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# Onset of the Gregory-Laflamme instability ν GL for black rings as a function of n for m = 8 (black lines), m = 12 (blue lines), m = 20 (purple lines) and m = 50 (red lines) using first order blackfold approach (dashed lines) and second order blackfold approach (solid lines).

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A bstract
We initiate the study of dynamical instabilities of higher-dimensional black holes using the blackfold approach, focusing on asymptotically flat boosted black strings and singly-spinning black rings in D ≥ 5. We derive novel analytic expressions for the growth rate of the Gregory-Laflamme instability for boosted black strings and its onse...

## Contexts in source publication

**Context 1**

... The full expression for the onset is provided in the ancillary Mathematica file. In fig. 7 we exhibit the onset of the instability ν GL as a function of n for different values of m, in particular m = 8 (black line) up to m = 50 (red line) as predicted by the first order approximation (dashed lines) and second order approximation (solid lines). It is clear from fig. 7 that the behaviour of the onset is qualitatively similar ...

**Context 2**

... for the onset is provided in the ancillary Mathematica file. In fig. 7 we exhibit the onset of the instability ν GL as a function of n for different values of m, in particular m = 8 (black line) up to m = 50 (red line) as predicted by the first order approximation (dashed lines) and second order approximation (solid lines). It is clear from fig. 7 that the behaviour of the onset is qualitatively similar to that of the boosted black string of fig. 4. One observes that as m increases, the onset ends at thiner and thiner rings, in agreement with fig. 6. These analytic results consist of the first analytic determination of ν GL ...

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## Citations

... An interesting way of gaining such additional knowledge is to perform a stability analysis of the configurations obtained here. This can be done by studying the linear spectrum of perturbations around the equilibrium states discussed in section 2. A similar analysis was carried out for black rings [44] and D3-NS5 branes in Klebanov-Strassler [45]. Our expectation is that a Gregory-Laflamme-type of instability will be present for certain values of the charges but that there will be a relatively large regime of stability. ...

A bstract
We use the long-wavelength effective theory of black branes (blackfold approach) to perturbatively construct holographic duals of the vacua of the $$ \mathcal{N} $$ N = 1* supersymmetric gauge theory. Employing the mechanism of Polchinski and Strassler, we consider wrapped black five-brane probes with D3-brane charge moving in the perturbative supergravity back-grounds corresponding to the high- and low-temperature phases of the gauge theory. Our approach recovers the results for the brane potentials and equilibrium configurations known in the literature in the extremal limit, while away from extremality we find metastable black D3-NS5 configurations with horizon topology ℝ ³ × 𝕊 ² × 𝕊 ³ in certain regimes of parameter space, which cloak potential brane singularities. We uncover novel features of the phase diagram of the $$ \mathcal{N} $$ N = 1* gauge theory in different ensembles and provide further evidence for the appearance of metastable states in holographic backgrounds dual to confining gauge theories.

... An interesting way of gaining such additional knowledge is to perform a stability analysis of the configurations obtained here. This can be done by studying the linear spectrum of perturbations around the equilibrium states discussed in section 2. A similar analysis was carried out for black rings [44] and D3-NS5 branes in Klebanov-Strassler [45]. Our expectation is that a Gregory-Laflamme-type of instability will be present for certain values of the charges but that there will be a relatively large regime of stability. ...

We use the long-wavelength effective theory of black branes (blackfold approach) to perturbatively construct holographic duals of the vacua of the $\mathcal{N}=1$* supersymmetric gauge theory. Employing the mechanism of Polchinski and Strassler, we consider wrapped black five-brane probes with D3-brane charge moving in the perturbative supergravity backgrounds corresponding to the high and low temperature phases of the gauge theory. Our approach recovers the results for the brane potentials and equilibrium configurations known in the literature in the extremal limit, while away from extremality we find metastable black D3-NS5 configurations with horizon topology $\mathbb{R}^3\times \mathbb{S}^2\times\mathbb{S}^3$ in certain regimes of parameter space, which cloak potential brane singularities. We uncover novel features of the phase diagram of the $\mathcal{N}=1$* gauge theory in different ensembles and provide further evidence for the appearance of metastable states in holographic backgrounds dual to confining gauge theories.

... 13 It is possible that instabilities observed for the polarised sphere in the small/finite radius regime (small M ) do not persist in the large radius regime (large M ). An analogous example of this picture can be found in the study of black rings where an instability (elastic mode instability) is observed for fat rings, i.e. rings with small curvature radius, but is not observed for thin rings, i.e. rings with large curvature radius, [40,41]. ...

... In certain cases, the leading order blackfold equations can detect the onset of a fragmentation instability.For example, the onset of the Gregory-Laflamme (GL) instability in black strings[36,37] and black rings[38] whose end point is fragmentation[39,40] can be observed by an analogous blackfold stability analysis[20,41]. ...

A bstract
We construct analytically a perturbative supergravity solution that captures the backreaction of a metastable state of anti-branes in the background of a particular modification of the Klebanov-Strassler throat in a long-wavelength approximation. Our solution, which has no unphysical singularities, describes how non-supersymmetric spherical NS5-branes with dissolved anti-D3 brane charge backreact in a fluxed throat geometry. It supports previous claims that there is a well-behaved supergravity description of the metastable state of wrapped NS5-branes proposed years ago by Kachru, Pearson, and Verlinde.

... 12 In certain cases, the leading order blackfold equations can detect the onset of a fragmentation instability. For example, the onset of the Gregory-Laflamme (GL) instability in black strings [36,37] and black rings [38] whose end point is fragmentation [39,40] can be observed by an analogous blackfold stability analysis [20,41]. 13 It is possible that instabilities observed for the polarised sphere in the small/finite radius regime (small M ) do not persist in the large radius regime (large M ). ...

... 13 It is possible that instabilities observed for the polarised sphere in the small/finite radius regime (small M ) do not persist in the large radius regime (large M ). An analogous example of this picture can be found in the study of black rings where an instability (elastic mode instability) is observed for fat rings, i.e. rings with small curvature radius, but is not observed for thin rings, i.e. rings with large curvature radius, [40,41]. ...

We construct analytically a perturbative supergravity solution that captures the backreaction of a metastable state of anti-branes in the background of a particular modification of the Klebanov-Strassler throat in a long-wavelength approximation. Our solution, which has no unphysical singularities, describes how non-supersymmetric spherical NS5-branes with dissolved anti-D3 brane charge backreact in a fluxed throat geometry. It supports previous claims that there is a well-behaved supergravity description of the metastable state of wrapped NS5-branes proposed years ago by Kachru, Pearson, and Verlinde.

... To avoid repetition, here and in the main text, we shall not discuss these common properties but instead focus on exploring the exotic pattern of thermal 3 It is interesting to note that, in certain cases, the leading order blackfold equations can detect the onset of a fragmentation instability. For example, the onset of the Gregory-Laflamme (GL) instability in black strings [44,45] and black rings [46] whose end point is fragmentation [47,48] can be observed by an analogous blackfold stability analysis [8,49]. 4 As the regime of validity of the analysis in [35,43] is that of small/finite size NS5 sphere, it is possible that the instability observed there does not persist in the large sphere regime. An analogous example of this picture can be found in the study of black rings where a fragmentation instability (elastic mode instability) is observed for fat rings but is not observed for very thin rings [48,49]. ...

... 4 As the regime of validity of the analysis in [35,43] is that of small/finite size NS5 sphere, it is possible that the instability observed there does not persist in the large sphere regime. An analogous example of this picture can be found in the study of black rings where a fragmentation instability (elastic mode instability) is observed for fat rings but is not observed for very thin rings [48,49]. 5 p denotes the number of the anti-M2 branes andM the strength of the CGLP background flux. ...

... Subsequently, we present the derivation of the blackfold perturbation equations used in the main text. For further discussions on variational properties of embedding geometry or blackfold perturbation equation, see e.g.[23,49]. ...

Despite their consequential applications, metastable states of antibranes in warped throats are not yet fully understood. In this thesis, we provide new information on various aspects of these metastable antibranes through applications of the blackfold effective theory for higher-dimensional black holes. As concrete examples, we study the conjectured metastable state of polarised anti-D3 branes at the tip of the Klebanov-Strassler (KS) throat in type IIB supergravity and the analogous state of polarised anti-M2 branes at the tip of the Cvetic-Gibbons-Lu-Pope (CGLP) throat in eleven-dimensional supergravity. For anti-D3 branes in KS throat, we provide novel evidence for the existence of the metastable state exactly where no-go theorems are lifted. In the extremal limit, we recover directly in supergravity the metastable states originally discovered by Kachru, Pearson, and Verlinde (KPV). Away from extremality, we uncover a metastable wrapped black NS5 state. We observe that such metastability is lost when the wrapped NS5 is heated sufficiently that its horizon geometry resembles that of a black anti-D3. We study the classical stability of the KPV state under generic long-wavelength deformations. We observe that, with regards to considered perturbations and regime of parameters, the state is classically stable. A study of anti-M2 branes in CGLP throat reveals many similarities to that of the anti-D3 branes. We recover directly in supergravity the Klebanov-Pufu (KP) state at extremality, and our finite temperature results fit suggestively well with known, complementary no-go theorems. However, we discover an unexpected, exotic pattern of thermal transitions of the KP state different from that of the KPV. This thesis contains also a pedagogical introduction to the blackfold formalism, focusing on aspects immediately relevant to applications to metastable antibranes.

... One might use blob rings to study black ring dynamics at large D as has been done for Myers-Perry black holes. It is known that thin rings are unstable to Gregory-Laflamme (GL) type non-axisymmetric fluctuations [5,46,47]. In the large D limit, the same instability was found in the effective theory on the ring coordinate [25]. ...

... The elastic instability numerically found in D = 5 [5] is another interesting phenomenon, which deforms the ring keeping its thickness almost unchanged, in contrast to the GL instability. Recently, it was shown that thin black rings are free of this type of instability within the scope of the blackfold approximation [47]. 15 It is interesting to see if blob rings admit the elastic-type instability with or without the non-perturbative correction caused by the blob-blob interaction. ...

... Here, the terms 'thin' and 'fat' are used regarding the applicability of the blob approximation.15 The elastic instability found in the large D ring[27] was questioned in ref.[47]. ...

A bstract
In the large dimension ( D ) limit, Einstein’s equation reduces to an effective theory on the horizon surface, drastically simplifying the black hole analysis. Especially, the effective theory on the black brane has been successful in describing the non-linear dynamics not only of black branes, but also of compact black objects which are encoded as solitary Gaussian-shaped lumps, blobs . For a rigidly rotating ansatz, in addition to axisymmetric deformed branches, various non-axisymmetric solutions have been found, such as black bars, which only stay stationary in the large D limit.
In this article, we demonstrate the blob approximation has a wider range of applicability by formulating the interaction between blobs and subsequent dynamics. We identify that this interaction occurs via thin necks connecting blobs. Especially, black strings are well captured in this approximation sufficiently away from the perturbative regime. Highly deformed black dumbbells and ripples are also found to be tractable in the approximation. By defining the local quantities, the effective force acting on distant blobs are evaluated as well. These results reveal that the large D effective theory is capable of describing not only individual black holes but also the gravitational interactions between them, as a full dynamical theory of interactive blobs, which we call brane blobology .

... Subsequently, we present the derivation of the blackfold perturbation equations used in the main text. For further discussion on embedding geometry and blackfold perturbation equation, see [27][28][29]. ...

A bstract
Using the blackfold approach, we study the classical stability of the KPV (Kachru-Pearson-Verlinde) state of anti-D3 branes at the tip of the Klebanov-Strassler throat. With regards to generic long-wavelength deformations considered, we found no instabilities. We comment on the relation of our results to existing results on the stability of the KPV state.

... One might use blob rings to study black ring dynamics at large D as has been done for Myers-Perry black holes. It is known that thin rings are unstable to Gregory-Laflamme (GL) type non-axisymmetric fluctuations [5,46,47]. In the large D limit, the same instability was found in the effective theory on the ring coordinate [25]. ...

... The elastic instability numerically found in D = 5 [5] is another interesting phenomenon, which deforms the ring keeping its thickness almost unchanged, in contrast to the GL instability. Recently, it was shown that thin black rings are free of this type of instability within the scope of the blackfold approximation [47]. 15 It is interesting to see if blob rings admit the elastic-type instability with or without the non-perturbative correction caused by the blob-blob interaction. ...

... Here, the terms 'thin' and 'fat' are used regarding the applicability of the blob approximation.15 The elastic instability found in the large D ring[27] was questioned in ref.[47]. ...

In the large dimension ($D$) limit, the Einstein's equation reduces to the effective theory on the horizon surface, drastically simplifying the black hole analysis. Especially, the effective theory on the black brane was successful in describing the non-linear dynamics not only on the black brane, but also on compact black objects which are encoded as solitary Gaussian-shaped lumps, blobs. In the rigidly rotating ansatz, in addition to axisymmetric deformed branches, various non-axisymmetric solutions as black bars are found by using this blob approximation, which only stay stationary at the large $D$ limit. In this article, we demonstrate the blob approximation has a wider range of applicability by formulating the interaction between blobs and subsequent dynamics. We identify that this interaction occurs via thin necks connecting blobs. Especially, black strings are captured quite well by this approximation sufficiently away from the perturbative regime. Highly deformed black dumbbells and ripples are also found to be tractable in the approximation. By defining the local quantities, the effective force acting on distant blobs are evaluated as well. These results reveal a new aspect of the large $D$ effective theory as the dynamics of interactive blobs -- brane blobology.

... Subsequently, we present the derivation of the blackfold perturbation equations used in the main text. For further discussion on embedding geometry and blackfold perturbation equation, see [27][28][29]. ...

Using the blackfold approach, we study the stability of the KPV (Kachru-Pearson-Verlinde) metastable state of anti-D3 branes at the tip of the Klebanov-Strassler throat. With regard to long-wavelength deformations observable to blackfold, in the regime $p/M \in (0, p_{new})$ with $p_{new} \approx 0.0801446$, we found no classical instabilities. However, in the regime $p/M \in (p_{new}, p_{crit})$ with $p_{crit} \approx 0.080488$ the original metastable threshold, we observe that the KPV state is tachyonic in the radial direction. We comment on the relation of this result to existing results on the stability of the KPV state

Recent progress in taking the large dimension limit of Einstein’s equations is reviewed. Most of the analysis is classical and concerns situations where there is a black hole horizon, although various extensions that include quantum gravitational effects are discussed. The review consists of two main parts: the first is a discussion of general aspects of black holes and effective membrane theories in this large dimension limit, and the second is a series of applications of this limit to interesting physical problems. The first part includes a discussion of quasinormal modes that leads naturally into a description of effective hydrodynamiclike equations that describe the near-horizon geometry. There are two main approaches to these effective theories, a fully covariant approach and a partially gauge-fixed one, which are discussed in relation to each other. In the second part the applications are divided up into three main categories: the Gregory-Laflamme instability, black hole collisions and mergers, and the anti–de Sitter/conformal field theory correspondence (AdS/CFT). AdS/CFT posits an equivalence between a gravitational theory and a strongly interacting field theory, allowing the spectrum of applications to be extended to problems in hydrodynamics, condensed matter physics, and nuclear physics. The final, shorter part of the review describes further promising directions where there have been, as yet, few published research articles.