Pathologies in Asymptotically Lifshitz Spacetimes

Keith Copsey; Robert Mann

REVIEW

Pathologies in Asymptotically Lifshitz Spacetimes

Keith Copsey, Robert Mann

High Energy Physics - Theory · Submitted 15 Nov 2010 · 49 references

Pith's verdict: The naked-singularity calculation is solid and the energy analysis is careful, but the headline claim that the initial value problem is 'generically' ill-posed is asserted across a gap the paper does not close.

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arXiv ID1011.3502 (version 2)

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CitationKeith Copsey, Robert Mann. "Pathologies in Asymptotically Lifshitz Spacetimes." arXiv:1011.3502 (2010). https://pith.science/paper/LFGSZU7A

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Editor's note

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Two things worth your time here, and one that needs pushing back on. The PPON calculation in Section 2 is clean work: Eq. 2.12 shows the tidal components diverge as r to 0 for z not equal to 1, and since r = 0 is a null surface reachable by both ingoing and outgoing rays, the singularity is naked. Part of this was in the air already (they credit [8] and [3] for the tidal divergence and the null-singularity picture), but putting it together with the Penrose diagram and the static-frame argument is useful. The static-frame observation is the sharp bit: curvature invariants are constant throughout the space, so alpha-prime corrections cannot blow up near r = 0 without blowing up everywhere, asymptotics included. That is a plausibility argument, not a proof, and they say so. Fine by me as stated. I would also flag Eqs. 3.32 to 3.37 as better than the abstract lets on. Showing that the energy needs the vector-field surface term to stay invariant under r to r(1 + alpha r^{-z-2}) is the kind of bookkeeping people skip, and they do it properly. Now the soft spot, and it is the load-bearing one. The abstract's 'generically not well-posed' (abstract:s3, s4) rests on two legs, both thinner than advertised. Eq. 3.53 is an initial-acceleration argument, and the authors themselves worry it may be contaminated by orders of limits. Section 3.6 then assumes Lifshitz asymptotics hold at all times, so it characterizes the solutions that already maintain them; it is a necessary condition, not a demonstration that generic constraint-satisfying data falls outside that class. The bridge, that the interior-regularity-fixed coefficient of r^{-z} in W generically differs from what the expansion demands, is asserted. The exact solutions at Eqs. 3.45 and 3.47 are the tuned z = 2 and z = 4 cases, so they do not probe the generic failure either. Note also that the classifier found no stated limitations; that is wrong. The authors flag the order-of-limits worry and the transverse-symmetry restriction explicitly, and frame the stringy argument as conservative. They are more upfront than the abstract suggests. Send it out. The singularity section will survive review; the initial-value section needs a referee who will make them either close the generic step or soften the quantifier. Worth citing for the singularity and the mass formula (Eqs. 3.72, 3.73).

Referee report

Significance

If correct, this is an important negative result for the program of constructing gravitational duals to anisotropically scaling field theories. The strengths are real: the PPON calculation (Eq. 2.12) is explicit and checkable; the constraint solution (Eqs. 3.38–3.43) is carried out in closed form; the asymptotic expansion (Eqs. 3.66–3.68) is worked to the order that matters; and the mass formula (Eqs. 3.72–3.73) is derived from the on-shell Hamiltonian with no fitted parameters. The known exact Lifshitz black holes (Eqs. 3.45, 3.47) are recovered as special cases, which is a useful consistency check. The paper is also careful to hedge where it should, for example restricting the interior-regularity analysis to d≤7 and noting the absence of embeddings of high-dimensional Lifshitz solutions into ten- or eleven-dimensional supergravity. The main open questions are ones of scope and framing rather than of the underlying derivations.

Major objections (6)

[Abstract (s3); §3.5–3.6] The headline phrase 'the initial value problem itself ... turns out to generically not be well-posed' uses 'well-posed' in a nonstandard sense. Hadamard well-posedness of a Cauchy problem concerns existence, uniqueness, and continuous dependence on initial data for the evolution equations. What §§3.5–3.6 actually establish is that the constraint equations, together with interior regularity and the Lifshitz boundary conditions, overconstrain the data: generic constraint-satisfying data is not in the class of solutions that maintain the asymptotics at all times. That is a statement about the compatibility of boundary conditions with the constraints, not about the well-posedness of the evolution. I would ask the authors either to define precisely the sense in which they use 'well-posed' or to rephrase the abstract and closing remarks in terms of incompatibility of generic data with the boundary conditions.
[§3.5, Eq. (3.53); §3.6, Eqs. (3.66)–(3.73)] The bridge between the two prongs of the ill-posedness argument is asserted rather than demonstrated. Section 3.5 explicitly flags that the initial-acceleration computation 'may be contaminated by some subtlety involving orders of limits for large r and small times' and defers to §3.6. Section 3.6, however, is a necessary-condition analysis: it characterizes solutions that already maintain Lifshitz asymptotics at all times and finds that the mass is fixed by the asymptotic vector field (Eqs. 3.72–3.73). The step that closes the argument — that the coefficient of ρ^{-z} in W(ρ), fixed by interior regularity, is generically independent of the coefficient of ρ^{-z} in f(ρ) and hence fails to cancel it — appears in a single sentence on p.30 ('For generic regular initial data the coefficient of W(ρ) at order ρ^{-z} is independent of the coefficient of f(ρ) at order ρ^{-z}'). This is a codimension-one cancellation argument and should be stated as such, with an explicit count of the free data versus the conditions. As written, the exact solutions (3.45) and (3.47) are precisely the tuned cases and do not probe the generic failure.
[§2, pp.12–13 (discussion of the RG-flow solution of [8])] The paper asserts that 'the naively smooth solution which tries to interpolate between Lifshitz asymptotics, including the renormalization group flow solution of [8], has a naked singularity very much like the naive ground state discussed before.' This is a claim about a specific published solution, and it rests on the assumed near-origin behavior (2.24)–(2.25) feeding into the PPON divergence (2.28). The manuscript does not verify that the Kachru–Liu–Mulligan flow solution actually has that near-origin behavior. Either an explicit check of the small-r expansion of that solution or a numerical evaluation of the PPON components (2.28) on it would make the claim secure; as it stands it is an assertion about someone else's result.
[§2, Eqs. (2.9)–(2.12)] The argument that the r=0 singularity cannot be resolved by α' corrections is that all curvature invariants contracted into the static frame (2.9) are constant, so stringy corrections cannot grow near r=0 without growing everywhere. But the static frame degenerates at r=0, and the divergence of the PPON components (2.12) is exactly the statement that the boost relating the static frame to a freely falling frame diverges there. Bounded curvature scalars therefore do not by themselves exclude a stringy resolution of the kind that is believed to operate for singular plane waves [22], where invariants are also bounded and yet the singularity is often regarded as resolvable. The paper should either explain why the plane-wave precedent does not apply here or soften the abstract's 'not resolvable by any known stringy effect' to reflect the scope of the argument.
[§3.2, Eqs. (3.72)–(3.73)] The statement that the z=2 Lifshitz black holes of [11,12,14] are 'exactly massless, despite having a wide range of sizes' is a strong claim that depends on the choice of standard Henneaux–Teitelboim boundary conditions and on setting K1=0 (i.e., forbidding asymptotic logarithmic terms). Footnote 4 already acknowledges a known disagreement between the background-subtraction and holographic-renormalization definitions of normalizability for z≤2. The masslessness statement should be presented explicitly as scheme-dependent, with a sentence saying what happens under the alternative definition, so that readers do not take it as a statement about the physical mass of those solutions.
[Abstract (s3, s4, s6); §3.2] The abstract's 'generically' is unqualified, but the entire initial-data analysis is restricted to maximal transverse symmetry (planar, spherical, or hyperbolic sections) with a specific massive-vector action. Section 3.2 justifies the restriction with 'it is reasonably clear from the constraint (3.16) that any other contributions will only serve to increase the energy,' which is an assertion rather than a demonstration. Since the abstract and closing remarks present the conclusion as a general statement about asymptotically Lifshitz spacetimes, the symmetry restriction should be stated there, and the claim that generic non-symmetric data cannot evade the conclusion should either be argued or flagged as a conjecture.

Minor objections (7)

[References] References [29] and [32]–[43] do not appear to be cited anywhere in the text. That is a block of thirteen references (ADM, Lovelock, and topological black hole literature) that appear to be left over from an earlier draft. Please cite them or remove them.
[Throughout] There are a number of typographical errors that should be corrected: 'beomes' (p.7), 'prpo rtional' (p.12), 'correspon' (p.12), 'similiar' (p.20), 'diffeormorphism' (p.21), 'Heannaux-Teitelboim' (p.19), 'simplicitly' (p.26), 'asympotics' (p.28), 'pertubations' (p.32), 'whether whether' (p.33), and 'thory' in the title of ref. [35].
[Notation, §2–§3] The symbols f(r) in Eq. (3.25), f1(r) in Eq. (2.17), and f0(r) in Eq. (3.28) are easily confused, and W(r) is used in both §2 and §3 with different meanings. Consider renaming one of the metric functions to avoid ambiguity.
[Figure 1] The Penrose diagram should label the past and future null singularities and the asymptotic boundary explicitly; the caption is a single line and does not identify the regions shown.
[Eq. (2.12)] It would help to state explicitly that the frame (2.10) is orthonormal only for r>0, and that the divergence as r→0 reflects the degeneracy of the static frame rather than a pathology of the frame itself.
[§3.6, p.28] The sentence 'the above equations have to be interpreted with a bit of care' could be sharpened to say precisely which powers in the expansion are reliable and which require the non-linear completion.
[§2, p.8] The claim that the singularity is 'naked' would benefit from one explicit sentence noting that there is no horizon at r=0 for z≠1, so the singularity is visible to observers at any r>0.

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The paper

The paper claims

The most-studied Lifshitz spacetime, proposed as the gravitational dual of anisotropically scaling field theories, has a naked null curvature singularity at its putative ground state and an initial value problem that is not generically well

The reading

Lifshitz spacetimes are the proposed gravitational duals of field theories that scale space and time differently, with a dynamical exponent $z$. The best-studied example, the flat-section metric with only a massive vector field and a cosmological constant, has been widely treated as the ground state of such a dual. Copsey and Mann argue that this state is not a legitimate ground state: $r=0$ is a null surface, both ingoing and outgoing rays reach it in finite affine parameter, and tidal forces measured by a freely falling observer diverge there for any $z>1$, so it is a naked singularity. They then argue the divergence cannot be cured by string tension or loop corrections, because the curvature measured in the static frame is constant throughout the spacetime; any correction large enough to matter at the singularity would be equally large in the asymptotic region where the dual field theory lives. Turning to stability, they set up the Hamiltonian initial data problem and find that imposing Lifshitz asymptotics at all times overconstrains the interior: the mass is fixed entirely by the asymptotic behavior of the vector field, so generic normalizable data either evolve into a state that breaks the asymptotics or must be excluded.

What carries the argument

Two tools carry the argument. The first is the parallelly propagated orthonormal frame (PPON): a frame built along a geodesic so that its components are what a freely falling observer actually measures. Curvature invariants computed in a static frame stay finite at $r=0$, which is why the singularity is easy to miss; the PPON tidal components diverge, which is what makes it real. The second is the Hamiltonian decomposition of the Einstein-plus-massive-vector action, in which the constraints are solved on a spacelike slice and the on-shell value of the Hamiltonian, including its surface terms, defines the energy. The surface-term analysis shows that for $z>1$ the energy is not diffeomorphism invariant unless the asymptotic value of the vector potential is included, and that the resulting mass is fixed by the asymptotic falloff of the electric field rather than by any interior data.

Extended reading notes

Core claim

The paper makes two linked claims. First, the flat-section Lifshitz metric with $z>1$ is not a smooth geometry with a Poincare-like horizon at $r=0$; it is a spacetime with a naked null curvature singularity. The authors show geodesic incompleteness directly, then compute the Riemann tensor in a parallelly propagated orthonormal frame along radial geodesics and find components such as $R_{0i0i}$ and $R_{1i1i}$ diverging as $r^{-2z}$ (Eq. 2.12), so the divergence is physical rather than a coordinate artifact. Because the curvature invariants built from a static orthonormal frame are constant everywhere, they argue no known stringy correction can resolve the singularity locally without also altering the asymptotics. Second, they examine the initial value problem for asymptotically Lifshitz data with maximal transverse symmetry. Solving the Hamiltonian, momentum, and gauge constraints leaves the radial profile of the vector field essentially free, but the time evolution of that data produces acceleration in the metric that, for generic interior behavior, fails to fall off fast enough to preserve Lifshitz asymptotics. The asymptotic expansion confirms the reverse direction: enforcing the asymptotics at all times ties the normalizable metric coefficient to the normalizable coefficient of the vector field, so the mass of any such solution is determined by the asymptotic vector field alone (Eqs. 3.72-3.73) and is independent of the deep interior.

Load-bearing premise

The claim that the initial value problem is generically ill-posed is derived only within maximal transverse symmetry, a fixed massive-vector action, and standard normalizability conditions; the step from that symmetric sector to fully generic data is asserted rather than demonstrated.

Editorial extensions

If this is right

Flat-section Lifshitz spacetimes cannot serve as the ground state of a holographic dual in the way asymptotically AdS spacetimes do, because the state that would play that role is singular.
The candidate regular interiors examined in the paper, a Poincare-type horizon and an extremal black hole, are each shown to be incompatible with the field equations or to hide a singularity of the same kind, at least up through seven dimensions.
If the mass of a transversally symmetric asymptotically Lifshitz solution is fixed by the asymptotic vector field, then perturbations that change the mass while preserving the asymptotics of the matter field are forbidden.
Any holographic construction that flows in the interior to a Lifshitz region should be checked for the same class of instability and for singularities of the type found in flat slicing.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The argument suggests a general diagnostic: for any proposed non-AdS holographic asymptotics, one should check whether the asymptotic expansion leaves an undetermined coefficient at the same order that interior regularity fixes. If it does not, the boundary theory may have no independent stress-tensor data to fluctuate.
The observation that curvature in the static frame is constant everywhere could be turned into a sharper no-go statement: any higher-derivative correction that is perturbatively small asymptotically is also perturbatively small at the singularity, so the singularity is either resolved everywhere or nowhere.
The paper's transverse-symmetry restriction means the ill-posedness result is established in a sector with no local gravitational degrees of freedom. Whether adding tensor modes restores a well-posed problem, as it does in AdS, is the natural next calculation and is not settled here.

Original abstract

There has been significant interest in the last several years in studying possible gravitational duals, known as Lifshitz spacetimes, to anisotropically scaling field theories by adding matter to distort the asymptotics of an AdS spacetime. We point out that putative ground state for the most heavily studied example of such a spacetime, that with a flat spatial section, suffers from a naked singularity and further point out this singularity is not resolvable by any known stringy effect. We review the reasons one might worry that asymptotically Lifshitz spacetimes are unstable and employ the initial data problem to study the stability of such systems. Rather surprisingly this question, and even the initial value problem itself, for these spacetimes turns out to generically not be well-posed. A generic normalizable state will evolve in such a way to violate Lifshitz asymptotics in finite time. Conversely, enforcing the desired asymptotics at all times puts strong restrictions not just on the metric and fields in the asymptotic region but in the deep interior as well. Generically, even perturbations of the matter field of compact support are not compatible with the desired asymptotics.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.30 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.21 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

There has been significant interest in the last several years in studying possible gravitational duals, known as Lifshitz spacetimes, to anisotropically scaling field theories by adding matter to distort the asymptotics of an AdS spacetime. We point out that putative ground state for the most heavily studied example of such a spacetime, that with a flat spatial section, suffers from a naked singularity and further point out this singularity is not resolvable by any known stringy effect. We review the reasons one might worry that asymptotically Lifshitz spacetimes are unstable and employ the initial data problem to study the stability of such systems. Rather surprisingly this question, and even the initial value problem itself, for these spacetimes turns out to generically not be well-posed. A generic normalizable state will evolve in such a way to violate Lifshitz asymptotics in finite time. Conversely, enforcing the desired asymptotics at all times puts strong restrictions not just on the metric and fields in the asymptotic region but in the deep interior as well. Generically, even perturbations of the matter field of compact support are not compatible with the desired asymptotics.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	There has been significant interest in the last several years in studying possible gravitational duals, known as Lifshitz spacetimes, to anisotropically scaling field theories by adding matter to distort the asymptotics of an AdS spacetime.	background 1.00	0.04	0.02	asserted 0.97
s1	We point out that putative ground state for the most heavily studied example of such a spacetime, that with a flat spatial section, suffers from a naked singularity and further point out this singularity is not resolvable by any known stringy effect.	result 0.94	0.91	0.38	asserted 1.00
s2	We review the reasons one might worry that asymptotically Lifshitz spacetimes are unstable and employ the initial data problem to study the stability of such systems.	contribution 0.60	0.58	0.06	asserted 0.48
s3	Rather surprisingly this question, and even the initial value problem itself, for these spacetimes turns out to generically not be well-posed.	result 1.00	0.85	0.36	asserted 1.00
s4	A generic normalizable state will evolve in such a way to violate Lifshitz asymptotics in finite time.	result 1.00	0.81	0.35	asserted 1.00
s5	Conversely, enforcing the desired asymptotics at all times puts strong restrictions not just on the metric and fields in the asymptotic region but in the deep interior as well.	result 0.90	0.75	0.14	asserted 0.99
s6	Generically, even perturbations of the matter field of compact support are not compatible with the desired asymptotics.	result 0.98	0.79	0.16	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 0.75.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Penrose diagram for flat Lifshitz spacetimes

Figures reproduced from arXiv:1011.3502 by the authors.

References

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M. H. Dehghani and R. B. Mann,“Lovelock-Lifshitz Black holes”, JHEP 1007 (2010) 019 [arXiv:1004.4397]

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G. Dotti, J. Oliva, and R. Troncoso,“Exact solutions fo r the Einstein-Gauss- Bonnet thory in ﬁve dimensions: Black holes, wormholes and s pacetime horns”, Phys. Rev. D 76 (2007) 064038 [arXiv:0706.1830]

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E. Ayon-Beato, A. Garbarz, G. Giribet and M. Hassaine, “Lifshitz black hole in three Dimensions”, Phys. Rev. D 80 (2009) 104029 [arXiv:0909.1347]

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E. Ayon-Beato, A. Garbarz, G. Giribet and M. Hassaine, “ Analytic Lifshitz black holes in higher dimensions”, JHEP 1004 (2010) 030 [arXiv:1001.2361]

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Forward citations

Cited by 12 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Non-Relativistic Chern-Simons Theories and Three-Dimensional Horava-Lifshitz Gravity
hep-th 2016-04 conditional novelty 6.0 of 10

Three-dimensional Horava-Lifshitz gravity is shown to equal a Chern-Simons gauge theory on extended Bargmann, Newton-Hooke, and Schroedinger algebras, with the Schroedinger case yielding a new z=2 Lifshitz-vacuum gravity.
Holographic Models for Theories with Hyperscaling Violation
hep-th 2012-12 accept novelty 6.0 of 10

A dilaton plus massive-vector gravity model reproduces the full family of hyperscaling-violating Lifshitz geometries, including the theta=1, z=3/2 case, without instability in the radial sector.
Towards non-AdS holography in 3-dimensional higher spin gravity
hep-th 2011-12 conditional novelty 6.0 of 10

A boundary term added to the Chern-Simons action of 3D higher spin gravity admits non-anti-de Sitter asymptotic geometries, demonstrated with explicit spin-4 Schrodinger, Lifshitz and warped AdS connections.
Lifshitz Singularities
hep-th 2011-11 accept novelty 6.0 of 10

Test strings crossing the null singularity of a Lifshitz spacetime become infinitely excited, so the geometry is singular in string theory and cannot describe the deep infrared of a Lifshitz critical point.
Field theories with anisotropic scaling in 2D, solitons and the microscopic entropy of asymptotically Lifshitz black holes
hep-th 2011-07 conditional novelty 6.0 of 10

A high/low temperature duality for Lifshitz algebras yields a generalized Cardy formula that reproduces the entropy of asymptotically Lifshitz black holes from the mass of a gravitational soliton.

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Thermodynamics and Instabilities of a Strongly Coupled Anisotropic Plasma
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A regular finite-temperature supergravity solution dual to an anisotropic N=4 plasma yields a conserved stress tensor with a conformal anomaly and a phase diagram in which the homogeneous anisotropic phase can be unst...
Black holes and black branes in Lifshitz spacetimes
hep-th 2011-05 accept novelty 6.0 of 10

Exact analytic Lifshitz black hole and black brane solutions are constructed for arbitrary z and dimension, and their phase diagrams are shown to change qualitatively at z = 2.
Resolving Lifshitz Horizons
hep-th 2012-02 conditional novelty 5.0 of 10

In magnetic Einstein-Maxwell-dilaton black branes, loop corrections to the gauge kinetic function stabilize the dilaton and replace the singular Lifshitz horizon with a regular AdS2 x R2 geometry in the deep infrared.
Black hole chemistry: thermodynamics with Lambda
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Treating the cosmological constant as pressure in black hole thermodynamics yields an extended dictionary with enthalpy, thermodynamic volume, and chemical-like phase transitions including Van der Waals behavior, reen...
Lifshitz holography
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A structured review of the holographic dictionary for Lifshitz backgrounds, arguing that the bulk matter supporting the geometry sources the non-relativistic energy-momentum complex plus scalar operators rather than a...
Horizons, holography and condensed matter
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Charged asymptotically AdS spacetimes fall into two families, those whose electric flux emerges from a horizon and those where charged matter sources it, and this split tracks the field theory distinction between frac...
Ho\v{r}ava Gravity at a Lifshitz Point: A Progress Report
gr-qc 2017-01 conditional novelty 1.0 of 10

Horava gravity remains an unfinished but actively studied route to quantum gravity, with four main versions whose viability depends on narrow parameter choices and unresolved strong-coupling and cosmological tests.

How the literature uses it

65 arXiv papers cite this in 87 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 66 sentences without a confident label

Contests it 1 paper

The reason why the analysis from [ref] does not apply to this particular flow is that we allow for the presence of leading logs.

On the Marginally Relevant Operator in z=2 Lifshitz Holography · The special nature of $z=d_s$ · contest 0.71

Uses its method or tool 1 paper

Through a compactification of the spacetime similar to that of AdS or Lifshitz spacetimes [ref] , it is possible to identify the light ray kinematics and portray the existence of a singularity.

Anisotropic Schr\"odinger black holes with hyperscaling-violation · Basics on hyperscaling violating Schrödinger spacetimes · 0.81

Compares with it 5 papers

This agrees with the analysis of [ref] .

Penrose Limit and Non-relativistic geometries · Hyperscaling violating Lifshitz spacetimes · 0.85
This result is reminiscent of the behaviour of the bulk singularity, where curvature diverges as $1/τ^2$ along the worldline of geodesics which approach the singularity (where $τ$ is the proper time), with a coefficient which vanishes as $z →1$ [ref] .

Scattering amplitudes in Lifshitz spacetime · Introduction · 0.90
These singularities are of the same type as those found in Lifshitz space-times [ref] .

Classification of certain asymptotically AdS space-times with Ricci-flat boundary · Gravity solutions of type II and type III: null deformations · 0.95

Cites it as support 2 papers

As noted in [ref] , this is reflected in a divergence of the components of the Riemann tensor in a parallely propagated orthonormal frame as $r →0$.

Flows involving Lifshitz solutions · Asymptotically AdS flows · 0.98
There exists another motivation to consider the asymptotically AdS spacetime: in a recent study using the bottom up approach, it is shown that the Lifshitz spacetimes are unstable [ref] and [ref] .

Approximate strange metallic behavior in AdS · Introduction · 1.00

Cites it as background 12 papers

One of the important issues being investigated currently is the stability of such Lorentz violating geometries [ref] .

Stability of Bianchi attractors in Gauged Supergravity · Introduction · 0.88
Before turning to the holographic Green's function, we recall that Lifshitz spacetimes are characterized by a tidal singularity at the horizon [ref] .

Spatial Anisotropy in Nonrelativistic Holography · Singularities · 0.92
In [ref] , possible singularities in hyperscaling violating Lifshitz spacetimes were exhaustively studied and constraints due to null energy condition were analyzed.

Penrose Limit and Non-relativistic geometries · Hyperscaling violating Lifshitz spacetimes · 0.92

What it cites · 31 sentences without a confident label

Compares with 1 paper

Hence it appears futile to appeal to string loop effects to resolve this singularity.[This feature distinguishes the present case from extremal Dp-branes for $p > 3$ where one again has a null singularity in the metric [ref] but one also has a diverging dilaton and so stringy corrections can resolve the singularity.] Note the proposition that one loses control of the calculation throughout a low-curvature, weakly coupled region, especially considering the embeddings of these solutions and action into honest supergravity solutions, would actually be quite radical.

Naked Black Holes · Flat Lifshitz solutions and interior regularity · 0.93

Cites as support 2 papers

In particular, explicit lifts involving a constant dilaton [ref] and a dilaton that is independent of the Lifshitz directions [ref] have both been presented.

Lifshitz solutions in supergravity and string theory · Flat Lifshitz solutions and interior regularity · 0.97
In particular, explicit lifts involving a constant dilaton [ref] and a dilaton that is independent of the Lifshitz directions [ref] have both been presented.

Lifshitz Solutions of D=10 and D=11 supergravity · Flat Lifshitz solutions and interior regularity · 0.97

Cites as background 6 papers

An extended class of vacuum solutions for a sort of higher-dimensional dilaton gravity with general $z$ appeared as an early example [ref] ; shortly afterward black hole solutions, both exact (for $z=2$) [ref] and numerical (for more general values of $z$) [ref] , were discovered.

Black holes in asymptotically Lifshitz spacetime · Introduction · 0.92
An extended class of vacuum solutions for a sort of higher-dimensional dilaton gravity with general $z$ appeared as an early example [ref] ; shortly afterward black hole solutions, both exact (for $z=2$) [ref] and numerical (for more general values of $z$) [ref] , were discovered.

Lifshitz Topological Black Holes · Introduction · 0.92
An extended class of vacuum solutions for a sort of higher-dimensional dilaton gravity with general $z$ appeared as an early example [ref] ; shortly afterward black hole solutions, both exact (for $z=2$) [ref] and numerical (for more general values of $z$) [ref] , were discovered.

Black holes in asymptotically Lifshitz spacetimes with arbitrary critical exponent · Introduction · 0.92

Discussion

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