We present and analyze a new non-perturbative radiative solution of Horndeski

gravity. This exact solution is constructed by a disformal mapping of a seed

solution of the shift-symmetric Einstein-Scalar system belonging to the

Robinson-Trautman geometry describing the gravitational radiation emitted by a

time-dependent scalar monopole. After analyzing in detail the properties of the

seed, we show that while the general relativity solution allows for a

shear-free parallel transported (PT) null frame, the disformed solution can

only admit parallel transported null frames with a non-vanishing shear. This

result shows that, at the nonlinear level, the scalar-tensor mixing descending

from the higher-order terms in Horndeski dynamics can generate shear out of a

pure scalar monopole. We further confirm this analysis by identifying the

spin-0 and spin-2 polarizations in the disformed solution using the Penrose

limit of our radiative solution. Finally, we compute the geodesic motion and

the memory effects experienced by two null test particles with vanishing

initial relative velocity after the passage of the pulse. This exact radiative

solution offers a simple framework to witness nonlinear consequences of the

scalar-tensor mixing in higher-order scalar-tensor theories.

According to the article, a new non-perturbative radiative solution of Horndeski gravity has been presented and analyzed. This solution is constructed by a disformal mapping of a seed solution belonging to the Robinson-Trautman geometry describing gravitational radiation emitted by a time-dependent scalar monopole.

The analysis of the seed solution reveals that while the general relativity solution allows for a shear-free parallel transported null frame, the disformed solution can only admit parallel transported null frames with a non-vanishing shear. This indicates that the scalar-tensor mixing in Horndeski dynamics can generate shear out of a pure scalar monopole at the nonlinear level.

The analysis is further confirmed by identifying the spin-0 and spin-2 polarizations in the disformed solution using the Penrose limit of the radiative solution. This provides evidence for the nonlinear consequences of scalar-tensor mixing in higher-order scalar-tensor theories.

As for the roadmap for readers, they can expect further exploration and research in the following areas:

- Investigation of other potential solutions and mappings in Horndeski gravity
- Exploration of the implications and effects of scalar-tensor mixing in higher-order scalar-tensor theories
- Application of the exact radiative solution to real-world scenarios and phenomena
- Study of the geodesic motion and memory effects experienced by null test particles after the passage of the pulse
- Development of frameworks and models to better understand and analyze the dynamics of scalar-tensor systems

Challenges and opportunities on the horizon include:

- Complexity of higher-order scalar-tensor theories and the need for new mathematical tools and techniques to deal with nonlinear dynamics
- Integration of the radiative solution into existing cosmological and astrophysical models
- Potential applications of the findings in gravitational wave astronomy and cosmology
- Collaboration and interdisciplinary efforts to further explore the implications of scalar-tensor mixing

In conclusion, the presented radiative solution in Horndeski gravity opens up avenues for studying the nonlinear consequences of scalar-tensor mixing. Further research and analysis in this field have the potential to deepen our understanding of gravity and its effects in various contexts.