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Mass Tensor

  1. Jan 25, 2004 #1
    There is often talk of mass and energy in this forum. I tend to use the notion of relativistic mass myself since I believe it to be the most precise and consistent way of defining mass. However I also hold that the complete description of relativistic mass is a tensor of rank 2. I call it the mass tensor, M. MTW discuss something very similar to this in their text as well but they call it the inertial mass tenso. That mass must be a tensor is what Einstein meant when he wrote
    The term energy-tensor, refering to Tuv, is not a necessary term. The same phenomena can be described in terms of a mass tensor. This follows from relativity due to the equivalence of mass and energy. The two differ only by the constant c2 but are defined differently. For instance, the T0k is energy flux in the kth direction while M0k is the kth component of momentum density. Due to the equivalence of mass and energy

    Tuv = Muvc2

    This too is discussed in MTW. They use the equality Energy = Mass, where "Mass" refers to "relativistic mass", to prove that Tuv is a symnetric tensor.

    The page which describes the mass tensor is here

  2. jcsd
  3. Jan 26, 2004 #2


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    stress energy tensor

    In modern relativity mass refers to an invariant scalar, not a matrix, and not a tensor. Unfortunatly MTW's terminology is not 100% self consistent. This is not surprising given it had three authors. The stress energy tensor is better called just that. I could multiply it by any constants I want to change units and define all kinds of strange names for each result, but this would be complication beyond reason.
  4. Jan 26, 2004 #3
    Re: stress energy tensor

    That quite is incorrect. It's well known that relativistic mass is a precisely defined quantity which is very meaningful.

    That can't be said for arbitrary quanties which yo want to create. The claim you just made for the reason you made is a poor one. Energy is defined one way. Mass is defined another way. It is proved that they are proportional. That's always been a fact that you've had trouble understanding.
  5. Jan 26, 2004 #4


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    Re: Re: stress energy tensor

    And the fact that you change the units of energy and call it by another name "relativistic mass" is complicating things beyond reason. Just call energy by energy and mass by mass.

    But that is exactly what you have done. You have changed the inits of something in order to call it something else. You then say you have a new way to present the field equations which is really nothing more than an older textbook presentation modified to accomidate your renamed stress energy tensor.


    They are not proportional. In SR they are related by
    [tex]E^2 = p^{2}c^2 + m^{2}c^4[/tex].
    It is only in the zero momentum frame that they are proportional and reduce to
    [tex]E_{0} = mc^2[/tex].
  6. Jan 26, 2004 #5
    Re: Re: Re: stress energy tensor

    re You have changed the inits ..

    That is incorrect as has been explained to you on many occasions already. I'll explain it once again

    relativistic mass, is defined as the quantity m such that mv is a conserved quantity in collisions etc.(v = 3-velocity). For precise definition see

    The Classical and Relativistic Concepts of Mass, Erik Eriksen and Kjell Voyenli, Foundations of Physics, Vol. 6, No. 1, (1976).

    Or see definition in Eq. (7) in

    As seen in this page it if follows that the mass of a tardyon is given by

    [tex]m = \gamma m_{0}[/tex]

    where m0 = m(0)

    E is defined as the sum of kinetic energy and rest energy, i.e.

    [tex]E = K + E_{0} = K + m_{0}c^{2}[/tex]

    It can be shown that

    [tex]K = (\gamma - 1)m_{0}c^{2}[/tex]

    It then follows that

    [tex]E = mc^{2}[/tex]

    Thus you're claim that this is simply changing units is unfounded. Such a claim implies that m is defined as

    [tex]m = \frac{E}{c^{2}}[/tex]

    Such a claim is invalid. This has already been explained to you on numerous occasions over the last 4 years. Are you having trouble following the derivations? If so then simply explain what part of the derivation you're having trouble with and I'll be more than happy to explain.

    Otherwise please don't simply repeat your claim that this is merely a change of units with no proof to back up your claim given the proof that is here in front of you to the contrary.
  7. Jan 27, 2004 #6


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    Re: Re: Re: Re: stress energy tensor

    No, m is the mass and is invariant. See

    Should read, energy of a tardyon is given by [tex]E = \gamma mc^2[/tex] where [tex]m = E_{o}/c^2[/tex]

    Should read, [tex]E = K + E_{0} = K + mc^2[/tex]

    Should read, [tex]K = (\gamma - 1)mc^2[/tex]

    Should read, [tex]E = \gamma mc^2[/tex], and the it follows that statement is circular.

    No its not. All you did was change the units by dividing the energy by [tex]c^2[/tex] and inappropriately call that mass.

    Then stop claiming it.

    I snipped a lot of flames at the end.
  8. Jan 27, 2004 #7
    Re: Re: Re: Re: Re: stress energy tensor

    dw - please back up your claims rather than simply repeat them over and over and over again.

    re I snipped a lot of flames at the end.

    There is a huge difference between constructive criticism and flaming.


    Criticism: a critical observation or remark

    Constructive Criticism: Criticism which directs further refinement of, in this case, an arguement.

    Flame: an angry, hostile, or abusive electronic message

    If you read something as angry or hostile here then you read it incorrectly. I suggest that if that is your assumption then ask for further clarification. As for abusive - that is a personal thing since you might feel abused by criticism of your arguement while somone else to whom a comment was directed to would not.
  9. Jan 27, 2004 #8


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    I don't have a good feeling about this thread, or about your claims, Arcon. First we have a discussion between DW and you about your techniques, and it gets down to the point where he asserts you just developed a circular argument by dividing energy by [tex]c^2[/tex] and later multiplying it again (or the reverse). Your immediate next post is not about math at all, certainly not a defense of your procedure, but a complaint about posting manners. That is a classic technique by somebody with a weak argument, change the subject and attack your opponent personally.

    You have come across as a good guy, knowledgable (at least at the advanced student level) and helpful. It is sad to see this kind of thing from you.
  10. Jan 27, 2004 #9
    What about the derivations that I've posted do you think are "claims" and not a fundamental facts of relativity, which can be found in most modern relativity texts? Is it the name of the tensor that concerns you? Is it the proof that you question? Please elaborate and I will clarify/explain

    I have no desire to discuss anything with dw. I'm simply correctin his claims.

    He is incorrect as I've proved.
    That is incorrect. Recall what my response was to his first post in this thread regarding his claim -
    Please show me where this has anything to do with posting style?

    dw repeated himself in a follow with the same claim that all that was done was to change units. I then explained to dw how they are defined differently and it was a matter of proof that they are related by E = mc2. I then outlined the proof.

    Over the last 5 years "dw" has responded with an identical response everytime this subject has come up. No arguement, no hint that he understood the derivation and disagreed with it, etc. Just a repeat of the same claim. A rough count of the number of times he's done that would be about 500 times. Since he did it twice in this thread I requested that instead of repeating the claim again that he backed himself up.

    Agree that is what normall happens. But that is clearly not what has happened in this thread. I'm at a loss to understand why you think so. I clearly backed up everthing with a very clear and detailed explaination as well as a detailed proof in the link which is the topic of this thread.

    Thanks. But please read these posts over again. Notice the flow

    Summary - (I'm paraphrasing here)

    Arcon Post 1) Posted detailed explaination. Rel-mass defined one way. Inertial energy defined another way. Matter of proof, not just change units, that E = mc2.

    dw Post 1) No. mass is a scalar not a matrix. MTW is wrong. All you did was change units. I can do that randomly and get junk. rel-mass is not meaningful. -- dw basis this on his personal choice of what "mass" should mean an not on the topic of this thread, i.e. rel-mass. Complains about subscript claiming that it is wrong to have a subscript to denote that it is proper mass.

    Arcon Post 2) That quite is incorrect. Rel-mass has a precise and consistent definition. Energy is defined one way. Mass is defined another way. It is a matter of proof that they are proportional.

    Note - Since dw has done this several hundred (if not thousands) of times in the past I explained to him that he is having trouble understanding this point. He has followed me from forum to forum to forum doing this for 5 years. I hope you can understand how irritating it can get over the years. I was being very polite by simply making a request. I even said "please" :-)

    dw post 2) Repeats his earlier comments. No comment was made regarding the derivation I provided. Nothing different from first dw post.

    Arcon Post 3) I suspected that he was not reading any derivation so I posted the exlicity details within this thread as to how relativists (i.e. Rinlder, Mould, D'Inverno, etc) arive at these conclusions in new relativity texts. Since he's merely repeated what he's stated several hundred of times over the last 5 years I simply asked him not to repeat himself but to prove what he was claiming. Once a derivation is posted and a claim is made against it then the the burden of proof rest on the perspon who claims the derivation is wrong. I saw no such proof.

    dw post 3) Repeats himself in a manner which is identical to the first. Repeats that there shoud not be a subscript. Claims that I'm flaming him.
  11. Feb 6, 2004 #10

    The first thing you should realize is that energy-momentum is a relativistic four-vector, and not energy-mass. The distinction between mass and momentum is crucial to a proper understanding of relativity theory. Stuck in the back of your head is the notion that p=mv (a Newtonian expression), and so therefore relativistic momentum, [itex]$p = \gamma mv$[/itex] must mean that
    [itex]$\gamma m$[/itex] is some relativistic quantity.

    Rather, you should view that term as [itex]$\gamma p_0$[/itex], and refrain from pulling out the mass. Think of momentum as a defining characteristic of the motion. Stop thinking in Newtonian terms.

    Second, as has been pointed out, mass is a scalar quantity, not a tensor. Third, multiplying something by c[itex]$^2$[/itex] means nothing. If I choose to set c=1, then presto!, your two tensors are one and the same.
  12. Feb 6, 2004 #11
    Re: Re: Mass Tensor

    As I explained above, the mass I refer to is relativistic mass. The subject of this thread is not on the proper mass of a particle. It is on the relativistic mass of continuous media.

    Regarding 4-momentum of a particle. If anything an alternate term of 4-momentum would be "mass-momentum 4-vector" not "mass-energy" 4-vector since each name should reflect a component. But I prefer the term 4-momentum. There are two ways in which this 4-vector has been described. Some people choose to define it as

    [tex]P^{\mu} = (E/c,\mathbf{p})[/tex]

    (E.g. Taylor/Wheeler)

    while others choose to define it as

    [tex]P^{\mu} = (mc,\mathbf{p})[/tex]

    where "m" is relativistic mass (E.g. Rindler, Jammer)

    Nope. Stuck in the back of my head is Einstein's theory of relativity. There is no reason to refer to p = mv as Newtonian just as there is no reason to refer to f = dp/dt as Newtonian. Both are well defined quantities in relativity. Definitions do not neccesarily belong to a theory. Especially in this case. In fact it may loosely be said that p = mv defines relativisitc mass, m.

    Relativistic mass, m, is related to proper mass, m0 as

    [tex]m = \frac {m_{0}} { \sqrt{1-v^{2}/c^{2}} }[/tex]

    This then gives a value to the relativistic momentum of

    [tex]\mathbf{p} = m\mathbf{v} = \frac {m_{0}\mathbf{v}} { \sqrt{1-v^{2}/c^{2}} }[/tex]

    dw already expressed this opinion. I was well aware of this opinion over 10 years ago. The proper mass of a particle is a tensor of rank 0. However this thread is not about proper mass.

    In a sense relativistic mass, being the time component of a 4-vector, is a scalar (i.e. tensor of rank 0). When speaking of the components of a tensor one is speaking of a particular Lorentz observer. That is the only time when the concept of 3-momentum, p, and relativistic mass mass, m, and energy, E, exists.

    Introducing a Lorentz frame splits a 4-vector into a scalar and a 3-vector. As such relativistic mass is the projection of the 4-momentum onto the time axis corresponding to that Lorentz observer. Such a projection is a scalar quantity and is invariant under coordinate transformtion. Different Lorentz observers will have different time axes and thus different projections. For those who find this confusing see details at


    That is incorrect. E = mc2 is not a definition of relativistic mass, m. It is an equality of two quantities which have completely different defintions. E.g. If f is the frequency of a photon and E is the energy of the photon then E = hf where h is Planck's constant. That doesn't mean that E and f have the same physical meaning.

    If you would like to learn more on the role relativistic mass plays in GR/cosmology please see

    Cosmological Principles, Peacock, Cambridge Univ. Press,

    Note that on page 18 he explains that T00 = c2x(mass density)

    Thank you for your opinion
  13. Feb 6, 2004 #12
    I know you are all professionals in Physics. I am in no position to make any judgements on this issue. The mass tensor does look like the energy-momentum-stress tensor divide by c^2. We might all agreed mass is related to Energy somehow.

    But Arcon did point out an intresting implication that mass, if it's any meaningful, shall be a tensor rather a scalar, because a tensor more likely exihibit its invariance through any coordinate translations.
  14. Feb 6, 2004 #13
    Yup. I agree. Einstein established that fact a very long time ago. That's why he said that light has mass, e.g.

    The Principle of Conservation of the Center of Gravity and the Inertia of Energy, Albert Einstein, Annalen der Physik, 20 (1906): 626-633.
    The Evolution of Physics, Einstein,Infeld page 221

    I find it strange that people focus all their attention on Einstein's 1905 paper and then ignore all the physics in the 100 years which followed!?!?!?
  15. Feb 6, 2004 #14

    If we assume there are internal motions ( momentum and stress ) inside the confinement, are we able to integrate this tensor and show that there is an externalized apparent mass without other components?
  16. Feb 6, 2004 #15
    I'm sorry but I don't understand your question. What does "externalized apparent mass" mean? What do you mean by "other components" in that context?
  17. Feb 6, 2004 #16


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    Arcon - your PM box is full
  18. Feb 6, 2004 #17

    I was refering to a modified Einstein's box, instead of containing lights, now containing assumed continuous moving perfect fluid that could be defined as tensors over a certain volume of (t,x,y,z) event space. At each point, there would be a mass tensor representing its mass density quantity and momenum/stress state.

    If we can somehow integrate it, the result will be more likely the mass density * volume + momentum * somethings. Given certain conditions, we can cancel out the momentum and stress component in the integration and the outcome will be a matrix, if not tensor, where only the M00 has value and the other 15 components, or the 9 independent components, will be cancelled out. M00 will be perceived as mass instead of mass density now by a much bigger outsider.

    Further, if the other 9 independent components were not completly cancelled out, their value and the mass value vibrate slightly through time, this could be viewed as a model of wave functions we saw in particles.

    It's just a crazy thought. I just looked for your opinion. If you think this inappropriate, just disregard me.

  19. Feb 6, 2004 #18
    Something must be wrong with the software. I have zero PM messages so how could it be full?
  20. Feb 6, 2004 #19


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    Did you check your "sent" box? If that's not the problem, then Greg can check into it. thx
  21. Feb 6, 2004 #20
    Actually I didn't even know I had a "sent" box. I've deleted everything in it.

    Thanks Phobos

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