## Astronomy Without A Telescope – Gravitational Waves

[/caption]

Gravitational waves have some similar properties to light. They move at the same speed in a vacuum – and with a certain frequency and amplitude. Where they differ from light is that they are not scattered or absorbed by matter, in the way that light is.

Thus, it’s likely that primordial gravitational waves, that are speculated to have been produced by the Big Bang, are still out there waiting to be detected and analyzed.

Gravitational waves have been indirectly detected via observations of pulsar PSR 1913+16, a member of a binary system, the orbit of which decays at the rate of approximately three millimetres per orbit. The inspiraling of the binary (i.e. the decay of its orbit) can only be explained by an invisible loss of energy, which we presume to be the result of gravitational waves transporting energy away from the system.

Direct observation of gravitational waves currently escapes us – but seems at least feasible by monitoring the alignment of widely separated test masses. Such monitoring systems are currently in place on Earth, including LIGO, which has test masses separated by up to four kilometres – that separation distance being monitored by lasers designed to detect tiny changes in that distance, which might result from the passage of a gravitational wave initiated from a distant point in the universe.

The passing of a gravitational wave should stretch and contract the Earth. This is not because it strikes the Earth and imparts kinetic energy to it – like an ocean wave hitting land. Instead, the Earth – which sits within space-time – has its geometry altered, so that it continues to fit the momentarily stretched and then contracted space-time within which it sits, as a gravitational wave passes.

Gravitational waves are thought to be unaffected by interaction with matter and they move at the speed of light in a vacuum, regardless of whether or not they themselves are in a vacuum. They do lose amplitude (wave height) over distance, but only through attenuation. This is similar to the way that a water wave, emanating from the point of impact of a pebble dropped into a pond, loses amplitude proportionally to the square of the radius of the growing circle that it forms.

Gravity waves may also decline in frequency (i.e. increase in wavelength) over very large distances, due to the expansion of the universe – in much the same way that the wavelength of light is red-shifted by the expansion of the universe.

Given all this, the exceedingly tiny effects that are expected of the gravitational waves that may routinely pass by Earth create a substantial challenge for detection and measurement – since these tiny space-time fluctuations must be distinguished from any background noise.

The noise background for LIGO includes seismic noise (i.e. intrinsic movements of the Earth), instrument noise (i.e. temperature changes that affect the alignment of the detection equipment) and a quantum-level noise, also known as Johnson-Nyquist noise – which arises from the quantum indeterminacy of photon positions.

Kip Thorne, one of the big names in gravity wave theory and research, has apparently ironed out that last and perhaps most troublesome effect through the application of quantum non-demolition principles – which enable the measurement of something without destroying it, or without collapsing its wave function.

Nonetheless, the need for invoking quantum non-demolition principles is some indication of the exceedingly faint nature of gravitational waves – which have a generally weak signal strength (i.e. small amplitude) and low frequency (i.e. long, in fact *very long*, wavelength).

Where visible light may be 390 nanometres and radio light may be 3 metres in wavelength – gravitational waves are more in the order of 300 kilometres for an average supernova blast, up to 300,000 kilometres for an inspiraling black hole binary and maybe up to 3 billion light years for the primordial echoes of the Big Bang.

So, there’s a fair way to go with all this at a technological level – although proponents (as proponents are want) say that we are on the verge of our first confirmed observation of a gravitational wave – or otherwise they reckon that we have already collected the data, but don’t fully know how to interpret them yet.

This is the current quest of citizen science users of [email protected] – the third most popular BOINC distributed computing project after [email protected] (spot an alien) and [email protected] (fold a protein).

*This article follows a public lecture delivered by Kip Thorne at the Australian National University in July 2011 – where he discussed plans for LIGO Australia and also the animated simulations of black hole collisions described in the paper below – which may provide templates to interpret the waveforms that will be detected in the future by gravitational wave observatories.*

**Further reading:** Owen et al (*including Thorne, K.*) Frame-Dragging Vortexes and Tidal Tendexes Attached to Colliding Black Holes: Visualizing the Curvature of Spacetime.

Given the wavelength and amplitude gravitational waves, can’t see any real progress in detecting them until the ESA’s LISA is launched.

(LISA was originally a joint project with NASA science until they sacrificed pretty much everything else to keep the JWST project alive – and we all know how that’s going). Thanks ESA for Planck, Herchel and good luck with LISA.

Given the wavelength and amplitude gravitational waves, can’t see any real progress in detecting them until the ESA’s LISA is launched.

(LISA was originally a joint project with NASA science until they sacrificed pretty much everything else to keep the JWST project alive – and we all know how that’s going). Thanks ESA for Planck, Herchel and good luck with LISA.

It is sort of a back-burner joint venture, as US separated but kept a LISA science project.

Coincidentally, for those who enjoy preprints, today there is Graviton mass bounds from space-based gravitational-wave observations of massive black hole populations for your reading pleasure.

This carries over from a response to Xplorer. String theory gives a quadratic form of the curvature. This is sometimes called Gauss-Bonnet gravity. A central part of this is the Bel-Robinson tensor of the form R_{abef} R_{cd}^{ef}. The trace of the BR tensor R^{abcd}R_{abcd} enters into the Lagrangian and gives the action for dynamics

S = ?[(1/2?)R + ?’R^{abcd}R_{abcd} + L]sqrt{-g}d^4x

for L the Lagrangian for everything else compactified on a Dp-brane for p = 0, 2, 4 6. The trace of the BR tensor R_{abef} R_{cd}^{ef} is multipled by ?’, which is the string parameter. The extremization of this (calculus of variations which gives dynamics) gives

?S = 0 = ?[(1/2?)[?R/?g^{ab} +2?R^{abcd}?R_{abcd}/?g^{ab} + ?L/?g^{ab}]sqrt{-g}d^4x

+ ?[[(1/2?)R + ?’R^{abcd}R_{abcd} + L]/sqrt{-g}] ?{-g}/?g^{ab}d^4x

Most of this is fairly straight forwards, except for the BR tensor part. We can derive a modified form of the Einstein field equation

R_{ab} – (1/2)Rg_{ab} + ?’R_{abcd}g^{cd} = ?T_{ab}.

The correction may be derived by other means, and in particular according to linear fields. Consider the metric (see other posts in this blog entery where I indicate more on this) of the form

g_{ab} = ?_{ab} + h_{ab}.

However, this is a classical expression. The Bel-Robinson tensor corresponds to quantum corrections on the order of the parameter ?. We then consider these as due to fields ?^a_b so that the quantum correction to the metric is

g_{ab} = ?_{ab} + h_{ab} + ?^c_a?_{bc},.

where we can regard ?^c_a?_{bc} = ?h_{ab}, which physically is a quantum correction to the classical gravity wave h_{ab}.

With the math over with, the graviton which emerges from the BR tensor with the string parameter ?’ has mass term. So gravitons at this early stageof the universe may have had a mass component. These gravitons would have not been light speed moving, or on null light cones. They may have also focused matter in a way similar to what I indicated in my post to Xplorer. The random formation of caustic may then have given rise to the structures we find with SDSS.

LC

The only thing that bugs me about the Astronomy Without A Telescope series is the title. The series is not about astronomical discoveries that were made without telescopes, and while it’s true that you don’t need a telescope in order to read it, the same is true of other Universe Today articles as well. Shouldn’t the title of a series reflect the salient features of its content?

It’s kind of whimsical rather than definitive.

A weekly column (rather than series I think) offering random subject material rather than breaking news. It’s unfocused 🙂

It is appreciated.

I am kinda living for this at the moment. For me this has been a fascinating series exploring such (sometimes controversial) topics as DM/DE / Quantum Fields, Gravity, etc. etc. What I really love is that I don’t have a high level of math or physics understanding but mostly I get most of it (Although I need to use google as my translator for LC’s and Torbjorn’s comments – but it is a price worth paying for the learning 😉 You can call the series “Poodles in Space” if you want and I will still read it :))

Minor nitpick at the twelfth paragraph: “Where visible light may be 300 nanometres…”

Actually, the visible spectrum wavelength ranges from about 380 nm to about 740 nm.

Thanks – I shouldn’t do these things from memory.

Primordial gravity waves are not likely to be detected by LIGO or LISA. Primordial gravity waves are due to gravitons which decoupled from the rest of quantum fields in the inflationary time period. Some ballpark numbers can give an idea of wavelengths we might expect to observe them at. A quantum graviton near the string scale in the earliest universe had a wavelength on the order of 10^{-31}cm to 10^{-25}cm. Inflation was a period where in about 10^{-20} seconds the universe expanded exponentially by 63 e-folds. This means scale factors increased by e^{63}, e = 2.71828, and e^{63}~ 2.3×10^{27}. So a bunch of gravitons moving in the save direction will have their wavelengths expanded to about 10^{-5}cm to 10cm. Since then the universe has expanded more or less at a uniform rate, and without going into details the scale factor has expanded about 10^{26}. So these primordial gravitons will have expanded to about 10^{21}cm, which means that now they have a wavelength on the scale of 1000 to 10^9 light years! As a rule with a wave detector you want a ¼ wave stack, and LIGO and LISA are nowhere near that scale.

So how could we detect primordial gravitons or gravity waves? To think about this some, we need to understand a little bit about wave mechanics. Further, since these waves are gravity waves we should connect up with that.

The Einstein field equation connects curvature of spacetime to mass-energy distributions. The equation is

R_{ab} – ½ Rg_{ab} = (8?G/c^4)T_{ab}

The g_{ab} is the metric tensor. This object is used to construct distances in spacetime. So a distance is s = g_{ab}x^ax^b, and for flat spacetime this is

s = g_{00}(x^0)^2 – g_{11}(x^1)^2 – g_{22}(x^2)^2 – g_{33}(x^3)^2

or

s = t^2 – x^2 – y^2 – z^2.

This is a form of the Pythagorean theorem for distance in spacetime. The R_{ab} is the Ricci curvature tensor, and R = R_{cd}g^{cd} is the Ricci scalar curvature. The T_{ab} is all the sources of curvature, matter etc.

The gravity waves we are most interested in are weak. This means the metric for flat spacetime is perturbed by some small variation. Therefore g_{ab} = ?_{ab} + h_{ab}, where ?_{ab} is the flat spacetime metric and h_{ab} is the small variation. We compute that Ricci curvature with this. The Ricci curvature scalar has all sorts of nonlinear terms. These nonlinear terms are on the order of the field strength squared. So we can ignore them, for the square of a small thing is really small. The Ricci stuff for a wave moving in the z direction turns out to be

R_{ab} – ½ Rg_{ab} = (?_z^2 – ?_t^2)h^{tl}_{ab} = ?h^{tl}_{ab}

The little superscript “tl” means traceless part, which is due to a bit of a math maneuver not shown here. The box symbol is the d’Alembertian. If we set all the sources to zero T_{ab} = 0 this is a plane wave moving in pure vacuum. So for those who have a little bit of undergraduate physics this reduces to something familiar.

The solution is from a 4×4 tensor, and there is a 2×2 subtensor that gives terms that are h_{xx} = h_{yy} which are the ++ polarizations and then terms h_{xy} = h_{yx} which are the xx polarizations. This is one departure from electromagnetic waves which have one direction of polarization. This is why these detectors have two arms. The gravity wave induces a motion on the two, which are due to the two polarization directions.

So if we reinsert that T_{ab} term we can ask how gravity waves are generated and how they interact with material. This is going to be serious rough shod work here with approximations. These are complicated equations, and even in a linear setting they are coupled Liénard-Wiechert potential equations used in electromagnetism, which is something covered in a first year graduate level electromagnetism course. A very approximate formula which can be used is

h_{ab} ~ (2G/c^4)(Df^2),

where D is the physical dimension of the mass-energy source region and f is the frequency of the radiation or the quadrupole oscillation of a driving source. This approximate formula can be derived. So for two black holes on an inspiral within 10 Schwarzschild radii of each other, the princpal mode of a linear gravity wave would more or less be of that wavelength. So for two 10 solar mass black holes that would be around 100km. There will of course be subharmonics as well of shorter wavelength. As the inspiral happens D decreases to around 20km. This is gravity wave physics detectable by LIGO.

What about primordial gravity waves? The best detector is the CMB. The two polarizations will leave an imprint of the h_{++} and h_{xx} polarizations on the radiation in the CMB. The plasma at the time decoupled from with imprints of the two polarization directions given approximately by (2G/c^4)(Df^2), where D is 10^3-10^9 light years. This will then be a 2-polarization signature on the CMB. The shorter wavelength or higher frequency stuff will have a larger signature, but smaller. The CMB is a sphere 46 billion light years out. The CMB is a total area of 2.1×10^{21} square light years and the stringy high frequency gravitons might leave an imprint that is 5×10^{-16} of the entire 4? steradian solid angle of sky. That is very small. The billion light year footprint would be 5×10^{-4} and of course there is a spectrum in between. These data might show up in the Planck spacecraft data.

LC

Agreed. You might catch them primordial GWs with pulsar timing analysis also.

I think the recipe is:

– LIGO: supernova, neutron stars

– LISA: inspiraling compact binaries

– Pulsar timing analysis: black hole mergers, Big Bang echo

– CMB analysis: Big Bang echo

LIGO and LISA should get gravity waves from black hole mergers. A black hole merger is the perfect source for a gravity wave. The wavelength should be on the order of 10-100km. Also the coalescence converts a huge amount of the mass of the two black holes into gravity waves. If you have two black holes with equal mass M they have a radius R = 2GM/c^2. The final state has a mass M’ and an event horizon radius at R’ = 2GM’/c^2. the radius of the event horizon of the coalesced black hole is

4?R’^2 = 16?^GM’/c^2)^2 > 32?^GM/c^2)^2

where the left hand side is for the coalesced black hole and the right hand side is the horizon area of the two initial black holes. So the final mass of the coalesced black hole is M’ > sqrt{2}M. The deficit in energy is then 2M – sqrt{2}M = .5858M. So out of the two masses M for the initial black hole up to 29.3% of that mass energy goes into gravitational radiation.

This is an estimate that plays rough with issues of ADM energy and localizability of energy and so forth, but it is a good estimate. That is a huge amount of mass-energy. However, the coupling constant 8?G/c^4 = 2.7×10^{-48}s^2/g-cm. Three solar masses of energy equivalents 10^{33} g or 10^{53}erg in a volume of 1000km ~ will induce a curvature of about 10^{38}erg/cm^3*10^{-48}s^2/g-cm ~ 10^{-17}cm^{-2}. This an enormous curvature actually, where Earth’s curvature of gravity is about 10^{-28}cm^{-2} at the surface. So the force of gravity that pulls apart two objects would be 10 billion Earth gravity “gees.” However, if I scale that up to 10^6 light years that curvature is about 10^{-35}cm^{-2} and at a 10^8 light years it is 10^{-41}cm^{-2}. This is in part why detecting gravity waves is so tough. One can estimate the distance this should move the arms and it is about 10^{-13}cm — about a distance equal to the radius of a nucleus. I think 10^{8} light years is about the upper limit of LIGO.

Pulsar timing analysis is of course an emerging subject. A gravity wave passing through a double pulsar would cause a momentary desynchronization of the pulsar timing and the orbital period. That is a touchy thing to try to measure. Yet for a gravity wave that is on the order of 10^5 to 10^8km in size this would be a decent antenna length for their detection.

LC

The fact that Gravity Waves and gravity’s influence travel at the speed of light is for me mind-blowing and seems counter-intuitive at first. You all have probably hear the though experiment that if the Sun vanished it would take Earthlings 8 minutes to realize it (the time it takes light to travel 150 million Kilometers). However, would the earth immediately shoot off from its slightly elliptical orbit (without the sun to conserve its angular momentum) OR would it wait 8 minutes also before reacting or changing its orbit?? This part of the thought experiment really seems counter-intuitive to me… But it does indeed took as though Einstein was right – at least in terms of our best (to my understanding) confirmation to date –

http://www.csa.com/discoveryguides/gravity/overview.php#n2

Quoting and distilling –

“…Professor Kopeikin realized that Einstein’s theory could be reformulated in a way that made gravity analogous to electromagnetic radiation … In 1999 Kopeikin, who was then at the University of Jena in Germany, made a crucial breakthrough. To the surprise of physicists worldwide, he came up with an exact solution to these equations. …

In short, this means that it is possible to calculate the speed of light from measurements of the electric and magnetic field of a moving charge, without having to detect electromagnetic waves directly. In the same way, Kopeikin’s reworking of general relativity expresses the gravitational field produced by a moving body in terms of the mass of the body, its velocity, and the speed of gravity. This information could be used to work out the speed of gravity. Obtaining this information, though, is not easy. One obvious approach is to use “gravitational lensing”. This is the apparent (but not actual) shift in position of a distant celestial object that occurs when its light is deflected on the way to Earth as the rays pass through the gravitational field of a massive body. If that body is moving, measurement of the lensing effect should give us the information we require.

[END QUOTE from above article]

Any massless particle or field travels at the speed of light. Special relativity indicates that distance in spacetime (space plus time) is

s^2 = (ct)^2 – x^2 – y^2 – z^2

This is a form of the Pythagorean Theorem. Now let us reduce this to a 2 dimensional case by setting y = z = 0. Then s^2 = (ct)^2 – x^2. Now if x = vt for a moving particle then s^2 = (c^2 – v^2)t^2. There are a number of things one can now observe. This can be written as s^2 = c^2(1 – (v/c)^2)t^2, where you might recognize the gamma factor ? = 1/sqrt(1 – (v/c)^2). Also if the particle is traveling the speed of light the distance s = 0. Physics has a sort of dual description; one in space or spacetime and the other in momentum space. In wave mechanics this is tied up with the Fourier transform. In relativity the momentum space is a momentum-energy space. The same distance measure in momentum energy space is

(mc^2)^2 = E^2 – (pc)^2

for a particle of mass m, momentum p and energy E. A massless particle m = 0 is the momentum-energy case of a particle moving at the speed of light.

There is an associated wave equation with this. The duality between energy and time and duality between momentum and space converts this to a wave equation for a wave field moving at the speed of light if m = 0. Photons are massless and the wave equation can be derived by Maxwell equations for the electromagnetic field. From general relativity the gravity wave equation can be similarly derived.

Gravity waves, or what is probably the more proper nomenclature gravitational radiation, were first derived back in the 1930s. There was considerable controversy over them. Research in general relativity went into dormancy from about 1939-49 due to various events in the world. In the 1950s Feynman used a simple argument to show that gravity wave could transfer energy and were thus physically real. The solution space for gravity waves were found in the 1960s by Petrov, with some work by Penrose and Pirani, which are a solution type determined by an eigenvalued system on the Weyl curvature. This is rather advanced stuff in general relativity, so I will leave the discussion at this level. The solution type, called type N, have 4 Killing vectors that are all degenerate or equal. In more recent time gravity wave solutions have been worked according to a power series on general relativity. The terms in this series are an expansion in 1/c, where to 1/c^4 the Einstein field equations give linear gravity waves. This interest has been to understand what the LIGO and other detectors might find. Kip Thorne has developed this into a pretty refined art.

LC

” it is possible to calculate the speed of light from measurements of the electric and magnetic field of a moving charge”

I think James Clarke Maxwell figured this out in the late 19th century.

Clearly the Earth would change relative trajectory at the same instant the information (that the Sun had disappeared) arrived – since physical effect and information would travel at the same speed (i.e. that of light). No real mystery there.

The ‘physical effect’ of the thought experiment is a kind of gravitational wave, since the space-time curvature created by the Sun’s mass would progressively flatten out if the Sun suddenly disappeared.

This is so awesome. This would change our understanding of the universe. http://www.iampleasant.com

Detecting gravitational waves has been described as being akin to “standing in the surf at Big Sur and listening for a kiss blown across the Pacific.”

Directly observing such phenomena would answer Einstein’s 1916 General Relativity prediction and somewhat harmonize string and other universal theories described by one physicist as: “The most overwhelming events in the universe.”

Despite budgetary constraints the ESA continues to revise the mission concepts for the laser inferometer space antenna (LISA).

http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=48728

I still have a problem discerning between gravity waves and atmospheric gravitational waves, are they the same thing? Atmospheric Gravitational waves can be seen in clouds, maybe a swarm of nano or pico satellites might be able to detect.

They are gravity waves – not gravitational waves. Whole different ball game. See here:

http://en.wikipedia.org/wiki/Gravity_wave#Atmosphere_dynamics_on_Earth

For a cool detection of gravity waves, tsunamis push the atmosphere out into the ionosphere to cause airglow.* Note that it could possibly happen from other events as well according to the follow up comment by the author:

“Ordinary ocean waves do cause gravity waves to propagate up into the atmosphere, but because their motion is random those atmospheric disturbances interfere destructively and essentially vanish. Jonathan Makela told me that only when you have a basin-wide signal of waves march-stepping in unison across the ocean do the resulting atmospheric gravity waves interfere constructively and hence are able to propagate all the way up to the ionosphere.”

—————

* Which effect of course is now considered to be added to tsunami early warning systems. Satellites to the rescue!

What about any gravitational wave focusing phenomena which can enhance their intensity for us to detect, like gravitational lensing effect observed in visible band!!!!!

The enhancement of focusing by gravity waves is something I and a couple of other physicists are seriously considering. The SDSS data on filaments and domain walls can be seen at

http://www.sdss.org/includes/sideimages/sdss_pie2.html

The appearance of these filaments may represent caustics or the focusing of relativistic matter in the early universe. A massive particle that is moving with a huge amount of energy E >> mc^2 approximates a massless particle or is similar to a photon. This appearance of filaments and domain wall is then remarkably similar to the appearance of focused light or caustics on the bottom of a swimming pool. The water waves and ripples on the surface set up surfaces that deviate from the horizontal and act as lenses. These ripples on the water surface are a sort of model of a distribution of gravity waves, and the light is a case of radiation and ultra-relativistic particles in the very early universe.

The problem involves optical chaos, and in some calculations it appears that general relativity will amplify chaos. This works for dynamical systems of planets (satellites), where the chaotic drift of a small satellite close to the main gravitating body due to a second larger satellite further out is enhanced if there are general relativistic effects with the smaller satellite. This carries over to optics, where electromagnetic waves that are deviated by a random distribution of gravitational scattering sources. Such sources could be gravitational waves.

I will now carry this over to another post here by Terry G below. This is a natural break point, and this leads to the issue of whether the graviton can have mass.

LC

In my mind gravity has no analog to electromagnetic radiation.

It does not ‘radiate’ at all but manifests as a stress field surrounding mass in volume.

The greater mass in the lesser volume equals more stress(gravity).

The only property that could propagate is a change in the magnitude of the stress.

We have no idea of the damping effect of local objects (the sun and earth) nor do we actually have an idea of it’s propagation velocity. A passing gravity wave will perturbate the very mass of the detecting instrument as well as the volume of space that contains it. Thus the rub!

I only wish I could follow the equations of LC et all and they had a more elegant way of posting them sans all those ascii limitations. Bravo.

This may be a very silly question so please go easy on me. If like electromagnetic waves gravity also travels at the speed of light is there a mechanism analogous to red or blue shift that applies to a moving gravity well?

@John,

This is a fair question John. The path to the answer might be clear after we ask a few more questions.

A gravity well which gives off ‘gravity waves’ would have the following if comparable to EM waves.

What is the frequency of the gravity radiation — where on that full spectrum for example would we find some source so as to calibrate our instruments.

What is the intensity of the gravity radiation, and for how long. What is the hysteresis of any system we use to make those measurements.

How would we discern which object is emitting the radiation from a crowded field of players; some might be approaching others retreating.

Another thought –what method of ‘splitting’ the ‘gravity light’ will we use, how to focus or guide it to where we can work with it correctly.

Myself, for splitting the gravity radiation I vote for the gravity inversion bar used in so many of the shows about geeks and isometrics. It supports substantial weight without having to be permanently installed by a professional, you can take it with you everywhere you go –don’t leave home without it; and finally, it looks cool in the living room, with the boots hanging on a peg next to the doorway, and that is important, right?

Now the last is a bit of fun of course but we need a lot more money, time, and skills, to work with the very long baseline for the types of ‘gravrad’ (to coin a word) we expect to find. As LC says, 1/4 wavelength is better for our purposes and more likely to be within our abilities.

Mary

The answer is yes. It is a fair question — not dumb In fact you can see that if the gravitational waves are very weak and essentially linear. They behave very much like photons. So the Doppler shift works with gravity waves just as with electromagnetic waves or light.

LC

Thank you for the help, after reading your answers I believe I need to re think my question I realize that I am asking something that strictly speaking can probably only be expressed in a mathematical equation.