# The Wave-Front Equation of Gravitational Signals in Classical General Relativity

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## Abstract

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## 1. Introduction

- As consistency tests for the Einstein field equations specifically in reference with the determination of its appropriate Hamiltonian variational representation. This is true especially in view of the experimental evidence of gravitational waves reached after intensive and long-time efforts by experimental astrophysics [6]. The matter concerns in particular highly non-linear phenomena of GR that could be explored by studies of gravitational waves produced in compact binaries mergers [7,8,9,10].
- As theoretical frameworks for the interpretation of observational features characterizing gravitational waves, with particular reference to the constraints that can be placed on their speed of propagation [13], in turn permitting the implementation of additional confirmations to SF-GR or restriction of the validity of alternative gravity theories [14,15,16,17,18].

- Its Hamiltonian density and canonical variables are respectively identified with a 4-scalar and 4-tensor with respect to the group of local point transformations which preserve the differential-manifold structure of $\left\{{\mathbf{Q}}^{4},\widehat{g}\right\}$;
- $g\equiv \left\{{g}_{\mu \nu}\right\}=\left\{{g}_{\nu \mu}\right\}$ is a symmetric second-order tensor, generally different from the background metric tensor $\widehat{g}\equiv \left\{{\widehat{g}}_{\mu \nu}\right\}\equiv \left\{{\widehat{g}}_{\nu \mu}\right\}$, while$$\pi \equiv \left\{{\Pi}_{\mu \nu}^{\alpha}\right\}\equiv \phantom{\rule{4pt}{0ex}}\left\{{\Pi}_{v\mu}^{\alpha}\right\}$$
- It is frame-independent; i.e., it holds for arbitrary coordinate systems, and therefore does not require a preliminary foliation of the space-time $\left\{{\mathbf{Q}}^{4},\widehat{g}\right\}$;
- It is constraint-free, in the sense that the continuum canonical variables are treated as independent.

## 2. Lagrangian Formalism

- A smooth real function of the generalized coordinate-field $g\equiv \left\{{g}_{\mu \nu}\right\}$ and its covariant derivative $\widehat{\nabla}g\equiv \left\{{\widehat{\nabla}}^{\alpha}{g}_{\mu \nu}\right\}$, with ${\widehat{\nabla}}^{\alpha}$ denoting the covariant derivative operator in which the Christoffel symbols are associated to the background field $\widehat{g}$;
- Coordinate-independent, in the sense that it does not explicitly depend on the 4-position $r\equiv \left\{{r}^{\mu}\right\}$ (however, the implicit dependence through g and $\widehat{\nabla}g$ still remains).

## 3. Hamilton and Hamilton–Jacobi Representations

## 4. Induced Hamilton–Jacobi Equation on $\left\{\widehat{\mathit{g}}\right\}$

## 5. Reduced Hamilton–Jacobi Theory on $\left\{\widehat{\mathit{g}}\right\}$

## 6. The Klein–Gordon Gravitational Wave-Front Equation

**Theorem**

**1 (Klein–Gordon gravitational wave-front equation).**

**Proof.**

## 7. Radiation-Field Solutions

- In both cases the wave-front propagates along the corresponding null geodetics, so that its speed of propagation coincides with the speed of light c.
- The non-local contribution to the wave-front solution of the Klein–Gordon equation in curved space-time can be absorbed into an analogous non-local contribution defining the teleparallel transformation to flat space-time, where the wave solution has a local character.

## 8. Concluding Remarks

- First, an exact GR-non-linear gravitational wave-front equation has been determined for the gravitational field, which has been identified with the 4-scalar Klein–Gordon equation for the wave-front surface.
- Second, at the same time, the mathematical proof that for arbitrary gravitational signals these wave-fronts propagate—both in vacuum (i.e., in the absence of sources) and in an arbitrary curved space-time—at the speed of light, has been reached.
- Third, the theory developed here holds also in the case of a non-vanishing cosmological constant, to be considered as a classical universal constant 4-scalar.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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Cremaschini, C.; Tessarotto, M.
The Wave-Front Equation of Gravitational Signals in Classical General Relativity. *Symmetry* **2020**, *12*, 216.
https://doi.org/10.3390/sym12020216

**AMA Style**

Cremaschini C, Tessarotto M.
The Wave-Front Equation of Gravitational Signals in Classical General Relativity. *Symmetry*. 2020; 12(2):216.
https://doi.org/10.3390/sym12020216

**Chicago/Turabian Style**

Cremaschini, Claudio, and Massimo Tessarotto.
2020. "The Wave-Front Equation of Gravitational Signals in Classical General Relativity" *Symmetry* 12, no. 2: 216.
https://doi.org/10.3390/sym12020216