# Einstein's:Mass increase resulting from Acceleration increase

#### Spin_Network

I am looking for the ultimate reference to the notion by Einstein, that increase in Acceleration will result in increase in Mass?

Any links or handwaving would be much appreiciated, as I intend to place a fundemental number of questions, in order to expand my knowledge on LQG, thanks.

Related Special and General Relativity News on Phys.org

#### Aer

Spin_Network said:
I am looking for the ultimate reference to the notion by Einstein, that increase in Acceleration will result in increase in Mass?

Any links or handwaving would be much appreiciated, as I intend to place a fundemental number of questions, in order to expand my knowledge on LQG, thanks.
I may be totally off base here (and I am sure others here will correct me if that is the case), but (I have been told) most physists have abandoned the idea of increasing mass with acceleration. Instead an objects mass is considered constant even though the objects total energy increases with velocity, that is e=f(m,c,v) (I do not know the exact equation). Of course this equation reduces to e=mc^2 for v<<c.

#### Chronos

Gold Member
Confusion sets in when you don't subtract the energy it takes to boost the relativistic mass from the local inertial reference frame.

#### pervect

Staff Emeritus
Aer said:
I may be totally off base here
Nope, you're on base, see for example the link:

Does mass change with velocity?

Relativistic mass changes with velocity, but the usefulness of the concept is questionable. Mass (without anyh qualifiers) is taken to be invariant mass, which does not change with velocity.

Acceleration changing mass is "right out".

#### Garth

Gold Member
It depends how you measure it, whether from a frame dependent observer's point of view or from the frame independent space-time 4D point of view.

From the POV of the observer with a definite preferred frame of reference, their own, foliated into 3D space + time, a moving mass appears to increase with velocity:
$$m=m_0\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$
where $$m_0$$ is the object's rest mass measured in its 'co-moving' rest frame.
The 'faster it goes the harder it is to push'. The force necessary to accelerate the object:
$$F=m\frac{d^2x}{dt^2}$$
tends to infinity as v tends to c.

However from the perspective of 4D space-time the four force is given by;
$$F^\mu=m_0\frac{d^2x^\mu}{d\tau^2}$$ where
$$d\tau^2=dt^2 -\frac{1}{c^2}(dx^2+dy^2+dz^2)$$ in a SR Minkowski space-time.
and the mass remains invariant $$m_0$$. The effect of the 'increase' in mass this time is accounted for by the definition of time; from the 4D perspective the invariant proper time $$\tau$$ is used not the observer dependent t.

It may be considered more 'pure' in relativity to use frame independent invariant mass and invariant time, however as real observers we are locked into a preferred time, our own, and we have to look out into the universe from that frame of reference. To any observer the masses of other moving objects appears to increase with velocity as above.

The choice of whether to use an observer dependent 3D + time or a frame independent 4D perspective is closely connected with how we want to account for the frame dependent concept of energy, as the observer dependent 'relativistic' mass subsumes the classical concept of kinetic energy as $$E = mc^2$$.

I hope this helps.

Garth

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#### Dr.Brain

Mass of a moving object deviates from the rest mass $m_o$ as:

$m= \frac {m_o}{\sqrt 1- \frac {v^2}{c^2} }$
As velocity increases , mass increases and with increasing velocity, it increases more rapidly.

BJ

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#### Garth

Gold Member
I think I have just said that....
But again note: only from the observer's frame dependent POV.

Garth

#### pmb_phy

pervect said:
...Mass (without anyh qualifiers) is taken to be invariant mass, which does not change with velocity.
It is incorrect to make this assertion since not all authors use the term in that way. MTW is one example. There are places in their text in which "mass" means relativistic mass. Its only in certain fields where it is easier to mean "rest mass" when someone uses the term "mass." However most people grasp what the term means either from the author stating it outright or when its obvious from context. This is true of other terms in relativity too. Take time dilation as an example; When people say that "the neutron has a half life of 15 minutes" it is assumed that they are referring to the proper life when they use that term. However nobody who knows relativity would say that time does not altered with speed.

Its quite easy to make errors if one does not completely understand the definitions correctly. In fact I know of one SR text which makes a mistake because of his usage of the term "mass" as proper mass.

Let me give you an example: Let S be an inertial frame of referanc in flat spacetime. If a magnetic field has a value of B and the electric field has a value of E = 0 then what is the mass density of the field in frame S? I'll get back next week to follow up with the answer.

Pete

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#### pmb_phy

Aer said:
I may be totally off base here (and I am sure others here will correct me if that is the case), but (I have been told) most physists have abandoned the idea of increasing mass with acceleration.
I can't see how that could be determined unless you took a poll. The usage of the term can be found in online university lecture notes, in new relativity texts and in the physics literature.

The term "mass" cannot be used as "invariant mass" in all generality since it can only be used under certain circumstances, namely for point particles or bodies wich are under stress. E.g. consider an electric dipole in an E field. Since the net charge on an electric dipole is zero the particle will have a rest frame. Rel-mass is defined as m = p/v. Some people claim that relativistic mass is defined as "m = E/c^2". However in this dipole example E/c^2 does not equal p/v so the terms cannot be defined to be equal/same since the definitions are different and since they are not always equal.

Pete

#### pmb_phy

Spin_Network said:
I am looking for the ultimate reference to the notion by Einstein, that increase in Acceleration will result in increase in Mass?
You must be refering to velocity, not acceleration, right? If so then the article was published in either 1906 (or 1907??). Sorry that I don't recall the name of the paper or the name of the journal. In the first section of this article Einstein shows how energy carries mass with it. In the second section he uses the term to speak of the mass of radiation. He also uses this notion on page 101 in his text The Meaning of Relativity when he's addressing the changing of a bodies mass with a change in the gravitational field in which it sits.

Pete

#### pervect

Staff Emeritus
pmb_phy said:
It is incorrect to make this assertion since not all authors use the term in that way. MTW is one example. There are places in their text in which "mass" means relativistic mass.
As I recall, I did not agree with this assertion the last time you made it, though I don't recall offhand which sections of MTW you interpreted as making this statement. You can quote them if you like, but I'm pretty sure they will not be clear-cut, and that I will disagree with your intepretation of what MTW actually said. If you feel like re-hashing this old point, though, go ahead.

I *really* do think that it is not too much to ask that people who mean "relativistic mass" should actually SAY "relativistic mass", not "mass".

It's quite simple - mass does not increase with velocity, relativistic mass does increase with velocity (I have no problem with that statement).

I'm willing to give older textbooks a "grandfather clause", on this point, for that matter, some of them may be written in such a confusing way. (MTW is pretty old, but it is modern enough not to suffer from this sort of confusion).

#### Integral

Staff Emeritus
Gold Member
I am going out on a limb here.

Perhaps the best way to prevent a recurrence of an old and ongoing quibble.

Let us define, for conversations in these forum, the meaning of the word "mass" to be REST MASS vs Relativistic mass.

Can we now direct the conversation toward the meaning and interpretation of the relativistic mass, rather then pointless argument over which is the correct viewpoint.

My question:
If applied energy is not increasing the mass, what is it increasing?

#### Aer

Integral said:
My question:
If applied energy is not increasing the mass, what is it increasing?
I cannot answer your question without risk being banned from physicsforums. As the answer (unfounded, no experimental data to prove or disprove it that I have found anyway) would imply one of the postulates of Special Relativity to be in slight error.

#### Integral

Staff Emeritus
Gold Member
I am not looking for, nor do I want an explantion which lies outside of the bounds of General Relativity.

#### pervect

Staff Emeritus
Integral said:
I am going out on a limb here.

Perhaps the best way to prevent a recurrence of an old and ongoing quibble.

Let us define, for conversations in these forum, the meaning of the word "mass" to be REST MASS vs Relativistic mass.

Can we now direct the conversation toward the meaning and interpretation of the relativistic mass, rather then pointless argument over which is the correct viewpoint.

My question:
If applied energy is not increasing the mass, what is it increasing?
If you apply a force to a single particle to accelerate it, you increase it's total energy (as measured in some particular inertial coordinate system). It's rest energy (rest mass) would remain the same, being an invariant. I would suggest that we say that the energy "went into" the kinetic energy of the object.

At least that's what we'd say if we were doing special relativity. If we were doing GR, we'd start asking about the presence of timelike Killing vectors, or the asymptotic flatness of space-time, and lose at least 90% of our audience in the process :-)

Applying energy to a system of particles to increase the temperature of said system yields a different result. Here, the energy of the system increases, and the system is still at rest, so we say that the rest mass (rest energy) of the system increases. This is true with any definition of energy or mass that I'm aware of.

People tend to think that if we have a system with a "relativistic mass" of mr, we can apply Newton's law to it simply by replacing mass with relativistic mass. This is false for a moving system (the only case when relativistic mass is different from invariant mass). The acceleration of a moving system will depend on the direction of the force as well as the "relativistic mass". The acceleration is NOT in general equal to a = F/mr. We can get around this by talking about the "transverse mass" of the system. In the past some authors did in fact do this (I believe Einstein has done this). But now we have _three_ kinds of mass to worry about. (And you thought things were confused with only two :-)).

People also tend to think that the gravitational field of a system with a "relativistic mass" of mr is -G mr /r^2 and that it's uniform in all directions. This is another all-too-common false notion that is caused by people assuning that relativistic dynamics is just a matter of substituting "relativistic mass" wherever "mass" occurs.

#### Aer

Integral said:
I am not looking for, nor do I want an explantion which lies outside of the bounds of General Relativity.
Nor do I want to give you such an explanation, although I said nothing of general relativity - special relativity is a "special" case of general relativity and it is only this special case that that the idea would challenge. Anyway, I don't understand your question. Isn't "relativistic" mass a fundamental part of the current understanding of relativity?

#### pervect

Staff Emeritus
"relativistic mass" is not particularly fundamental. Some people like it, some people don't. Read the sci.physics.faq

Does mass change with velocity?

#### pmb_phy

pervect said:
As I recall, I did not agree with this assertion the last time you made it, though I don't recall offhand which sections of MTW you interpreted as making this statement. You can quote them if you like, but I'm pretty sure they will not be clear-cut, and that I will disagree with your intepretation of what MTW actually said. If you feel like re-hashing this old point, though, go ahead.
I'm simply stating a fact. I'm not attempting more that that but its quite clear that MTW is doing what I said. I know relativity far too well to make such a silly mistake that you assumed I did.

However all one has to do is open the book and read it and that's all that is required. If someone has MTW and wants the reference then I'll post it.

Pete

#### pmb_phy

pervect said:
People ....
Who are these "people"? Are they students? Teachers?

...tend to think ...
As with any subject, if a student makes a mistake like this then they will have to study more.
...that if we have a system with a "relativistic mass" of mr, we can apply Newton's law to it simply by replacing mass with relativistic mass.
What is mr"? This is false for a moving system (the only case when relativistic mass is different from invariant mass). The acceleration of a moving system will depend on the direction of the force as well as the "relativistic mass". The acceleration is NOT in general equal to a = F/mr.
That's no even correct in Newtonian mechanics. For a particle with a force being applied to it the acceleration is given by F = ma -> a = F/m.

For a student to learn relativity correctly they cannot assume that a quantity will reduce to the Newtonian quantity if they simply rearrange the terms. I.e. If p = mv then it can't be shown that m = F/a. There is no reason to assume that it can at all. You've chosen to use a relation, i.e. m = F/a, which is even wrong in Newtonian mechanics. Force does not have that value. Force is given by F = dp/dt where p = mv (m = rel-mass). The m in this relation comes from Weyl's definition of mass whereas F = ma comes from Euler's definition of mass. To make a guess we turn to the fact that, in 3+1 language, we turn to the relationship for force. Force is always a function of velocity in special relativity. The gamma factor is not 1/sqrt(1-v^c/c^2) but has the value gamma = dt/dT. Relativistic mass then has the correct value and will be a function of velocity. Now take a hint from EM. The EM force is a function of velocity. Now jump to GR. Make the assumption that the force will be a function of velocity and you'll get the right answer.

People also tend to think that the gravitational field of a system with a "relativistic mass" of mr is -G mr /r^2 and that it's uniform in all directions.
Students will then need to hit the books then.

At this web site I explain much of what I mentioned above
http://www.geocities.com/physics_world/gr/grav_force.htm
This is another all-too-common false notion that is caused by people assuning that relativistic dynamics is just a matter of substituting "relativistic mass" wherever "mass" occurs.
Again, they will need to hit the books again and learn it right. I've seen grad students make mistakes in relativity by using false assumptions. The one I have in mind is from a grad student from MIT. He thought that the reason for clocks slowing down was that to get the clock to slow down you need to exert forces on it and it is those forces which damage" the clock and thus is the reason for time dilation (Wrong!)

Confusion like this is found elsewhere in physics. For example; consider the term momentum as used in, say, quantum mechaniocs. When this is seen in the quantum mechanics literature people will have to be careful. When it is meant by "momentum" in quantum mechanics is that the momentum is really canonical momentum, not linear mechanical momentum. These two momenta are not the same. E.G. in EM one has to add a term to linear mechanical momentum in order to get the canonical momentum.

Pete

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#### pervect

Staff Emeritus
pmb_phy said:
Who are these "people"? Are they students? Teachers?
As with any subject, if a student makes a mistake like this then they will have to study more.
After a few zillion times (I exaggerate slightly, perhaps) of trying to tell people that, it is wrong to subsitute mr (relativisitc mass) for m in Newton's laws (either the F=ma variety, or the F=gmM/r^2 variety), I get just a little bit tired of repeating myself.

So by people, I mean people on physics forums, many of which are not particularly knowledgable about relativity.

Some people don't seem to listen, either. (Not much that can be done in that case, I suppose, but it doesn't help the frustration level). I could point to a recent example or two on the general physics forum of this phenomenon, BTW, but I think it would be better if I kept things less personal and didn't single out any particular individuals in this manner.

The sci.physics.faq also comments on the large number of people who get confused, rather than enlightened, by relativistic mass.

In spite of this, I'm not on any campaign to stamp out the concept. If people like the idea of relativistic mass, that's fine with me, as long as they use it correctly. Part of correct usage is to identify that one is in fact talking about relativistic mass, rather than invariant mass. Making this distinction manifest would avoid the age-old confuison about the fact that a photon has no invariant mass (rest mass), though it does have energy (and so has "relativistic mass").

What I object to is when people use "mass" with no qualifiers to mean "relativistic mass". This is not done in modern textbooks, and I certainly am not aware of it being done anywhere in MTW's "Gravitation".

#### pmb_phy

pervect said:
After a few zillion times (I exaggerate slightly, perhaps) of trying to tell people that, it is wrong to subsitute mr (relativisitc mass) for m in Newton's laws (either the F=ma variety, or the F=gmM/r^2 variety), I get just a little bit tired of repeating myself.
Then it'd be more clear if you expressed it as m_r/

So by people, I mean people on physics forums, many of which are not particularly knowledgable about relativity.

The sci.physics.faq also comments on the large number of people who get confused, rather than enlightened, by relativistic mass.
Are you also saying that these same people don't get other relativisitic concepts wrong such as time dilation. Lorentz contraction etc?

Part of correct usage is to identify that one is in fact talking about relativistic mass, rather than invariant mass.
That's not quite correct. It definitely cannot be said that when someone who uses the term "mass" to mean rel-mass is doing so incorrectly.
Making this distinction manifest would avoid the age-old confuison about the fact that a photon has no invariant mass (rest mass), though it does have energy (and so has "relativistic mass").
It would be best to teach the concept well enough so that no tudent would make such a mistake. In all cases that I am aware of authors state what "m" means just once and then use it as such. Then there will be no problems.
What I object to is when people use "mass" with no qualifiers to mean "relativistic mass".
I feel the same way when people use the term "mass" to mean "proper mass"
This is not done in modern textbooks, and I certainly am not aware of it being done anywhere in MTW's "Gravitation".
You should hit the library and scan the relativity text. What you'll see is that there are several new texts (new = published within last 10 years) which do this. As for MTW - Turn to the page where the authors state "Mass is the source of gravity" or something like that. Also turn to the derivation which they give which shows that the energy-momentum tensor is symmetric.

Pete

#### pervect

Staff Emeritus
pmb_phy said:
Are you also saying that these same people don't get other relativisitic concepts wrong such as time dilation. Lorentz contraction etc?
Nope. But anyone who uses the term "relativistic mass" is particlularly likely to be severly confused. It's a trend I've noticed. In fact, I can't think of anyone other than you (pmb_phy) who is reasonably knowledgable who uses the term "relativistic mass" and isn't confused. In a way it's quite useful, I know automatically to turn the math level way way down when responding to anyone who asks a question with the words "relativistic mass" in them (with currently one exception to this general rule).

That's not quite correct. It definitely cannot be said that when someone who uses the term "mass" to mean rel-mass is doing so incorrectly.
At the very best, they are being ambiguous.

And the sci.physics.faq says the following on the topic:

Although the idea of relativistic mass is not wrong, it often leads to confusion, and is less useful in advanced applications such as quantum field theory and general relativity. Using the word "mass" unqualified to mean relativistic mass is wrong because the word on its own will usually be taken to mean invariant mass. For example, when physicists quote a value for "the mass of the electron" they mean its invariant mass.
The utility and the usefullness of the sci.physics.faq to all those confused newbies out there is extremely high (assuming they can be motivated to read it). Rather than say something a zillion times, possibly not explaining it very well, I can point people to a well-writen document that goes through the commonly-asked questions in great detail, with references for further reading. The utility of your objections on this point is very low (you wanting to make a shortcut, basically). Apparently you actually think about physics in such a way that you find the concept useful. That's fine. Feel free to use the term "relativistic mass" all you like. Tout its advantages. Show the world how it makes problems easier to approach (if you can).

The only thing the FAQ says is that since the term "mass" has multiple meanings, if you mean relativistic mass you should actually say "relativistic mass". This strikes me as being quite reasonable.

If you would just go along with the standard usage, and not try and be such a flaming martar, it would be nice, because I really have learned a lot of useful things from talking to you on quite a large number of issues.

#### pmb_phy

pervect said:
Nope. But anyone who uses the term "relativistic mass" is particlularly likely to be severly confused.
Why? It has never confused me. There are lots of things in physics which is potentially confusing to students. But we don't get rid of them.
In fact, I can't think of anyone other than you (pmb_phy) who is reasonably knowledgable who uses the term "relativistic mass" and isn't confused.
Thanks. But keep in mind that hardly anyone will take the time out to learn the most basic of relativity problems. These include things like extended bodies and the use of the energy-momentum tensor. In fact people will tend to want to forget about this tensor if they're working strictly with SR. But these kinds of problems are the most general. There are some troublesome areas for those who use "mass = proper mass."
In a way it's quite useful, I know automatically to turn the math level way way down when responding to anyone who asks a question with the words "relativistic mass" in them (with currently one exception to this general rule).

Feel free to use the term "relativistic mass" all you like. Tout its advantages. Show the world how it makes problems easier to approach (if you can).
Its useful when someone is speaking of things like a extendewd charged object in an E-field. One cannot use something like a 4-vector such as 4-momentum to describe the mass of this body. Its also useful in cases like that of a radiating body which is not a point particle. If you attempt to use the formulas you're familiar with (magnitude of 4-momentum) you'll run into a problem.
The only thing the FAQ says is that since the term "mass" has multiple meanings, if you mean relativistic mass you should actually say "relativistic mass". This strikes me as being quite reasonable.
Later on this summer the relativity lecture notes under the Harvard University physics web site will be changed. The author held that energy and relativistic mass are the same thing. I sent the author and e-mail explaining the nature of his error. He agreed and will be changing it later on this summer. If a Harvard professor who teaches relativity gets "mass = proper mass" wrong then that says something about the usefulness of saying "energy = relativistic mass. A similar thing happened in Ohanian's new SR text. Ohanian actually made an error when he was speaking of the mass density of a magnetic field. If Ohanian can get this wrong then it also says something about "mass = proper mass"

Here is the page I was going to post last week
http://www.geocities.com/physics_world/misc/relativistic_mass

The MTW entry is there as are other examples. Here is a web page where I pointed out other things.
http://www.geocities.com/physics_world/sr/invariant_mass.htm

Pete

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#### robphy

Homework Helper
Gold Member
Just to chime in:

Taylor-Wheeler (1963, 1966 editions) - p. 108
"Mass most usefully defined as the velocity-independent factor in the momentum". (see also p. 137, as quoted in the URL given by pervect: http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html )

MTW - p. 53
"Consider the 4-momentum p of a particle, an electron, for example. To spell out
one concept of momentum, start with the 4-velocity, $${\sf u}=d{\cal P}/d\tau$$ , of this electron ("spacetime displacement per unit of proper time along a straightline approximation of the world line"). This is a vector of unit length. Multiply by the mass m of the particle to obtain the momentum vector $${\sf p} = m {\sf u}$$." [By the way, in this context, $${\cal P}$$ is a worldline [curve], parametrized by proper time $$\tau$$.]

MTW - p. 141 (§5.7. SYMMETRY OF THE STRESS-ENERGY TENSOR)
"...Consider first the momentum density (components T^{j0}) and the energy flux (components (T^{0j}). They must be equal because energy = mass ("E = Mc^2 = M")."
It can argued that this use of "mass" is a loose one. It seems that, in most places in MTW, they are careful to use the words rest-mass and mass-energy.

It seems to me (and to others), in this age of various conventions and "schools of thought" (e.g. particle physics vs. relativity(SR/GR) ), that one simply needs to clearly define one's terms. Pardon the pun... may I suggest the "PF signature convention"... to make clear what you might mean by "mass", include its definition in your PF signature.

Certainly, SR forces us to refine our understanding and use of words like "mass", "energy", "length", "time", "momentum", "force", "velocity", "acceleration", etc...

Ideally, one should try to be consistent. Observe (up to factors of c)
$$\begin{tabular}{llll} vector & square-norm & temporal component & spatial component \\ \hline Euclidean vector & (hypotenuse)^2 & (hyp)*\cos\theta & (hyp)*\sin\theta \\ \hline Minkowskian 4-vector & square-interval & (hyp)*\cosh\theta=(hyp)\gamma & (hyp)*\sinh\theta=(hyp)\beta\gamma \\ timelike-displacement & (proper-time)^2 & apparent-time & apparent-distance \\ spacelike-displacement & -(proper-length)^2 & apparent-time & apparent-distance \\ 4-momentum {\small of a point particle}& (rest-mass)^2 & relativistic-mass (relativistic-energy) & relativistic-momentum \end{tabular}$$

Observe that "relativistic mass" is an observer-dependent concept... a vector component, whereas "rest mass" is an observer-independent concept... a scalar.

It is my preference (guided by the precision and conciseness of geometrical coordinate-independent concepts and calculations) to work with scalars (such as "rest mass") and vectors (such as "4-momentum") as much as possible. It is (in my opinion) more elegant to draw a vector, then (if necessary) draw axes then project out components... as opposed to first drawing axes and components, then trying to find a transformation to someone else's axes and components (possibly reconstituting the vector that they both describe). Said another way, I prefer working with [observer-independent] geometry than with [observer-dependent] coordinates.

It is also my preference to do away with the "relativistic" adjective altogether... as if we at some point "switch-on relativity". The range of applicability of SR encompasses that of Galilean/Newtonian physics! On the assumption that SR/GR and quantum physics are more-correct descriptions of nature [in spite of their mutual inconsistencies], our "classical" concepts are merely "Newtonian approximations" to their truer SR/GR or quantum descriptions. In those "component" columns above, I feel the "apparent" adjective is better at suggesting the observer-dependence of the quantity under discussion.

Concerning the redefinition of "mass".
• I do not think that "mass" should mean "relativistic mass"...As suggested above, I would prefer "apparent-mass" over "relativistic-mass", again emphasizing the observer-dependence.
• If "mass" is redefined to mean "rest mass" or "proper mass", I feel that this must be done consistently. If we drop "rest" or "proper", then "length" should always mean "rest length" or "proper length", "time" should always mean "proper time", etc...
• Given the potential ambiguities and the care that must be taken to generalize concepts to extended objects, it's best to retain "rest" , "proper", "apparent", etc... for clarity.

Symbolically, I prefer $$m$$ or $$m_0$$ for the invariant "rest-mass"... that is $$m^2={\sf p}\cdot{\sf p}$$.
I prefer $$m_{\vec V}=m_0\gamma=m_0\gamma_{\vec V}=p_t={\sf p}\cdot{\sf t}$$ (where $${\sf t}$$ is the observer's 4-velocity). The last expressions are the most descriptive, again emphasizing the observer-dependence.

My \$0.03.

#### pervect

Staff Emeritus

The example on pg 141 of MTW does not convince me that MTW embraces the concept of relativistic mass. You will note that the term "relativistic mass" does not appear anywhere in the equation you wrote.

If we proceed to the preceding pages for some context, on pg 140, we see
However, for a general perfect fluid, density $\rho$ of mass-energy as measured in the fluid's rest-frame includes not only rest mass plus kinetic energy of particles, but also energy of compression, energy of nuclear binding, and all other sources of mass energy.
Thus when MTW says on the very following page

$T^{0j}$ = (energy flux) = (energy density) * (mean velocity of energy flow) = (mass density)*(mean velocity of mass flow)

We should interpret their remarks on pg 141 in light of the previous definitions on pg 140.

Robphy's example on pg 53 is a fine example of where the term mass is used in MTW to indicate invariant mass.

The moral of the story in my opinion is that the term "mass" by itself is ambiguous. By taking a short section of MTW out of context, you make it appear that MTW embraces the concept of relativistic mass, where nothing could be further from the truth.

The safest thing to do is undoubtedly to write down "invariant mass" or "relativistic mass" every single time the phrase is used. This could become quite tedious, so I have no objection to an author writing the term mass(invariant mass) or mass (relativistic mass) to define what sort of mass he or she is talking about, with the understanding that further usages of the term will be consistent with the definition originally offered.

The only two dangers with this approach are deliberate "nit-picking", where people go out of their way to look for things to complain about, - and unintentional confusion when quotations get trimmed. Hopefully neither of these will be a large problem. I have a large flame-thrower reserved for case 1, "deliberate nit-picking" :-).

I do not expect the layman or new person on the forums to ask their questions so carefully and precisely, BTW. Many times it takes quite a bit of back-and-forth communication to find out exactly what a layman's question really is, it's unfortunately quite easy for misscommunication to occur.

I do agree that the issue of the mass of an extended body is not one that is adequately covered in MTW. I don't think that this is particularly related to the question of "relativistic mass" though.

Substitution of the flat space metric $g_{ab} = \eta_{ab}$ into the equation for mass 11.2.10 on pg 289 of Wald is one (tedious) way of getting the correct definition of mass in flat-space time (the equation is for static space-times, flat space-times are automatically static) which properly includes the pressure terms. The result is

M = $2 \int(T_{00}-T/2) dV$, T=$T^a{}_a$.

Or equivalently, for an orthonormal coordinate system,
M = $\int(T_{00} + T_{11} + T_{22} + T_{33})dV$

Compare and contrast this to the expression on 449 on MTW, which unfortunately appears to be quite wrong, which ommits the pressure terms. I believe that the pressure terms will average out to zero when the body is not rotating, but it seems obvious that a rotating body will have a net tension required to hold it together.

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