# Speed Increases 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?

Related Special and General Relativity News on Phys.org

#### DW

Dook said:
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

Dook said:
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.

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#### Dook

Not everyone can read Greek.

#### turin

Homework Helper
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

Gold Member
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 possess 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

Homework Helper
Dook said:
... 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

Dook said:
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

Gold Member
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

jcsd said:
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

Gold Member
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.

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#### DW

jcsd said:
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

Gold Member
hmm, I didn't consider the directional dependence.

#### pmb_phy

Dook said:
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 web site (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

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#### DW

pmb_phy said:
...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

jcsd said:
hmm, I didn't consider the directional dependence.
There is no directional dependance 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

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

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#### DW

pmb_phy said:
There is no directional dependance 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.

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#### pmb_phy

mijoon said:
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.

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#### DW

pmb_phy said:
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

DanielK said:
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 developement 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 realised 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 thier 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.

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#### DanielK

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

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