Ah, this makes a lot of sense, indeed. I was expecting Line 8 to be correct, since the rest of the proof follows through with this value, but couldn't find the error. Thanks a lot, @TSny, for spotting the error before that.
Thanks @berkeman
Here is the page you requested (the attached version seems to be of rather poor quality). I don't think the previous page is needed, but if you think it could be useful, please let me know.
The original, good quality image is available at...
I'm trying to follow the proof given in Box 5.3, page 235, of the MCP book regarding the Van der Waals grand potential. It seems to me that there is a missing factor (2l−1)!/(l−1)! in the last term of Equation (8). What am I doing wrong?
One needs the 1/2 prefactor since the term under the sum is symmetric in ##i## and ##j##, so one needs to counterbalance for the double counting. At least, this is how I make sense of it...
Assume that ##\partial M_{ab}/\partial \hat{n}_c## is completely symmetric in ##a, b## and ##c##. Then, it is stated in the book I read that the divergence of the traceless part of ##M## is proportional to the gradient of the trace of ##M##. More precisely,
$$ \partial /\partial \hat{n}_a...
Hi,
I stumbled upon an identity when studying tensor perturbations in cosmology. The formula states that
$$
\int d^2\hat{p} f(\hat{p}.\hat{q})\hat{p}_i \hat{p}_k e_{jk}(\hat{q}) = e_{ij}(\hat{q})/2 \int d^2\hat{p} f(\hat{p}.\hat{q})(1-(\hat{p}.\hat{q})^2),
$$ where ##e_{ij}(\hat{q})## is any...
Hi,
I stumbled upon an identity when studying tensor perturbations in cosmology. The formula states that
$$
\int d^2\hat{p} f(\hat{p}.\hat{q})\hat{p}_i \hat{p}_k e_{jk}(\hat{q}) = e_{ij}(\hat{q})/2 \int d^2\hat{p} f(\hat{p}.\hat{q})(1-(\hat{p}.\hat{q})^2),
$$ where ##e_{ij}## is a symmetric...
Hi kimbyd,
I already checked, and there are no errors mentioned on that page in the errata list.
I'll try to make sense of this factor when reading the following pages, since this notion is used again. Also, since ##\rho_{EQ}## is the common density of matter and radiation where they are...
Hi George,
I also managed to get it, up to this ##1/\sqrt{2}## factor :)
I'm not sure this is relevant, but this relation is obtained in the case where one tries to model the perturbations to density, speed and so on outside the horizon and also using the adiabatic mode, in which all...
Hello Kimbyd,
I typed too fast my translation to LaTeX: indeed, ##a_{EQ}## is a constant. Sorry for the confusion.
I did try to use the chain rule already, but to no avail; in particular, the square root in the numerator eludes me. There are no additional explanations in the text, although...
Sure, no problem: $$ \frac{d}{dt} = \frac{H_{EQ}}{\sqrt{2}}\frac{\sqrt{1+y}}{y}\frac{d}{dy},$$ with $$y = \frac{a(t)}{a_{EQ}(t)}.$$ ##H## is the Hubble "constant", of course: ##H = (da(t)/dt)/a(t)##.
Thanks for any help that might come :)
Bye,
Pierre
Hi everyone,
I'm unable to understand how to derive Formula (6.3.11) in Weinberg's cosmology book. It's a relation between time-related derivation (d/dt) and RW-scale-factor-related derivation (d/dy, where y = a(t)/aEQ, a(t) is the RW scale factor in the metric and the EQ subscript denotes the...
Hello George,
I have no problem deriving Eq. 3.1.4. My issue has to do with the comment in the footnote on Page 151 that states that Eq. 3.1.7 can be derived _from_ Eq. 3.1.4 and also Eq. 3.1.6 (the derivation of which is simple too). The sheet you provided doesn't seem to help in that regard...