Tuesday, June 20, 2017

Gibbons–Hawking ansatz,lets discusss about it.. By Rkmht

In mathematics, the Gibbons–Hawking ansatz is a method of constructing gravitational instantons introduced by Gary Gibbons and Stephen Hawking (19781979). It gives examples of hyperkähler manifolds in dimension 4 that are invariant under a circle action...

In mathematics, the Gibbons–Hawking ansatz is a method of constructing gravitational instantons introduced by Gary Gibbons and Stephen Hawking (19781979). It gives examples of hyperkähler manifolds in dimension 4 that are invariant under a circle action.



In mathematical physics and differential geometry, a gravitational instanton is a four-dimensional complete Riemannian manifold satisfying the vacuum Einstein equations. They are so named because they are analogues in quantum theories of gravity of instantons in Yang–Mills theory. In accordance with this analogy with self-dual Yang–Mills instantons, gravitational instantons are usually assumed to look like four dimensional Euclidean space at large distances, and to have a self-dual Riemann tensor. Mathematically, this means that they are asymptotically locally Euclidean (or perhaps asymptotically locally flat) hyperkähler 4-manifolds, and in this sense, they are special examples of Einstein manifolds. From a physical point of view, a gravitational instanton is a non-singular solution of the vacuum Einstein equations with positive-definite, as opposed to Lorentzian, metric.
There are many possible generalizations of the original conception of a gravitational instanton: for example one can allow gravitational instantons to have a nonzero cosmological constant or a Riemann tensor which is not self-dual. One can also relax the boundary condition that the metric is asymptotically Euclidean.
There are many methods for constructing gravitational instantons, including the Gibbons–Hawking Ansatz, twistor theory, and the hyperkähler quotient construction.

  • A four-dimensional Kähler–Einstein manifold has a self-dual Riemann tensor.
  • Equivalently, a self-dual gravitational instanton is a four-dimensional complete hyperkähler manifold.
  • Gravitational instantons are analogous to self-dual Yang–Mills instantons.
By specifying the 'boundary conditions', i.e. the asymptotics of the metric 'at infinity' on a noncompact Riemannian manifold, gravitational instantons are divided into a few classes, such as asymptotically locally Euclidean spaces (ALE spaces), asymptotically locally flat spaces (ALF spaces). There also exist ALG spaces whose name is chosen by induction.

It will be convenient to write the gravitational instanton solutions below using left-invariant 1-forms on the three-sphere S3 (viewed as the group Sp(1) or SU(2)). These can be defined in terms of Euler angles by

Taub–NUT metric

Eguchi–Hanson metric

The Eguchi–Hanson space is important in many other contexts of geometry and theoretical physics. Its metric is given by
where . This metric is smooth everywhere if it has no conical singularity at . For  this happens if  has a period of , which gives a flat metric on R4; However for  this happens if  has a period of .
Asymptotically (i.e., in the limit ) the metric looks like
which naively seems as the flat metric on R4. However, for  has only half the usual periodicity, as we have seen. Thus the metric is asymptotically R4 with the identification , which is a Z2 subgroup of SO(4), the rotation group of R4. Therefore the metric is said to be asymptotically R4/Z2.
There is a transformation to another coordinate system, in which the metric looks like
where 
(For a = 0, , and the new coordinates are defined as follows: one first defines  and then parametrizes  and  by the R3 coordinates , i.e. ).
In the new coordinates,  has the usual periodicity 
One may replace V by
For some n points i = 1, 2..., n. This gives a multi-center Eguchi–Hanson gravitational instanton, which is again smooth everywhere if the angular coordinates have the usual periodicities (to avoid conical singularities). The asymptotic limit () is equivalent to taking all  to zero, and by changing coordinates back to r,  and , and redefining , we get the asymptotic metric
This is R4/Zn = C2/Zn, because it is R4 with the angular coordinate  replaced by , which has the wrong periodicity ( instead of ). In other words, it is R4 identified under , or, equivalently, C2 identified under zi ~  zi for i = 1, 2.
To conclude, the multi-center Eguchi–Hanson geometry is a Kähler Ricci flat geometry which is asymptotically C2/Zn. According to Yau's theorem this is the only geometry satisfying these properties. Therefore this is also the geometry of a C2/Zn orbifold in string theory after its conical singularity has been smoothed away by its "blow up" (i.e., deformation).

Gibbons–Hawking multi-centre metrics

where
 corresponds to multi-Taub–NUT,  and  is flat space, and  and  is the Eguchi–Hanson solution (in different coordinates).

Hawking energy,,lets discuss about it and its implication in life

The Hawking energy or Hawking mass is one of the possible definitions of mass in general relativity. It is a measure of the bending of ingoing and outgoing rays of light that areorthogonal to a 2-sphere surrounding the region of space whose mass is to be defined.


Let  be a 3-dimensional sub-manifold of a relativistic spacetime, and let  be a closed 2-surface. Then the Hawking mass  of  is defined to be
where  is the mean curvature of 

n the Schwarzschild metric, the Hawking mass of any sphere  about the central mass is equal to the value  of the central mass.
A result of Geroch implies that Hawking mass satisfies an important monotonicity condition. Namely, if  has nonnegative scalar curvature, then the Hawking mass of  is non-decreasing as the surface  flows outward at a speed equal to the inverse of the mean curvature. In particular, if  is a family of connected surfaces evolving according to
where  is the mean curvature of  and  is the unit vector opposite of the mean curvature direction, then
Said otherwise, Hawking mass is increasing for the inverse mean curvature flow
Hawking mass is not necessarily positive. However, it is asymptotic to the ADM[4] or the Bondi mass, depending on whether the surface is asymptotic to spatial infinity or null infinity.

Black hole thermodynamics. . . .

Black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons. As the study of the statistical mechanics of black body radiation led to the advent of the theory of quantum mechanics, the effort to understand the statistical mechanics of black holes has had a deep impact upon the understanding of quantum gravity, leading to the formulation of the holographic principle.In physicsblack hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons. As the study of the statistical mechanics of black body radiation led to the advent of the theory of quantum mechanics, the effort to understand the statistical mechanics of black holes has had a deep impact upon the understanding of quantum gravity, leading to the formulation of the holographic principle.


The second law of thermodynamics requires that black holes have entropy. If black holes carried no entropy, it would be possible to violate the second law by throwing mass into the black hole. The increase of the entropy of the black hole more than compensates for the decrease of the entropy carried by the object that was swallowed.
Starting from theorems proved by Stephen Hawking, Jacob Bekenstein conjectured that the black hole entropy was proportional to the area of its event horizon divided by the Planck area. In 1973 Bekenstein suggested  as the constant of proportionality, asserting that if the constant was not exactly this, it must be very close to it. The next year in 1974, Hawking showed that black holes emit thermal Hawking radiation corresponding to a certain temperature (Hawking temperature).Using the thermodynamic relationship between energy, temperature and entropy, Hawking was able to confirm Bekenstein's conjecture and fix the constant of proportionality at :
where  is the area of the event horizon, calculated at  is Boltzmann's constant, and  is the Planck length. This is often referred to as theBekenstein–Hawking formula. The subscript BH either stands for "black hole" or "Bekenstein–Hawking". The black hole entropy is proportional to the area of its event horizon . The fact that the black hole entropy is also the maximal entropy that can be obtained by the Bekenstein bound (wherein the Bekenstein bound becomes an equality) was the main observation that led to the holographic principle.
Although Hawking's calculations gave further thermodynamic evidence for black hole entropy, until 1995 no one was able to make a controlled calculation of black hole entropy based on statistical mechanics, which associates entropy with a large number of microstates. In fact, so called "no hair" theorems appeared to suggest that black holes could have only a single microstate. The situation changed in 1995 when Andrew Strominger and Cumrun Vafa calculated the right Bekenstein-Hawking entropy of a supersymmetricblack hole in string theory, using methods based on D-branes and string duality. Their calculation was followed by many similar computations of entropy of large classes of otherextremal and near-extremal black holes, and the result always agreed with the Bekenstein-Hawking formula. However, for the Schwarzschild black hole, viewed as the most far-from-extremal black hole, the relationship between micro and macro-states is expected to be clarified from the string theoretical viewpoint. Various studies are in progress, but this has not yet been elucidated.
In Loop quantum gravity (LQG) it is possible to associate a geometrical interpretation to the microstates: these are the quantum geometries of the horizon. LQG offers a geometric explanation of the finiteness of the entropy and of the proportionality of the area of the horizon. It is possible to derive, from the covariant formulation of full quantum theory (Spinfoam) the correct relation between energy and area (1st law), the Unruh temperature and the distribution that yields Hawking entropy. The calculation makes use of the notion of dynamical horizon and is done for non-extremal black holes. There seems to be also discussed the calculation of Bekenstein-Hawking entropy from the point of view of LQG.

The laws of black hole mechanics

The four laws of black hole mechanics are physical properties that black holes are believed to satisfy. The laws, analogous to the laws of thermodynamics, were discovered byBrandon Carter, Stephen Hawking, and James Bardeen.

Statement of the laws

The laws of black hole mechanics are expressed in geometrized units.

The Zeroth Law

The horizon has constant surface gravity for a stationary black hole.

The First Law

For perturbations of stationary black holes, the change of energy is related to change of area, angular momentum, and electric charge by:
where  is the energy,  is the surface gravity,  is the horizon area,  is the angular velocity,  is the angular momentum,  is the electrostatic potential and  is the electric charge.

The Second Law

The horizon area is, assuming the weak energy condition, a non-decreasing function of time,
This "law" was superseded by Hawking's discovery that black holes radiate, which causes both the black hole's mass and the area of its horizon to decrease over time.

The Third Law    :-

It is not possible to form a black hole with vanishing surface gravity.  = 0 is not possible to achieve.

Discussion of the laws

The Zeroth Law

The zeroth law is analogous to the zeroth law of thermodynamics which states that the temperature is constant throughout a body in thermal equilibrium. It suggests that the surface gravity is analogous to temperature. T constant for thermal equilibrium for a normal system is analogous to  constant over the horizon of a stationary black hole.

The First Law

The left hand side, dE, is the change in energy (proportional to mass). Although the first term does not have an immediately obvious physical interpretation, the second and third terms on the right hand side represent changes in energy due to rotation and electromagnetism. Analogously, the first law of thermodynamics is a statement of energy conservation, which contains on its right hand side the term T dS.

The Second Law

The second law is the statement of Hawking's area theorem. Analogously, the second law of thermodynamics states that the change in entropy in an isolated system will be greater than or equal to 0 for a spontaneous process, suggesting a link between entropy and the area of a black hole horizon. However, this version violates the second law of thermodynamics by matter losing (its) entropy as it falls in, giving a decrease in entropy. Generalized second law introduced as total entropy = black hole entropy + outside entropy.

The Third Law

Extremal black holes have vanishing surface gravity. Stating that  cannot go to zero is analogous to the third law of thermodynamics which states, the entropy of a system at absolute zero is a well-defined constant. This is because a system at zero temperature exists in its ground state. Furthermore, ΔS will reach zero at 0 kelvins, but S itself will also reach zero, at least for perfect crystalline substances. No experimentally verified violations of the laws of thermodynamics are known.

Interpretation of the laws

The four laws of black hole mechanics suggest that one should identify the surface gravity of a black hole with temperature and the area of the event horizon with entropy, at least up to some multiplicative constants. If one only considers black holes classically, then they have zero temperature and, by the hair theorem, zero entropy, and the laws of black hole mechanics remain an analogy. However, when quantum mechanical effects are taken into account, one finds that black holes emit thermal radiation (Hawking radiation) at a temperature
From the first law of black hole mechanics, this determines the multiplicative constant of the Bekenstein-Hawking entropy which is

Beyond black holes>....>

Hawking and Page have shown that black hole thermodynamics is more general than black holes—that cosmological event horizons also have an entropy and temperature.
More fundamentally, 't Hooft and Susskind used the laws of black hole thermodynamics to argue for a general Holographic Principle of nature, which asserts that consistent theories of gravity and quantum mechanics must be lower-dimensional. Though not yet fully understood in general, the holographic principle is central to theories like the AdS/CFT correspondence.
There are also connections between black hole entropy and fluid surface tension.

Finding funds: On COP28 and the ‘loss and damage’ fund....

A healthy loss and damage (L&D) fund, a three-decade-old demand, is a fundamental expression of climate justice. The L&D fund is a c...