What is the true relationship between speed and mass?

[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 explanation 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, p_x, p_y, p_z) 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 length 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, argument 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 }

 

[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]

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]

You are talking to a physicist. ME. And I am telling you otherwise.

Perhaps we should ask more than one?

http://www.Newton.dep.anl.gov/askasci/phy99/phy99186.htm

http://www2.slac.stanford.edu/vvc/theory/relativity.html

http://www.cavendishscience.org/bks/rel/mass.htm

http://www.fnal.gov/pub/inquiring/more/light/light_page15.html

http://www.egglescliffe.org.uk/physics/relativity/relmass/relmass.html

http://www.bartleby.com/65/ma/mass.html

 

[GRQC]

Perhaps we should ask more than one?

http://www.Newton.dep.anl.gov/askasci/phy99/phy99186.htm

http://www2.slac.stanford.edu/vvc/theory/relativity.html

http://www.cavendishscience.org/bks/rel/mass.htm

http://www.fnal.gov/pub/inquiring/more/light/light_page15.html

http://www.egglescliffe.org.uk/physics/relativity/relmass/relmass.html

http://www.bartleby.com/65/ma/mass.html

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.

 

[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?

 

[selfAdjoint]

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}.

 

[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]

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 achieve according to an observer who didn't accelerate at all.

 

[reilly]

One more once about mass

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 you 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?

My e-mail address is pmb61@hotmail.com

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

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 going to class the place up!

 

[pmb_phy]

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]

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.