Introduction
Near-horizon symmetries and black hole
Introduction
In Kerr space-time, while in Boyer-Lindquist (BL) coordinates and φ can be separated, r and θ remain connected due to lack of symmetry [39]. In this paper, we (i) construct these basis functions, (ii) prove orthogonality in geodesically complete coordinates, and (iii) demonstrate the separation of variables in differential equations for some physical systems.
Kerr and the NHEK limit
In global coordinates we can similarly find four (different) generators which are KVFs of the NHEK spacetime. The induced action of g on tensors is therefore Lie derivation along the representation of the Lie algebra element.
The highest- (lowest-) weight method
In Poincaré coordinates we focus on the basis functions that form the highest weight module because their expressions are simpler. For the scalar case, for example, if we apply the increment operator to a scalar in the module with the highest weight, we get
Orthogonality in global coordinates
This step is identical if we consider scalars/vectors/tensors, since the argument of the Lie derivative contracted all indices. Similarly when k0 < k, we can extract all reduction operators from the bra and increase the weight of the conditions in the ket, which will end up increasing the highest weight condition.
Separation of variables
The linearized Einstein operator G(1)[h] can be written in terms of the derivative of the background covariant∇ as. We substitute the highest weight metric perturbation into the left-hand side of the linearized Einstein equation, and the result is given by Eq. 2.47) Again note that the dependence (T,Φ,R) is directly factorized through the differential operator, resulting in ten coupled ODEs for the tenC functions, which we collect together as C(u).
Conclusions and future work
We choose the three vector bases with the highest weight VbB(m h0) and the six tensor bases with the highest weight WabAB(m h0) so that the metric perturbation with k =0 can be written as Eq. Therefore, we conclude that with these scalar, vector and tensorbases we can separate variables in the linearized Einstein system in NHEK.
Appendix
Finally, we calculated the corrections to the horizon regions and the macroscopic entropies of the extreme black hole solutions in D2CS and D2GB. In the vicinity of the horizon, the s = −2 SN function, i.e. the one associated with ψ4, can be written as.
![Table 2.1: The coefficient table that gives the expressions of D A (m,h) [C(u)], A ∈ {T, Φ, R, u} in Maxwell systems.](https://thumb-ap.123doks.com/thumbv2/123dok/10401444.0/47.918.161.763.102.509/table-coefficient-table-gives-expressions-φ-maxwell-systems.webp)
Metric deformations from extremal Kerr black holes
Introduction
In the slow rotation limit, BH solutions [64, 65] have been found for dynamical Chern-Simons theory [66]. In this chapter we find BH solutions in the near-horizon extreme limit for theories over GR.
Einstein-dilaton-Gauss-Bonnet and dynamical Chern-Simons gravity 40
We also expect a double CFT description of the extreme entropy of the black hole for theories beyond the GR in the decoupling limit. The enhanced symmetry due to the near-horizontal extreme boundary allows us to separate variables in the linearized Einstein equation (LEE) in NHEK spacetime [36].
Solving for the metric deformations
Therefore, the metric perturbations on the LHS of the LEE have only stationary base asymmetric components, either for D2CS or D2GB. By requiring regularity at the two poles and reflection symmetry of the distorted metric, we state.
Properties of solutions
The horizon areas calculated here will be used in the next subsection to calculate the entropy of the two deformed solutions. A combination of the horizon area calculations given by Eq. 3.48), the entropies of both deformed NHEK solutions can be written as
Discussion and future work
The shaded area consists of the infalling matter that all went into the black hole. In Einstein gravitation, the entropy of the BTZ black hole is S =4πr+, and the curve of constant entropy is given by.
Gedanken experiments to destroy the event
Introduction
Recently, significant progress was made for the general proof of the WCCC by Sorce and Wald [17] who adopted a general relativistic formulation of the energy conservation that can work for general forms of matter obeying the Zero Energy Condition (NEC). Moreover, their method of investigating the WCKK also provides a systematic framework for general theories other than Einstein-Maxwell.
EFTs, black-hole solutions, and extremality condition
In low-energy effective field theory (EFT), these quantum corrections can leave low-energy relics in the form of higher-order derivative terms beyond the Einstein-Maxwell terms, modifying the black hole solutions, as well as the relativistic laws of conservation of energy momentum. This calls for a further examination of WCCC conservation in the second-order variant for the EFTs considered1.
Gedanken experiment to destroy the horizon
The advantage of starting at the extreme solution is:. 4.11) is violated, then any infinitesimally small w will lead to destruction of the horizon and we can limit ourselves to linear perturbations. In this way, the violation of the condition (4.11), or the destruction of the extreme horizon, relies on the possibility of reducing the area in the linear order.
Test particle
More specifically, if we denote the horizon region by A(m,q), we can show that. The latter inequality is a consequence of ~uH · ξ~ ≤ 0: the 4-velocity of the particle must be directed towards the future when the particle crosses the horizon.
Sorce-Wald method for generic matter
6 We note that in general Σ1 is only part of a "Cauchy surface" - with the remainder filled in by some future null infinity. This inequality serves as a constraint on the changes in black hole mass and infall charge. process, and will be used to check WCCC by comparing with condition (4.11).
Parameter bounds from WCCC
In our case, we simply have c00 = c0, so the WCCC is preserved regardless of how we choose the coupling coefficients. There is therefore no limit that one can place on these coefficients using the WCKK.
Extension to more general theories
Finally, no matter how we change the behavior of the EFTs, the condition for the case falling into the extreme black hole always coincides with the condition for WCCC. However, for near-extreme black holes one must consider second-order variations of black hole mass and charge.
Appendix
We have focused on the reflection-type echo, i.e., the echo created by the reflection of the main wave on the ECO surface. We then derive the boundary conditions for the Teukolsky equations in terms of the tidal responses of the ECO in the FIDO framework.
Gedanken experiments to destroy a BTZ black hole: second-order
Introduction
Based on an earlier development of the second-order variation of black hole mechanics [102], they went beyond the first-order analysis in [16] and derived the following inequality. We can transform the above third law into the thermodynamics of the black hole if we adopt Bekenstein and Hawking's point of view.
BTZ black hole and variational identities
For the time-like Killing field ∂/∂t and the rotational Killing field ∂/∂ϕ, the above integration results in a variation of the total mass M and the total angular momentum J . The right-hand side of (5.42) would be related to the energy and momentum tensor of matter.
Gedanken experiment to destroy an extremal BTZ
We consider a perturbation δφ whose initial data for both fields δea and δωa on Σ0 vanish in the vicinity of the intersection between Σ0 and the horizon. As we will show later, this identity restricts the sign of f(λ). 5.58), the integral in the second line is not positive definite because of the spin angular momentum term and its coupling to torsion.
Gedanken experiment to destroy a near-extremal BTZ
Hence, extreme BTZ black hole cannot be destroyed in three-dimensional Einstein gravity, leaving WCCC preserved. Our gedanken experiment cannot destroy a near-extreme BTZ black hole in three-dimensional Einstein gravity, leaving WCCC preserved.
Conclusions and discussions
These metric perturbations can then be used to correlate the ECO's response to external perturbations. It has been proposed that the reflectivity of ECOs can be modeled by changing the impedance of the ECO surface.
Searching for near-horizon quantum struc-
Set up of the problem
When the spatial extent of the GWs in each direction is compacted within the hoop, the event horizon forms. The conventional estimate for the time lapse between the first echo and the start of the hang-up signal is given by
Estimates based on the hoop conjecture
6.1, at the moment when all the incoming energy is compressed into a "ring", an event horizon is formed. If we consider the null packet carrying the ringing energy (6.1), the location of the ECO surface must meet the conditions to prevent the formation of the horizon. 6.3) This means that stable, static ECOs cannot be very compact - or ∆ are far from the Planck scale.
In-going Vaidya spacetime
A static ECO scenario where the ECO surface is outside the event horizon and can cause GW echoes. If the ECO surface always remains outside the event horizon, the incoming rays can lead to GW echoes, as shown by the dashed green lines.
Back reaction: static ECO with future incoming pulse
The closest incoming spatial beam, marked by the dashed green line, is reflected at the potential barrier and the ECO surface, leading to a time delay ∆techo between the main wave and the echo as observed at spatial infinity. A static ECO scenario where part of the input GW energy is reflected at the ECO surface and creates echoes down to the "last runaway rays".
Back reaction: expanding ECO
Implications for GW-echo phenomenology
The reflected GW will first oscillate and then be "frozen" due to ECO gravitational collapse. Since the low-frequency component of the reflected GW cannot propagate to infinity due to the filtering of the frequency-dependent potential barrier, the distant observer will only see an attenuated QNM waveform.
Discussions
Appendix
Now we are prepared to discuss the reflecting boundary condition of the Teukolsky equations in terms of the tidal response of the ECO. In this subsection, we consider another scenario where some effective matter fields exist in the vicinity of the horizon.
Tidal response and near-horizon boundary conditions for spin-
Introduction
A black hole (BH) is characterized by the event horizon, the boundary of the space-time region within which a future zero-infinity cannot be reached. In this paper, we will continue with the membrane paradigm and propose a physical definition of ECO reflectivity.
The reflection boundary condition from tidal response
For the moment, let us assume that the linear perturbation theory holds in the entire outer Kerr space of ECO. Furthermore, the response of the ECO may not be instantaneous, but instead may depend on the history of the tidal disturbance exerted.
Wave propagation in the vicinity of the horizon
This ˆM``0mω directly shows the mixing of modes due to the phasesδ(θ) and β(θ), which arise due to the non-spherical nature of the ECO surface. Physically speaking, µ is the linear response of matter to external perturbations, which is similar to the permeability of gravitational waves in matter.
Boundary condition in terms of various functions
If we restrict ourselves to the simple case where the `- and `0- modes do not mix, we can simply write Eq. Since most of the previous literature on gravitational wave echoes based their models on Sasaki–Nakamura (SN) reflection functions, one might ask how tidal reflection can be related to SN reflection.
Waveforms and quasi-normal modes of the ECO
Using the symmetry of the real-valued Teukol sky equation, it is straightforward to show that homogeneous solutions have the symmetry that Following the same steps as in the last subsection, it is straightforward to show that the echo solution for the Teukolsky equation at infinity is given by .
Conclusions
Appendix
Fast-spinning black holes in Chern-Simons dynamical gravity: Boundary decoupling solutions and decay. Slow-rotating black holes in Einstein-Dilaton-Gauss-Bonnet gravity: Quadratic order in rotational solutions.
A matrix
B matrix
Part I of C matrix
Part II of C matrix
Part III of C matrix
