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In numerous discussion of the special conformal transformation, they cite the finite transformation as

$${x^\mu}'=\frac{x^\mu-b^\mu x^2}{1-2x\cdot b+b^2 x^2}$$

This can be found from integrating the infinitesimal conformal transformation

$$\delta x^\mu =2(b\cdot x)x^\mu-x\cdot x b^\mu$$

I found the derivation given as an answer on this site. I completely understand what they did, but at the end of the day they get the answer to be

$$x(t)=\frac{x_0-x^2_0 (tb)}{1-2x_0(tb)+x_0^2(tb)^2}$$

Their starting point was the differential equation $\dot{x}=2(b\cdot x)x-x^2 b$.

Why, exactly, is there a $t$ in the second equation but no $t$ in the first? Is $tb=b^\mu$? Also, where in the derivation I've cited do they only consider the case of $\mu=0$ (i.e., time)? Any clarification would be greatly appreciated.

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  • $\begingroup$ @CosmasZachos That was a typo--thank you for pointing that out. I've edited the post appropriately. I followed the derivation out completely and I believe I understand it, but I don't understand why the final result given in that answer differs from the result in my textbooks. $\endgroup$ Feb 10, 2018 at 16:54
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    $\begingroup$ $t=|b|$, that is, $b^\mu\equiv \hat{b}^\mu t$. $\endgroup$ Feb 10, 2018 at 17:00
  • $\begingroup$ But is it different? why did you not reinstate the 4-indices μ in the last displayed equation you wrote, so $x(t)^\mu =...$ and the parametric diff eqn (s) that led to it? $\endgroup$ Feb 10, 2018 at 17:05
  • $\begingroup$ Also, Josh is not considering $\mu=0$ only. He is suppressing the indices, so the vector equation $x=y$ is equivalent to $x^\mu=y^\mu$. $\endgroup$ Feb 10, 2018 at 17:09
  • $\begingroup$ @CosmasZachos I was quoting the result from the linked answer which didn't have the superscripts. That's why I was confused--they solved the equation in a different notation, and that's what's confusing me. Also, in their solution they have $\dot{y}=-b$, so $y=y_0-bt$. Would this still mean that $b=|t|$? Are all derivatives technically taken with respect to $|b|$? $\endgroup$ Feb 10, 2018 at 17:16

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@Trimok solved the problem most elegantly in his comment to the question cited, and since you are troubled by @Josh's simplifying changes of variables, $b^\mu\equiv \hat{b}~ t$, I'll avoid them to merely integrate the variation $$\delta x^\mu =2(b\cdot x)x^\mu-x\cdot x~ b^\mu$$ directly. It immediately implies $$\delta \left (\frac{ x^\mu}{x^2}\right) = \frac{ \delta x^\mu}{x^2} -2 x^\mu \frac{ x\cdot \delta x }{x^4} = -b^\mu ~.$$

That is, the vector $x^\mu/x^2$ evolves by shifting along $-\hat{b}^\mu$, linearly in the magnitude $|b|$ of $b^\mu$, so with constant unit speed in this "pseudotime" $|b|$. Integrating this simplest of advections for finite pseudotime, we immediately get $$ \frac{ x'^\mu}{x'^2}= \frac{ x^\mu}{x^2} -b^\mu . $$ Square both sides, to get the normalization, $$ \frac{ 1}{x'^2}= \frac{1}{x^2} +b^2 -2\frac{ x\cdot b}{x^2}= \frac{1-2x\cdot b+b^2 x^2 }{x^2}, $$ which divides the above vector equation to yield your conventional form for it, ∴ $${x^\mu}'=\frac{x^\mu-b^\mu x^2}{1-2x\cdot b+b^2 x^2}~.$$

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