Wikipedia:Reference desk/Archives/Mathematics/2018 January 2

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January 2

Newton approximation numbers for sqrt(3) and special relativity

I got side tracked while answering this question. I want to stick to the math here, but just for context, if you look at a proton from a frame moving at sqrt(3)/2 times the speed of light, it seems to have (nay, it has) twice the mass. If you look at it from a frame moving that fast compared to the last one you looked at it from, it has seven times the mass...

Working it out, I realize that if you number your frames n=0,1,2,3,4... then g(n) is 1,2,7,26,97,362,1351,5042,18817,70226,262087,978122... The rule for adding these is that g(n1+n2) = g(n1)*g(n2) + sqrt(g1^2*g2^2 - g1^2 - g2^2 + 1). To iterate, use n1=1 ; g(1) = 2 so g(n+1) = 2*g(n1) + sqrt(3*(g(n1)^2 - 1).

These turn out to be iterations in a Newton's method for approximating the square root of 3, apparently. [1] I should note that both numerator and denominator appear in the speeds of the proton in all those different frames, which divides the square root of 3 by those approximations to approximate but never reach the speed of light from the relativistic frames.

Despite all these things, I haven't managed to figure out much about the series...

1) I don't know how to calculate g(n) from n without iterating.

2) It is in no way restricted to integers. However, it seems like g(n) is always integral for integer n, which means that g^2-1 is always 3*N^2, where N is an integer. From the iteration I was doing it is not obvious to me why this is the case.

3) This goes beyond the scope, but this value n seems like it should be useful for describing "speed" (in a loose sense) under relativity in a way that can just be added and subtracted between multiple frames of reference.

Wnt (talk) 14:49, 2 January 2018 (UTC)

The OEIS is your friend. This turns out to be A001075 in the OEIS. From the description, you'll see that this is actually a second-order, linear, homogeneous recurrence relation. Finding explicit formulas for these is straightforward, and would probably help shed some light on what's going on. Also, for future reference, please use some kind of markup (whether it's <math>...</math> or just regular wiki markup with templates) when asking questions; it's a bit tough to try to read as is. –Deacon Vorbis (carbon • videos) 15:42, 2 January 2018 (UTC)

You might look at the article Lucas sequence since your sequence is a simple modification (divide by 2) of V_n(4,1), see A003500. (The OEIS isn't always so good at pointing out relationships between sequences.) Lucas sequence are generalizations of Fibonacci & Lucas numbers and many of the interesting properties of those sequences have similar versions for Lucas Sequences. Also, I'm not sure what you meant by saying this was an application of Newton's method, but the link was about the continued fraction convergents of √3, which is related but not the same. Related to this, and point #2 above, is Pell's equation with n=3. Basically the solutions of Pell's equation are given by the continued fraction convergents of √3, which should help explain the 'why' of point #2. Much of this would covered in a course on elementary number theory, but the connection to relativity is new to me. --RDBury (talk) 16:19, 2 January 2018 (UTC)

PS. The addition formula above seems to be related to the hyperbolic addition formula $\cosh(x+y)=\cosh(x)\cosh(y)+\sinh(x)\sinh(y)$ where the square roots come in because $\sinh(u)={\sqrt {\cosh ^{2}(u)-1}}$. So to answer point #3, you can use hyperbolic cosine and it's inverse to convert between ordinary addition and the 'relativistic addition' of the formula. --RDBury (talk) 16:49, 2 January 2018 (UTC)

See the article on rapidity. Rapidity w corresponding to velocity v is defined as $w=\operatorname {artanh} {\frac {v}{c}},$. Rapidities are additive and $\cosh w=\cosh \left(\operatorname {artanh} {\frac {v}{c}}\right)={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}=\gamma$. Also from the article: Rapidities $\mathbf {w} _{1},\mathbf {w} _{2}$ with directions inclined at an angle $\theta$ have a resultant norm $w\equiv |\mathbf {w} |$ (ordinary Euclidean length) given by the hyperbolic law of cosines, $\cosh w=\cosh w_{1}\cosh w_{2}+\sinh w_{1}\sinh w_{2}\cos \theta .$

-Modocc (talk) 17:29, 2 January 2018 (UTC)

By no means do I understand everything here yet, but I should thank @RDBury: for a key step forward. Both the U_n and V_n sequences are relevant here for the (4,1) Lucas series. From a note in the OEIS entry, the V sequence can be written as g(n) = ((2+sqrt(3))^n + (2-sqrt(3))^n)/2 . (I know, I should get better at math tags, but they are so laborious and in the end I type all this into the R console anyway) Wnt (talk) 20:23, 2 January 2018 (UTC)

For your information:

$g(n)={(2+{\sqrt {3}})^{n}+(2-{\sqrt {3}})^{n} \over 2}$

Bo Jacoby (talk) 21:34, 2 January 2018 (UTC).

Continuing, I agree that g = cosh(artanh(x)), where x = v/c. There is some coincidence of form with the above equation, because, using w = artanh(x) = $\ln {\sqrt {\frac {1+x}{1-x}}}$,

$g=coshw={\frac {e^{w}+e^{-w}}{2}}={\frac {e^{\ln {\sqrt {\frac {1+x}{1-x}}}}+e^{\ln {\sqrt {\frac {1-x}{1+x}}}}}{2}}$ = ${\frac {{\sqrt {\frac {1+x}{1-x}}}+{\sqrt {\frac {1-x}{1+x}}}}{2}}$ = ${\frac {1}{\sqrt {1-x^{2}}}}$. Indeed, I must note that for x = ${\frac {\sqrt {3}}{2}}$, the shift between frames used above for iterations, the two components being added are indeed 2 + sqrt(3) and 2 - sqrt(3)! (yes, it took forever to debug the code above) On further consideration, I realize that this definitely is the rapidity. For the value of w where gamma = 2 and v = sqrt(3)/2, we get w = $\ln {\sqrt {\frac {2+{\sqrt {3}}}{2-{\sqrt {3}}}}}$ ... however, by multiplying this by the numerator, we can simplify this to $\ln {(2+{\sqrt {3}})}$. With cosh we take e to that power, so for n*w we get $(2+{\sqrt {3}})^{n}$ for the +w portion. So it checks out, once I remember that hyperbolic angle isn't measured in radians! Wnt (talk) 04:56, 3 January 2018 (UTC)

Just as an FYI, it looks like there are similar series any time the ratio of mass to rest mass is an integer. For ratio 2 get the above 1, 2, 7, 26, 97, but for ratio 3 the series is 1, 3, 17, 99, ... and for ratio 4 the series is 1, 4, 31, 244, ... . All the entries are integers which relates to the fact that cosh nx is a polynomial is cosh x. --RDBury (talk) 18:23, 3 January 2018 (UTC)

@RDBury: Going to OEIS [2] I see the first sequence is described as "Lucas-balancing numbers" with P=6, Q=1. It is also V_n/2 (6,1); the rule is the same but the starting values 1, 3 are not used for the Lucas numbers proper. The second accordingly I think can be recognized as half of 2, 8, 8*8-2... [3] Both are linked to Diophantine equations also (x^2 - 8*y^2 = 1, x^2 - 15*y^2 = 1) and therefore can be linked to "corresponding y values" [4] and [5]. The ratio of these values approximate sqrt(8) and sqrt(15), respectively. Wnt (talk) 01:09, 7 January 2018 (UTC)

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