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Gravity is one of the four fundamental forces of nature and, despite being the first one to be described successfully by a mathematical model (Newton’s theory), it remains arguably the least understood one: t¬¬he problems of dark matter, dark energy and quantum gravity are still vexing, and many researchers pursue possib¬¬le modifications of our current best understanding of gravity, which is Einstein’s theory of spacetime, General Relativity (GR).

But great progress is being made. One of the most interesting recent developments is first direct detection of gravitational waves, announced by LIGO in spring 2016. Until then, basically all of our knowledge about the distant universe has been derived from light, that is, electromagnetic radiation; now the discovery of gravitational radiation will open up an entirely new, gravitational, sky, and this will also enable unprecedented tests of our theory of gravity. For instance, GR predicts that gravitational radiation and electromagnetic radiation have the same speed; following its confirmation by recent observations, several proposed modifications of GR can now be ruled out. Japan is also going to contribute to this effort soon, as the gravitational wave detector KAGRA is being completed at Kamioka in Gifu (Fig. 1) and is expected to join the third observation run, together with LIGO and the European counterpart, Virgo, in 2019. Thus, it is an excellent time to investigate possible modifications of GR, and ways to test them observationally.

From the theoretical point of view, modified gravity theories are usually introduced by stipulating the dynamics of spacetime from the outset. In the case of GR, this is called the Einstein-Hilbert action, and one can generate new theories of gravity by adding extra geometrical fields to it, and study how spacetimes would behave in consequence; we might call such an approach `top-down’. Now while this top-down technique allows researchers to quickly generate modified theories, it suffers from the problem that spacetime behaviour is implicit in the stipulated action and may not have some indispensable physical properties.

One such indispensable property is predictivity: indeed, one might argue that a hallmark of any physical theory should be its ability to predict future states from the present state. In mathematical terms, this means the theory should be hyperbolic, that is, have a well-posed Cauchy initial value problem. We can determine whether this is the case by considering the so-called hyperbolicity cones of the test matter probing the spacetime geometry. In the case of a metric geometry, these are the famous light cones, and the standard theory does indeed have the desired hyperbolicity property (Fig. 2). But once we consider modified theories with a more complicated, possibly non-metric, spacetime geometry, it turns out that the mathematical criterion for predictivity needs to be refined, and this is called bihyperbolicity.

But there is more. So far, I have described predictivity only as a local spacetime property, which may be called the kinematics of spacetime. However, we are particularly interested in its dynamics, its global behaviour – that is, the gravity theory. Now, astonishingly, it turns out that spacetime dynamics can actually be derived from the underlying spacetime kinematics, assuming merely that the theory be predictive, or bihyperbolic. Building on earlier work in geometrodynamics, my colleague Frederic P. Schuller and his group at Erlangen University, Germany, have developed this technique since 2012. In this `bottom-up’ approach, the resulting theory is guaranteed to be predictive, by construction – hence, we call it constructive gravity.

Thus, we are now in a position not only to derive GR in metric spacetime geometry (which Einstein had guessed, correctly), but derive gravity theories for non-metric spacetime geometries, too. Unsurprisingly, those can exhibit qualitatively new effects, for example birefringent propagation of radiation, which are well constrained by observations and hence provide strong tests. Last year, Dr Schuller and I worked out the first concrete astrophysical consequence of such a theory, the first derived, predictive gravity theory beyond Einstein’s GR. This July, I organized a session on constructive gravity at the Marcel Grossmann Meeting in Rome, one of the field’s major international conferences. Anticipating more results from constructive gravity, exciting times lie ahead.

View along one tunnel of KAGRA, the Japanese gravitational wave observatory, in Gifu Prefecture [MCW].

Three snapshots of the time evolution of space in geometrodynamics, with light cones governing predictivity [FPS].

（まーくす Ｃ わーなー）

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