# Dimensional analysis for gravitational radiation expression

on this paper, please refer to equation 2.117 for the power emitted for a rotating mass system:

$$P = - \frac{128}{5} G M^2 R^4 \Omega^6$$

power in cgs should be (g is grams, m is meter, s is seconds):

$$g m^2 s^{-3}$$

now, $G$ is in $m^3 g^{-1} s^{-2}$ and $\Omega$ is in $s^{-1}$ so 2.117 right hand side is

$$m^7 g s^{-8}$$

so, i'm going to infer that the right hand side is missing a factor of $\frac{1}{c^5}$, so the dimensionally accurate expression for power (without weird normalized units) is;

$$P = - \frac{128}{5 c^5} G M^2 R^4 \Omega^6$$

is that dimensional analysis accurate?

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## 1 Answer

Your dimensional analysis is correct. Note, however, that units are irrelevant, when it comes to this kind of analysis; only the dimension matters.

One thing that you may have overlooked are dimensions hidden in the 128/5 constant. It would take a while for me to work through the paper to see where that comes from, but it would be easy to overlook those.

While you're correct that the missing dimensions would be accounted for by a velocity raised to the fifth power, it could also be accounted for by any number of combinations of length and time (volume times velocity squared times frequency cubed?).

So, in short, the analysis is correct, the conclusions: suspect.

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The linked paper mentions they use units in which $c=1$ twice (in the appendix), so the concern is unnecessary. – alemi Aug 8 '14 at 4:02