Do Einstein's Theories of Relativity Contradict Each Other?

[Garth]

Was Einstein inconsistent between his theories of Special and General Relativity?

In the theory of Special Relativity we learn that energy and mass are interchangeable E = mc^2.

In the theory of General Relativity we learn that because of Einstein's equivalence principle (EEP) the mass of a particle is invariant. When a uranium atom undergoes fission, the energy released is only the energy of the system, bound up in the atom, that is being re-allocated; the masses of all the constituent particles making up the atom remain invariant.

Are these two theories therefore mutually contradictory?

 

[DW]

No. Mass is invariant in both theories, not just general relativity.

 

[pmb_phy]

Was Einstein inconsistent between his theories of Special and General Relativity?

No.

In the theory of Special Relativity we learn that energy and mass are interchangeable E = mc^2.

True.

In the theory of General Relativity we learn that because of Einstein's equivalence principle (EEP) the mass of a particle is invariant.

The proper mass (aka rest mass) is invariant. That is not a result of relativity. Its a fact of nature which relativity never changed.

When a uranium atom undergoes fission, the energy released is only the energy of the system, bound up in the atom, that is being re-allocated; the masses of all the constituent particles making up the atom remain invariant.

The energy released is not the only energy of the system. The energy released is the Q of the system and the Q of the system is only part of the energy of the system. See
http://www.geocities.com/physics_world/sr/nuclear_energy.htm

Are these two theories therefore mutually contradictory?

Not that I've seen.

Pete

 

[DW]

No.

True.

The proper mass (aka rest mass) is invariant. That is not a result of relativity. Its a fact of nature which relativity never changed.

The energy released is not the only energy of the system. The energy released is the Q of the system and the Q of the system is only part of the energy of the system. See
http://www.geocities.com/physics_world/sr/nuclear_energy.htm

Not that I've seen.

Pete

Your own personal sight is not an independent reference. As always you are spamming for Planck's outdated concept of mass. You are wrong.

 

[quantumdude]

Your own personal sight is not an independent reference. As always you are spamming for Planck's outdated concept of mass. You are wrong.

So what if it's outdated? That doesn't mean it's wrong.

Incidentally, it's not outdated. That concept of mass is still alive and well among those who work in nuclear power.

 

[Garth]

Mass is invariant in both theories, not just general relativity.

The mass of a body as measured by an observer depends on the velocity of the the body in the observer's frame of reference.

 

[quantumdude]

The mass of a body as measured by an observer depends on the velocity of the the body in the observer's frame of reference.

That is just one convention. We aren't obligated to adopt it, and indeed most physicists don't. The convention adopted by most physicists is that there is only one mass: the invariant mass. That quantity is the norm of the 4-momentum. But as I said before, the concept of mass that says m=γm₀ isn't wrong, it's just out of style.

 

[pmb_phy]

That is just one convention. We aren't obligated to adopt it, and indeed most physicists don't.

That is misleading. The majority of particle physicists don't use it. The majority of GRists and cosmologists do use it.

But the way, what are you basing that assumption on?

Pete

 

[quantumdude]

That is misleading. The majority of particle physicists don't use it.

Misleading? Correct me if I'm wrong, but I think that the community of particle physicists is the majority of physicists who use relativity. Factor in those solid state physicists who use relativisitc quantum mechanics or QED, and it's no contest.

The majority of GRists and cosmologists do use it.

Really? Every textbook I have teaches the concept of mass as the invariant norm of the 4-momentum, and they are written by relativists (Taylor and Wheeler, Ohanian and Ruffini, et al). What books do use it? And are there publications in the arxiv that use it?

But the way, what are you basing that assumption on?

All my undergraduate and graduate coursework.

 

[pmb_phy]

Hi Tom

For my response to be logical it turned out to be too long for a post so I started a new thread. See the new thread Those who use relativistic mass and why

Pete

 

[DW]

So what if it's outdated? That doesn't mean it's wrong.

Incidentally, it's not outdated. That concept of mass is still alive and well among those who work in nuclear power.

It is outdated whether it is being used or not and it is wrong. It was a guess that just happened to put \gamma in the place that it needed to be in a momentum equation to yield dynamics consistent with special relativity, but the mass term in that equation is NOT where it comes from. It comes from time dilation in the time differential in the velocity term. This missassociation of the factor with the mass is why it is wrong and the modern understanding of where the term comes from in terms of the four vector law is why it is outdated.

 

[DW]

The mass of a body as measured by an observer depends on the velocity of the the body in the observer's frame of reference.

No. Mass is invariant.

 

[Garth]

What we have here is a conflict of conventions of definition of terms.

The question of whether or not the mass of a particle can vary or not ought to be a matter of observation not definition. If we define mass to be invariant then we are blinding ourselves to the fact that it might be otherwise.

In the “classical interpretation” of the Einstein’s equivalence principle (EEP) mass is invariant. Therefore we have masses on the one hand and energies on the other, and although energy has a mass equivalent, they cannot transform one into the other. Yet at a fundamental level a particle seems to be a string, or whatever, of vibrating energy, and sufficiently energetic photons can transform into a particle and its anti-particle and vice versa.

My original question was to question this convention, is it not inconsistent with the precept of SR? Incidentally SR says nothing about the invariance of mass, that has been read in later.

In my theory of self creation I choose to define mass to be able to include potential energy and it leads to some very interesting observational consequences; one of which is a heterodox prediction for geodetic precession, which is about to be measured by the Gravity Probe B satellite.

 

[pmb_phy]

What we have here is a conflict of conventions of definition of terms.

If dw posted what I think he did then I agree 100%.

The question of whether or not the mass of a particle can vary or not ought to be a matter of observation not definition. If we define mass to be invariant then we are blinding ourselves to the fact that it might be otherwise.

There are two definitions in common use.

Let v = 3-velocity. Then when m is defined such that mv is conserved then this is an implicit definition of m and is commonly referred to as inertial mass (aka relativistic mass).

Let U = 4-velocity. Then when m₀ is defined such that m₀U is conserved then this is an implicit definition of m₀ and is commonly referred to as proper mass (aka rest mass).

When people use the term mass, some of them are referring to m while others are referring to m₀.

And that's the whole story on the concept of mass as it pertains to definition.

In the “classical interpretation” of the Einstein’s equivalence principle (EEP) mass is invariant.

Please provide a definition of classical interpretation.

Thanks

Pete

 

[quantumdude]

It was a guess that just happened to put \gamma in the place that it needed to be in a momentum equation to yield dynamics consistent with special relativity, but the mass term in that equation is NOT where it comes from.

So what?

In p=γmv, is γ multiplied by m? Answer: Yes.

Does the law of associativity under multiplication still hold? Answer: Yes.

Can I associate (γm) together and call it something else? Answer: Yes.

Does the quantity have the dimensions of mass? Answer: Yes.

Is there anything wrong with giving that mass a name? Answer: No.

 

[DW]

So what?


Can I associate (γm) together and call it something else? Answer: Yes.

Is there anything wrong with giving that mass a name? Answer: No.

Concerning the first question here you are not calling that just "something" else. You are calling it something that it does not mean. Your last question here has a wrong hidden statement. You state that the something you want to name is mass. That is what is wrong.

 

[quantumdude]

Concerning the first question here you are not calling that just "something" else. You are calling it something that it does not mean.

It means "relativistic mass" if I define it to mean that. That is the nature of a definition.

 

[DW]

It means "relativistic mass" if I define it to mean that. That is the nature of a definition.

A missnomer is a better word for it.

 

[quantumdude]

A missnomer is a better word for it.

You do realize that this is just your personal opinion, right?

 

[Garth]

Classical interpretation of mass: "rest mass", i.e. the mass of an object measured in a co-moving frame of reference in which the object is at rest, is equal to the norm of the 4-momentum vector and is invariant. It is a direct consequence of the EEP (see for example Weinberg) and therefore GR.

 

[pmb_phy]

Classical interpretation of mass: "rest mass", i.e. the mass of an object measured in a co-moving frame of reference in which the object is at rest, is equal to the norm of the 4-momentum vector and is invariant. It is a direct consequence of the EEP (see for example Weinberg) and therefore GR.

Why do you use the term "classical" here as a qaulifier for "interpretation"? What is it supposed to refer to? Classical in what sense of the word?.

Where in Weignberg's text do you see Weignberg say "It is a direct consequence of the EEP ... and therefore GR."?

Thanks

Pete

 

[Garth]

"Classical": just my term for "normal convention", there are others.
The conservation of the norm of the 4-momentum vector is true under Lorentz transformations in the absence of gravitation; by the EEP it is also true in the presence of gravitation, see Weinberg's development in "Gravitation and Cosmology" pg. 44, and the definition of the EEP which states that "at every space-time point in an arbitrary gravitational field it is possible to choose a "locally inertial coordinate system" such that, within a sufficiently small region of the point in question, the laws of nature take the form as in unaccelerated Cartesian coordinate systems in the absence of gravitation" .(Weinberg pg. 68)

 

[pmb_phy]

"Classical": just my term for "normal convention",..

You have the privilege of being the first person to use that term in that way in this forum.

The conservation of the norm of the 4-momentum vector is true under Lorentz transformations in the absence of gravitation;

You have to be very careful how you say that. The magnitude of the 4-momentum is not always a conserved quantity. That is only true for closed systems. E.g. if you have a particle which emits radiation then the magnitude of the 4-momentum of that particle changes and is therefore not a conserved quanity. In general it is a function of the proper time of the particle. For details please see Invariant vs. Time Independent at
http://www.geocities.com/physics_world/sr/invariant_mass.htm

You really have to be careful when you add 4-momenta too. Its only meaningful to add them when the particles interact only through contact forces.

...by the EEP it is also true in the presence of gravitation, see Weinberg's development in "Gravitation and Cosmology" pg. 44, and the definition of the EEP which states that "at every space-time point in an arbitrary gravitational field it is possible to choose a "locally inertial coordinate system" such that, within a sufficiently small region of the point in question, the laws of nature take the form as in unaccelerated Cartesian coordinate systems in the absence of gravitation" .(Weinberg pg. 68)

You didn't answer my question. I asked you where in Weinberg hge said that rest mass = mag of 4-momentum is a It is a direct consequence of the EEP (see for example Weinberg) and therefore GR. He does not say that in those pages. Yes, its true what he says on those pages but that rest mass = mag of 4-momentum is not a direct result of EEP. As I explained to you before, rest mass was constant before SR and GR and they (SR/GR) didn't change it or prove it. Just because its true in SR/GR it doesn't imply that the EEP proved it.

Pete

 

[Garth]

There are two uses of the word "invariant" - invariant under coordinate transformation and invariant under particle and/or force interaction. I was using the first meaning of that term.

We do not know whether (rest) mass is/was constant unless it can be measured or compared with something other than rest mass! I suggest that when the energy of a photon, cosmologically a photon taken from the peak intensity of the MBR, is compared to rest masses those masses will be seen to be secularly increasing. To do so however would be to violate the EEP.

 

[pmb_phy]

There are two uses of the word "invariant" - invariant under coordinate transformation and invariant under particle and/or force interaction. I was using the first meaning of that term.

Why do you mention this? I was commenting on your comment "The conservation of the norm of the 4-momentum vector is true under Lorentz transformations in the absence of gravitation". I believe that you used the term "conservation" when you mean "invariance". Did you not?

 

[Garth]

If an observer in one inertial frame observes a particle in another, which is accelerating relative to the observer's frame because of gravitational forces , the four-momentum of the particle is observed to be constant over time, and also equal to its value in the particle's rest frame. It is therefore invariant and conserved. Of course any energetic interactions will change its value but that is an added complication not addressed in my post above.

 

[mijoon]

You do realize that this is just your personal opinion, right?

I believe that this was also the personal opinion of a certain Albert Einstein .

 

[DW]

Classical interpretation of mass: "rest mass", i.e. the mass of an object measured in a co-moving frame of reference in which the object is at rest, is equal to the norm of the 4-momentum vector and is invariant. It is a direct consequence of the EEP (see for example Weinberg) and therefore GR.

Mass is not just equal to the norm of the 4-momentum, it is equivalent to it. As such mass does not depend on frame and as such need not be measured specifically from rest frame coordinates and as such is improper to qualify with the word rest.

 

[Garth]

Mass is not only something to be defined, it is something to be measured. In specifying how it is measured one cannot be too careful.

 

[pmb_phy]

Mass is not only something to be defined, it is something to be measured. In specifying how it is measured one cannot be too careful.

Yup. I quite agree Garth.

Pete

 

[DW]

Mass is not only something to be defined, it is something to be measured. In specifying how it is measured one cannot be too careful.

You are the one suggesting that it is to be measured from its rest frame. I am saying particle mass really never is. I have given an experiment with which one "measures" the mass of a charged particle. You haven't. So what is your point then really?

 

[Garth]

You are the one suggesting that it is to be measured from its rest frame. I am saying particle mass really never is. I have given an experiment with which one "measures" the mass of a charged particle. You haven't. So what is your point then really?

That if you define mass to be invariant then you are blind to any possible variation mass may actually experience.

The essential question, which nobody has challenged me on, yet, is if mass should vary how would you detect it?
Observation and the theory by which that observation is interpreted are inextricably bound up with each other.

Essentially measurement of mass is by comparison with a standard, a lump of platinum in a Paris safe. So if particle mass varies cosmologically, for example, the mass of the standard will vary with that of the object and you will not detect a variation. If however we have another standard, say the energy of a photon, cosmologically one that is sampled from the peak of intensity of the MBR, then all masses may be seen to vary.

In this view masses would be cosmologically increasing instead of light being cosmologically red shifted. As a result the rates of two types of clock: an 'atomic' clock (time interval = period of atomic vibration) and a 'photonic' clock (time interval = inverse of frequency) will diverge - there will be a time slip. A slip between atomic and ephemeris time would explain the Pioneer anomaly. [See Ostermann, P.: Dec 2002, arXiv:gr-qc/0212004. Relativity Theory and a Real Pioneer Effect.]

Incidentially did you know that after allowing for tidal effects the Earth is spinning up by 0.6 milliseconds/day/century? [See: Morrison, L. & Stephenson, F.R.:1998, Astronomy & Geophysics Vol. 39 October. The Sands of Time and the Earth’s Rotation and Stephenson, F.R.:2003, Astronomy & Geophysics Vol. 44 April. Historical eclipses and Earth’s rotation.]
What is the significance of this?
0.6 millisecs/day/century is Hubble's constant! Makes you think eh?

 

[DW]

That if you define mass to be invariant then you are blind to any possible variation mass may actually experience.

One can not be blind to something that can not happen by definition. You are trying to be insulting and instead have demonstrated a fault in your own thinking.

The essential question, which nobody has challenged me on, yet, is if mass should vary how would you detect it?

Since it doesn't vary with respect to speed by definition you can't.

If however we have another standard, say the energy of a photon, cosmologically one that is sampled from the peak of intensity of the MBR, then all masses may be seen to vary. ...

You can't use a zero mass particle as a mass standard.
(snipped a lot irrelevent)

 

[pmb_phy]

The essential question, which nobody has challenged me on, yet, is if mass should vary how would you detect it?

Depends on the definition again. Once you properly define a quantity you can learn its properties. If you define mass as the ratio of momentum to speed then you can measure its mass as a function of speed in various ways depending on the particulars (e.g. is it charged? Etc).

By "vary" do you mean "function of time" or "function of speed"?

re - "Essentially measurement of mass is by comparison with a standard, a lump of platinum in a Paris safe."

That is one standard. Others exist. Such as the atomic mass unit which is defined as 1/12 the the mass of a Carbon-12 atom.

re - " So if particle mass varies cosmologically, .."

Please explain what "varies cosmologically" means?

Pete

 

[Garth]

DW - "One can not be blind to something that can not happen by definition"
Thank you for proving my point.
Should not science advance by observations/experiments that challenge and possibly falsify previous definitions and theories?

Pete - "Please explain what "varies cosmologically" means?"
That the mass of an object be a function of cosmological time, such as in Hoyle and Narlikar's conformal relativity theories (and self creation too I might add).

But you do need a standard, which by convention and definition is invariant, to compare it against. It is just a question of identifying the correct standard and the underlying principle which determines it to be invariant.

 

[DW]

DW- "One can not be blind to something that can not happen by definition. "
Thank you for proving my point.

I didn't prove your point true. I proved your point incorrect.

Should not science advance by observations/experiments challenging and possibly falsifying previous definitions and theories?

No. One never falsifies a definition. That is irrelevant to falsification of a theory.

Pete "Please explain what "varies cosmologically" means?"
That the mass of an object be a function of cosmological time, such as in Hoyle and Narlikar's conformal relativity theories (and self creation too I might add).

But you do need a standard, which by convention/definition is invariant, to compare it against. It is just a question of identifying the correct standard and the principles by which you think that it is invariant.

Invariant is not synonymous with conserved.

 

[Garth]

DW - The definition may be falsified in the sense that it is proven to be inadequate.

A standard of measurement is invariant; the principle by which it is defined so is a conservation law.

 

[quantumdude]

DW - The definition may be falsified in the sense that it is proven to be inadequate.

It is indeed true that definitions cannot be falsified. But this makes me wonder why DW thinks he has falsified the definition of relativistic mass.

 

[DW]

It is indeed true that definitions cannot be falsified. But this makes me wonder why DW thinks he has falsified the definition of relativistic mass.

I suppose falsified is a bad choice of words. What I clearly stated was that it was in the "sense" that I showed it to be inadiquate. How much more simply can I state that so you needn't wonder what I think?

 

[DW]

DW - The definition may be falsified in the sense that it is proven to be inadequate.

And the relativistic mass definition is inadiquate. In that "sense" the definition is falsified.

...invariant...the principle by which it is defined so is a conservation law.

Since when?

 

[quantumdude]

I suppose falsified is a bad choice of words. What I clearly stated was that it was in the "sense" that I showed it to be inadiquate. How much more simply can I state that so you needn't wonder what I think?

You can clearly state that all you like, but when you say (as you have repeatedly) that such a concept of mass is "blatantly wrong" or "proven incorrect", it sends an uniquivocal message that you believe that such a definition has been falsified in the fullest sense of the word.

 

[pmb_phy]

You can clearly state that all you like, but when you say (as you have repeatedly) that such a concept of mass is "blatantly wrong" or "proven incorrect", it sends an uniquivocal message that you believe that such a definition has been falsified in the fullest sense of the word.

Its quite the opposite in fact. I've already shown why mass = proper mass is inadequate in general, i.e. in more complex situations. Althought I don't recall mentioning it here, at least not recently. Any definition of mass should hold under all circumstances and be meaningful. If it is to be meaningful in special relativity then it should also be measurbable from all frames of references, not just measurable from the rest frame, e.g. you can't do that for a photon. The concept of mass = proper mass does not have that property.

In more complex systems the system itself is spatially extended and that complicates matters. When taking measurements of such a system then one needs to take measurements of parts which are spatially separated "at the same time." But "at the same time" in one frame is not "at the same time" in another frame. This implies that defining mass by adding the 4-momenta of all the particles in the system and then adding then will fail when the system is not closed. E.g. the mass of a rod in space which is cooling or the mass of two particles moving in a cyclotron.

For proof see the sections labeled Invariant Mass of a System of Particles - Non-Closed System and An Incorrect Application of Invariant Mass at
http://www.geocities.com/physics_world/sr/invariant_mass.htm

Tom - This is a good example of where mass = proper mass doesn't work and where a particle physicist would never run into this problem. After all, how many times does a particle physicist consider the mass of a moving rod which is cooling?

An acquantance of mine, who is an expert in GR, told me that this notion was originally spoken of by Pauli in his relativity text. I've just ordered it and will get back as far as what Pauli says about all this.

Pete

 

[Garth]

DW - My use of the word "falsified" when applied to definitions may have been an unwise one written in haste, but the meaning is clear, at least to me! Scientific definitions in the past have been proven inadequate (falsified?) as the result of further experiments and observations. For example, the definition of the aether as being the fluid through which light is transmitted.
If science is to progress then we have to allow the possibility that the definitions and theories that we work with today may be proven inadequate/falsified in the future.
Let us have open minds.

Garth - "A standard of measurement is invariant; the principle by which it is defined so is a conservation law."

Since when?

Sorry if the following is pedantic, but here goes!
1. A principal conservation law on which GR is based is the conservation of energy-momentum.
2. The energy-momentum tensor (Weinberg convention) is conserved with respect to covariant differentiation if no external forces are acting on the system.
3. The EEP states that in an arbitrary gravitational field it is possible to choose a locally inertial coordinate system, the freely falling frame of reference, such that in a sufficiently small region the laws of nature take the same form as in an unaccelerated coordinate system.
4. There are no forces acting on such a sufficiently small system, thus the energy-momentum tensor is conserved.
5. By the principle of general covariance, generally covariant physical equations hold in a general gravitational field.
6. The process of transformation of coordinates from the freely falling frame into a general one can now be subsumed by a transformation of the affine connection. As a consequence Newtonian gravitational forces are "explained" by the curvature of the manifold.
7. The (rest) mass of a particle cannot be assumed or defined in these new coordinates but it has to be calculated from the energy-momentum tensor.
8. You will find this calculation in Weinberg G&C, culminating in equation 9.3.2, and the statement "this may be regarded as the law of conservation of mass" in the P.N.A. [At higher energies, in the P.P.N.A., particles may enter into energetic interactions and their mass not be conserved.]
10. If mass is 'conserved' then we can also say it is 'invariant' in the sense that the value we put on that mass does not vary. [I do not want a spat on the use of the two words invariant/conserved as it is a waste of time.]
11. Therefore the invariance of mass is a direct consequence of the law of the conservation of energy-momentum and the EEP.

However this is not the only approach.
If E = mc^2 is a fundamental relationship then there are two other possibilities:
One is that it is energy that is conserved, not energy-momentum, and the EEP is modified - this is the approach of self creation cosmology. [Note: although the EEP is violated in the theory, experimentally in Eotovos type experiments it would only be violated by one part in 10^-17, three orders of magnitude smaller than present experimental limits].
The other possibility is that it is c that varies - the VSL cosmologies, in which both energy and energy-momentum have to be carefully re-defined. As c is a comparison of length and time, measured by rulers and clocks, then it is the fine structure constant that would vary. [Note: There are a number of observations that claim to detect this, see a paper revised today, http://arxiv.org/abs/gr-qc/0403013, v3 28 Jul 2004.]

As I have said before if you insist on using the definition that mass is invariant then you would be blind to the fact that it might be otherwise.

Let us understand the present position but have minds open to other possibilities.

 

[russ_watters]

DW - "One can not be blind to something that can not happen by definition"
Thank you for proving my point.
Should not science advance by observations/experiments that challenge and possibly falsify previous definitions and theories?

I think you may be missing the difference between a definition and a postulate (assumption). A postulate is an assumption based on evidence and a definition is an entirely human construct with the purpose of being able to verbalize (or mathematize) a theory. A definition is not part of a theory. Ie, Einstein postulated that the speed of light is constant for all observers. He didn't define it that way. An awful lot of people think that it was a definition and he just as easily could have defined it as variable, eliminating uncomfortable realities such as time dilation. No so. Similarly:

Scientific definitions in the past have been proven inadequate (falsified?) as the result of further experiments and observations. For example, the definition of the aether as being the fluid through which light is transmitted.

That's not a definition, that's a postulate. Definitions must be invariant, otherwise two people could be saying the same thing and meaning two different things. Consider how much trouble the words "two," "too," and "to" cause for people learning English.

That all said, the argument on the first page was on a definition - but like Tom said, as long as people are using the same definition, both work fine. Postulates don't work that way (assuming something to be true doesn't make it true). Both parties in that argument know the other definition is functional, they just disagree on which is more often used.

 

[DW]

1. A principal conservation law on which GR is based is the conservation of energy-momentum.

Few people seem to know this, but GR does not generally conserve energy-momentum. It generally conserves an energy parameter and momentum parameters. It is merely that in many cases this corresponds to energy-momentum conservation. See-
http://www.geocities.com/zcphysicsms/chap5.htm#BM5_5

2. The energy-momentum tensor is conserved with respect to covariant differentiation if no external forces are acting on the system.

I think you mean stress-energy tensor. That does correspond to a "local" conservation of energy-momentum, but in fact that it is covariant differentiation does lead to what is globally not a true conservation of energy-momentum because the nonzero affine connections set this apart from an ordinary divergence of zero. See-
http://www.geocities.com/zcphysicsms/chap6.htm#BM78

4. There are no forces acting on such a sufficiently small system, thus the energy-momentum tensor is conserved.

I think you mean a free fall system. It is local free fall frames according to which the affine connections vanish and the stress-energy tensor obeys what's interpereted as a conservation equation.

7. The (rest) mass of a particle cannot be assumed or defined in these new coordinates but it has to be calculated from the energy-momentum tensor.

Since when? "Particles" are not ever calculated from the stress-energy tensor. It is the other way around. The stress-energy tensor is a density tensor which means that it is made of local averages from a virtually continuous distribution of particles. It sounds like you are now actually referring to the energy-momentum four-vector, but then your statements above don't make sense. Anyway, the mass can be calculated from the momentum four-vector for arbitrary gravitation with ease.

10. If mass is 'conserved' then we can also say it is 'invariant' in the sense that the value we put on that mass does not vary. [I do not want a spat on the use of the two words invariant/conserved as it is a waste of time.]

Maybe you should because it does sound like in places you are using the word invariant to mean conserved.

However this is not the only approach.
If E = mc^2 is a fundamental relationship

It is not. You are missing the momentum term.

As I have said before if you insist on the invariance of mass by definition then you would be blind to the fact that it might be otherwise.

And you have been told in responce already why that statement was wrong.

 

[DW]

You can clearly state that all you like, but when you say (as you have repeatedly) that such a concept of mass is "blatantly wrong" or "proven incorrect", it sends an uniquivocal message that you believe that such a definition has been falsified in the fullest sense of the word.

As you subtly admit here, I am not really arguing against a definition at all. I am arguing against a "CONCEPT". It is the "relativistic mass" paradigm that is blatantly wrong as I have already proven.

 

[DW]

Its quite the opposite in fact. I've already shown why mass = proper mass is inadequate in general, i.e. in more complex situations.

No you haven't.

If it is to be meaningful in special relativity then it should also be measurbable from all frames of references, not just measurable from the rest frame, e.g. you can't do that for a photon.

Since when? Oh you must be missrepresenting my definitions again, pretending that I define momentum in terms of four-vector velocity again instead of in terms of a wavelength k-vector that applies to both massless and massive particles as I do.

The concept of mass = proper mass does not have that property.

Yes, unlike mass as invariant, it does. You state here by the term proper that you define it in terms of the "proper" frame for which none exists for a photon.

This implies that defining mass by adding the 4-momenta of all the particles in the system and then adding then will fail when the system is not closed. ...

No it doesn't. It just implies that the sum is not simultaneous. But, one doesn't "define" it that way for a system anyway.

For proof see the sections labeled Invariant Mass of a System of Particles - Non-Closed System and An Incorrect Application of Invariant Mass at
http://www.geocities.com/physics_world/sr/invariant_mass.htm

Your site is not a reference.

 

[Garth]

I think you may be missing the difference between a definition and a postulate (assumption).

as long as people are using the same definition, both work fine. Postulates don't work that way (assuming something to be true doesn't make it true). Both parties in that argument know the other definition is functional, they just disagree on which is more often used.

Thank you for that. I was picking up on the contributions already posted.
There has seemed to have been a general confusion between the two.
Garth

 

[Garth]

For the record, as there does seem to be confusion in this thread:

I use the word "invariant" as an adjective to qualify a quantity that does not vary.

I use the word "an invariant or an invariance" as a noun to mean those quantities (such as the speed of light) or geometric objects (such as a body's energy-momentum) that are not coordinate or frame dependent.

I use the word "conserved" to mean without source or sink, such that a conserved quantity is an invariance in a physical process, or under a transformation of frame of reference.

If the statement "the mass of an object is invariant" is a definition then that creates a convention that is to be used consistently throughout. It does not rule out the use of other definitions of mass that create other conventions, so long as they are used consistently throughout too. An argument in one language is still the same argument if translated into another. However there may be nuances that appear in one language and not the other and likewise different definitions may have different merits.

If the statement "the mass of an object is invariant" is a postulate then it is to be tested and possibly falsified, it is a matter of observation not dogma. Of course that raises the all-important question, "How is it measured?" The answer is inevitably theory dependent and that may bring us back to the statement being a definition again. This is where the definition/postulate confusion has arisen. So long as consistent theories define how such a measurement may be made the statement remains a testable postulate.

The idea that invariances are intimately bound up with conservation laws is not new; in fact it is self-evident. A popular account by a front-line cosmologist may be found in John Barrow's 'The World within the World', "Unchanging properties are called conserved quantities and the statements that these quantities remain unchanged in Nature are called conservation laws." (page 114) He is prepared to consider other possibilities, other conventions than the standard one, such as a varying speed of light VSL theory. Such theories may just be playing with words', rather 'equations', but when they lead to testable, or observed, predictions then they enter the realm of hard science.

I do continue to argue for open minds able and willing to examine and critically discuss these possibilities. It is a way to understand the standard model more deeply and who knows? It might even lead to new physics.

 

[pmb_phy]

For the record, as there does seem to be confusion in this thread:

I use the word "invariant" as an adjective to qualify a quantity that does not vary.

That still doesn't define what you mean by the term. You have to say with what it does not vary with respect to. Does it vary with respect to a general coordinate transformation? Does it vary with respect to time/proper time?

Pete