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Lorents Factor Expansion

  1. May 14, 2015 #1
    simply put, objects with mass cannot be accelerated to exactly the speed of light as the Lorentz factor will diverge to infinity. If you take the Maclaurin expansion of the Lorentz factor

    12197b735002b40d712c535cb843c586.png
    And then allow the velocity of the object to hypothetically be equal to the speed of light or let B=1
    then your left with a series
    =1+1/+2+3/8+5/16+35/128......
    which is divergent.
    My question is, does this series have an analytic continuation and if so what is the value assigned to that continuation ?
     
  2. jcsd
  3. May 14, 2015 #2

    wabbit

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    The analytic continuation is that of the function ##\frac{1}{\sqrt{1-\beta^2}}##, it is singular at ##\beta=1## so you can assign it the value ##\infty## but that's it - you can't remove a pole from a function by using a series expansion.
     
  4. May 14, 2015 #3
    So there is no sum ability method for this series ? Abel Cesàro ??
     
  5. May 14, 2015 #4

    wabbit

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    Oh I don't know about that, if I recall correctly you can sum the series 1+2+3+4+... with the right method. But the interpretation of such results is delicate, and I don't see what this can tell you about the Lorentz factor: that factor goes to infinity as you approach light speed, and this is not going to go away by magic.
     
  6. May 14, 2015 #5
    That's what I'm trying to get at, there are several methods to assign values to divergent sums, I've looked through Hardy's divergent Series but its not explicit on which method to use for which series. I need the infinite product in terms of integrals, I know it involves factorials so maybe the gamma function ?? thoughts?
     
  7. May 14, 2015 #6

    wabbit

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  8. May 14, 2015 #7
    Interesting article, this is a very sensitive subject up to interpretation. I felt after reading about divergent sums with assignable values that this was an immediate extension, bosonic string theory makes use of the analytic continuation of the natural number sum
    f98fa2eae8ba3eeea13695e76d746d5d.png
    and QFT use renormalization methods to evaluate path integrals. Why not apply similar methods to evaluate singularities in other areas ?
     
  9. May 14, 2015 #8

    Nugatory

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    It sounds as if you're trying for a more mathematically sophisticated way of seeing what happens "if we could" just set ##v=c## in the Lorentz transforms. It's an interesting thought, but what do you expect the answer might tell you? The comparison with QFT renormalization is misleading in one very important way:

    That's sort of backwards. It might be better to say that because no massive object can be accelerated to exactly the speed of light, there's no particular reason to be surprised that equations describing the behavior of massive objects refuse to predict travel at the speed of light; or that because the Lorentz transform are derived under assumptions that are inconsistent with travel at the speed of light, they cannot be applied to that situation. Either way, the singularity at ##v/c=1## isn't what's stopping us from traveling at lightspeed; instead the singularity is there because there is no consistent way of combining the principle of relativity, the laws of E&M, and massive bodies moving at the speed of light.
     
  10. May 14, 2015 #9
    I am very unsure as to weather or not the answer will have any meaning at all or even if there is an answer. If there is an answer I expect it will be nonsensical much like other divergent series assigned unintuitive values. It is certainly possible that there is no summation method applicable to this series. I will admit the idea is much more mathematically motivated rather than from a physics perspective. QFT renormalization has roots in regularization theory, perturbation theory and asymptotic analysis, Feynman diagrams are basically a pictorial description of evaluating infinites i was merely reasoning by analogy.
     
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