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Register a new userSparling two-forms, the conformal factor and the gravitational energy density of the teleparallel equivalent of general relativity
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J. W. Maluf
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It has been shown recently that within the framework of the teleparallel equivalent of general relativity (TEGR) it is possible to define the energy density of the gravitational field. The TEGR amounts to an alternative formulation of Einsteins general relativity, not to an alternative gravity theory. The localizability of the gravitational energy has been investigated in a number of space-times with distinct topologies, and the outcome of these analises agree with previously known results regarding the exact expression of the gravitational energy, and/or with the specific properties of the space-time manifold. In this article we establish a relationship between the expression for the gravitational energy density of the TEGR and the Sparling two-forms, which are known to be closely connected with the gravitational energy. We also show that our expression of energy yields the correct value of gravitational mass contained in the conformal factor of the metric field.
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This work generalizes the treatment of flat spin connections in the teleparallel equivalent of general relativity. It is shown that a general flat spin connection form a subspace in the affine space of spin connections which is dynamically decoupled from the tetrad and the matter fields. A translation in the affine subspace introduces a torsion term without changing the tetrad. Instead, the change in the torsion is related to the introduction of a global acceleration field term that introduces Lorentz inertial effects in the reference frame. The dynamics of the gravitationally coupled matter fields remains however equivalent regardless of the flat spin connection chosen. The implications of the break of this invariance by a general $f(T)$ and $f(R)$ is discussed.
Spaniol and Andrade introduced grvitoelectromagnetism in TEGR by considering superpotentials, times the determinant of tetrads, as the gravitoelectromagnetic fields. However, since this defined gravitoelectromagnetic field strength does not give rise to a complete set of Maxwell-like equations, we propose an alternative definition of the gravitoelectromagnetic field strength: instead of superpotentials, torsions are taken as the gravitoelectromagnetic field strengths. Based on this new proposal we are able to derive a complete set of Maxwell-like equations. We then apply them to obtain explicit expressions of the gravitoelectromagnetic fields both in Schwarzchilds spacetime and for gravitational waves.
In cite{Bahamonde:2019zea}, a spherically symmetric black hole (BH) was derived from the quadratic form of $f(T)$. Here we derive the associated energy, invariants of curvature, and torsion of this BH and demonstrate that the higher-order contribution of torsion renders the singularity weaker compared with the Schwarzschild BH of general relativity (GR). Moreover, we calculate the thermodynamic quantities and reveal the effect of the higher--order contribution on these quantities. Therefore, we derive a new spherically symmetric BH from the cubic form of $f(T)=T+epsilonBig[frac{1}{2}alpha T^2+frac{1}{3}beta T^3Big]$, where $epsilon<<1$, $alpha$, and $beta$ are constants. The new BH is characterized by the two constants $alpha$ and $beta$ in addition to $epsilon$. At $epsilon=0$ we return to GR. We study the physics of these new BH solutions via the same procedure that was applied for the quadratic BH. Moreover, we demonstrate that the contribution of the higher-order torsion, $frac{1}{2}alpha T^2+frac{1}{3}beta T^3$, may afford an interesting physics.
Recently, Choquet-Bruhat and York and Abrahams, Anderson, Choquet-Bruhat, and York (AACY) have cast the 3+1 evolution equations of general relativity in gauge-covariant and causal ``first-order symmetric hyperbolic form, thereby cleanly separating physical from gauge degrees of freedom in the Cauchy problem for general relativity. A key ingredient in their construction is a certain wave equation which governs the light-speed propagation of the extrinsic curvature tensor. Along a similar line, we construct a related wave equation which, as the key equation in a system, describes vacuum general relativity. Whereas the approach of AACY is based on tensor-index methods, the present formulation is written solely in the language of differential forms. Our approach starts with Sparlings tetrad-dependent differential forms, and our wave equation governs the propagation of Sparlings 2-form, which in the ``time-gauge is built linearly from the ``extrinsic curvature 1-form. The tensor-index version of our wave equation describes the propagation of (what is essentially) the Arnowitt-Deser-Misner gravitational momentum.
Gravitational wave observations of compact binary coalescences provide precision probes of strong-field gravity. There is thus now a standard set of null tests of general relativity (GR) applied to LIGO-Virgo detections and many more such tests proposed. However, the relation between all these tests is not yet well understood. We start to investigate this by applying a set of standard tests to simulated observations of binary black holes in GR and with phenomenological deviations from GR. The phenomenological deviations include self-consistent modifications to the energy flux in an effective-one-body (EOB) model, the deviations used in the second post-Newtonian (2PN) TIGER and FTA parameterized tests, and the dispersive propagation due to a massive graviton. We consider four types of tests: residuals, inspiral-merger-ringdown consistency, parameterized (TIGER and FTA), and modified dispersion relation. We also check the consistency of the unmodeled reconstruction of the waveforms with the waveform recovered using GR templates. These tests are applied to simulated observations similar to GW150914 with both large and small deviations from GR and similar to GW170608 just with small deviations from GR. We find that while very large deviations from GR are picked up with high significance by almost all tests, more moderate deviations are picked up by only a few tests, and some deviations are not recognized as GR violations by any test at the moderate signal-to-noise ratios we consider. Moreover, the tests that identify various deviations with high significance are not necessarily the expected ones. We also find that the 2PN (1PN) TIGER and FTA tests recover much smaller deviations than the true values in the modified EOB (massive graviton) case. Additionally, we find that of the GR deviations we consider, the residuals test is only able to detect extreme deviations from GR. (Abridged)
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