# Gravitational wave energy

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

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""Electromagnetic energy can be related to it's frequency via E = h x nu."" This is not entirely wrong, but misleading. The cited relation is for a quantum of electromagnetic energy, not for the energy radiated! – Georg Nov 18 '11 at 9:32

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.

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Thank you Luboš. Is there an a relationship that forecasts the frequencies/wavelengths of orbiting bodies like two neutron stars or a star obiting a black hole? – Michael Luciuk May 4 '11 at 17:51
Dear Michael, the frequency of the gravitational waves is $1/T$ where $T$ is the period of one orbit, as measured from infinity. However, $E=h\nu$ is just the energy of one graviton - $h$ is a tiny number. Orbiting stars or black holes emit a gigantic number of gravitons. The question what is the energy carried by gravitational waves from a binary star is a totally different question. – Luboš Motl May 4 '11 at 18:52
Luboš: 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? – Michael Luciuk May 4 '11 at 19:51
Dear Michael, I have already answered this question. A single quantum always carries $E=h\nu$ but the total energy or frequency of a gravitational wave from a binary system has nothing whatsoever to do with that. The predominant - really only possible - wavelength of a binary system is $\lambda = c / \nu$ (universal for all waves!) where $\nu=1/T$ where $T$ is just the period of the orbiting. It would be one year, 365.2422 days, for the Sun-Earth system. The orbital period is different for different binary systems but it can't be derived just from the laws of physics. Got it? – Luboš Motl May 6 '11 at 4:50
It should be added that seeing any evidence of the quantum nature of gravitational radiation in LISA or LIGO is extremely, extremely unlikely. – Jerry Schirmer Apr 9 '12 at 13:13

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.