Gravitational wave energy

Question

Electromagnetic energy can be related to it's frequency via $E=h\nu$. Is there a comparable relationship between gravitational wave energy and frequency?

Answer 1

Yes, it is $E=h\nu$, too. This relation is totally universal across the world of quantum mechanical theories. It holds for photons, gravitons, electrons, Higgs bosons, or any other particle. It's because of the very general Schrödinger equation that says $$ i\hbar \frac{d}{dt} |\psi\rangle = H | \psi \rangle $$ The operator on the left hand side adds $\hbar\omega = h\nu$, the operator (Hamiltonian) on the right hand side adds $E$ if the state $|\psi\rangle$ is an energy eigenstate.

Answer 2

Is there an a relationship that forecasts the frequencies/wavelengths of orbiting bodies like two neutron stars or a star obiting a black hole?

Yes, the frequency of the gravitational waves is generally twice the orbital frequency of the binary pair. You can use normal Newtonian gravity to get a good estimate of the properties of a given orbit.

My question on forecasts for the frequencies/wavelengths of orbiting bodies like two neutron stars or a star obiting a black hole relates to what LIGO or LISA would expect to encounter for different situations. Would there be a predominant wavelength given the specific conditions like Wein's displacement law, or would the wavelengths be quite random?

Because the frequency of the gravitational wave (during the slow decay of a binary orbit) is simply given by the orbit of the bodies, the spectrum of expected gravitational waves is given by the expected populations of astrophysical objects of various masses.

Electromagnetic energy can be related to it's frequency via E=hν. Is there a comparable relationship between gravitational wave energy and frequency?

This question is a bit subtle. We don't ever need to talk about gravitons when talking about gravitational waves--we don't have a quantum mechanical theory of gravity--so the equivalent formula is mostly irrelevant.

Here I'll quote Sean Carroll (from Chapter 6 of his Lecture notes on General Relativity):

It is natural at this point to talk about the energy emitted via gravitational radiation. Such a discussion, however, is immediately beset by problems, both technical and philosophical. As we have mentioned before, there is no true local measure of the energy in the gravitational field. Of course, in the weak field limit, where we think of gravitation as being described by a symmetric tensor propagating on a fixed background metric, we might hope to derive an energy-momentum tensor for the fluctuations hµν , just as we would for electromagnetism or any other field theory. To some extent this is possible, but there are still difficulties. As a result of these difficulties there are a number of different proposals in the literature for what we should use as the energy-momentum tensor for gravitation in the weak field limit; all of them are different, but for the most part they give the same answers for physically well-posed questions such as the rate of energy emitted by a binary system.

Further reading

• Peter Saulson, "Physics of Gravitational Wave Detection"
• Sean Carroll, "Lecture notes on General Relativity"
• Bernard Schutz, "A First Course in General Relativity"