On the relationship between charge and mass in General Relativity
Vineet Kumar
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This is a quick draft submitted to arxiv for preliminary comments and feedback, which would be much appreciated. More writing and figures are intended to be added
This paper demonstrates a relationship between mass and charge through explicit construction of exact Einstein-Maxwell spacetimes by embedding the Schwarzschild and Kerr instantons in 5 dimensions. It is shown further how by varying only the mass of an object, its electric field or apparent "charge" changes, while making it possible for two different mass values to yield the same electric field strength. The 5-dimensional solution, however, reduces to the familiar Coulomb limit $1/r^2$ for large distances.
A general prescription for constructing quasi-local conserved quantities in General Relativity is proposed. The construction is applied to BMS symmetry generators in Newman-Unti gauge, so as to define quasi-local BMS charges. It is argued that the zero mode of this BMS charge is a promising definition of quasi-local energy.
We extend the Non-Relativistic formulation of General Relativity (NRGR) given in Goldberger et al. (Phys Rev D 73:104029, 2006) to incorporate the effects of electromagnetic charge of the constituents of the binary. We incorporate the photon field in NRGR by giving the field decomposition and power counting rules. Using these, we develop the Feynman rules to describe photon and graviton interactions with point particle worldline. We then find the corrections to the Einstein-Infeld-Hoffmann Lagrangian (Einstein et al. in Ann Math 39:65-100, 1938) due to the presence of the photon field and elec…
In this paper, we make a close comparison of a covariant definition of an energy/entropy in general relativity, recently proposed by a collaboration including the present authors, with existing definitions of energies such as the one from the pseudo-tensor and the quasi-local energy. We show that existing definitions of energies in general relativity are conserved charges from the Noether's 2nd theorem for the general coordinate transformation, whose conservations are merely identities implied by the local symmetry and always hold without using equations of motion. Thus none of existing defini…
We show that the asymptotic charges associated with Lorentz symmetries on past and future null infinity match in the limit to spatial infinity in a class of asymptotically-flat spacetimes. These are spacetimes that obey the Ashtekar-Hansen definition of asymptotic flatness at null and spatial infinity and satisfy an additional set of conditions which we lay out explicitly. Combined with earlier results on the matching of supertranslation charges, this shows that all Bondi-Metzner-Sachs (BMS) charges on past and future null infinity match in the limit to spatial infinity in this class of spacet…
We use the formalism developed by Wald and Zoupas to derive explicit covariant expressions for the charges and fluxes associated with the Bondi-Metzner-Sachs symmetries at null infinity in asymptotically flat spacetimes in vacuum general relativity. Our expressions hold in non-stationary regions of null infinity, are local and covariant, conformally-invariant, and are independent of the choice of foliation of null infinity and of the chosen extension of the symmetries away from null infinity. While similar expressions have appeared previously in the literature in Bondi-Sachs coordinates (to wh…
The ring-down phase of black-hole perturbations is governed by the Quasi-Normal modes (QNM) and offer valuable insight into the nature of the objects emitting them. In General relativity, we identify a dimensionless parameter that can estimate the charge of a black-hole from the energetics of quasi-normal modes. We then extend the analysis for f(R) theories of gravity. Using Isaacson's prescription to modified theories of gravity, we obtain an accurate and robust method to estimate the difference in the radiated energies in the scalar and vector modes due to modifications to general relativity…
In this manuscript, we show that the external Schwarzschild metric can be a good approximation of exact black hole solutions of entangled relativity. Since entangled relativity cannot be defined from vacuum, the demonstrations need to rely on the definition of matter fields. The electromagnetic field being the easiest (and perhaps the only) existing matter field with infinite range to consider, we study the case of a charged black hole -- for which the solution of entangled relativity and a dilaton theory agree -- as well as the case of a pure radiation -- for which the solution of entangled r…
Geometrically the phase space of a mechanical system involves the co-tangent bundle of the configuration space. The phase space of a relativistic field theory is infinite dimensional and can be endowed with a symplectic structure defined in a perfectly co-variant manner that is very useful for discussing symmetries and conserved quantities of the system. In general relativity the symplectic structure takes Darboux form and it is shown in this work that the presence of a cosmological constant does not change this conclusion. For space-times that admit time-like Killing vectors the formalism can…
It is known that two Reissner-Nordstrom black holes or two overextreme Reissner-Nordstrom sources cannot be in physical equilibrium. In the static case such equilibrium is possible only if one of the sources is a black hole and another one is a naked singularity. We define the notion of physical equilibrium in general (stationary) case when both components of a binary system are rotating and show that such system containing a Kerr-Newman black hole and a Kerr-Newman naked singularity also can stay in physical equilibrium. The similar question about the system of two charged rotating black hole…
A general Hamiltonian approach to black hole thermodynamics is used to study entropy and conserved charges for Kerr-AdS solutions in general relativity. These thermodynamic variables are first consistently defined by choosing suitable boundary conditions, and then, they are shown to satisfy the first law of black hole dynamics
We identify a symplectic potential for general relativity in tetrad and connection variables that is fully gauge-invariant, using the freedom to add surface terms. When torsion vanishes, it does not lead to surface charges associated with the internal Lorentz transformations, and reduces exactly to the symplectic potential given by the Einstein-Hilbert action. In particular, it reproduces the Komar form when the variation is a Lie derivative, and the geometric expression in terms of extrinsic curvature and 2d corner data for a general variation. The additional surface term vanishes at spatial …
In a nonlinear theory, such as General Relativity, linearized field equations around an exact solution are necessary but not sufficient conditions for linearized solutions. Therefore, the linearized field equations can have some solutions which do not come from the linearization of possible exact solutions. This fact can make the perturbation theory ill-defined, which would be a problem both at the classical and semiclassical quantization level. Here we study the first and second order perturbation theory in cosmological Einstein gravity and give the explicit form of the integral constraint, w…
We investigate the stability of motion and the thermodynamics in the case of spherically symmetric solutions in $f(T)$ gravity using the perturbative approach. We consider small deviations from general relativity and we extract charged black hole solutions for two charge profiles, namely with or without a perturbative correction in the charge distribution. We examine their asymptotic behavior, we extract various torsional and curvature invariants, and we calculate the energy and the mass of the solutions. Furthermore, we study the stability of motion around the obtained solutions, by analyzing…
The Bondi-van der Burg-Metzner-Sachs (BMS) group, which uniquely describes the symmetries of asymptotic infinity and therefore of the gravitational waves that propagate there, has become increasingly important for accurate modeling of waveforms. In particular, waveform models, such as post-Newtonian (PN) expressions, numerical relativity (NR), and black hole perturbation theory, produce results that are in different BMS frames. Consequently, to build a model for the waveforms produced during the merging of compact objects, which ideally would be a hybridization of PN, NR, and black hole pertur…
We analyze the asymptotic symmetries and their associated charges at spatial infinity in $4$-dimensional asymptotically-flat spacetimes. We use the covariant formalism of Ashtekar and Hansen where the asymptotic fields and symmetries live on the $3$-manifold of spatial directions at spatial infinity, represented by a timelike unit-hyperboloid (or de Sitter space). Using the covariant phase space formalism, we derive formulae for the charges corresponding to asymptotic supertranslations and Lorentz symmetries at spatial infinity. With the motivation of, eventually, proving that these charges ma…
The existence of stationary solutions of the Einstein-Vlasov-Maxwell system which are axially symmetric but not spherically symmetric is proven by means of the implicit function theorem on Banach spaces. The proof generalises the methods of an article by Håkan Andréasson, Markus Kunze and Gerhard Rein where a similar result is obtained for uncharged particles. Among the solutions constructed in this article there are rotating and non-rotating ones. Static solutions exhibit an electric but no magnetic field. In the case of rotating solutions, in addition to the electric field, a purely poloidal…
