A lecture given 10/27/08 by Professor Freeman Dyson of the Institute of Advanced Studies at Princeton, in honor of the 100th anniversary of the founding of the University of California, Davis.

$$E=\left(\frac{c^{2}}{32\pi G}\right)\omega^{2}f^{2}$$

is the energy per gravity wave, where f is the dimensionless amplitude/strain.


is the energy per graviton, taken from \(\hbar\omega\) energy times \(\frac{\omega^3}{c^3}\) density


is the strain per graviton.


Gives the linear displacement per graviton.

Note that spherical objects can’t radiate gravitational waves, and that binary stars produce kilohertz gravity waves.

LIGO’s threshold is therefore \(10^{37}\) gravitons.

$$M\delta^{2}\geq\hbar T$$

is the uncertainty in position and velocity.


(from combining previous two equations)

$$\delta^{2}\geq\frac{\hbar D}{M_{s}}$$

Which exceeds the Schwarzschild radius, so impossible.

Then the Bohr-Rosenfeld argument is:

$$\Delta E\_{x}(1)\Delta E_{x}(2)\approx\hbar\left|A(1,2)-A(2,1)\right|$$

where A(2,1) is the field from dipole 2 at location 1.

The detector is described by:


where a is the initial state, b is the final state, and m is the detector mass.


is the logarithmic average taken over the graviton cross section.


Now consider the gravitophotoelectric effect, where the graviton removes an electron.

$$Q=\int\left|\left(x\frac{\partial}{\partial y}+y\frac{\partial}{\partial x}\right)\Psi_{a}\right|^{2}d\tau$$
$$Q=\frac{\int\bar{r}^{4}\left[f'(r)\right]^{2}d\bar{r}}{\int r^{2}\left[f(r)\right]^{2}dr}$$
$$\int r^{4}\left[f'+\left(\frac{3}{2}r\right)f\right]^{2}dr>0$$

This means that if you take a detector the mass of the Earth, squash it into a large flat sheet, and run it for the lifetime of the universe, you’ll detect 4 gravitons.

From the Sun, there are \(10^{8}\)W of gravitons and \(10^{25}\)W of neutrinos, and we can detect gravitons about \(10^{-35}\) less than neutrinos.

Special thanks to MathJAX and this post on how to use MathJax in Blogger!

N.B. There’s a good followup post on Cosmic Variance, along with an earlier entry giving some good background information.