Understanding Mass in Relativity Theory: Definitions and Conflicting Views

In summary, the term 'mass' is the source of many conflicting opinions among the authors writing on relativity theory. Different authors denote by this term different concepts that are not always consistent with one another. For instance, when introducing his famous diagrams Richard Feynman used the concept of invariant mass of a particle defined by equation ##m^2=p^2##, where ##p## is four-momentum. But later in his Feynman Lectures on Physics he preferred to define mass by the equation ##E=mc^2##. Thus defined mass ##m## obviously increases with increase of total energy ##E## and hence of speed of a particle.
  • #1
levokun
Gold Member
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The term "mass" is the source of many conflicting opinions among the authors writing on relativity theory.
Different authors denote by this term different concepts.
Quite often even the same author denotes by mass different concepts in his different writings.
For instance when introducing his famous diagrams Richard Feynman used the concept of invariant mass of a particle defined by equation ##m^2=p^2##, where ##p## is four-momentum. But later in his Feynman Lectures on Physics he preferred to define mass by the equation ##E=mc^2##. Thus defined mass ##m## obviously increases with increase of total energy ##E## and hence of speed of a particle.
The equation ##E=mc^2## usually referred to as the super-famous Einstein equation, though Einstein himself preferred another definition: ##E_0=mc^2##, where ##E_0## is the rest energy, or energy of a particle at rest. To a certain extent the partial source of confusion was the term "rest mass" used by him.
 
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  • #2
Is there a question here? The term 'mass' in SR most commonly refers to the rest mass of a particle, which is a Lorentz scalar.
 
  • #3
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  • #4
There is historical reason why people use relativistic mass, but
nowadays most people use [itex]E=\gamma mc^2[/itex], where [itex]\gamma[/itex] is the Lorentz factor. Here [itex]m[/itex] is the rest mass, as WannabeNewton said.
 
  • #5
This discussion might give you some further perspective:

http://en.wikipedia.org/wiki/Rest_mass

In SIX EASY PIECES ( 1997), pages 87 to 91, Feynman seems to say repeatedly 'mass' DOES increase with speed. [He is using the older 'relativistic' mass..which includes kinetic energy.

For example:
The mass of the object which is formed when two equal objects collide must be twice the mass of the objects which come together...the masses have been enhanced over the masses they would have been if standing still...the mass they form must be greater than the rest masses of the objects even though the objects are at rest after the collision!
and separately:
when we put energy into the gas molecules move faster and so the gas gets heavier...kinetic energy does not affect the mass according to Newton's laws...but there is no place in relativity for strictly inelastic collisions...conservation of energy must go along with conservation of momentum in the theory of relativity...because of the kinetic energy involved in the collision, the resulting object will be heavier, therefore it will be a different object...

there is more than one definition of "mass" in relativity.

- invariant mass, or rest mass, or proper mass, which excludes the kinetic energy of the object's center of momentum
- relativistic mass, sometimes called inertial mass, which includes the kinetic energy of the object's center of momentum.

Feynman was referring to relativistic mass, but many users in this forum prefer the modern convention of referring to invariant mass, that is, rest mass. The prior post uses that convention.
 
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  • #6
levokun said:
The term "mass" is the source of many conflicting opinions among the authors writing on relativity theory.
Different authors denote by this term different concepts.
Quite often even the same author denotes by mass different concepts in his different writings.
For instance when introducing his famous diagrams Richard Feynman used the concept of invariant mass of a particle defined by equation ##m^2=p^2##, where ##p## is four-momentum. But later in his Feynman Lectures on Physics he preferred to define mass by the equation ##E=mc^2##. Thus defined mass ##m## obviously increases with increase of total energy ##E## and hence of speed of a particle.
The equation ##E=mc^2## usually referred to as the super-famous Einstein equation, though Einstein himself preferred another definition: ##E_0=mc^2##, where ##E_0## is the rest energy, or energy of a particle at rest. To a certain extent the partial source of confusion was the term "rest mass" used by him.

My take : there's only one 'mass'. If one needs or feels the urge to put an adjective next to it, then <invariant> would be the best, but I don't advise it. In the community, when one speaks about mass, he either refers to the Newtonian definition (quantity of substance, as in chemistry), or to the 0-th component of the 4-momentum, in units where c=1 and the Minkowskian metric is mostry minus.

The subject in the original post has been pretty much exhausted by the Russian physicist Lev B. Okun. Specifically, he wrote a book, <Energy and Mass in Relativity Theory>, a summary of own articles and talks on the 2 concepts (amazon link: https://www.amazon.com/dp/9812814116/?tag=pfamazon01-20) where he shared his teacher's (?) - Lev Landau - views and presented them from all sorts of angles. To a non-geometrist, Landau's textbook supplemented by Okun's writings serve as a solid foundation of Relativity Theory. Then you can, if you wish, add the somewhat abstract flavor of differential geometry.
 
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  • #7
The relativistic increase in mass is due to the fact that at high speeds (u->c) the mechanics you use are relativistic.
The term invariant mass is the correct one -
a question: how would you define mass in a proper textbook format?
E.g I would say that mass is the energy of an object observed at its rest frame
 
  • #8
You can define it in a manifestly Lorentz invariant manner as ##m^{2} = -p_{a}p^{a}##. In the instantaneous rest frame this reduces to ##m = E##.
 
  • #9
levokun said:
Richard Feynman used the concept of invariant mass of a particle defined by equation ##m^2=p^2##, where ##p## is four-momentum.
WannabeNewton said:
You can define it in a manifestly Lorentz invariant manner as ##m^{2} = -p_{a}p^{a}##
This is my preferred definition also. I also prefer the term "invariant mass" over the term "rest mass". Rest implies a frame where the entire system is at rest, which may not exist or may not be inertial. Invariant implies that any frame is fine for calculating it.
 
  • #10
Yeah invariant mass is definitely a better way to phrase it. I do agree with the OP though that the term "mass" in SR isn't always used consistently across different texts. For example in chapter 5 of Purcell's "Electricity and Magnetism", the term mass is constantly used to refer to the relativistic mass.
 
  • #11
Entirely consistent with post#8, #9, I believe:


from other discussions in these forums...

Einstein:
"It is not good to introduce the concept of the mass M ... of a moving body for which no clear definition can be given. It is better to introduce no other mass concept than the ’rest mass’ m. Instead of introducing M it is better to mention the expression for the momentum and energy of a body in motion."

Taylor & Wheeler:

"The concept of "relativistic mass" is subject to misunderstanding. That's why we don't use it. First, it applies the name mass - belonging to the magnitude of a 4-vector - to a very different concept, the time component of a 4-vector. Second, it makes increase of energy of an object with velocity or momentum appear to be connected with some change in internal structure of the object. In reality, the increase of energy with velocity originates not in the object but in the geometric properties of spacetime itself."


And there is another good reason to view 'mass' as invariant mass: because that's the notion of gravitational curvature sourced from the stress energy momentum tensor.


I believe the last sentence is an oblique [as I see it] reference to the fact that an increase of velocity of a center of mass is NOT associated with an increase in gravitational attraction; in other words, the SET [the source of gravity] acts in such a way as to reflect invariant mass.
You can tell relative velocity should not change gravity because gravitational curvature produced by an object" is frame-invariant; it doesn't matter what your state of motion is relative to the object...hence one does NOT want to get other frame dependent energy sources, such as kinetic energy, mixed up with 'mass'...nor 'gravity'...
 
  • #12
Naty1 said:
This discussion might give you some further perspective:

http://en.wikipedia.org/wiki/Rest_mass

In SIX EASY PIECES ( 1997), pages 87 to 91, Feynman seems to say repeatedly 'mass' DOES increase with speed. [He is using the older 'relativistic' mass..which includes kinetic energy.

For example:

and separately:


there is more than one definition of "mass" in relativity.

- invariant mass, or rest mass, or proper mass, which excludes the kinetic energy of the object's center of momentum
- relativistic mass, sometimes called inertial mass, which includes the kinetic energy of the object's center of momentum.

Feynman was referring to relativistic mass, but many users in this forum prefer the modern convention of referring to invariant mass, that is, rest mass. The prior post uses that convention.

It is more consistent to use the term mass for both massive and massless particles than to use the term rest mass for massless particles, which are never at rest.
 
  • #13
Welome to the Forum, Lev. It's a pleasure to have you here.

In case you are interested, I've made several references to your paper on this topic in a number of discussions in this forum. This is one such example:

https://www.physicsforums.com/showthread.php?t=642188

On the other hand, Frank Wilczek has written a lengthy discussion (and not easy to follow) on the origin of mass.

http://arxiv.org/abs/1206.7114

It may not follow in your theme for this thread here, but I'm curious as to what you make of it.

Cheers!

Zz.
 
  • #14
from Wilczek's Conclusions:

As reviewed here, we have achieved profound insight into the origin of mass for standard matter, and we may be set to crown, with the discovery of the Higgs particle, a compelling account of the origin of mass for W and Z bosons. Those origins are distinct, though
there is an attractive conceptual connection between their mechanisms, and between both
mechanisms and superconductivity. That's the good news.

The bad news is that nothing in these ideas explains the origin of the mass of the Higgs
particle itself...

The ugly news is that the origin of most of the mass in the universe, that in dark matter
and dark energy, remains deeply mysterious. ...

We've passed some milestones, but the end of the road is not in sight.
 
  • #15
ZapperZ said:
Welome to the Forum, Lev. It's a pleasure to have you here.

In case you are interested, I've made several references to your paper on this topic in a number of discussions in this forum. This is one such example:

https://www.physicsforums.com/showthread.php?t=642188

On the other hand, Frank Wilczek has written a lengthy discussion (and not easy to follow) on the origin of mass.

http://arxiv.org/abs/1206.7114

It may not follow in your theme for this thread here, but I'm curious as to what you make of it.

Cheers!




Zz.

Cheers, Zz!

It is nice to see on this thread people who can explain more clearly than I can
how the term mass should be used.
As for the lengthy article by Wilczek, it is obviously lacking
references to the papers of Steve Weinberg
on masses of hadrons.

Best regards
Lev
 
  • #16
levokun said:
For instance when introducing his famous diagrams Richard Feynman used the concept of invariant mass of a particle defined by equation [itex]m^2=p^2[/itex], where [itex]p[/itex] is four-momentum. But later in his Feynman Lectures on Physics he preferred to define mass by the equation [itex]E=mc^2[/itex]. Thus defined mass [itex]m[/itex] obviously increases with increase of total energy [itex]E[/itex] and hence of speed of a particle.

Lev,

In The Feynman Lectures on Physics Feynman does not precisely "define mass by the equation [itex]E=mc^2[/itex]." In fact he defines velocity-dependent mass as a modification (made by Einstein) of Newton's law, starting on the very first page of the book, page 1-1 of Volume I, where it says,

For example, the mass of an object never seems to change: a spinning top has the same weight as a still one. So a “law” was invented: mass is constant, independent of speed. That “law” is now found to be incorrect. Mass is found to increase with velocity, but appreciable increases require velocities near that of light. A true law is: if an object moves with a speed of less than one hundred miles a second the mass is constant to within one part in a million. In some such approximate form this is a correct law.​

And again when special relativity is introduced, in chapter 15,

Newton’s Second Law, which we have expressed by the equation

[itex]F = d(mv)/dt [/itex],

was stated with the tacit assumption that m is a constant, but we now know that this is not true, and that the mass of a body increases with velocity. In Einstein’s corrected formula [itex] m[/itex] has the value

[itex]m = m_0 / \sqrt(1 − v^2/c^2)[/itex],

where the “rest mass” [itex] m_0[/itex] represents the mass of a body that is not moving and [itex]c[/itex] is the speed of light, which is about [itex]3×10^5[/itex] km· sec[itex] ^{−1}[/itex] or about [itex]186,000[/itex] mi · sec[itex] ^{−1}[/itex].

For those who want to learn just enough about it so they can solve problems, that is all there is to the theory of relativity—it just changes Newton’s laws by introducing a correction factor to the mass.​

(From this he eventually derives [itex]\Delta E = \Delta (mc^2)[/itex], at the end of the chapter.)

I have only found one clue that might explain why Feynman used the (pedagogically poor) velocity-dependent mass as the basis for his lectures on special relativity in FLP, in Jagdish Mehra's biography, The Beat of a Different Drum, which includes interviews of Feynman Mehra made only a few weeks before Feynman's death. In one of these interviews Feynman says,

As for the lectures on physics, I have put a lot of thought into these things over the years. I've always been trying to improve the method of understanding everything. I had already tried to explain the results of relativity theory in my own way to my girlfriend, Arline, and then I used the same explanations in my lectures. These things are very personal, my own way of looking at things and I recognize them. I did everything—all of it—in my own way.​

Arline was Feynman's first wife. They married when they were young - Feynman was 23 - but they had already been girlfriend and boyfriend for 10 years. It is probable that Feynman studied relativity theory at age 13 or 14 (when he was studying calculus) and he would probably have studied from a book borrowed from a public library, one of the old books (this would have been 1933 or 1934) that use velocity-dependent mass. Furthermore, even if Feynman at that tender age realized mass was an invariant 4-scalar, it is very unlikely he would try to explain relativity theory to his non-scientist girlfriend in terms so abstract as Minkowski spacetime. It's more likely he would choose the less correct but more appealing to "common sense" way of teaching relativity theory, involving [itex] m(v)[/itex] . Thirty years later, when Feynman composed his freshmen lectures on relativity, it seems he simply ignored what he had learned about mass since he was 13, and for emotional reasons reconstructed his old lectures in Arline's memory (who died of tuberculosis not long after they were married, while Feynman was working at Los Alamos on the first atom bomb). At least that would explain the inconsistency between his more mature work, in which mass is a constant, and his freshmen relativity lectures, in which [itex] m(v)[/itex] plays a pivotal role.

Mike Gottlieb
Editor, The Feynman Lectures on Physics, New Millennium Edition
 
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  • #17
codelieb said:
Lev,

In The Feynman Lectures on Physics Feynman does not precisely "define mass by the equation [itex]E=mc^2[/itex]." In fact he defines velocity-dependent mass as a modification (made by Einstein) of Newton's law, starting on the very first page of the book, page 1-1 of Volume I, where it says,

For example, the mass of an object never seems to change: a spinning top has the same weight as a still one. So a “law” was invented: mass is constant, independent of speed. That “law” is now found to be incorrect. Mass is found to increase with velocity, but appreciable increases require velocities near that of light. A true law is: if an object moves with a speed of less than one hundred miles a second the mass is constant to within one part in a million. In some such approximate form this is a correct law.​

And again when special relativity is introduced, in chapter 15,

Newton’s Second Law, which we have expressed by the equation

[itex]F = d(mv)/dt [/itex],

was stated with the tacit assumption that m is a constant, but we now know that this is not true, and that the mass of a body increases with velocity. In Einstein’s corrected formula [itex] m[/itex] has the value

[itex]m = m_0 / \sqrt(1 − v^2/c^2)[/itex],

where the “rest mass” [itex] m_0[/itex] represents the mass of a body that is not moving and [itex]c[/itex] is the speed of light, which is about [itex]3×10^5[/itex] km· sec[itex] ^{−1}[/itex] or about [itex]186,000[/itex] mi · sec[itex] ^{−1}[/itex].

For those who want to learn just enough about it so they can solve problems, that is all there is to the theory of relativity—it just changes Newton’s laws by introducing a correction factor to the mass.​

(From this he eventually derives [itex]\Delta E = \Delta (mc^2)[/itex], at the end of the chapter.)

I have only found one clue that might explain why Feynman used the (pedagogically poor) velocity-dependent mass as the basis for his lectures on special relativity in FLP, in Jagdish Mehra's biography, The Beat of a Different Drum, which includes interviews of Feynman Mehra made only a few weeks before Feynman's death. In one of these interviews Feynman says,

As for the lectures on physics, I have put a lot of thought into these things over the years. I've always been trying to improve the method of understanding everything. I had already tried to explain the results of relativity theory in my own way to my girlfriend, Arline, and then I used the same explanations in my lectures. These things are very personal, my own way of looking at things and I recognize them. I did everything—all of it—in my own way.​

Arline was Feynman's first wife. They married when they were young - Feynman was 23 - but they had already been girlfriend and boyfriend for 10 years. It is probable that Feynman studied relativity theory at age 13 or 14 (when he was studying calculus) and he would probably have studied from a book borrowed from a public library, one of the old books (this would have been 1933 or 1934) that use velocity-dependent mass. Furthermore, even if Feynman at that tender age realized mass was an invariant 4-scalar, it is very unlikely he would try to explain relativity theory to his non-scientist girlfriend in terms so abstract as Minkowski spacetime. It's more likely he would choose the less correct but more appealing to "common sense" way of teaching relativity theory, involving [itex] m(v)[/itex] . Thirty years later, when Feynman composed his freshmen lectures on relativity, it seems he simply ignored what he had learned about mass since he was 13, and for emotional reasons reconstructed his old lectures in Arline's memory (who died of tuberculosis not long after they were married, while Feynman was working at Los Alamos on the first atom bomb). At least that would explain the inconsistency between his more mature work, in which mass is a constant, and his freshmen relativity lectures, in which [itex] m(v)[/itex] plays a pivotal role.

Mike Gottlieb
Editor, The Feynman Lectures on Physics, New Millennium Edition

It seems just a question of naming, so perhaps he also had in mind another lesson his father taught him?

http://www.haveabit.com/feynman/2
 
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  • #18
codelieb said:
Lev,

In The Feynman Lectures on Physics Feynman does not precisely "define mass by the equation [itex]E=mc^2[/itex]." In fact he defines velocity-dependent mass as a modification (made by Einstein) of Newton's law, starting on the very first page of the book, page 1-1 of Volume I, where it says,

For example, the mass of an object never seems to change: a spinning top has the same weight as a still one. So a “law” was invented: mass is constant, independent of speed. That “law” is now found to be incorrect. Mass is found to increase with velocity, but appreciable increases require velocities near that of light. A true law is: if an object moves with a speed of less than one hundred miles a second the mass is constant to within one part in a million. In some such approximate form this is a correct law.​

And again when special relativity is introduced, in chapter 15,

Newton’s Second Law, which we have expressed by the equation

[itex]F = d(mv)/dt [/itex],

was stated with the tacit assumption that m is a constant, but we now know that this is not true, and that the mass of a body increases with velocity. In Einstein’s corrected formula [itex] m[/itex] has the value

[itex]m = m_0 / \sqrt(1 − v^2/c^2)[/itex],

where the “rest mass” [itex] m_0[/itex] represents the mass of a body that is not moving and [itex]c[/itex] is the speed of light, which is about [itex]3×10^5[/itex] km· sec[itex] ^{−1}[/itex] or about [itex]186,000[/itex] mi · sec[itex] ^{−1}[/itex].

For those who want to learn just enough about it so they can solve problems, that is all there is to the theory of relativity—it just changes Newton’s laws by introducing a correction factor to the mass.​

(From this he eventually derives [itex]\Delta E = \Delta (mc^2)[/itex], at the end of the chapter.)

I have only found one clue that might explain why Feynman used the (pedagogically poor) velocity-dependent mass as the basis for his lectures on special relativity in FLP, in Jagdish Mehra's biography, The Beat of a Different Drum, which includes interviews of Feynman Mehra made only a few weeks before Feynman's death. In one of these interviews Feynman says,

As for the lectures on physics, I have put a lot of thought into these things over the years. I've always been trying to improve the method of understanding everything. I had already tried to explain the results of relativity theory in my own way to my girlfriend, Arline, and then I used the same explanations in my lectures. These things are very personal, my own way of looking at things and I recognize them. I did everything—all of it—in my own way.​

Arline was Feynman's first wife. They married when they were young - Feynman was 23 - but they had already been girlfriend and boyfriend for 10 years. It is probable that Feynman studied relativity theory at age 13 or 14 (when he was studying calculus) and he would probably have studied from a book borrowed from a public library, one of the old books (this would have been 1933 or 1934) that use velocity-dependent mass. Furthermore, even if Feynman at that tender age realized mass was an invariant 4-scalar, it is very unlikely he would try to explain relativity theory to his non-scientist girlfriend in terms so abstract as Minkowski spacetime. It's more likely he would choose the less correct but more appealing to "common sense" way of teaching relativity theory, involving [itex] m(v)[/itex] . Thirty years later, when Feynman composed his freshmen lectures on relativity, it seems he simply ignored what he had learned about mass since he was 13, and for emotional reasons reconstructed his old lectures in Arline's memory (who died of tuberculosis not long after they were married, while Feynman was working at Los Alamos on the first atom bomb). At least that would explain the inconsistency between his more mature work, in which mass is a constant, and his freshmen relativity lectures, in which [itex] m(v)[/itex] plays a pivotal role.

Mike Gottlieb
Editor, The Feynman Lectures on Physics, New Millennium Edition

Dear Michael,

Your reply added two important statements about ##m(v)## which were missing in my original thread, namely:

1) the inconsistent and illegal attempt to keep for the relativistic particles the non-relativistic definition of momentum ##p=mv##;

2) the emotional personal reason for Feynman in memory of his first wife Arline to use ##m(v)##
via Newtonian definition of momentum as was stressed on page 488 of the book of Mehra.

I agree with what you have written about the drawbacks of this approach.What I cannot agree with is that you (and Feynman whom you quote) ascribe the concept of velocity-dependet mass ##m(v)## to Einstein. In fact Einstein criticized this concept, though not irrerprochably consistently. (See the article by Carl Adler "Does mass depend on velocity Dad?" which was published in 1987, but was not noticed by Feynman in the last year of his life.) In 1921 Einstein had cast his famous equation in the form ##E_0=mc^2##, where ##E_0## is the energy of a particle at rest. But he never wrote it in the invariant form ##p^2=m^2##. Note that in this form it was first written in 1941 in the first edition of textbook "Field Theory" by Lev Landau and Eugene Lifgarbagez. However for unknown reasons they preferred not to introduce notation ##E_0## for the rest energy. This tradition of avoiding ##E_0## is kept in the modern (tenth) edition of their course.

Best regard.
Lev
 
  • #19
levokun said:
What I cannot agree with is that you (and Feynman whom you quote) ascribe the concept of velocity-dependet mass ##m(v)## to Einstein.

Dear Lev,

Feynman (and his coauthors) ascribe ##m(v)## to Einstein in The Feynman Lectures on Physics. I do not. (I am aware of Einstein's ambivalence towards ##m(v)##, mostly through proofreading your papers on the concept of mass, and the research I conducted on your behalf in the Einstein Papers at Caltech.)

But are you telling me that before 1941 there was not a single publication (book or article) in which it was stated that mass is the magnitude of the 4-momentum (or something equivalent, like E^2 - p^2 = m^2)? I find that remarkable. The basis for this was laid out by Minkowski in 1908. Surely, it must have been recognized long before 1941? I suppose one could only come to your conclusion by examining every publication between 1908 and 1941 that discusses relativity theory... but that would probably be impossible, so I think you must either be assuming that you have looked at a sufficiently large sample to infer your conclusion (of course such an inference can never be certain) or you are repeating what you have read elsewhere, in which case I would like to know your source.



Mike
 
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  • #20
levokun said:
The term "mass" is the source of many conflicting opinions among the authors writing on relativity theory.
Different authors denote by this term different concepts.
Quite often even the same author denotes by mass different concepts in his different writings.
For instance when introducing his famous diagrams Richard Feynman used the concept of invariant mass of a particle defined by equation ##m^2=p^2##, where ##p## is four-momentum. But later in his Feynman Lectures on Physics he preferred to define mass by the equation ##E=mc^2##. Thus defined mass ##m## obviously increases with increase of total energy ##E## and hence of speed of a particle.
The equation ##E=mc^2## usually referred to as the super-famous Einstein equation, though Einstein himself preferred another definition: ##E_0=mc^2##, where ##E_0## is the rest energy, or energy of a particle at rest. To a certain extent the partial source of confusion was the term "rest mass" used by him.
If you search this forum then you'll find a number of interesting discussions on that topic, with quite some references. And the physics FAQ gives IMHO a fair summary:
http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html
What do you think?
 
  • #21
levokun said:
What I cannot agree with is that you (and Feynman whom you quote) ascribe the concept of velocity-dependet mass ##m(v)## to Einstein. In fact Einstein criticized this concept, though not irrerprochably consistently.

Since Einstein did not criticize it consistently, why can the concept not be ascribed to Einstein in certain of his writings? Presumably you are thinking of http://www.fourmilab.ch/etexts/einstein/E_mc2/www/ ?
 
  • #22
dextercioby said:
The subject in the original post has been pretty much exhausted by the Russian physicist Lev B. Okun. Specifically, he wrote a book, <Energy and Mass in Relativity Theory>,
You may not have noticed it but Lev B. Okun is the person who created this thread.
 
  • #23
WannabeNewton said:
Is there a question here? The term 'mass' in SR most commonly refers to the rest mass of a particle, which is a Lorentz scalar.

Just a quick note on terminology - The proper mass of a particle is more properly referred to as simply a scalar. The term Lorentz scalar refers to any quantity which remains invariant under a Lorentz transformation. Any quantity which remains invariant under a general spacetime transformation is referred to simply as a scalar. Likewise any quantity which remains invariant under an orthogonal transformation (i.e. a "rotation" of the coordinates about the origin) is called a Cartesian scalar. Thus distance and volume are Cartesian scalars.
 
  • #24
The rest mass of a time-like particle is a Lorentz scalar because it transforms under the trivial representation of the proper Lorentz group. The Lorentz transformations preserve the inner product between vectors hence ##p_a p^a## transforms under the identity element.
 
  • #25
Phy_Man said:
Any quantity which remains invariant under a general spacetime transformation is referred to simply as a scalar.]
This depends entirely on what you mean by invariant. Isometries preserve the metric tensor under the induced pullback hence the metric tensor remains invariant under isometries in that sense. It is obviously not a scalar field. If by "general space-time transformation" you mean arbitrary diffeomorphisms that are endomorphisms of space-time then this is also not true because diffeomorphisms can move points around: ##f:\mathbb{R}\rightarrow \mathbb{R},x \mapsto x + a ## does not leave invariant the scalar field ##g:\mathbb{R}\rightarrow \mathbb{R},x \mapsto x^2 ## because ##g(f(x)) = g(x + a) = (x + a)^{2}\neq x^{2} = g(x)##. If you mean coordinate transformations then yes scalar fields are the ones which unequivocally remain the same under coordinate transformations (this is because coordinate transformations are a special class of diffeomorphisms called passive diffeomorphisms-they fix points, they don't move them around).
 
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  • #26
WannabeNewton said:
This depends entirely on what you mean by invariant.
The meaning of the term invariant is determined by the context in which it's used, just like any other term of course. In the context in which I used it it means that the quantity's numerical value is not altered by a coordinate transformation. I was pointing out the various types of scalars which are defined by the class of coordinate transformation which defines them.
 
  • #27
WannabeNewton said:
The rest mass of a time-like particle is a Lorentz scalar because it transforms under the trivial representation of the proper Lorentz group. The Lorentz transformations preserve the inner product between vectors hence ##p_a p^a## transforms under the identity element.
The purpose of my comment was to note that there is a much larger class of spacetime coordinate transformations which leaves the magnitude of the particle's 4-momentum invariant. I appologize if that was unclear.
 
  • #28
There are some things that concern me about concepts of mass as discussed in threads such as this and I would appreciate it if someone could answer the following :

1. I have the impression that the majority of physicists who use relativity do not favour the equation:

M=MoL (L= Lorentz factor)

. I am a bit familiar with the other equations but is it considered that the above equation is archaic or misleading or incorrect in some way?

. If people reject the equation is it because of the terminology sometimes used? It seems to me that it is accepted that Mo can be referred to as the mass,or invariant mass and sometimes rest mass and that it is unnecessary to use the subscript o.
What doesn't seem to be accepted is that M (or E as it is sometimes written) should be referred to as the total mass where the total mass is the sum of the invariant mass plus the mass equivelent of the kinetic energy .If it is not accepted then what is wrong in calling it total mass and what, if anything, should it be called instead?

. Are there physicists who favour the use of the equation and if so are there examples of where the equation is more useful than any alternatives?



A big concern is that it is a requirement of some A2 physics courses in the UK that some relativity,including the above equation and the consequences of it, be taught ( eg in AQA A level physics unit 5). The students here are usually from age 16 to 18 and only a tiny minority will go on to further study in physics. If the equation is not generally favoured then what should be taught in its place?
 
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  • #29
1. I have the impression that the majority of physicists who use relativity do not favour the equation:

M=MoL (L= Lorentz factor)

I am a bit familiar with the other equations but is it considered that the above equation is archaic or misleading or incorrect in some way?

yes, that has been the consensus in these forums...see my post #11 in this thread.


Good discussion here:

Does the speed of a moving object curve spacetime?
https://www.physicsforums.com/showthread.php?t=602644
 
  • #30
To expand on my explanation in post #11, an easy way to see a 'problem' introduced by velocity dependent 'mass' [relativistic mass] is that one can infer a REALLY fast moving particle with lots of 'relativistic mass' [really kinetic energy] can turn into a black hole.

That a fast moving particle buzzing by earth, for example, CANNOT induce enough gravitational curvature to form a black hole can be understood from the frame of the particle: From that perspective it is the Earth that has the KE, not the particle...and that type of frame dependency is not gravitational curvature which is independent of coordinates. KE is observer dependent.
 
  • #31
Dadface said:
A big concern is that it is a requirement of some A2 physics courses in the UK that some relativity,including the above equation and the consequences of it, be taught ( eg in AQA A level physics unit 5). The students here are usually from age 16 to 18 and only a tiny minority will go on to further study in physics. If the equation is not generally favoured then what should be taught in its place?

Most of the current introductory textbooks that I have at hand (for college/university level in the US) simply use equations that are written in terms of invariant mass (your m0, but usually simply called m). That is, they write e.g. ##p = mv / \sqrt{1 - v^2/c^2} = \gamma mv## instead of p = mv. They do not mention the so-called "relativistic mass" at all, except sometimes as a historical footnote for the benefit of students who have seen it elsewhere.

The only exception in my admittedly small collection is French's "Newtonian Mechanics" which I think is still somewhat popular even though it was written over forty years ago.
 
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  • #32
Naty1 said:
yes, that has been the consensus in these forums...see my post #11 in this thread.


Good discussion here:

Does the speed of a moving object curve spacetime?
https://www.physicsforums.com/showthread.php?t=602644

Naty1 said:
To expand on my explanation in post #11, an easy way to see a 'problem' introduced by velocity dependent 'mass' [relativistic mass] is that one can infer a REALLY fast moving particle with lots of 'relativistic mass' [really kinetic energy] can turn into a black hole.

That a fast moving particle buzzing by earth, for example, CANNOT induce enough gravitational curvature to form a black hole can be understood from the frame of the particle: From that perspective it is the Earth that has the KE, not the particle...and that type of frame dependency is not gravitational curvature which is independent of coordinates. KE is observer dependent.

Thank you Naty1. I have some reservations. Take for example the "fast moving particle" as observed from the Earth frame. From such a frame it may be the case that some may (incorrectly) judge that the particle has or can gain enough KE to form a black hole but what particles are being considered here?
As far as I know such high energy particles don't pre exist but in theory certain particles can be accelerated to the necessary high energies. If the latter is the case then a system is needed to accelerate the particle, the energy gains of the particle being balanced by energy losses of certain parts of the rest of the system that the particle interacts with.
The result is that although mass may be moved from one place to another the total mass remains fixed. There is not the overall increase of mass necessary to produce the black hole.
 
  • #33
If the latter is the case then a system is needed to accelerate the particle, the energy gains of the particle being balanced by energy losses of certain parts of the rest of the system that the particle interacts with...

Conservation of mass/energy does not apply in GR...that is, in curved spacetime.

There is no universal frame to even define velocity...or time...precisely...

Anyway, you don't need a lot of mass to form a black hole... Gravitational collapse occurs when an object's internal pressure is insufficient to resist the object's own gravity. That could be a pea sized mass...or smaller...
 
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  • #34
As far as I know such high energy particles don't pre exist but in theory certain particles can be accelerated to the necessary high energies.

I kind of overlooked that premise: I disagree as stated.

No single particle can be accelerated enough to form a black hole. That's because the source of gravity, the stress energy tensor, is independent of the particle's lateral velocity [kinetic energy].

But there is a possibly scenario where we CAN possibly increase gravity with KE:

http://en.wikipedia.org/wiki/Black_hole#Gravitational_collapse

Gravitational collapse is not the only process that could create black holes. In principle, black holes could be formed in high-energy collisions that achieve sufficient density.
 
  • #35
Dadface said:
There are some things that concern me about concepts of mass as discussed in threads such as this and I would appreciate it if someone could answer the following :

1. I have the impression that the majority of physicists who use relativity do not favour the equation:

M=MoL (L= Lorentz factor)

. I am a bit familiar with the other equations but is it considered that the above equation is archaic or misleading or incorrect in some way?

. If people reject the equation is it because of the terminology sometimes used? It seems to me that it is accepted that Mo can be referred to as the mass,or invariant mass and sometimes rest mass and that it is unnecessary to use the subscript o.
What doesn't seem to be accepted is that M (or E as it is sometimes written) should be referred to as the total mass where the total mass is the sum of the invariant mass plus the mass equivelent of the kinetic energy .If it is not accepted then what is wrong in calling it total mass and what, if anything, should it be called instead?

. Are there physicists who favour the use of the equation and if so are there examples of where the equation is more useful than any alternatives?



A big concern is that it is a requirement of some A2 physics courses in the UK that some relativity,including the above equation and the consequences of it, be taught ( eg in AQA A level physics unit 5). The students here are usually from age 16 to 18 and only a tiny minority will go on to further study in physics. If the equation is not generally favoured then what should be taught in its place?

The use of relativistic mass was quantified to a certain extent in On the Use of Relativistic Mass in Various Published Works by Gary Oas which is online at http://arxiv.org/abs/physics/0504111
Abstract - A lengthy bibliography of books referring to special and/or general relativity is provided to give a background for discussions on the historical use of the concept of relativistic mass.
If you look at the last table you'll see that relativistic mass is used about twice as much as proper mass. See also On the Abuse and Use of Relativistic Mass by Gary Oas at http://arxiv.org/abs/physics/0504110
Abstract - The concept of velocity dependent mass, relativistic mass, is examined and is found to be inconsistent with the geometrical formulation of special relativity. This is not a novel result; however, many continue to use this concept and some have even attempted to establish it as the basis for special relativity. It is argued that the oft-held view that formulations of relativity with and without relativistic mass are equivalent is incorrect. Left as a heuristic device a preliminary study of first time learners suggest that misconceptions can develop when the concept is introduced without basis. In order to gauge the extent and nature of the use of relativistic mass a survey of the literature on relativity has been undertaken. The varied and at times self-contradicting use of this concept points to the lack of clear consensus on the formulation of relativity. As geometry lies at the heart of all modern representations of relativity, it is urged, once again, that the use of the concept at all levels be abandoned.
His assertion that the oft-held view that formulations of relativity with and without relativistic mass are equivalent is incorrect. is wrong. It's based on the incorrect assumption that everything that relativity can describe can be completely, accurately and meaningfully described in geometric terms, which is incorrect. All of the every day objects that we encounter in life cannot be treated as the point objects that are described using 4-vectors since such objects have a finite extent in spacetime, are subject to stress and are not closed systems. As such they can't be correctly and completely described using 4-vectors. It's a glaring error that I've seen everyone make who supports this viewpoint.

Physicists often ignore the inertia of stress, and often the stress-energy-momentum tensor, in textbooks which teach special relativity. Wolgang Rindler is a good exception. He does a fine job at describing this in is SR/GR text. Schutz does a good job at explaining the inertia of pressure. Many others ignores these all too important facts. I'm not sure why though.
 
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<h2>1. What is the definition of mass in relativity theory?</h2><p>In relativity theory, mass is defined as the measure of an object's resistance to acceleration. It is a fundamental property of matter and can be thought of as the amount of matter contained within an object.</p><h2>2. How does mass change in relativity theory?</h2><p>In relativity theory, mass is not a constant quantity and can change depending on the frame of reference. This is known as the principle of mass-energy equivalence, where mass can be converted into energy and vice versa.</p><h2>3. What is the difference between inertial and gravitational mass?</h2><p>Inertial mass is a measure of an object's resistance to acceleration, while gravitational mass is a measure of the strength of an object's gravitational pull. In classical mechanics, these two types of mass are considered to be equivalent, but in relativity theory, they are distinct concepts.</p><h2>4. How does relativity theory conflict with classical views of mass?</h2><p>In classical mechanics, mass is considered to be a constant and absolute property of an object. However, in relativity theory, mass is not constant and can change depending on the frame of reference. This concept of relative mass is one of the key differences between classical and relativistic views of mass.</p><h2>5. How does the concept of mass contribute to our understanding of the universe?</h2><p>The concept of mass is a fundamental building block in our understanding of the universe. It helps us explain the behavior of objects in motion, the effects of gravity, and the relationship between matter and energy. Without the concept of mass, many of the principles and theories in physics, including relativity, would not exist.</p>

1. What is the definition of mass in relativity theory?

In relativity theory, mass is defined as the measure of an object's resistance to acceleration. It is a fundamental property of matter and can be thought of as the amount of matter contained within an object.

2. How does mass change in relativity theory?

In relativity theory, mass is not a constant quantity and can change depending on the frame of reference. This is known as the principle of mass-energy equivalence, where mass can be converted into energy and vice versa.

3. What is the difference between inertial and gravitational mass?

Inertial mass is a measure of an object's resistance to acceleration, while gravitational mass is a measure of the strength of an object's gravitational pull. In classical mechanics, these two types of mass are considered to be equivalent, but in relativity theory, they are distinct concepts.

4. How does relativity theory conflict with classical views of mass?

In classical mechanics, mass is considered to be a constant and absolute property of an object. However, in relativity theory, mass is not constant and can change depending on the frame of reference. This concept of relative mass is one of the key differences between classical and relativistic views of mass.

5. How does the concept of mass contribute to our understanding of the universe?

The concept of mass is a fundamental building block in our understanding of the universe. It helps us explain the behavior of objects in motion, the effects of gravity, and the relationship between matter and energy. Without the concept of mass, many of the principles and theories in physics, including relativity, would not exist.

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