What is the true relationship between speed and mass?

[Dook]

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.

Where does this extra mass come from?

Matter/particles can be created in intense energy fields so is the energy that is added to the particle by accelerating it to EMR somehow converted to matter that adds to the particles mass making it harder to reach light speed?

 

[DW]

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.

Where does this extra mass come from?

Matter/particles can be created in intense energy fields so is the energy that is added to the particle by accelerating it to EMR somehow converted to matter that adds to the particles mass making it harder to reach light speed?

No, mass is invariant. Read section 1 at the link.

http://www.geocities.com/zcphysicsms/chap3.htm

 

[Dook]

Doesn't that seem to violate E=MC^2 then? Why do particles in cyclotrons seem to gain mass as they approach light speed then?

 

[DW]

Doesn't that seem to violate E=MC^2 then? Why do particles in cyclotrons seem to gain mass as they approach light speed then?

They don't seem to. You need a better text. And $$E_{R}$$ doesn't equal $$mc^2$$. It equals $$\gamma mc^2$$ as is all thoroughly explained in section 1 at the link I provided if you would please just read it.

 

[Dook]

Not everyone can read Greek.

 

[turin]

The extra factor that DW is including is a relativistic factor (and a Greek letter "gamma"). It depends in an ever-so-slightly complicated way on speed. To first order approximation, this factor is what gives you the familiar KE = (1/2)mv². The factor depends on speed in such a way that, as a particle approaches the speed of light, the energy of the particle diverges.
Note: both speed and energy are frame dependent quantities.

 

[Chronos]

Think rest mass. Mass does not increase, just the force necessary to accelerate it as it approaches 'c'. This creates much confusion. The problem originates in confusing inertial resistance with rest mass. The mass being moved always 'weighs' the same [according to its reference frame] no matter what velocity it appears to possesses relative to any other observer.

 

[mijoon]

Follow the link which DW gave and meditate on eq. 3.1.8 . Understanding this equation and the text relavant to it will greatly clearify many things.

 

[Dook]

For me the math just makes it more confusing.

If the particle does not increase mass then there must be something preventing it from further accelerating to the speed of EMR. Whatever it is it seems to be somehow connected to the speed of EMR since this effect, this resistance, comes on only as the particle exceeds .5 c. It's virtually unnoticed below that speed. Is this correct?

 

[turin]

... this resistance, comes on only as the particle exceeds .5 c. It's virtually unnoticed below that speed. Is this correct?

No. It can be noticed at any speed. At 0.5c, the effect cannot be ignored. At 0.05c, for instance, the effect can be ignored for certain approximations, but cannot be ignored for more exact considerations.

 

[mijoon]

For me the math just makes it more confusing.

If the particle does not increase mass then there must be something preventing it from further accelerating to the speed of EMR...

Correct!
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.

 

[jcsd]

It's inertial mass increases as it approaches c, you can see this simply follows on from Chronos's last post. As in special relativity problems are usually approached from the pouint of viuew of spacetime, it's considered better to have a defitnion of mass that is Lorentz invaraint (i.e. invaraint with relative speed).

 

[DW]

It's inertial mass increases as it approaches c, you can see this simply follows on from Chronos's last post. As in special relativity problems are usually approached from the pouint of viuew of spacetime, it's considered better to have a defitnion of mass that is Lorentz invaraint (i.e. invaraint with relative speed).

Most people define inertial mass as the resistence a particle has to acceleration. As an equation that is expressed as the m in
F = mA. In relativity the m in four force is m time four acceleration IS the Lorentz invariant mass. The factors of $$\gamma$$ one observes in the relation between ordinary force and coordinate acceleration have nothing to do with the mass. They have everything to do with time dilation as pertaining to the time derivatives.

 

[jcsd]

Poin taken, but in a certain frame it is as if the inertial mass increases with relative speed.

That is to say from a purely 3-force and 3-accelartion point of view inertial mass increases with relative speed as I don't see a way, if only using 3-vectors, of having a frame dependnet defintion of inertial mass.

 

[DW]

Poin taken, but in a certain frame it is as if the inertial mass increases with relative speed.

That is to say from a purely 3-force and 3-accelartion point of view inertial mass increases with relative speed as I don't see a way, if only using 3-vectors, of having a frame dependnet defintion of inertial mass.

Its not as if mass increased with speed. Its as if time dilation increased with speed. There is no function of velocity at all for which ordinary force is that function of velocity times coordinate acceleration for arbitrary forces. In other words, in general the equation
$$f^{i} = [M(v)]a^{i}$$ is wrong, but it is the mass in that position in such an equation that one calls inertial mass. Again, in relativity that equation of motion is wrong. The closest thing to it for ordinary force and coordinate acceleration is equation 3.2.10 at
http://www.geocities.com/zcphysicsms/chap3.htm#BM29
but you can see from it that in the relativistic expression that force isn't even always in the same direction as the acceleration. This means that if you wanted to define a relativistic inertial mass that was velocity dependent and related ordinary force and coordinate acceleration as above that it would have to be a matrix. see last section-
http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html
You certainly aren't proposing that are you?
No, correct way to do the physics is to recognise that the mass as in the four vector general relativistic law of motion F = mA is invariant and that the factors of $$\gamma$$ in the special relativistic expression 3.2.10 did not come from the mass, but from time dilation related to the time derivatives in the steps that led to that equation. It is not mass dilation that one is seeing in accelerators. It is time dilation.

 

[jcsd]

hmm, I didn't consider the directional dependence.

 

[pmb_phy]

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.

Where does this extra mass come from?

There are two electrodes in the cyclotron which are called "Dee's". When the charge particle enters the region between the Dee's, where there is an electric field, the particle accelerates. The mass of the particle due to an increase in speed comes from the energy gained as the particle is accelerated.

Matter/particles can be created in intense energy fields ..

Not that I know of.

..so is the energy that is added to the particle by accelerating it to EMR somehow converted to matter that adds to the particles mass making it harder to reach light speed?

The increase in mass is a direct result of both time dilation and Lorentz contraction. The proper mass (aka rest mass) of the particle never changes.


There are a few links to cyclotrons at the bottom of this page

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

re - "Think rest mass. Mass does not increase, .." - What that means is "Please don't ask that question." However the weight of a moving particle increases with speed. The weight is given by W = mg where m is a function of the particle's speed.

The very meaning of inertial mass in relativity comes from the notion of momentum. By this I mean that in special relativity, inertial mass is defined so that momentum is conserved. I.e. mass is the m in p = mv. That's called Weyl's definition of inertial mass. Once you define mass in this way, and you hold the postulates of special relativity to be true, then, for a particle with non-zero proper mass is a function of the particle's speed. The true reason is rooted in time dilation and length contraction and isn't that easy to explain in a few paragraphs with no math.

The derivation is here - http://www.geocities.com/physics_world/sr/inertial_mass.htm

re - "... this resistance, comes on only as the particle exceeds .5 c. It's virtually unnoticed below that speed. Is this correct?"

No. It is not correct. In fact there was an article to this effect published in the American Journal of Physics in 1995.

Relativistic mass increase at slow speeds, Gerald Gabrielse, Am. J. Phys. 63(6), 568 (1995).

Its online under Harvard's website (where the author is from) at -- http://hussle.harvard.edu/~gabrielse/gabrielse/papers/1995/RelativisticMassAJP.pdf

Its been measured at speeds so low that gamma deviates from 1 by a factor of 10^-9

re - "As in special relativity problems are usually approached from the pouint of viuew of spacetime, it's considered better to have a defitnion of mass that is Lorentz invaraint (i.e. invaraint with relative speed)."

Its appropriate to have definitions of proper quanties which are Lorentz invariant. Relativistic mass is not such a quantity. Neither is coordinate time and neither is inertial energy, neither of which are Lorentz invariant. Relativistic mass is the time component of the particles 4-momentum where as proper mass is the magnitude of the 4-momentum. (if you wish you can say that inertial energy is the time component and rest energy is the magnitude). Wishiing to have all quantities in relativity be Lorentz invariant would literally mean that we'd no longer be able to speak of time dilation and length contraction. What would we call the components?

The proper time interval between two events is the magnitude of the spacetime displacement when those events have a spacelike spacetime seperation. The time component of the spacetime displacement is not a Lorentz invariant. That quantity is called the "Time interval between the two events."

The proper distance between two events is the magnitude of the spacetime displacement when those events have a spacelike spacetime seperation. The space component of the spacetime displacement is not a Lorentz invariant. That quantity is called the "spatial displacement between the events."

Attempts have been made to rid relativity of the notion or relativistic mass but all such attempts use flawed logic.

Pete

 

[DW]

...The increase in mass is a direct result of ... The proper mass (aka rest mass) of the particle never changes...

No, the mass is invariant. An invariant is the same according to every frame. The mass is the same for the proper frame as for any other coordinate frame. Invariants should not be called by proper or rest because they are the same for all frames, not any different for the proper frame Vs any other frame. There is no mass dilation. You are missassociating factors of gamma that come from time dilation with the mass. That is modern relativity like it or not. Planck's relativity is dead.

 

[pmb_phy]

hmm, I didn't consider the directional dependence.

There is no directional dependence on mass. You're incorrectly thinking of longitudinal and transverse mass which is based on the relation m = |F|/|a|.

For a definition and derivation see
http://www.geocities.com/physics_world/sr/long_trans_mass.htm

Its based on an incorrect definition of force as F = ma. It was Euler, not Newton, who tried to define momentum that way. Newton used F = dp/dt as does everyone else today. The original post seemed to speak of relativistic mass, i.e. the mass the most people (e.g. Cern's web site, etc) use when they say that mass increases with speed. It's also the mass that appears in the cyclotron relation. E.g. see

http://www2.fpm.wisc.edu/safety/Radiation/2000 Manual/chapter12.pdf


Inertial mass is defined as m = |p|/|v|. This is the momentum and mass that appears in F[/b] = dp/dt.

 

[DW]

There is no directional dependence on mass. You're incorrectly thinking that mass is defined as m = F/a. That is incorrect. Mass is defined as m = p/v.

He isn't "incorrectly thinking". Inertial mass is the resistence something has to acceleration. As an equation that means the constant m in F = mA. and m is not p/v. The mass is the m in p = mU. You are the one who is wrong.

 

[pmb_phy]

Correct!
If the mass is constant (as Einstein believed) , ..

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.

However, as it accelerates, it begins to "see" spacetime differently.

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?

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.

Particle's don't make measurements. Observers do. Hence the observer dependant nature of mass etc.

So we must use it's PROPER velocity, not its reference frame velocity.

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.

 

[DW]

Actually Einstein held that the mass of a particle will be altered in a gravitational field. ... Its quite possible to do all calculations in special relativitywithout even knowing what proper velocity is.

No, he held that its energy varied. Its mass as defined in modern relativity is something that he never ever claimed varied in the presence of gravitation. And who cared if one can mindlessly punch numbers?

 

[DanielK]

I don't see the point of this debate. According to relativity mass is invariant because it is the "length" of the energy-momentum four vector (mass-shell condition). That is rest mass is what relativity calls "mass". "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.

 

[DW]

According to relativity mass is invariant because it is the "length" of the energy-momentum four vector (mass-shell condition). That is rest mass is what relativity calls "mass". "Mass increasing", "longitudinal and transverse mass" are concepts that come into consideration when one wants to insist on the usual, Newtonian sense of mass, ... or one misinterprets the mass-energy equivalence by thinking that it holds for a moving mass, too, thereby attributing speed dependency to mass.

Right.

I don't see the point of this debate.

Relativity wasn't developed overnight and the mathematics it involves were not all developed by one person. To understand this debate one must have some history. Einstein originally in special relativity used mass or m without any subscripting to mean what it means in modern relativity which is mass as an invariant. In his 1905 paper he derived the motion for a charge in the presence of an electromagnetic field, but did so by making reference to the proper frame ordinary force on the particle because he did not yet have an expression corresponding to Newton's second law of motion for an arbitrary frame. Planck, Lewis and Tolman later developed the "relativistic mass" concept by shear luck discovering that if one uses
$$M = \gamma m$$ in $$p^{i} = Mu^{i}$$ that the ordinary force defined by $$f^{i} = \frac{dp^{i}}{dt}$$ resulted in a law of motion that agreed with Einstein's postulates from his 1905 paper and agreed with Einstein's results for the equation of motion for the charged particle in an electromagnetic field given that the ordinary force from the field for an arbitrary frame would be $$\vec{f} = q(\vec{E} + \vec{v}\times \vec{B})$$. At that point it was not understood "why" that worked. For a short while after that discovery Einstein also made use of that "relativistic mass" concept. Later Einstein developed general relativity. Like special relativity it was also not fully developed with the publication of a single paper. In its development it was found that the general laws of relativistic physics were tensor laws. In other words the laws of relativistic physics obey tensor equations. I think it was when he realized the significance of this that he got away from Planck Lewis and Tolman's concept of relativistic mass in favor of an invariant mass that satisfied the tensor law $$p^{\lambda } = mU^{\lambda }$$. It is this tensor law that reveals why they were able to describe the motion as they did. In relating the four-vector velocity to coordinate velocity using time dilation a factor of $$\gamma$$ comes to sit in the exact place same place that it would in their expression:$$p^{\lambda } = \gamma mu^{\lambda }$$. So now in modern relativity we see that the factor of $$\gamma$$ that they discovered had to be in that expression actually doesn't come from the mass at all as they had thought, but from time dilation. The major problem is that even though most authors have switched over to the modern invariant mass concept, there are remnants of the old relativistic mass concept especially in public relations articles and introductory level texts. The debate is that pmb has for some reason made it his lifes goal to resurect such outdated concepts, not only concerning mass but other things including completely Newtonian paradigms as well.

 

[DanielK]

OK, I just wanted to emphasize that the concept of mass is clear in relativity and there is nothing to worry about.

 

[mijoon]

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..

DW answered this.

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]

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]

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.

re - "You have answered your own question."

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.

 

[DanielK]

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.

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.

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.

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.

 

[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