Plane-fronted Waves Research Articles

Plane-fronted electromagnetic waves and an asymptotic limit of Liénard–Wiechert fields

Colliding or noncolliding plane-fronted electromagnetic or gravitational waves are the asymptotic limit of Robinson–Trautman spherical electromagnetic or gravitational waves. Noncolliding plane-fronted waves contain no information about their sources whereas colliding waves contain information about possibly the motion of their sources. As a first step to investigate the latter phenomenon, we construct an asymptotic limit of Liénard–Wiechert electromagnetic fields in the context of Minkowskian spacetime. This has the advantage that the source is well known and the calculations can be carried out in full detail. The final result is an algebraically general Maxwell field which consists of colliding plane-fronted waves in a subregion of Minkowskian spacetime and an interesting byproduct is a novel perspective on a Maxwell field originally discovered by Bateman.

Canonical almost-Kähler metrics dual to general plane-fronted wave Lorentzian metrics

In the compact setting, Aazami and Ream (Lett Math Phys 112(4):17, 2002) proved that Riemannian metrics dual to a class of Lorentzian metrics, called (compact) general plane-fronted waves, are almost-Kähler. In this note, we explain how to construct extremal and second-Chern–Einstein non-Kähler almost-Kähler metrics dual to those general plane-fronted waves.

Colliding plane fronted waves and a gravitoelectromagnetic searchlight

We present a formulation of Einstein--Maxwell vacuum fields due to plane fronted electromagnetic waves sharing their wave fronts with gravitational waves. This is based on a recent geometrical reconstruction of plane fronted wave fields by the authors which clearly identifies the cases in which the wave fronts collide or do not collide. In the former case our construction suggests an explicit example of a searchlight beam, accompanied by gravitational radiation, which sweeps across the sky. This gravito--electromagnetic searchlight and its properties are described in detail.

Simulating gravitational waves passing through the spacetime of a black hole

We investigate how GWs pass through the spacetime of a Schwarzschild black hole using time-domain numerical simulations. Our work is based on the perturbed $3+1$ Einstein's equations up to the linear order. We show explicitly that our perturbation equations are covariant under infinitesimal coordinate transformations. Then we solve a symmetric second-order hyperbolic wave equation with a spatially varying wave speed. As the wave speed in our wave equation vanishes at the horizon, our formalism can naturally avoid boundary conditions at the horizon. Our formalism also does not contain coordinate singularities and, therefore, does not need regularity conditions. Then, based on our code, we simulate both finite and continuous initially plane-fronted wave trains passing through the Schwarzschild black hole. We find that for the finite wave train, the wave zone of GWs is wildly twisted by the black hole. While for the continuous wave train, unlike geometric optics, GWs cannot be sheltered by the black hole. A strong beam and an interference pattern appear behind the black hole along the optical axis. Moreover, we find that the back-scattering due to the interaction between GWs and the background curvature is strongly dependent on the direction of the propagation of the trailing wavefront relative to the black hole. Finally, for a realistic input waveform generated by binary black holes, we find that the lensed waveform in the merger and ringdown phases is much longer than that of the input waveform due to the effect of back-scattering.

Rigidity results for complete spacelike submanifolds in plane fronted waves

New rigidity results for complete non-compact spacelike submanifolds of arbitrary codimension in plane fronted waves are obtained. Under appropriate assumptions, we prove that a complete spacelike submanifold in these spacetimes is contained in a characteristic lightlike hypersurface. Moreover, for a complete codimension two extremal submanifold in a plane fronted wave we show sufficient conditions to guarantee that it is a (totally geodesic) wavefront.

Memory effect of the pp waves with torsion

We analyse the motion of test particles in the spacetime of the plane-fronted (pp) waves with torsion in four-dimensions. We conclude that there is a velocity memory effect in the direction of advanced time and along radial direction, while we have rotation of particles in angular direction. The velocity memory effect in the aforementioned directions is severely affected by the value of the tordion mass and probably it is not observable. A very interesting, probably observable effect, steams from the rotation, which is insensitive to the tordion mass.

Plane fronted limit of spherical electromagnetic and gravitational waves

We demonstrate how plane fronted waves with colliding wave fronts are the asymptotic limit of spherical electromagnetic and gravitational waves. In the case of the electromagnetic waves we utilize Bateman's representation of radiative solutions of Maxwell's vacuum field equations. The gravitational case involves a novel form of the radiative Robinson--Trautman solutions of Einstein's vacuum field equations.

Mean curvature of spacelike submanifolds in a Brinkmann spacetime

Several geometric properties of complete spacelike submanifolds, with codimension at least two, in a Brinkmann spacetime are shown from natural assumptions involving the mean curvature vector field H of the spacelike submanifold. Especially, we get sufficient conditions that assure that a spacelike submanifold is contained in a leaf of the foliation of the Brinkmann spacetime defined by the orthogonal vectors to the parallel lightlike vector field. When this vector field is the gradient of a smooth function, a characterization of arbitrary codimension spacelike submanifolds contained in a leaf of this foliation is given. In the case of plane fronted wave spacetimes, relevant examples of Brinkmann spacetimes that generalize pp-waves spacetimes, several uniqueness results for codimension two spacelike submanifolds are obtained. In particular, it is proven that any compact codimension two spacelike submanifold with H = 0 in a plane fronted spacetime wave must be a (totally geodesic) front of wave.

Gravitational waves in metric-affine gravity theory

We derive the exact gravitational wave solutions in a general class of quadratic metric-affine gauge gravity models. The Lagrangian includes all possible linear and quadratic invariants constructed from the torsion, nonmetricity and the curvature. The ansatz for the gravitational wave configuration and the properties of the wave solutions are patterned following the corresponding ansatz and the properties of the plane-fronted electromagnetic wave.

Velocity memory effect without soft particles

We study the behavior of geodesics in the plane-fronted wave background of the three-dimensional gravity with propagating torsion, which possesses only massive degrees of freedom. We discover the velocity memory effect, in contrast to the current belief that its existence is due to the presence of soft particles.

Einstein-Maxwell fields with vanishing higher-order corrections

We obtain a full characterization of Einstein-Maxwell $p$-form solutions $(\boldsymbol{g},\boldsymbol{F})$ in $D$-dimensions for which all higher-order corrections vanish identically. These thus simultaneously solve a large class of Lagrangian theories including both modified gravities and (possibly non-minimally coupled) modified electrodynamics. Specifically, both $\boldsymbol{g}$ and $\boldsymbol{F}$ are fields with vanishing scalar invariants and further satisfy two simple tensorial conditions. They describe a family of gravitational and electromagnetic plane-fronted waves of the Kundt class and of Weyl type III (or more special). The local form of $(\boldsymbol{g},\boldsymbol{F})$ and a few examples are also provided.

Gravitational waves in Poincaré gauge gravity theory

We derive the exact gravitational wave solutions in a general class of quadratic Poincar\'e gauge gravity models. The Lagrangian includes all possible linear and quadratic invariants constructed from the torsion and the curvature, including the parity odd terms. The Ansatz for the gravitational wave configuration and the properties of the wave solutions are patterned following the corresponding Ansatz and the properties of the plane-fronted electromagnetic wave.

Rainich conditions in (2 + 1)-dimensional gravity

In (3 + 1) spacetime dimensions, the Rainich conditions are a set of equations expressed solely in terms of the metric tensor which are equivalent to the Einstein-Maxwell equations for non-null electromagnetic fields. Here we provide the analogous conditions for (2 + 1)-dimensional gravity coupled to electromagnetism. Both the non-null and null cases are treated. The construction of these conditions is based upon reducing the problem to that of gravity coupled to a scalar field, which we have treated elsewhere. These conditions can be easily extended to other theories of (2 + 1)-dimensional gravity. For example, we apply the geometrization conditions to topologically massive gravity coupled to the electromagnetic field and obtain a family of plane-fronted wave solutions.

Travelling waves in expanding spatially homogeneous space–times

Some classes of the so-called ‘travelling wave’ solutions of Einstein and Einstein–Maxwell equations in general relativity and of dynamical equations for massless bosonic fields in string gravity in four and higher dimensions are presented. Similarly to the well known plane-fronted waves with parallel rays (pp-waves), these travelling wave solutions may depend on arbitrary functions of a null coordinate which determine the arbitrary profiles and polarizations of the waves. However, in contrast with pp-waves, these waves do not admit the null Killing vector fields and can exist in some curved (expanding and spatially homogeneous) background space–times, where these waves propagate in certain directions without any scattering. Mathematically, some of these classes of solutions arise as the fixed points of Kramer–Neugebauer transformations for hyperbolic integrable reductions of the above mentioned field equations or, in other cases, after imposing the ansatz that these waves do not change the part of the spatial metric transverse to the direction of wave propagation. It is worth noting that the strikingly simple forms of all the solutions presented prospectively make possible the consideration of the nonlinear interaction of these waves with the background curvature and singularities, as well as the collision of such wave pulses with solitons or with each other in the backgrounds where such travelling waves may exist.

Monochromatic plane-fronted waves in conformal gravity are pure gauge

We consider plane-fronted, monochromatic gravitational waves on a Minkowski background, in a conformally invariant theory of general relativity. By this we mean waves of the form: $g_{\mu\nu}=\eta_{\mu\nu}+\epsilon_{\mu\nu}F(k\cdotx)$, where $\epsilon_{\mu\nu}$ is a constant polarization tensor, and $k_\mu$ is a lightlike vector. We also assume the coordinate gauge condition $|g|-1/4\partial_\tau(|g|1/4g^{\sigma\tau})=0$ which is the conformal analog of the harmonic gauge condition $g^{\mu\nu}\Gamma_{\mu\nu}^\sigma=-|g|-1/2\partial_\tau(|g|1/2g^{\sigma\tau})=0, where $\det[g_{\mu\nu}]\equivg$. Requiring additionally the conformal gauge condition $g=-1$ surprisingly implies that the waves are both transverse and traceless. Although the ansatz for the metric is eminently reasonable when considering perturbative gravitational waves, we show that the metric is reducible to the metric of Minkowski space-time via a sequence of coordinate transformations which respect the gauge conditions, without any perturbative approximation that \in{\mu}{\nu} be small. This implies that we have, in fact, exact plane-wave solutions; however, they are simply coordinate/conformal artifacts. As a consequence, they carry no energy. Our result does not imply that conformal gravity does not have gravitational wave phenomena. A different, more generalized ansatz for the deviation, taking into account the fourth-order nature of the field equation, which has the form $g_{\mu\nu}=\eta_{\mu\nu}+B_{\mu\nu}(n\cdotx)G(k\cdotx)$, indeed yields waves which carry energy and momentum [P.D. Mannheim, Gen. Relativ. Gravit. 43, 703 (2010)]. It is just surprising that transverse, traceless, plane-fronted gravitational waves, those that would be used in any standard, perturbative, quantum analysis of the theory, simply do not exist.

AdS waves as exact solutions to quadratic gravity

We give an exact solution of the quadratic gravity in D dimensions. The solution is a plane fronted wave metric with a cosmological constant. This metric solves not only the full quadratic gravity field equations but also the linearized ones which include the linearized equations of the recently found critical gravity. A subset of the solutions change the asymptotic structure of the anti-de Sitter space due to their logarithmic behavior.

Nonminimally coupled gravitational and electromagnetic fields:pp-wave solutions

We give the Lagrangian formulation of a generic non-minimally extended Einstein-Maxwell theory with an action that is linear in the curvature and quadratic in the electromagnetic field. We derive the coupled field equations by a first order variational principle using the method of Lagrange multipliers. We look for solutions describing plane-fronted Einstein-Maxwell waves with parallel rays. We give a family of exact solutions associated with a partially massless spin-2 photon and a partially massive spin-2 graviton.

Gravitational waves: just plane symmetry

In four spacetime dimensions gravitational plane waves (a special case of the plane-fronted waves with parallel rays) admit a 5 parameter isometry group. We generalize this group to n-dimensions and explore some special features of spacetimes admitting this isometry group. In particular, it is shown that every generally covariant rank-2 symmetric tensor field constructed from a metric with plane wave symmetry will vanish except multiples of the metric and Ricci tensors. We show that, in four spacetime dimensions, a particular enlargement of the plane wave symmetry group is enough to force the group-invariant metrics to satisfy all generally covariant vacuum equations. We comment on the construction of a “mini-superspace” description of spacetimes admitting plane wave symmetry.

Plane waves in metric-affine gravity

We describe plane-fronted waves in the Yang-Mills type quadratic metric-affine theory of gravity. The torsion and the nonmetricity are both nontrivial, and they do not belong to the triplet ansatz.

Kerr–Schild metric in topological massive (2+1) gravity

In the framework of the (2+1)-dimensional topologically massive theory of gravity put forward by Deser, Jackiw, and Templeton, we analyze the (2+1) Kerr–Schild metric, in order to search for exact solutions, and to obtain plane-fronted wave solutions, which in the standard (2+1) Einstein theory (whose dynamics is provided by a cosmological constant term) do not exist.