Astronomy Without A Telescope – Unreasonable Effectiveness

Article written: 5 Mar , 2011
Updated: 24 Dec , 2015
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Gravitational waves are apparently devilishly difficult things to model with Einstein field equations, since they are highly dynamic and non-symmetric. Traditionally, the only way to get close to predicting the likely effects of gravity waves was to estimate the required Einstein equation parameters by assuming the objects causing the gravity waves did not generate strong gravity fields themselves – and nor did they move at velocities anywhere close to the speed of light.

Trouble is, the mostly likely candidate objects that might generate detectable gravity waves – close binary neutron stars and merging black holes – have exactly those properties. They are highly compact, very massive bodies that often move at relativistic (i.e. close to the speed of light) velocities.

Isn’t it weird then that the ‘guesstimate’ approach described above actually works brilliantly in predicting the behaviors of close massive binaries and merging black holes. Hence a recent paper titled: On the unreasonable effectiveness of post-Newtonian approximation in gravitational physics.

So, firstly no-one has yet detected gravity waves. But even in 1916, Einstein considered their existence likely and demonstrated mathematically that gravitational radiation should arise when you replace a spherical mass with a rotating dumbbell of the same mass which, due to its geometry, will generate dynamic ebb and flow effects on space-time as it rotates.

To test Einstein’s theory, it’s necessary to design very sensitive detecting equipment – and to date all such attempts have failed. Further hopes now largely rest on the Laser Interferometer Space Antenna (LISA), which is not expected to launch before 2025.

The proposed Laser Interferometer Space Antenna (LISA) system using laser interferometry to monitor the fluctuations in the relative distances between three spacecraft, arranged in an equilateral triangle with five million kilometer sides. Hopefully, this will be sensitive enough to detect gravity waves. Credit: NASA.

However, as well as sensitive detection equipment like LISA, you also need to calculate what sort of phenomena and what sort of data would represent definitive evidence of a gravity wave – which is where all the theory and math required to determine these expected values is vital.

Initially, theoreticians worked out a post-Newtonian (i.e. Einstein era) approximation (i.e. guesstimate) for a rotating binary system – although it was acknowledged that this approximation would only work effectively for a low mass, low velocity system – where any complicating relativistic and tidal effects, arising from the self-gravity and velocities of the binary objects themselves, could be ignored.

Then came the era of numerical relativity where the advent of supercomputers made it possible to actually model all the dynamics of close massive binaries moving at relativistic speeds, much as how supercomputers can model very dynamic weather systems on Earth.

Surprisingly, or if you like unreasonably, the calculated values from numerical relativity were almost identical to those calculated by the supposedly bodgy post-Newtonian approximation. The post-Newtonian approximation approach just isn’t supposed to work for these situations.

All the authors are left with is the possibility that gravitational redshift makes processes near very massive objects appear slower and gravitationally ‘weaker’ to an external observer than they really are. That could – kind of, sort of – explain the unreasonable effectiveness… but only kind of, sort of.

Further reading: Will, C. On the unreasonable effectiveness of the post-Newtonian approximation in gravitational physics.



15 Responses

  1. Torbjorn Larsson OM says

    Ahem. I had no idea that the non-linear nature of Einstein’s equations translated to such distorted waves, though in retrospect maybe I should have.

    Anyway, quantization of general relativity yields an effective field theory that is applicable for the regions below Planck scales the paper discuss. Such gravitons should act linearly AFAIU, obeying classical superposition (which is why I’m surprised by asymmetries), and if so goes a long way towards making seemingly “extreme situations” less so. In effect, making “”strong” gravity” … “not as strong as one might think”, to quote the paper.

    Relatedly, another semiclassical approximation that works well are semiclassical world lines in cosmology and especially inflation. Perhaps relatedly, the Standard Model of particles works well.

    It is as if the world likes to behave reasonable. I blame environmental principles; it works reasonably well, because we need it to.

    • Member

      Agree NASA’s artwork is often illustrative rather than accurate.

      Otherwise my layperson understanding is that this is an area where GR seems to falter, but I am not sure the quantum camp has really cracked it either. While the quantum math may work great at Planck scales, can it be scaled it up to macro environments? And the reality of gravitons remains to be demonstrated?

      • Torbjorn Larsson OM says

        I am sorry, I didn’t mean to imply anything on the artwork, I was reflecting on my own naiveté.

        the quantum math may work great at Planck scales,

        Again I’m sorry, but this is turning quantum physics on its head in my opinion.

        When applied to fields, quantum mechanics works from low energies, in the form of so called effective theories, and up.

        That QM seems best for small scales, which scales needs high energies to resolve, is because at larger scales classical mechanics is mostly a good approximation. (And it is a bitch to derive that from integrating over quantum physics.) But quantum math works really best for small energies, such as electron-photon interactions in atoms (bounded systems), zero point energies of fields, uncertainty, et cetera.

        This is also the case for the quantized field theory I linked to, which specifically breaks down at Planck scales. I.e. its math stinks at such scales.

        To sum up, for low energies the gravitons of that theory should sum up to macro scales as regards gravity waves. Keeping in mind that the spacetime they act on is itself acted on by the masses that generated them in the first place. (And all other masses.) And it is the backreaction between masses and waves (gravitons) that feed the non-linearities of GR.

        And the reality of gravitons remains to be demonstrated?

        As for the reality of gravitons, I’m stumped on one of the characteristics. AFAIU all effective theories can be quantized (I really haven’t studied quantum field theory).

        [Btw, this makes *quantization* a great example of “unreasonable effectiveness” in physics!]

        The quanta are either quasiparticles (say phonons out of crystals collective vibrations) or “real” particles (say photons out of the EM part of electroweak interactions).

        Whether these gravitons are particles, or quasiparticles denoting distributed characteristics instead of localized, depends on the relation between the effective theory and the underlying theory. Photons turned out to be “real” enough, even if the actual theory was not EM but electroweak. (And it doesn’t stop there.)

        Here we need a theoretical physicist to tell us what kind of objects to expect, I think. I assume a string physicist would deem gravitons particles (closed strings), while loop quantum mathematicians would deem them quasiparticles (“fluctuations of quantum geometry in spin-foams” – phew!).

        What I do know is that the observability of the objects called gravitons remains to be demonstrated. Gravity waves should exist, it explains neutron stars slowing down. But even those are in need of more direct testing, as the post describes.

      • Member

        Sorry I thought ‘the quantum math may work great at Planck scales’ was your assertion. The paper you linked to talks about ‘new degrees of freedom appear at the Planck scale’. I may have misconstrued.

    • Lawrence B. Crowell says

      The issue of quantization does not directly apply to the astrophysics. The linearized equation, such as equation 2 in Clifford Will’s paper, can be easily quantized if the right hand side sources are equal zero. The wave equation is formally the same as the Klein-Gordon equation. However, the metric h_{ij} is a bit more complicated than a scalar field. In particular it has two spatial terms which are independent, sometimes denoted by h_{++} and h_{xx} which define two independent polarization directions for the wave. This is a bit different that electromagnetic waves which have one polarization direction. These two polarization states, or helicity = 2, is why the quantum form of this wave has a spin s = 2hbar.

      The free equation, with the right hand side set to zero, may be easily quantized, and the graviton behaves similar to a bi-photon state. A bi-photon is a quantum optical correlation condition between two photons so they act as one particle or boson. This happens with bunching or the HBT effect. Now if you turn on the source, the right hand side, for certain idealized and linear terms you can now get a some what more realistic theory. The graviton is terribly difficult to detect if you think about it. The gravity wave occurs on a scale G/c^4, which if you look these numbers up you can readily see this is very small. The graviton introduces the Planck unit of action into the scaling so the curvature is ~ G hbar/c^3. This is much smaller. So the graviton is far more difficult to detect than a classical gravity wave,

      The LIGO and future LISA detectors are for classical gravity waves. These might be thought of as many gravitons where the action S = N times hbar occurs for N very large (infinite in a continuum approximation), and the Planck unit of action converges to zero. This recovers the classical wave. The classical wave as it passes through a spherical shell of test particles causes the shell to distend into a prolate ellipsoid, which then oscillates into a. oblate ellipsoid (the football ends squash inwards), which then continues to oscillate this way. An interferometer with different arms will detect the changing arm length by the phase shift of interfering photons traveling along the two arms. Now since gravity is very weak, again that coupling constant G/c^4, particle motion gravity waves induce is very small. This is particularly the case for extremely weak linearized gravity waves.

      Can we ever detect the graviton? Maybe, or indirectly. There is a correspondence between the isometries of gravitation in the interior of an anti-de Sitter spacetime and the conformal structure of a quantum field on its boundary. Physically this does imply that certain chains of gluons, the gauge particle for quantum chromodynamics (QCD) and what binds quarks, in plasmas will have graviton amplitudes. In effect the gravitational constant G is renormalized to a much larger value, and there are some small amplitude for a graviton in a plasma of quarks and gluons. There are some possible signatures of this already from RHIC, and the LHC heavy ion experiments may get a better signal to noise ratio and indicate we have QCD correspondences with gravitons.

      LC

  2. Lawrence B. Crowell says

    I wish this web site was TeX enabled. The linearized equations come form the following. The metric g_{ij} (similar to a 4×4 matrix) defines a distance between spacetime vectors x^I as s = g_{ij}x^ix^j. The metric in general relativity defines a connection term, which we will call A, by differential operators, so A = dg — written cryptically. The equation of motion for a particle with a spacetime velocity v is then cryptically

    dv/ds + A*v*v = 0,

    where for weak gravity recovers Newton’s second law with gravity. The connection term has units of the gravitational constant G. This is important because curvature is computed by a commutator of a general D = d + A so that the commutator [d + A, d + A] is dA + [A, A] = R. Again this is cryptic, but it is easy to see dA is in units of G while [A, A] is G^2. This means that for comparatively weak gravity dA is significant and [A, A] ~= 0. Further, since A = dg this means the curvature is R ~= (&_i)^2g. This then feeds into the Einstein field equation, which leads to a linear wave equation.

    What is curious about this UT article is that there is a mixing of the idea of a weakly interacting gravity system and a strong gravity field. Two 10 solar mass black holes can orbit each other by 10 billion kilometers and the interacting system is essentially Newtonian. This is even if the gravity field near each black hole is extreme and there are general relativistic descriptions of them. If these two black holes are separated by a say 10^6 to 10^3km there are then departures from Newtonian mechanics. The post Newtonian approach expands the metric in orders of 1/c^2, where the first order is flat space, the next order recovers the perihelion shift of Mercury, and the next reproduces the linear wave equation. There are also higher order terms from there.

    The paper is by Clifford Will, who is one of the leading thinkers on observational tests of general relativity.

    LC

  3. lars says

    Whatever LISA finds is varying in its interferometer will be interesting, however there are more than numerous folks predicting that nothing will be found to be varying. Nevertheless it’s a good experiment to see what’s happening, but can we accept the results ?

  4. Peter says

    I am unqualified to do anything but ask questions here but if gravity is simply a result of the fabric of space-time being bent then what is and why would we require gravitons? I can see gravitational waves coming from binary systems because the space in bent intermittantly and the waves would naturally expand. However, the picture shows a galaxy and would that not approximate a static spherical object like a star because of its large scale symmetry? How close is the nearest binary system fast and heavy enough to create detectable waves? You’re not going to find them at all coming from the Centauri system.

    • Torbjorn Larsson OM says

      You don’t need quanta to understand all of physics, of course.

      The reason to expect gravitons here is because of what I mentioned in a comment above, you seem to always get them out of effective field theories if you try. And then you will also find them for real, again it seems always.

      And GR is an effective field theory, at least if you go by way of its Lagrangian as I linked above.

    • Member

      Hi Peter,

      Again the picture is more illustrative than accurate. You are right that an active galaxy with jets is not a strong candidate for producing detectable gravity waves (unless there was something going on within it that the picture does not illustrate).

      The closest compact binary I have been able to quickly google search is http://en.wikipedia.org/wiki/PSR_B1913%2B16 – (actually a star of bethlehem candidate apparently) but may be there are closer ones.

  5. Lawrence B. Crowell says

    As I indicated above on March 6, 2011 at 5:47 am, the graviton is difficult to detect. The graviton is a quantization of the gravity field, where that field is spacetime. Spacetime has a metric g_{ab}, where the indices a and b range from 0 to 3, where 0 stands for time. The submatrix g{ij} where I and j range from 1 to 3 pertains to the 3-dimensional space embedded in spacetime. This matrix element has a dual element called pi^{ij}, which is a momentum metric term. The metric g_{ij} and its dual pi^{ij} are related to each other the same way position and momentum are related to each other in mechanics. This analogue is also similar to the duality between the electric and magnetic fields. When quantized they are related by an uncertainty principle. So space, which dynamically evolves in spacetime, exhibits wave-like quantum behavior.

    I should be stressed that largely astrophysical gravity waves are classical. They are very weak, certainly by the time they reach us, and they fit entirely within a linearized form of general relativity. The source of gravity waves does not need to be symmetric, though that makes working these problems easier. The image indicates a galaxy with a supermassive black hole, where matter falling into it can generate gravity waves.

    LC

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