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Sergei Winitzki has a very nice copylefted GR book: http://sites.google.com/site/winitzki/index/topics-in-general-relativity (direct PDF link: http://sites.google.com/site/winitzki/index/topics-in-general-relativity/GR_course.pdf?attredirects=0&d=1 ) I'm puzzling over his treatment of conformal transformations.
On p. 85, near the bottom left corner of the page, he has
[tex]\tilde{r}(i^o)=\lim_{r\rightarrow\infty}e^\lambda r^2=\lim_{v\rightarrow -\infty, u\rightarrow\infty}f'(u)f'(v)\frac{(u-v)^2}{4}[/tex].
Comparing with the expressions near the top of the column, it seems to me that the [itex]e^\lambda[/itex] should clearly be [itex]e^{2\lambda}[/itex]. I emailed him about this, hoping that I wasn't just making some dumb mistake.
After this, there is another thing that seems wrong to me, but considering that my understanding of GR is far inferior to his, I'm a little hesitant to shoot off another email. He says suppose that f'(s) goes like s-n for large s; then the limit in the third expression above should approach zero if [itex]n\ge 2[/itex]. But shouldn't the condition be [itex]n\ge 1[/itex]??
If I'm right about the second error, then it's a more substantial hole in his analysis. Basically you want io to be pointlike (a 2-sphere of radius 0) and points on scri+ to be non-pointlike (each such point represents a 2-sphere of finite radius). The best motivation I know of for wanting these to be this way is that it allows you to make a tiling of Penrose diagrams to cover the static Einstein universe. (Is there some more fundamental reason?) So the condition on io requires [itex]n\ge 1[/itex], and the condition on scri+ forces [itex]n\le 2[/itex]. But then this leaves a whole range of possible exponents between 1 and 2 (contrary to Winitzki's analysis, which fixes the exponent uniquely at 2). Does this seem right?
One would then need some other reason to pick n=2. I think that if you want f' to be analytic and positive-definite, then that does rule out n=1, or any other value besides n=2. Does this make sense?
On p. 85, near the bottom left corner of the page, he has
[tex]\tilde{r}(i^o)=\lim_{r\rightarrow\infty}e^\lambda r^2=\lim_{v\rightarrow -\infty, u\rightarrow\infty}f'(u)f'(v)\frac{(u-v)^2}{4}[/tex].
Comparing with the expressions near the top of the column, it seems to me that the [itex]e^\lambda[/itex] should clearly be [itex]e^{2\lambda}[/itex]. I emailed him about this, hoping that I wasn't just making some dumb mistake.
After this, there is another thing that seems wrong to me, but considering that my understanding of GR is far inferior to his, I'm a little hesitant to shoot off another email. He says suppose that f'(s) goes like s-n for large s; then the limit in the third expression above should approach zero if [itex]n\ge 2[/itex]. But shouldn't the condition be [itex]n\ge 1[/itex]??
If I'm right about the second error, then it's a more substantial hole in his analysis. Basically you want io to be pointlike (a 2-sphere of radius 0) and points on scri+ to be non-pointlike (each such point represents a 2-sphere of finite radius). The best motivation I know of for wanting these to be this way is that it allows you to make a tiling of Penrose diagrams to cover the static Einstein universe. (Is there some more fundamental reason?) So the condition on io requires [itex]n\ge 1[/itex], and the condition on scri+ forces [itex]n\le 2[/itex]. But then this leaves a whole range of possible exponents between 1 and 2 (contrary to Winitzki's analysis, which fixes the exponent uniquely at 2). Does this seem right?
One would then need some other reason to pick n=2. I think that if you want f' to be analytic and positive-definite, then that does rule out n=1, or any other value besides n=2. Does this make sense?
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