Boris Nikolaevich Frolov – Professor of the Faculty of Physics of the Moscow State Pedagogical University, Doctor of Physical and Mathematical Sciences (1999).

There is an increased interest in the study of plane waves in the gravity theory in connection with the recent discovery of gravity waves predicted in the General Relativity (GR).

**What are plane waves in field and gravity physical theory?**

Plane waves are the simple object in field theory (similar to a material point in mechanics), which allows us to study the features of a specific field physical theory in a fairly simple way.

**What do you understand by affine-metric space in your research?**

There are reasons at present to consider a more complex space-time structure, containing additional geometric characteristics, such as torsion and non-metricity. Space-time in terms of its geometric properties no longer remains the Riemann space of the General Relativity, but it becomes the general affine-metric space in this case.

The appropriate gravity field equations, which generalize Einstein’s equations, show that torsion and non-metricity can propagate in the form of waves, in particular, in the form of plane waves at a great distance from the sources of these waves also.

We have defined an affine-metric space of the plane wave type as a space in which the metric, torsion, and non-metricity are equal to zero under the action of the Lie derivative concerning the vector generating the five-parameter invariance group of plane electromagnetic waves in the Minkowski space.

An affine-metric space is endowed with non-metricity also in addition to curvature and torsion. A special case of a general affine-metric space is the Cartan-Weyl space, in which the non-metricity is restricted by the Weyl condition. The Cartan–Weyl space is used, for example, in the Weyl–Cartan theory of gravity.

**What are the prerequisites for further complication of the space-time geometric properties?**

There are quite convincing arguments in favor of complicating the space-time geometric properties for the presence of non-metricity also along with torsion. Several arguments can be pointed out in favor of the existence of non-metricity as a physical entity and not just as a mathematical concept. Thus, a scalar field is used in some modern studies on cosmological inflation in the framework of the Jordan-Brans-Dicke theory, this scalar field, when interpreted geometrically, turns out to be a scalar field introduced by Weyl to ensure the theory scale invariance.

**The wave solutions properties both in the General Relativity (GR) and in modified gravity theories have been studied by many authors. What is your contribution to this field?**

We found that this type of wave can carry information also, like the recently discovered curvature waves, since their description contains retarded time arbitrary functions, which can be encoded in the source of these waves (by analogy with electromagnetic waves).

This condition is connected with a possible perspective of our research, which, however, can be realized only if non-metricity is discovered as a physical phenomenon and not only as a mathematical generalization of A. Einstein’s theory of relativity.

An important question in the plane waves theory is whether plane waves can transmit information from a field source to other spatial domains. A. Trautman formulated this problem, he explained that this is possible only if the plane wave mathematical description contains arbitrary functions that can be encoded in the wave source and then propagate in space unchanged, thereby transferring the encoded information to a given point in space-time. So, for example, the operation control of the electric dipole that generates electromagnetic waves makes it possible to determine arbitrarily the propagating wave amplitude and frequency in the case of an electromagnetic field.

It is known that plane electromagnetic waves have an invariance symmetry for the five-parameter motions’ group of space-time. Curvilinear coordinates and a basis formed by four vectors are introduced in space-time, two of which are isotropic and two are space-like. The vector is covariantly constant and directed along the wave ray. The coordinate has the meaning of retarded time and is interpreted as the wave phase. The coordinates parametrize the wave surface. The group leaves unchanged the isotropic hypersurface describing the front of a plane wave with a constant amplitude.

**How does the structure of non-metric plane waves allow them to be used to transmit information?**

The plane gravitational torsion waves structure was studied In the case of the Riemann–Cartan space and it was shown that plane torsion waves in such a space are determined by four arbitrary functions, two of which determine, the torsion trace and pseudo trance respectively and the remaining two determine the traceless part, irreducible for the Lorentz group torsion.

We have found that nonmetric plane waves are determined by five arbitrary functions.

The 1-form of nonmetricity can be represented as the sum of four irreducible parts that are invariant under Lorentz transformations. We use an equivalent representation of this decomposition. The first and second irreducible parts each have 16 components, and the third and fourth parts each have 4 components in four-dimensional Minkowski space. The 1-form partition into irreducible parts is implemented by the following 1-forms:

• Weyl 1-form

• trace 1- form

• 1-form of spin 2

• 1-form of spin 3

The presence of arbitrary functions that determine the propagation of nonmetric plane waves allows them to be arbitrarily encoded in these waves’ source. Thus, it becomes possible to transmit information using nonmetric waves.

What research are you planning to do in the future?

The main idea of this direction of work in studying the plane waves structure in modified gravity theories is to apply an analogy between the properties of plane electromagnetic waves and metrics, torsion, and non-metricity plane waves.

Continuing these works, we have determined the structure of nonmetricity plane waves in the general affine-metric space and its particular case, the Cartan-Weyl space.

We plan for further research to obtain the exact form of the generalized Einstein equations, the solution of which will be the nonmetricity plane waves studied by us.

**Interview: Ivan Stepanyan**