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Energy and Special Relativity

  1. May 13, 2003 #1


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    [SOLVED] Energy and Special Relativity

    For some reason the time component of the 4-momentum has the name "Energy." Many people even call it the "total energy." However the time component is the "free particle energy" and not the total energy. Total energy for a conservative system is, by definition, an integral of motion i.e. it is a constant. It has the value of the energy function. The energy function expressed in terms of canonical momentum, instead of the the derivatives of the postion variable, is the Hamiltonian. For a precise definition of "energy function" see


    h is the "total energy" of the particle. It is h which is the conserved energy - not E.

    Note: A conservative system is one in which the potential energy function V is time independant [I.e. V = V(r) not V(r,t)] and thus the Lagrangian is time independant and thus the partial derivative with respect to time is zero. Also note that the energy function is not always the energy so one has to be careful when calling the Hamiltonian "total energy". Goldstein has a nice example of this. Referance provided upon request (I'd have to look up the page # anbd have to know what edition one has)

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  3. May 14, 2003 #2
    For a noninteracting particle, that def works. If external fields are involved, then you have to take into account the 'energy of the field' which accounts for the difference. I think Goldstein might go into this in his SR chapter...
  4. May 14, 2003 #3


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    It is the external field that generates the potential and thus the force. Goldstein does do this. And he does this exactly as I've done it (or rather I've done it the exact same way Goldstein has) in


    This thread is about the total energy of a single particle moving in an external field. The total energy is an integral of motion.

    See Eq. (12) Notice the difference between the total energy W and the free-particle energy E (time component of 4-momentum).

  5. May 14, 2003 #4
    OK. Still, the reason for calling it the total energy of the particle is because that is in a sense the mass/energy the particle will behave as if it has. eg, if you move an electron into a region of higher uniform potential, it will not behave any differently in terms of inertia under acceleration.
  6. May 15, 2003 #5


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    Actually I understand the reason why some people call it that (I aksed and recieved e-mail from a few well known relativity authors). I simply disagree with it. It gives the wrong impression of what total energy is supposed to me. The text "Classical Electrodynamics," J.D. Jackson gets it right.

    I disagree with the naming convention because it can lead to error quite easily. In fact I just saw someone make this error and in a very predictable way. This is especially relavent when one is using the relativistic Hamiltonian for a charged particle in an EM field and when calculating the 4-momentum. If you have Jackson's text see pagre 575 (or I can scan it in and e-mail it for those who are interested).

  7. May 16, 2003 #6
    Definitions are not argued about.

    Definitions are just short-naming longer expressions by shorter symbol(s).
  8. May 16, 2003 #7


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    Definitions are very much argued about. In fact some of the more popular arguements regarding definition is with regards to the definition of the term "mass." Then there is the definition of the term "gravitational field" and then there is the definition of the term "force" and then there is the definition of the term "weight" and then there is the definition of the term "inertial frame" etc.

    In this particular case the definition of the term "Energy" is the subject. It can mean at least one of two things in relativity.

    The definition of all of these terms have appeared in the Physics literature - Especially in the American Journal of Physics.

    But without doubt it is most certainly true that definitions *are* argued about.


  9. May 16, 2003 #8
    Pete, sorry that I did not deliver my point. Definitions are not argued about. What I mean by this is ONCE you defined something, that is that. You can't THEN argue how good is this definition. You can only argue BEFORE that, when there is NO definition yet.

    Let me illustrate that. Suppose we define average or mean of two values x1 and x2 as follows <x> = (x1+x2)/2. That is that. Both parts of this equation are EQUAL, and shall be treated as such. The equation simply states than wherever you see a symbol <x> replace it by (x1+x2)/2. There is no difference in quantity <x> and quantity (x1+x2)/2. You CAN NOT argue that <x> and (x1+x2)/2 are DIFFERENT - they are ONE and SAME thing.

    Now, suppose someone came across and said: hey, I don't like this definition, and therefore I propose to use the following one: <x>=sqrt(x1x2) which I feel is "much better".

    Then what? One of 2 things will happen: either we shall cancel the "old" definition (and ALL other definitions wich follow from it or are related to it) and replace it by the "new" one (and also re-define many other related things)...,

    OR we shall simply find DIFFERENT symbol (word, expression, etc) for the "new" operation sqrt(x1x2). Say, we may call the "old" one as "arithmetic mean" and keep the symbol <x> for it, and call the "new" one as "geometric mean" and introduce DIFFERENT symbol for it, say {x} in order not to confuse two DIFFERENT quantities (actually operations).

    So, what exactly is your proposal? To cancel "old" definition of mass, energy, etc and to introduce new definition of mass, energy, etc which ACTUALLY will NOT mean old quantities, but something DIFFERENT?

    Or to introduce different symbols for NEW quantities (because E, m, etc symbols have already been reserved)?
    Last edited by a moderator: May 16, 2003
  10. May 16, 2003 #9


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    Hi Alexander
    Thanks for the clarification. To some extent I agree and to another extent I disagree.

    However when one defines certain things one has to make sure that the definition fits. I'm sure you know the following but for those who don't let me fill in th details while I give this example

    Example: After Einstein's 1905 paper Plank showed that you can write F = dP/dt if the momentum has the value p = Mv where M = gamma*m_o is relativistic mass. Later Tolman showed that if M = gamma*m_o then write the momemtum as P = Mv then momentum will then be conserved. So, I think it was Tolman who said something like "Thus the mass is given by M = gamma*m_o" or whatever. So before 1905 "mass" didn't depend on velocity. After 1905 it did. But it didn't stay that way all around. People the interpreted this P = Mv as being better explained by saying that P = MV is wrong. The "real" (and I don't like that word very much) explaination is that momentum is not mass times velocity but that momentum is gamma times mass is thus mass is not velocity dependant.

    So the arguement continued. So there is no universal definition. So if somone asks "What is mass" it has no absolutley unique answer. However if you say "Define mass as M = gamma*m_o" and then at that point there is no question as you indictated.

    This is the case with the term "energy" in this example. Energy is normally taken to mean "Total energy" when there is no qualifier. I.e. If someone says "energy" you know they tend not to mean "kinetic energy" or "potential energy" (although that does happen in important cases). In this case the E in E^2 - p^2 = m^2 the E is not energy but is the free-particle energy. Not total energy.

    But that is not the only two cases. You're assuming that everyone uses the same definition for things. That isn't true in this case. One person uses one definition and another person uses another definition.

    When it's actually me making that decision then I'll tell you. But in this case I wasn't posting this with the intent of complaining about the definition per se but merely commenting on it for those who might not have considered this consciously.

    I mean I always knew the definitions of each. However when it came to a rather advanced application when it was crucial to understand the difference I was a bit confused by it. Later when I realized what was going on it then dawned on me that the term was being overloaded. I then went back to the texts where I originally learned the concept. I was then surprised to see that all of the derivations were suspisous. So I sat down and did my own out and thought the whole thing through. Here is what I got


    See the part where it starts "The work done by the conservative force ..."

    That's part of it. Rest energy is another thing and I love Einstein's first derivation - as explained by Stachel that is. Have you read that paper by Stachel? Do you know the one of which I speak? It was in the Am. J. Phys. in the Early 80's. I don't recall the name at the moment. But Stachel explains Einstein's first derivation in a wonderfully logical way. Several well known papers and books were written which claimed that Einstein made an error in that he was "begging the question".

    If you'd like I can scan it and e-mail it to you - I;m getting a clear copy in the mail soon. My old one is a poor copy.

  11. May 16, 2003 #10
    Yeas, I'd like to take a look if possible.

    Full energy depends on system of reference. It is conserved within each reference system quantity, but not invariant in transformation between reference systems quantity.
  12. May 17, 2003 #11


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    I'll be happty to e-mail it. It should arrive in the mail today. How do I get the scanned images to you. You didn't list your e-mail address. The entire set of images might be over 1 meg, maybe ~1.2 meg maybe less.

    While its true that in many cases energy conservation in one frame means energy is conserved in anothe frame it is not always true. In fact it is not true in the case I started this thread with. Nor is it true in classical mechanics as a general rule. The reason being that a force might not do work in one frame but in a frame moving with respect to that the force might do work. Exmples include forces of contraint doing work on a system.

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