Theorem For Black Holes Research Articles

Tracing the geometry around a massive, axisymmetric body to measure, through gravitational waves, its mass moments and electromagnetic moments

The geometry around a rotating massive body, which carries charge and electrical currents, could be described by its multipole moments (mass moments, mass-current moments, electric moments, and magnetic moments). When a small body is orbiting around such a massive body, it will move on geodesics, at least for a time interval that is short with respect to the characteristic time of the binary due to gravitational radiation. By monitoring the waves emitted by the small body we are actually tracing the geometry of the central object, and hence, in principle, we can infer all its multipole moments. This paper is a generalization of previous similar results by Ryan. Ryan explored the mass and mass-current moments of a stationary, axially symmetric, and reflection symmetric, with respect to its equatorial plane, metric, by analyzing the gravitational waves emitted from a test body which is orbiting around the central body in nearly circular equatorial orbits. In our study we suppose that the gravitating source is endowed with intense electromagnetic field as well. Because of its axisymmetry the source is characterized now by four families of scalar multipole moments: its mass moments ${M}_{l}$, its mass-current moments ${S}_{l}$, its electrical moments ${E}_{l}$, and its magnetic moments ${H}_{l}$, where $l=0,1,2,\dots{}$. Four measurable quantities, the energy emitted by gravitational waves per logarithmic interval of frequency, the precession of the periastron, the precession of the orbital plane, and the number of cycles emitted per logarithmic interval of frequency, are presented as power series of the Newtonian orbital velocity of the test body. The power series coefficients are simple polynomials of the various moments. If any of these quantities are measured with sufficiently high accuracy, the lowest moments, including the electromagnetic ones, could be inferred and thus we could get valuable information about the internal structure of the compact massive body. The fact that the electromagnetic moments of spacetime can be measured demonstrates that one can obtain information about the electromagnetic field purely from gravitational-wave analysis. Additionally, these measurements could be used as a test of the no-hair theorem for black holes.

No black-hole theorem in three-dimensional gravity

A common property of known black-hole solutions in (2+1)-dimensional gravity is that they require a negative cosmological constant. To explain this, it is shown in this Letter that a (2+1)-dimensional gravity theory which satisfies the dominant energy condition forbids the existence of a black hole.

Self-force on a static scalar test charge outside a Schwarzschild black hole

The finite part of the self-force on a static scalar test-charge outside a Schwarzschild black hole is zero. By direct construction of Hadamard's elementary solution, we obtain a closed-form expression for the minimally coupled scalar field produced by a test-charge held fixed in Schwarzschild spacetime. Using the closed-form expression, we compute the necessary external force required to hold the charge stationary. Although the energy associated with the scalar field contributes to the renormalized mass of the particle (and thereby its weight), we find there is no additional self-force acting on the charge. This result is unlike the analogous electrostatic result, where, after a similar mass renormalization, there remains a finite repulsive self-force acting on a static electric test-charge outside a Schwarzschild black hole. We confirm our force calculation using Carter's mass-variation theorem for black holes. The primary motivation for this calculation is to develop techniques and formalism for computing all forces - dissipative and non-dissipative - acting on charges and masses moving in a black-hole spacetime. In the Appendix we recap the derivation of the closed-form electrostatic potential. We also show how the closed-form expressions for the fields are related to the infinite series solutions.

Positive mass theorem for black holes in Einstein-Maxwell axion-dilaton gravity

We present a proof of the positive mass theorem for black holes in Einstein-Maxwell axion-dilaton gravity, which is the low-energy limit of heterotic string theory. We show that the total mass of a spacetime containing a black hole is greater than or equal to the square root of the sum of the squares of the adequate dilaton-electric and dilaton-magnetic charges.

Bounded area theorems for higher-genus black holes

By a simple modification of Hawking's well known topology theorems for black hole horizons, we find lower bounds for the areas of smooth apparent horizons and smooth cross sections of stationary black hole event horizons of genus g>1 in four dimensions. For a negatively curved Einstein space, the bound is 4(g-1)/(-) where is the cosmological constant of the spacetime. This is complementary to the known upper bound on the area of g = 0 black holes in de Sitter spacetime. It also emerges that g>1 quite generally requires a mean negative energy density on the horizon. The bound is sharp; we show that it is saturated by certain extreme, asymptotically locally anti-de Sitter spacetimes. Our results generalize a recent result of Gibbons.

Gravitating brane systems: Some general theorems

Multidimensional gravity interacting with intersecting electric and magnetic p-branes is considered for fields depending on a single variable. Some general features of the system behavior are revealed without solving the field equations. Thus, essential asymptotic properties of isotropic cosmologies are indicated for different signs of spatial curvature; a no-hair-type theorem and a single-time theorem for black holes are proved (the latter makes sense in models with multiple time coordinates). The validity of the general observations is verified for a class of exact solutions known for the cases when certain vectors, built from the input parameters of the model, are either orthogonal in minisuperspace, or form mutually orthogonal subsystems. From the nonexistence of Lorentzian wormholes, a universal restriction is obtained, applicable to orthogonal or block-orthogonal subsystems of any p-brane system.

Is there a general area theorem for black holes?

The general validity of the area law for black holes is still an open problem. We first show in detail how to complete the usually incompletely stated textbook proofs under the assumption of piecewise C2-smoothness for the surface of the black hole. Then we prove that a black hole surface necessarily contains points where it is not C1 (called “cusps”) at any time before caustics of the horizon generators show up, like, e.g., in merging processes. This implies that caustics never disappear in the past and that black holes without initial cusps will never develop such. Hence black holes which will undergo any nontrivial processes anywhere in the future will always show cusps. Although this does not yet imply a strict incompatibility with piecewise C2 structures, it indicates that the latter are likely to be physically unnatural. We conclude by calling for a purely measure theoretic proof of the area theorem.

The positive mass theorem for black holes revisited

We present a rigorous proof of the positive mass theorem for black holes. Accordingly, in a four-dimensional Lorentz manifold satisfying the dominant energy condition, the mass of a three-dimensional asymptotically flat slice with boundary composed of a finite number of future or past trapped closed 2-surfaces is nonnegative. The proof uses the classical Witten argument and is valid even if only rather weak asymptotic conditions are imposed.

New no-scalar-hair theorem for black holes

A new no-hair theorem is formulated which rules out a very large class of nonminimally coupled finite scalar dressing of an asymptotically flat, static, and spherically symmetric black hole. The proof is very simple and based on a covariant method for generating solutions for nonminimally coupled scalar fields starting from the minimally coupled case. Such a method generalizes the Bekenstein method for conformal coupling and other recent ones. We also discuss the role of the finiteness assumption for the scalar field.

Novel "no-scalar-hair" theorem for black holes.

We formulate a new ``no-hair'' theorem for black holes in general relativity which rules out a multicomponent scalar field dressing of any asymptotically flat, static, spherically symmetric black hole. The field is assumed to be minimally coupled to gravity, and to bear a non-negative energy density as seen by any observer, but its field Lagrangian need not be quadratic in the field derivatives. The proof centers on energy-momentum conservation and the Einstein equations. One kind of field ruled out is the Higgs field with a double (or multiple) well potential. The theorem is also proved for scalar-tensor gravity.

ARE BLACK HOLES IN BRANS-DICKE THEORY PRECISELY THE SAME AS IN GENERAL RELATIVITY?

We study a three-parameter family of solutions of the Brans-Dicke field equations. They are static and spherically symmetric. We find the range of parameters for which this solution represents a black hole different from the Schwarzschild one. We find a subfamily of solutions which agrees with experiments and observations in the solar system. We discuss some astrophysical applications and the consequences on the “no hair” theorems for black holes.

4-Dimensional black holes from Kaluza-Klein theories

In this paper we consider generalizations in 4 dimensions of the Einstein-Maxwell equations which typically arise from Kaluza-Klein theories. We specify conditions such that stationary solutions lead to non-linearσ-models for symmetric spaces. Using both this group theoretic structure and some properties of harmonic maps we are able to generalize many of the known existence and uniqueness theorems for black holes in Einstein-Maxwell theory to this more general setting.

Positive mass theorems for black holes

We extend Witten's proof of the positive mass theorem at spacelike infinity to show that the mass is positive for initial data on an asymptotically flat spatial hypersurface Σ which is regular outside an apparent horizonH. In addition, we prove that if a black hole has electromagnetic charge, then the mass is greater than the modulus of the charge. These results are also valid for the Bondi mass at null infinity. Finally, in the case of the Einstein equation with a negative cosmological constant, we show that a suitably defined mass is positive for data on an asymptotically anti-de Sitter surface Σ which is regular outside an apparent horizon.

"No hair" theorem for black holes in the Mansouri-Chang theory of gravitation

We show that a static, spherically symmetric black hole in Mansouri-Chang (MC) gravity theory (where the Lagrangian contains up to fourth derivatives of the metric) will "swallow" all effects due to the higher derivatives, so that outside the event horizon the metric necessarily reduces to a Schwarzschild metric. Hence Pavelle's recent proof of the equivalence of MC and standard Einstein theory should be reinterpreted as a proof of a "no hair" theorem for MC black holes. For normal stars, MC and standard theory need not be equivalent. Our results apply to a wide class of higher-derivative theories in addition to MC theory. We also derive the "oscillator variable" decomposition for these theories.