In our last blog we derived the speed of gravity from the observational period and variation in the period of the binary pulsar PSR 1913+16 that was discovered by Taylor and Hulse. The luminosity spectrum they used went back to a 1963 paper of Peter and Mathews who took the rate of energy emission of Einstein's rotating dumbbell,

$$L=\frac{G}{5c^5}\left(\frac{d^3Q_{ij}}{dt^3}\frac{d^3Q_{ij}}{dt^3}-\frac{1}{3}\frac{d^3Q_{ii}}{dt^3}\frac{d^3Q_{jj}}{dt^3}\right),$$

and evaluated the quadrupole moments using a Keplerian ellipse

$$d=\frac{a(1-e^2)}{1+e\cos\psi},$$

and the angular "velocity"

$$\dot{\psi}=\frac{G(m_1+m_2)a(1-e^2)]^{1/2}}{d^2}.$$

However, $\dot{\psi}$ will only be the angular *speed*, which implies that

$$\dot{\psi}^2a^3=\frac{GM(1+e\cos\psi)^4}{(1-e^2)^3},$$

which should be independent of both the eccentricity $e$ and the true anomaly, $\psi$, for then it would reduce to Kepler's III.

Be that as it may, they find a luminosity given by

$$L=\frac{2}{5}\frac{GM^2}{c^5}a^4\omega^6,$$

where $\omega$ is the angular speed of a binary system of two equal masses, each with mass $M/2$ in a circular orbit of radius $a/2$ about the system's center of mass. The LIGO team took this as the "gravitational wave luminosity". How electromagnetic radiation became gravitational radiation is anyone's guess.

However, their expression is incompatible with the luminosity of a star considered as a perfect black body. Omitting numerical factors, the Luminosity is

$$L\sim\omega E\sim \omega\frac{(kT)^4a^3}{(\hbar c)^3},$$

where $E$ is the total radiated energy. In comparison their proposed luminosity can be written as

$$L\sim\omega\frac{GM}{a}\left(\frac{a\omega}{c}\right)^3.$$

In order that this be associated with gravitational waves, we must assume with Poincare' that gravitational waves travel at the speed of light. Over a decade later, Einstein made the same assumption but forgetting to reference the source.

Equating the two expressions, we come to a blatant contradiction: the absolute temperature cannot depend on the speed of light, or for that matter, any other speed. We have point out in a previous blog, "An error in the LIGO calculation of the 'chirp' mass" that the luminosity must be given by

$$L\sim\omega\frac{GM}{a}\left(\frac{a\omega}{c}\right)^3,$$

for in this case the temperature will be given by

$$kT=\left[\frac{GM}{a}(\hbar\omega)^3\right]^{1/4},$$

independent of $c$.

We recall from yesterday's blog that the cubic term in the expression for the luminosity is a third-order aberration effect. Einstein would raise it by a power of two. For then the temperature would be a function of $c$, and temperature measurements would allow the determination of the speed of light.

*Gravitational waves are not the same as electromagnetic waves confined to a black-body cavity! And even if it were, it would not be given by the Peters-Mathew expression. *

We may venture to derive the expression for the luminosity of gravitational waves that should (and must) propagate at a velocity $v_G$, which we have no reason to suppose is the same as $c$. The luminosity would be given by

$$L\sim\omega\frac{GM}{a}\frac{a\omega}{v_G}.$$

The corresponding thermal luminosity would be

$$L\sim\omega\frac{(kT)^2}{\hbar v_G}a.$$

This would correspond to a one dimensional system of dimension $a$. The temperature of such a system would be

$$kT=\left(\frac{GM}{a}\hbar\omega\right)^{1/2}.$$

We would also venture so far as the say that the dynamics of a rotating dumbbell is incompatible with Keplerian dynamics, as it is with black-body radiation.