The gravitational energy-momentum pseudo-tensor of Higher-Order Theories of Gravity

Abstract

We derive the gravitational energy momentum tensor ${\tau }_{\alpha }^{\eta }$ for a general Lagrangian of any order $L=L\left(g_{\mu \nu },g_{\mu \nu ,i_1},g_{\mu \nu ,i_1{i}_2},g_{\mu \nu ,i_1{i}_2{i}_3},\cdots ,g_{\mu \nu ,i_1{i}_2{i}_3\cdots i_n}\right)$ and in particular for a Lagrangian such as $L_g=(\overline{R}+a_0{R}^2+{\sum }_{k=1}^p{a}_{k}R{\square }^{k}R)\sqrt{-g}$. We prove that this tensor, in general, is not covariant but only affine, then it is a pseudo-tensor. Furthermore, the pseudo-tensor ${\tau }_{\alpha }^{\eta }$ is calculated in the weak field limit up to a first non-vanishing term of order $h^2$ where $h$ is the metric perturbation. The average value of the pseudo-tensor over a suitable spacetime domain is obtained. Finally we calculate the power per unit solid angle $\Omega$ carried by a gravitational wave in a direction $\hat{x}$ for a fixed wave number $\mathbf{k}$ under a suitable gauge. These results are useful in view of searching for further modes of gravitational radiation beyond the standard two modes of General Relativity and to deal with nonlocal theories of gravity where terms involving $\square R$ are present. The general aim of the approach is to deal with theories of any order under the same standard of Landau pseudo-tensor.

arXiv comment23 pages, to be published in Annalen der Physik

References in corpus (27)

• Observation of Gravitational Waves from a Binary Black Hole Merger
• Extended Theories of Gravity
• f(T) teleparallel gravity and cosmology
• A Generalized Uncertainty Principle in Quantum Gravity
• Extended Theories of Gravity and their Cosmological and Astrophysical Applications
• Universality of Quantum Gravity Corrections
• Probing Planck-scale physics with quantum optics
• Nonlocal Cosmology
• The Generalized Uncertainty Principle in (A)dS Space and the Modification of Hawking Temperature from the Minimal Length
• Black holes in an ultraviolet complete quantum gravity
• Superrenormalizable quantum gravity with complex ghosts
• Super-renormalizable or Finite Lee-Wick Quantum Gravity
• Reconstructing the Distortion Function for Nonlocal Cosmology
• Massive, massless and ghost modes of gravitational waves from higher-order gravity
• Universally Finite Gravitational & Gauge Theories
• Observational Viability and Stability of Nonlocal Cosmology
• Testing f(R)-theories using the first time derivative of the orbital period of the binary pulsars
• The Newtonian limit of fourth and higher order gravity
• Quadrupolar gravitational radiation as a test-bed for f(R)-gravity
• Nonlocality in string theory
• The Most General Form of Deformation of the Heisenberg Algebra from the Generalized Uncertainty Principle
• Generalized Uncertainty Principle as a Consequence of the Effective Field Theory
• Linearized Gravitational Waves in Nonlocal General Relativity
• Determining Cosmology for a Nonlocal Realization of MOND
• Nonlinear gravitational waves and their polarization
• Modelling Space with an Atom of Quantum Geometry
• The post-Minkowskian limit of f(R)-gravity

Cited by in corpus (21)

• Definition of Complexity Factor for Self-Gravitating Systems in Palatini $f(R)$ Gravity
• Weak Field Limit and Gravitational Waves in $f(T,B)$ Teleparallel Gravity
• Constraining Teleparallel Gravity through Gaussian Processes
• The gravitational energy-momentum pseudotensor: the cases of $f(R)$ and $f(T)$ gravity
• Analytic infinite derivative gravity, $R^2$-like inflation, quantum gravity and CMB
• Emergent Universe from Energy-Momentum Squared Gravity
• Gravitational Waves in Higher Order Teleparallel Gravity
• Quasinormal modes as a distinguisher between general relativity and f(R) gravity: Charged black-holes
• Weak field limit and gravitational waves in higher-order gravity
• Gravitational tensor and scalar modes in $f(Q,B)$ non-metric gravity
• Gravitational energy-momentum pseudo-tensor in Palatini and metric $f(R)$ gravity
• The energy-momentum complex in non-local gravity
• Energy-Momentum Complex in Higher Order Curvature-Based Local Gravity
• Cosmological perturbations in gravitational energy-momentum complex
• Monopolar and quadrupolar gravitational radiation from magnetically deformed neutron stars in modified gravity
• Gravitational waves in massive conformal gravity
• Magnetic black holes in Weitzenböck geometry
• Perturbations of the Gravitational Energy in the TEGR: Quasinormal Modes of the Schwarzschild Black Hole
• Localization of Energy and Momentum in an Asymptotically Reissner-Nordström Non-singular Black Hole Space-time Geometry
• On the occurrence of gauge-dependent secularities in nonlinear gravitational waves
• The gravitational energy-momentum pseudo-tensor in $f(Q)$ non-metric gravity