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Dimensional homogeneity

  1. Aug 16, 2010 #1
    I have two very specific questions.

    I was trying to read this paper :

    and i noticed that equation 3 and equation A6 were different :

    Eq3 : L(t)=Q(t)/£(t) (not dimensionally correct)


    Eq A6 : L(t)=Q(t)/£(t) * delta_t

    When you read under equation 3, you understand why this is like that ; they say :
    "As seen in Eq.(A6) of Appendix A, lifetime indeed has the dimension of time, being multiplied by the sampling time interval (which is here equal to unity–1 day)."

    Questions :

    1/ Is it ok to have a dimensionally wrong equation if you say why in the text right next to it (i.e. numbers have a dimension and here because the time interval chosen is unity ) ?

    2/ Is it common practice in Physics to do that ?

    Thank you !

    Note : i do not want a comment on the paper itself, just on this specific point.
  2. jcsd
  3. Aug 16, 2010 #2


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    So in other words it is dimensionally correct, and they merely left out the factor [tex]\Delta t[/tex] when they wrote Eq 3, but clearly state that they did so.

    The appendix has the derivation and eq A6 is the definition of the function. Equation 3 is not the definition, it's just A6 with the time interval equal to unity, as they say it is.

    I see nothing strange here.
  4. Aug 16, 2010 #3

    So you do agree that what they have done in this paper is common practice in any kind of research paper, right ?

    I know that 99% percent of theoretical physics paper are written this way as they all drop c, h,... you just have to keep track of them as authors always say what convention they're using. Same thing here.

    Correct ?

    Thanks for the quick reply.
    Last edited: Aug 16, 2010
  5. Aug 16, 2010 #4


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    I take the other point of view. Let's say a basic Physics equation for Work is Work = Force times distance, (W =Fd), where work is in units of Newton-meters, F is in Newtons, and distance is in meters. Let's call that equation A6. But then to say that equation 3, W = F, does indeed have units of Newton-meters because the distance travelled in that equation is unity (1 meter), is a bit non conventional, in my mind, and can lead to confusion. Just my opinion.
  6. Aug 16, 2010 #5

    Here i am not talking about a basic physics equations, but a research paper (so if someone with good and broad knowledge of the scientific literature can answer, that would be nice).

    The case I want to discuss is the very specific one given in the research paper in the first message i.e. a paper with an apparently non homogeneous equation but where it is clearly stated that some parameter was set to unity.

    To my knowledge this is very common practice in most research papers (for example in theoretical physics with the dropping of c, h,...).
    Am I correct that what is done in that paper is common practice in the scientific literature ? (cf my previous questions).

    Thanks again
    Last edited: Aug 16, 2010
  7. Aug 16, 2010 #6


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    I don't know if I'd call it common practice, but as I said, I don't see anything wrong with it. Delta-t is somewhat arbitrary and irrelevant to the main analysis. Choosing things such that the arbitrary factor is 1 is pretty common though.

    Slightly different, since those are constants; setting them to 1 changes the units, but not the dimension. (I.e. you're not actually removing them from a dimensional-analysis point of view, which is the same as here, but riskier since delta-t is not a constant)

    But to use PhanthomJay's example, one could write something like:
    "W = F*d. Since the distance in question is arbitrary, we let d = 1 m and have
    W = d. Blabla.."

    And I'd see nothing wrong with it. The only difference to the original example is the sequence, really. Anyway the final authority on the writing and such is the editor and reviewers of the journal in question, who obviously didn't have a problem with it; and this would typically be the kind of thing that they'd pick on if they'd seen a problem with it.
  8. Aug 16, 2010 #7
    You're correct to make a slight difference between a fundamental constant (which is always a constant) and a constant parameter (which is a constant for your study).

    It's obvious that you can only "drop" terms that are constant (as you have to assign them a numerical value when doing that).
    Here delta_t is a sampling time interval, it's not meant to vary (in other words it is a constant parameter -and that's why they can "drop" it-).

    I don't see anything wrong or confusing with this procedure either, as long as you clearly state what you are doing of course (like the way it's done in this paper).

    Thanks for your answer.
    Last edited: Aug 16, 2010
  9. Aug 16, 2010 #8


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    Evidently my opinions on this matter are not universally shared, but here they are for the record:

    Although in this case, they explain what they're doing, well enough that I can easily insert the proper factor to make the equation dimensionally correct. I'd mentally correct it to
    [tex]L(t) = \frac{Q(t)}{\zeta(t)}(1 \mathrm{day})[/tex]
    So even though I believe equation (3) is technically wrong, I wouldn't make a fuss about it.
    Not so much in my experience, although I do most of my reading in high-energy physics, where it's reasonably common to use unit analysis alone to figure out how many factors of some value to insert. So if you're not careful about the units, you could wind up with the entirely wrong formula. I guess that's less of an issue in this paper.
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