Lorentz Factor Expansion: Analytic Continuation Value

[Bradosgood]

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

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 ?

 

[wabbit]

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.

 

[Bradosgood]

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.

So there is no sum ability method for this series ? Abel Cesàro ??

 

[wabbit]

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.

 

[Bradosgood]

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.

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?

 

[wabbit]

No but perhaps https://cornellmath.wordpress.com/2007/07/28/sum-divergent-series-i/ and following, as well as references therein can give you some ideas.

Out of curiosity, what is your purpose in summing this series?

 

[Bradosgood]

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

and QFT use renormalization methods to evaluate path integrals. Why not apply similar methods to evaluate singularities in other areas ?

 

[Nugatory]

and QFT use renormalization methods to evaluate path integrals. Why not apply similar methods to evaluate singularities in other areas ?

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:

simply put, objects with mass cannot be accelerated to exactly the speed of light as the Lorentz factor will diverge to infinity.

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.

 

[Bradosgood]

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.

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.