Gravitational waves

Intuitively, a wave in a field theory is a disturbance of the field that moves through the space. Although this idea seems to be rather simple its implementation in the case of general relativity turned out to be very difficult - only more than 40 years after the birth of the theory relativists managed to formulate a proper definition of gravitational waves and to show that the waves carry energy and thereby are supposed to be detectable.

There were three main reasons because of which it was not easy to formulate the theory of gravitational waves: general covariance of general relativity, non-linearity of the Einstein equations and a view held by Einstein himself that the waves do not exist.

General covariance allows one to use all coordinate frames on a spacetime on an equal footing to express a spacetime metric $g$ which describe the gravitational field. It turns out that this freedom in expressing the metric can be used to produce apparent wave-like behavior of the gravitational field by means of a coordinate transformation: suppose that the components of the metric $g$ expressed in a coordinate frame $(t,x,y,z)$ do not reveal any such behavior. If $t$ is a time-like coordinate and $x$ a space-like one then one can introduce a new time-like coordinate $t'$ which differs from $t$ by a ``small'' disturbance proportional to $\sin(t'-x)$. Then components of the metric given by the frame $(t',x,y,z)$ contain terms (originating in that disturbance of the coordinate $t'$) which look like ``a disturbance of the field that moves through the space''. Thus the general covariance makes rather fruitless attempts to define gravitational waves by means of a particular behavior of the metric expressed in a coordinate frame and forced relativists to formulate a coordinate independent definition of gravitational waves.

Any gravitational wave must be a solution to the Einstein equations, which form a complicated system of non-linear partial differential equations (imposed on components of the spacetime metric). Obviously, these features of the equations make it even more hard to find a "good" definition of the waves.

Regarding the Einstein's view on nonexistence of gravitational waves, right after he formulated general relativity he found that the linearized version of his equations in vacuum are just standard wave equations imposed on a disturbance of the Minkowski metric. However, twenty years later while trying to find plane wave solutions of the full non-linear equations he encountered some obstacles i.e. he found some solutions of desired sort, but they appeared to be singular and thus unphysical and this made him to believe that his equations do not admit plane wave solutions at all. This belief had quite an impact on many other relativists of his time and discouraged them from searching for a theory of gravitational waves.

The problem of plane gravitation waves was solved finally in the end of the fifties in the last century by H. Bondi, F. Pirani and I. Robinson: they defined such a wave as a solution of the vacuum Einstein equations which possesses a five-dimensional group of isometries (since the dimension of the symmetry group of an electromagnetic plane wave is five) and showed that there exist metrics which satisfy their definition. Moreover, they proved that their plane waves carry energy and therefore, in principle, can be detected.

However, plane waves are unphysical in this sense that it is hard to expect that there exist in nature sources which would produce such waves. On the other hand non-linearity of the Einstein equations does not allow to superpose plane waves in order to get a wave of an arbitrary front. Therefore the theoretical proof by Bondi, Pirani and Robinson of the existence of plane gravitational waves does not guarantee that general relativity admits waves of closed fronts which can be produced by sources existing in nature. 

Thus some work had to be done to handle the problem of gravitational waves with non-planar fronts. This work has been done by A. Trautman and I. Robinson.

First Trautman formulated a definition of general (i.e. non-plane) gravitational waves. To this end he assumed that a gravitational wave should be a solution of the Einstein equations distinguished by a special asymptotic behavior and a positive amount of energy radiated to infinity. Regarding the special asymptotic behavior, he generalized Sommerfeld's radiation conditions and imposed them on the gravitational field as boundary conditions at infinity. Next, he introduced a formula defining a four-momentum of gravitational field at a given moment (i.e. on a fixed space-like hypersurface) and showed that if the gravitational field satisfies his boundary conditions then the energy of the field radiated to infinity is non-negative.

This last result was a bit too weak to claim that general relativity admits non-plane gravitational waves since for the radiation to be present the energy escaping to infinity must be positive. This problem was solved by Trautman and Robinson: they found a class of solutions to the Einstein equations which satisfy Trautman's boundary conditions and for which the energy radiated to infinity is positive, which means that the solutions satisfy Trautman's definition of gravitational waves. Moreover, the fronts of the waves found by Trautman and Robinson are closed which allows to treat the waves as ones emitted by bounded sources.

The article based on a paper by C. Denson Hill and Pawel Nurowski

 "How the green light was given for gravitational wave search" 


Contribution to Yang-Mills theory

Regardless a subject of his scientific studies, Trautman always tried to present it clearly in terms of modern mathematics. About 1970, physicists and mathematicians started to realize that, at the classical level, foundations of the Yang-Mills' theory of elementary interactions corresponded to the theory of fibre bundles in differential geometry. Trautman played a great role in propagation of this relation. The article Fibre bundles associated with space-time (Rep. Math. Phys. 1 (1970) 29) was his first of many papers and talks on application of the fibre bundles in the gauge theories. He included in them the Higgs mechanism and description of spacetime symmetries of Yang-Mills' fields. He also advocated for the fibre bundle approach to the theory of gravitation. In this case, an appropriate bundle is the bundle of linear frames which admits additional structure with respect to the Yang-Mills theories. Trautman paid a special attention to the Einstein-Cartan's theory of gravity which admits a nonvanishing torsion related to the spin of matter. This theory allows to avoid the cosmological singularity of Big Bang (Spin and torsion may avert gravitational singularities, Nature 242 (1973) 7).

From the didactic point of view, the best summary of Trautman's geometrical approach to the Yang-Mills fields and gravity is given in a little book Differential Geometry for Physicists edited by Bibliopolis in 1976. It is based on lectures given by Trautman in the State University of New York at Stony Brook, where he was invited by C. N. Yang. This book is still recommended to students and researches interested in an elegant geometrical description of the classical field theories.

On top of the general description, Trautman presented a number of more technical results in the Yang-Mills theory. For example, he was able to interpret known electromagnetic and gauge fields as connections on Hopf bundles (Solutions of the Maxwell and Yang-Mills equations associated with Hopf fibrings, Intern. J. Theor. Phys. 16 (1977) 561) and to construct solutions of the Yang-Mills equations by means of natural connections on Stiefel bundles (Natural connections on Stiefel bundles are source- less gauge fields, with J. Nowakowski, J. Math. Phys. 19 (1978) 1100). 

He also used his and Robinson's approach to the gravitational radiation in order to solve the Yang-Mills equations in curved spacetime (A class of null solutions to Yang-Mills equations, J. Phys. A: Math. Gen. 13 (1980) L1). Following his experience in description of physical quantities in gravity, Trautman proposed to define a total non-Abelian charge of a gauge field configuration His paper Can poles change color? (with J. Tafel, J. Math. Phys. 24 (1983) 1087) stimulated other people to study this issue. The article attracted attention of the international community of physicists also because of the political meaning of its title in time of the martial law in Poland.

Works in the field of spinors

Trautman interest in the effects of the spin of matter in gravity and gauge theories led him naturally to study the geometry of spinors, the carriers of spin. On the algebraic level he elegantly clarified and found new equalities for the intertwiners of various representations of Clifford algebras of complex or real vector spaces of arbitrary dimension equipped with quadratic forms of arbitrary index, and the corresponding bilinear or hermitian forms on spinors (with P. Budinich, The Spinorial Chessboard Springer-Verlag 1988). He also studied the link between totally isotropic (null) vector subspaces and pure spinors, with applications to the optical geometry. On the geometric level Trautman investigated spinors on hypersurfaces and on nonorientable manifolds. In particular he demonstrated that the product of two manifolds has a pin structure if, and only if, both are pin and at least one of them is orientable and studied the suitably modified Dirac operator (with M. Cahen and S. Gutt, J. Geom. Phys. 17 (1995) 283-297).