rest frame. Why would you want to call the time component something other than mass and then call that same time component "mass" when the frame of reference is just right?
The reason for why people call the magnitude of a timelike spacetime displacement the proper time is that it has no spacelike component in the frame of the observer. In other words, the spacetime interval, which is invariant, is decomposed into a clear timelike and a clear spacelike interval, from which the spacelike one in this special reference frame vanishes.

Why do you call the timelike component of a 4-momentum vector mass? As far as I know it is energy - this is where the other name of this quantity comes from: energy-momentum four vector. However, you may measure energy in mass units, see my other note below.

pmb_phy said:
By the way, it is not true that proper mass is defined that way. It would be a circular defintion.
Yes, that's exactly what I say, too.

pmb_phy said:
I'm sorry to inform you but that is a common misconception. In the first place it is very far from being a misconception. Especially if one knows how each term is defined. E = mc2 does not define relativistic mass. Far from it.
I didn't assert the contrary. What's more, I think the same. This definition would end in tautology.

pmb_phy said:
That is a relation which is derived from the definition of relativistic mass as the m in p = mv and the conservation of energy. I.e. m is defined such that p = mv is a conserved quantity. For a tardyon m is a function of speed, m = m(v). Proper mass is then defined as m0 = m(0).

Many relativists (e.g. J.D. Jackson, Hans C. Ohanian, etc) do this in a very similar way. They assume that p = Mv is a conserved quantity where p is 3-momentum and v is 3-velocity. M is assumed to be a function of speed, i.e. M = M(v) where v is the speed of the particle. They then define m = M(0) as the "mass" of the particle. Different strokes for different folks.

Inertial energy E is then defined as the sum of kinetic energy and rest energy. From those two definitions and the derivation of m = gamma*m0 for tardyon's then the relation E = mc2 is then derived. It would be a serious error to think that relativistic mass is defined as m = E/c2. It has never been defined that way in any serious treatment of special relativity. When Tolman and Lewis first proposed this definition (and its a definition - note that this is the topic of this thread so far - definition). I.e. Tolman and Lewis defined the mass of a particle to be the m such that p = mv is conserved. All derivations of the relationship for relativistic mass I've ever seen employ this notion.
I don't think that the definition of relativistic momentum has anything to do with the definition of mass. We call relativistic energy and momentum what we actually do because these are the quantities that are conserved in relativistic processes and these are the quantities that in a low speed approximation give back the familiar formulas used in nonrelativistic dynamics. Except for the rest energy term in the expression of total energy. This is new compared to what we are used to in nonrelativistic dynamics. This means that a body at rest contains an amount of energy proportional to its mass. This enables us to measure energy in mass units and vice versa. I guess this is what you do when you say that timelike component of 4-momentum vector is mass. But it is senseless to call the factor mass in front of v in momentum or c2 in energy. Especially, if you introduce inertial energy as a sum of two different kinds of energy, since in this step you accept that there are two different things accounting for measurable energy: the rest mass and relative motion. Why should we consider energy as something that arises from only a speed dependent mass? It's nonsense. (As the saying goes here, in Hungary: "an iron ring made from wood" ). Let's reverse it, keep the speed independent definition of mass and consider relativistic corrections to energy and momentum as coordinate effects. In the whole theory of relativity there is one basic principle that underlies all relativistic effects: the relativity of space and time. This one fact is responsible for mass increasing, too. Keeping this in mind, I found it somewhat incorrect to say that mass increases with speed. No, mass has no such inherent nature like space and time do, because mass is a quantity in our physics textbooks while space and time are fundamental concepts in our understanding of the universe and have been proved to be relative. Everything other is based on this. Moreover, speed dependent mass loses the right to be regarded as the measure of inertia because it can be different in different processes (longitudinal and transverse mass).

In my opinion, the question of what we should call mass in relativity is mainly phraseological. I just say that stating that mass is relative is attributing relative property to something that borrows this from spacetime.

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#### pmb_phy

So far people here have began all discussion with one basic starting thought/assumption - Mass is an inherent property of a particle. That is something being assuming and anything which follows is based on that assumption. However that means you've assumed a definition right from the start and are arguing that you can prove that mass does not depend on speed. That is an illogical approach to any discussion. There is no sense in discussing the properties of mass without a clear definition to begin with. Once you've chosen a definition the rest follows by derivation.

In relativity there are two masses currently in use: relativistic mass and proper mass.

Relativistic mass is defined as the coefficient of proportionality between 3-momentum and 3-velocity.

proper mass is defined as the coefficient of proportionality between 4-momentum and 4-velocity.

Relativistic mass has all the properties associated with the three aspects of mass: Inertial mass, passive gravitational mass and active graivtational mass.

Its incorrect to claim that anyone can deduce that "mass" is one or the other. All that is possible is to make a choice and do so by arbitrary, albeit personally preferred, convention.

I've posted an explaination to your first question but it was erased when I hit the wrong button and I'm not about to repeat it [damn computers! :)]. However it can be found in Jammer's book mentioned above on pages 49-50 (i.e. why mass is the time component of 4-momentum). Some Relativists (e.g. Rindler, Jammer) simply define 4-momentum that way (relativistic mass is the time component) and that is all there is to that. But if dX = (cdt, dx, dy, dz) and P = (cp0, px, py, pz) then it follows that p0 should be called the time component of P and in doing so it follows that

p0 = m = gamma*m0 = relativistic mass.

Assuming, of course, you don't mind callind "dt" the time component of dX? :-)

I've addressed all objections to everything on relativsitic mass in "On the concept of mass in relativity" which is online at - http://www.geocities.com/physics_world/ (I put it online for you).

I took it down because there are grammatical errors and typos in it. It will be proof read later this year and the correct version will be back up when my back is better and can sit at a computer for a lenght of time without hurting myself. I shouldn't be posting now but I wanted to address your comments and place that paper back on line.

I could give an answer to each objection posted on this newsgroup since I came here but it'd be short due to the room allowed here. Short answers are not complete. Complete answers are detailed and hence the article I wrote.

Please note that I'm not on a crusade. I'm just stating facts, facts that are in the relativity literature.

Pete

ps - At least consider reading Jammer's new book. It certainly can't hurt you.

#### pmb_phy

By the way, about the notion of mass having "always" been an inherent property of matter in classical mechanics. Consider Galileo's comment
Thus, there are mathematical properties, inherent in matter; but mass, although mathematizable, is not one of them, for it is another name for matter itself and distinguishes it from abstract matter which is geometry. Physical reality and mass are two names for the same thing which possesses inertial motion, whereas geometrical shapes do not posses it. Hence mass cannot be defined in terms of anything else; it is primum.
Pete

#### DW

pmb you have been told before, arguement by analogy is not a logical inference and stop accusing me of not understanding etc. The coordinates of time and space are what are relative in relativity. Physicaly complete real things are tensorial or invariant and as such do not depend on coordinates for their existence and as such are not relative. So coordinates in the displacement four-vector should not be treated as analogous to the length of the momentum four vector. Einstein's most fundamental postulate is that the laws of physics do not depend on frame. As such a complete expression for mass can not depend on frame. No one is arguing against the fact that the outdated references you use successfully modeled special relativistic behavior with Planck's relativistic mass concept. What you are being told is that it is not how modern relativity is done wherein mass does not depend on speed or on anything coordinate dependent for that matter. Saying mass is defined such that p = Mv is conserved is absurd for the following reason. If Mv is conserved then so is kMv which leaves whether one should use M or any arbitrary value cons*v as the mass so that cons*v is conserved utterly ambiguous. Instead modern relativity defines particle mass in the presence of a vector potential as the positive root for m in
$$m^{2}c^{2} = |g_{\mu }_{\nu }(P^{\mu } - qA^{\mu })(P^{\nu} - qA^{\nu })|$$ and has done such ever since Dirac's work on the matter. This does not depend on speed. This does not depend on position with respect to gravitational sources. This is invariant. Equivalent to this, the mass for a particle is the length of the momentum four vector of the first kind $$p^{\mu }$$, not its time element. From this The mass of a particle in special relativity is what Einstein said should be referred to as mass long after Planck Lewis and Tolman's work concerning relativistic mass. And, as for inertial mass I already explained in this very thread how that is defined. Your assertions like that are utter nonsense. For another example:
There is nothing Newtonian about F = dp/dt.
roflmao
Since when is there nothing Newtonian about Newton's second law?! And yes that absolutely is the Newtonian expression with the exception that you mixed up the four-vector F with the ordinary force f as differentiated by caps from the context already set in this thread. The relativistic version of Newton's second law is:
$$F^{\lambda } = \frac{Dp^{\lambda }}{d\tau }$$

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#### reilly

Talk to a particle physicist and, chances are he will say, "Sure, mass increases with speed." Push a little harder, and she will draw the distinction between rest mass and what most of us call relativistic mass -- gamma*(rest mass). But in doing kinematics, energy(kinetic) = gamma*(rest mass), so most of the time we use the term energy for gamma*(rest mass), just as momentum is gamma*v*(rest mass). Physicists are not always as rigorous as they could be with language and math. (When potential energy is involved, things get a little tricky. But all this stuff can be found in any intermediate or advanced text on relativity.) And, of course, the increase in relativistic mass, or energy comes from the work done by the force accelerating the particle.
Regards,
Reilly Atkinson

#### DW

reilly said:
Talk to a particle physicist and, chances are he will say, "Sure, mass increases with speed." Push a little harder, and she will draw the distinction between rest mass and what most of us call relativistic mass -- gamma*(rest mass). But in doing kinematics, energy(kinetic) = gamma*(rest mass), so most of the time we use the term energy for gamma*(rest mass), just as momentum is gamma*v*(rest mass). Physicists are not always as rigorous as they could be with language and math. (When potential energy is involved, things get a little tricky. But all this stuff can be found in any intermediate or advanced text on relativity.) And, of course, the increase in relativistic mass, or energy comes from the work done by the force accelerating the particle.
Regards,
Reilly Atkinson
You are talking to a physicist. ME. And I am telling you otherwise. Once more see-
http://www.geocities.com/zcphysicsms/chap3.htm

#### quartodeciman

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#### GRQC

quartodeciman said:
Those are all layman overviews of special relativity, not advanced modern treatments on the subject. I wouldn't take them as absolute gospel. Take it from a physicist who specializes in relativity: there is no such thing as "relativistic mass" nor is there such a thing as "rest mass". There is only one quantity -- "mass" -- which is invariant in all reference frames.

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#### Tim

If the mass is constant (as Einstein believed) , then the "something" is connected to the acceleration. When the particle is at rest in the observer's reference frame, it "sees" spacetime the same way as the observer does. However, as it accelerates, it begins to "see" spacetime differently. It undergoes the famous dilations and contractions. How it "sees" spacetime may be calculated using the Lorentz transforms. The faster it goes, the more differently it "sees" spacetime.
Remember that the force is being applied to the particle itself, not to the observers reference frame, so it is the particles system of measuring the universe that counts. So we must use it's PROPER velocity, not its reference frame velocity.This is why, it will get closer and closer to c, but never quite gets there.
DW's great site gives the correct formulas for calculating these things.

Would this be the same as when one is sitting in a car going at a certain speed. To the observer in the car things don't like as fast as to an observer outside the car seeing how fast the car is traveling?

Staff Emeritus
Gold Member
Dearly Missed
The particle sees length and time the same just as if it were at rest. It sees the LAB lengths shrunk and the lab time dilated. And the lab sees IT'S lengths shrunk ant time dilated. And this is a real effect not illusion. For example the lab sees the fast moving particle's lifetime before decay as thousands of times longer than the particle itself experiences, and this enables experiments on the particle that couldn't otherwise be done.

#### DW

The particle sees length and time the same just as if it were at rest. It sees the LAB lengths shrunk and the lab time dilated. And the lab sees IT'S lengths shrunk ant time dilated. And this is a real effect not illusion. For example the lab sees the fast moving particle's lifetime before decay as thousands of times longer than the particle itself experiences, and this enables experiments on the particle that couldn't otherwise be done.
Yes, but the proper velocity $$U^{i}$$ in $$p^{i} = mU^{i}$$ Is the proper time derivative of coordinate position. As he is saying it is the time according to the particle that is used in that velocity calculation and it is time dilation that then yields the relation $$p^{i} = \gamma mu^{i}$$. What is happening is that the law of momentum that the particle obeys is a tensor law expressed by the four-vector equation $$p^{\lambda } = mU^{\lambda }$$. This is the frame invariant law for a massive particle. Written explicitly in terms of the definition of four-vector velocity this is $$p^{\lambda } = m\frac{dx^{\lambda }}{d\tau }$$. Time dilation yields the factor of $$\gamma$$ from $$dt = \gamma d\tau$$ resulting in $$p^{\lambda } = \gamma m\frac{dx^{\lambda }}{dt} = \gamma mu^{\lambda }$$. That is where the $$\gamma$$ comes from, not from the mass. Without the four-vector law as the physics one could not even understand where the mass - rest energy equivalence even comes from because asserting $$p^{i} = \gamma mu^{i}$$ as the relativistic law and changing the meaning of the word mass to mean $$\gamma m$$ is not sufficient to imply that $$E_{0} = mc^{2}$$.

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#### quartodeciman

This all seems to be a debacle over canonical use of language, not about anything substantive. Some quantities are invariant and some are not. We have to deal with BOTH.

What conceivable harm arises by defining a "relativistic mass" by γm? No, it isn't invariant to a lorentz transformation. No problem. It all worked just fine back when.

#### DW

quartodeciman said:
What conceivable harm arises by defining a "relativistic mass" by γm?
For example of what harm arises just read the first post in this thread.

#### quartodeciman

Dook said in post #1: When particles in cyclotrons increase speed and near the speed of EMR they require much more energy for a further increase in speed. Einstein says this is because the object increases mass with speed.

Well, I reckon those statements are incorrect. The particles don't need more energy; the same amount would increase the speed. And I don't think Einstein said that mass increase prevents acceleration. I think Einstein said that a mass particle reaching lightspeed would be impossible since one would have necessarily raised the particle mass infinitely in doing so. That isn't the same thing as merely accelerating a fast particle.

Increasing a speed to, say, twice the speed has more to do with composition of particle velocities (lorentz boosts) than facts about mass values. A proposed increase of mass helps a little to justify that an equal change in speed is harder to acheive according to an observer who didn't accelerate at all.

#### reilly

DW -- I'm a physicist too, a retired high energy theoretician. Physicist's are pretty smart people, and I doubt many have much difficulty in distinguishing rest mass from realtivistic mass, even if they are a bit sloppy with their language. I note that in the piece by Philip Gibbs he suggests the two "mass camps" are distinguished primarily by semantic notions. See
http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html,

A very excellent, nonpolemical piece, by the way.

I learned a lot of relativity from Moller's book, including the notion of relativistic mass. I still find a certain stately elegance to his text, and, to the best of my knowledge, he did no damage to my study of physics.

You will note in my previous post that I point out high energy folk tend to use the word energy for relativistic mass -- among other things, the energy approach facilitates covariant mechanics and kinematics, much favored in high energy physics. Not to worry, the mass subversives have yet to make a dent in practical physics. I don't think there is much call for a counter revolution.
Regards,
Reilly Atkinson

#### quartodeciman

That is indeed a good piece on the subject of mass dependence upon velocity. I buy the point made that one might need more than one mass variable (longitudinal and transverse). This wasn't a new development but was right there in the decade of special relativity launching. Lorentzian electrodynamics had multiple special mass terms and these punctuated the debates over electron theory and the generation of electron mass during that same time period.

I could be happy if I had a really excellent elementary argument for momentum being m*v*γ. Given that, I can get to the energy-momentum-mass relations. The only elementary arguments I know are tap-ball-into-the-side-pocket-shot elastic action or head-on-cue-balls inelastic action. Or is it necessary to invoke Minkowski spacetime-mathemagic, or fiddle around with making up a suitable special relativity Lagrangian function?

#### reilly

q -- In beginning undergraduate courses, profs wave their hands a lot, and say "trust me". In more advanced courses, either an appeal to an invariant Lagrangian, or arguments that make F=dp/dt covariant are standard. Personally, I think working with E&M forces -- Lorentz's q(E + vXB) -- is the most clear. but this is not an elementary approach. Minkowski always seems to hang around with Einstein.

I'd be curious to know of a simpler way to get to relativistic mechanics.

Regards,
Reilly Atkinson

#### quartodeciman

"...a simpler way to get to relativistic mechanics."

I wish for this too.

Quart

#### pmb_phy

Reilly!!! How ya doing ??? Its me, Pete Brown. We used to work together in ther mid 90's in Waltham MA. Its GREAT to hear from you again and to see you posting here. You recall Joe Gibbs right? He used to work with us. He and I used to have this converationm way back even then.

Give me a call or e-mail me. Tonight I'm in the hospital at Brigham and Women's Hospital, Boston, MA, floor 10B room 31. I'm in for a slipped disk. Itd be nice to hear from you, expecially tonight to take my mind of my back. It'd be great to hear from you. Do you still keep in touch with any of those folks from our old workplace?

Do you recall that problem that I was working on regarding the rotating magnet? Well I solved it.

reilly said:
DW -- I'm a physicist too, a retired high energy theoretician. Physicist's are pretty smart people, and I doubt many have much difficulty in distinguishing rest mass from realtivistic mass, even if they are a bit sloppy with their language. I note that in the piece by Philip Gibbs he suggests the two "mass camps" are distinguished primarily by semantic notions. See
http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html,
dw has never been either or able or willing to understand that rather simple statement.

Many relativists use the term "mass" to refer to "inertial mass" (aka "relativistic mass".) A list of such relativists is at

www.geocities.com/physics_world/relativistic_mass.htm

A very excellent, nonpolemical piece, by the way.

I learned a lot of relativity from Moller's book, including the notion of relativistic mass. I still find a certain stately elegance to his text, and, to the best of my knowledge, he did no damage to my study of physics.

re - "Reilly Atkinson"

I used to work with a gentleman named "Reilly Atkinson" in the early 90's in Waltham MA. Is that you Reilly? Its me, Peter Brown.

By the way, Bernard F. Schutz just published a new book last year and uses the term "inertial mass" to mean relativistic mass

Other well known physicists uses the term "mass" to mean "relativistic mass". In fact even the author of dw's relativity text uses it, i.e. Wolfgang Rindler.

Reilly - See www.geocities.com/physics_world and take a look at "On the concept of mass in relativity". I'd enjoy your opinion.

Pete

ps - Again, really cool to see you posting here Rielly. You're sure gonna class the place up!!!

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#### pmb_phy

reilly said:
Talk to a particle physicist and, chances are he will say, "Sure, mass increases with speed." Push a little harder, and she will draw the distinction between rest mass and what most of us call relativistic mass -- gamma*(rest mass). But in doing kinematics, energy(kinetic) = gamma*(rest mass), so most of the time we use the term energy for gamma*(rest mass), just as momentum is gamma*v*(rest mass). Physicists are not always as rigorous as they could be with language and math. (When potential energy is involved, things get a little tricky. But all this stuff can be found in any intermediate or advanced text on relativity.) And, of course, the increase in relativistic mass, or energy comes from the work done by the force accelerating the particle.
Regards,
Reilly Atkinson
Particle physicists have their own lingo as do GR'ists, cosmologists etc. For example: If someone were to ask a particle physicist what the mass of a free neutron was he'd tell you 1.008665 u. Ask the very same particle physicist what the lifetime of a free neutron is he'll tell you its about 15 minutes. If that particle physicist does not ask you what speed of the neutron is does that mean that this particle physicist does not believe in time dilation? No. Of course not. It simply means that since you left out the speed then it was assumed to be the intrinsic property, i.e. that which is inherent and thefore d not need clarification.

However not all objects found in nature are particles. To be more general one has to use a second rank tensor to completely describe mass. Especially when you're interested only in part of a system or a system which is not closed.

Pete

#### DW

pmb_phy said:
Particle physicists have their own lingo as do GR'ists, cosmologists etc. For example: If someone were to ask a particle physicist what the mass of a free neutron was he'd tell you 1.008665 u. Ask the very same particle physicist what the lifetime of a free neutron is he'll tell you its about 15 minutes. If that particle physicist does not ask you what speed of the neutron is does that mean that this particle physicist does not believe in time dilation? No. Of course not. It simply means that since you left out the speed then it was assumed to be the intrinsic property, i.e. that which is inherent and thefore d not need clarification.

However not all objects found in nature are particles. To be more general one has to use a second rank tensor to completely describe mass. Especially when you're interested only in part of a system or a system which is not closed.

Pete
Mass does not need clarification of a speed because it does not depend on speed. It is the length of the momentum four vector according to any frame and that length has no speed dependence at all. Even for extended objects mass is not a tensor either. It is still an invariant. You are mistaking a density matrix for a mass, as well as a matrix for a tensor. The dimentions aren't even the same.

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