Spectra of gravitational waves

In Jim Arnold’s blog, we have been having an occasionally enlightening, occasionally exasperating discussion about whether gravitational waves (GWs) exist.

The evidence strongly supports the interpretation of the mathematics of general relativity that says gravitational waves do indeed exist and are, in principle if not yet in practice, observable.

That leads to a question that hasn’t come up in Jim’s blog but I’d like to raise here: What is the spectrum of gravitational radiation?

Burt Jordaan and SL, aka Scruffy, clearly know the mathematics better than I. Jim Arnold does, too, but I don’t think he’ll have anything useful to say here because he disputes the very existence of GWs.

I think my question is shared by many of the people who have been reading the discussion at Jim’s blog, and I hope some of them will chime in with related questions.

My understanding is that any accelerated mass changes the geometry of (perturbs) spacetime, and that perturbation propagates in all directions from that mass at the speed of light. If we want to visualize it, it is like a ripple in a pond. If we want to describe it strictly mathematically, that propagation fits the pattern of a wave. It has a wavelength and frequency. In fact, unless the disturbance is strictly sinusoidal, it has multiple frequencies, each of which with a different intensity–i.e., a spectrum.

Light spectra are typically described as continuous or line spectra. That’s a convenient distinction although we know that a continuous spectrum is just a set of line spectra with too small a spacing between wavelengths to distinguish between the lines. Or we could talk about its being made up of photons with energies at a continuum of values.

In any case, I have not seen any discussion of what the expected spectra are for GWs.

In the other discussion, we were focusing on a case of a pair of neutron stars in a tight orbit which is getting tighter due to the emission of GWs. Burt noted that the propagating GW is periodic in character, and that made me wonder about its period. Does its spectrum have a narrow peak at the orbital frequency (and smaller peaks–harmonics–at multiples of that frequency)? Or does it have a spectrum more like blackbody radiation in thermodynamics?

What about the spectra in these other cases:

  • An apple falls to Earth?
  • A supernova ignites?
  • Two black holes merge?
  • Other phenomena that create huge gamma-ray bursts?

    Will the new GW detectors being planned be able to give us much information about the spectra of such events? Will we ever be able to see a spectrum in enough detail to show the need for gravitons, just as Planck proposed the quanta we now call photons to explain the shape of the blackbody spectrum at high frequencies?

    As you can see, this post is more about questions than answers. I hope the comments will lead us toward some answers as well as to some more questions.

    Fred Bortz,
    Author of Physics: Decade by Decade, a history of Physics in the 20th century

  • 10 thoughts on “Spectra of gravitational waves”

    1. Hi Fred, it’s a mouthful of questions! You wrote:

      Does its spectrum have a narrow peak at the orbital frequency (and smaller peaks–harmonics–at multiples of that frequency)? Or does it have a spectrum more like blackbody radiation in thermodynamics?

      The simplest starting point is to look at “our” binary pulsars. They essentially just emit at the orbital frequency of the pair and that frequency is extremely low: around one cycle per 13,000 seconds (4 x 10^(-5) Hz), or a wavelength of ~23 km! This is pretty much immeasurable now or in the future.

      As the binaries spiral in, the frequency will gradually build up to more measurable proportions, but in the case of said binary pulsars, that will take many, many years. Further, binary pulsar GW amplitudes are predicted to be far below the sensitivity that we can achieve, unless they are very close to us.

      Two black holes that spiral in and coalesce are more promising candidates, with frequency in the range 10 Hz to 10 kHz, just about the audio frequency band. This is the band that LIGO is designed for. The only time where the amplitude may reach detectable amplitude is as the black holes merge, when they are predicted to transmit a “chirp” (rapid increase in both amplitude and frequency).

      As the black holes start to coalesce, they move at orbital speeds approaching c, with the chirp reaching a crescendo at a frequency roughly given by: 10 kHz divided by the number of solar masses of the combined binary black holes. The amplitude then “rings down” at (say) 1000 Hz in a few seconds to minutes, as the two bulges on the combined black hole disappear.

      The detection problem is that coalescing black holes are not common, especially in our neighborhood. They are thought to happen frequently in the cores of young galaxies, which are mostly very, very far away.

      A strong source could be asymmetrical supernova explosions, but sadly (or luckily!), local ones are also rare. They could very well emit a spectrum of gravitational waves, but I do not know what the probable spectrum looks like.

      The super-massive black hole at the center of the Milky Way may be a strong source (as it gobbles up stars), but the frequencies (micro-Hz) will be far below LIGO’s capabilities. Think about the period of a star orbiting around a super-massive black hole, even just outside of the event horizon.

      That’s where LISA (Laser Interferometer Space Antenna) will come in. With detector arms measured in millions of km, micro-Hz frequencies will be no problem…

      Apples falling from trees? Nah! Particles in free fall do not emit GWs…

      Burt Jordaan (www.Relativity-4-Engineers.com)

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