Perturbative Solutions Linearized in Chemical Potentials

A different kind of simplification arises when linearizing in the chemical potentials.

Expanding the metric (13.11) in terms of the chemical potentials,

gµν = ¯gµν+hµν+O(µ²_M, µ²_L, µMµL) (13.21) with the background line-elementg¯µνdx^µdx^ν given by the right hand side of (10.1), yields for the linear terms

h_µνdx^µdx^ν =2 MµM+NµL

du²+ r²µL+N µM

2 dudϕ−2µM drdu + 2 r µ⁰_L−µ⁰⁰_Mdu²−2r µ⁰_M dudϕ . (13.22) The terms in the second line vanish for constant chemical potentials.

Comparison with Holographic Dictionary

From a holographic perspective, the first two terms in the linearized solution (13.22) show the typical coupling between sources (chemical potentials) and vacuum ex-pectation values (canonical charges). The r²µLdudϕterm and the µMdrdu term correspond to the essential terms in the two towers of non-normalizable²solutions to the linearized equations of motion.

In the holographic dictionary, these non-normalizable contributions should be dual to sources of the corresponding operators in the dual field theory. Indeed, this is what happens as shown in [72]. Note, however, that [72] worked in Euclidean signature, restricted to zero mode solutions and imposed axial gauge for the non-normalizable solutions to the linearized Einstein equations on a flat space background, so a direct comparison is not straightforward. Exploiting the interpretation of constant chemical potentials as modifications of lapse and shift (see section13.4.2) one can interpret the results of [72] as follows (see their section 3.4): their quantityδξJ corresponds precisely to the (linearized) even chemical potential δξ_J ∼µL, and their quantity δξ_M corresponds to twice the (linearized) odd chemical potential,δξ_M ∼2µ_M. This identification is perfectly consistent with the holographic interpretation summarized above.

2Here and in the following the attribute “non-normalizable” always means “breaking the Barnich–

Compère boundary conditions” [51] or the corresponding spin-3 version [150,154].

13.5 Applications

After having introduced chemical potentials in spin-2 and spin-3 gravity in three dimensional flat space I now want to present some applications thereof, such as the entropy of flat space cosmologies with spin-3 charges and the corresponding free energies in the following section³.

13.5.1 Entropy

In this section I will determine the entropy of flat space cosmologies including spin-3 charges, by solving holonomy conditions of the connection Aaround the non-contractibleϕ-cycle. In addition, I will restrict myself to solutions with constant chemical potentials in order to carry out this calculation.

Using the hatted trace introduced in (12.2) one can write the entropy of a spin-3 charged flat space cosmology, similar to a spin-3 charged BTZ black hole [164] as

S= 2kβLTrb auaϕ

EOM=βL 2(1 +µM)Q_M+ 2µLQ_L+ 3µVQ_V+ 3µUQ_U. (13.23) The quantity βL is not necessarily the inverse temperature, but rather the length of the relevant cycle appearing in the holonomy condition below. The zero mode chargesQ_iare displayed in (13.6).

The holonomy condition I want to solve is given by exp iβLau

= 1l. (13.24)

This condition is completely analogous to the holonomy conditions for higher spin black holes in AdS [159]. To solve the holonomy condition (13.24) one can use the representation summarized in AppendixA.2.3in terms of9×9matrices. By a similarity transformation one can diagonalize the ad-part of a generic matrix of the form (A.22). A matrix of this form is easily exponentiated. Assuming that ad has zero as eigenvalue with geometric and algebraic multiplicitynand denotingv=A⁻¹odd yields

exp A⁻¹adA_8×8 A⁻¹odd_8×1

3The following section of applications is based on a collaboration with Grumiller, Gary and Rossel first presented in [III].

In the case at hand one has n = 2 which is the rank of sl(3,R)⁴. The holonomy condition (13.24) is then solved by the relations

λ_a= 0mod 2π

β_L, a= 1. . .6 ; v_m = 0, m= 1. . .2. (13.27) The first set of relations (13.27) is precisely the same as for one chiral half of AdS spin-3 gravity. Therefore, one must be able to represent these conditions in the same way as it was done in AdS. In fact, a plausible guess for the two holonomy conditions that follow from the first set of relations (13.27) is given by

1

4Tr a_ua_u

=0 =Mµ²_L + 24Vµ_Lµ_U+⁴₃M²µ²_U = 4π²

β_L² , (13.28)

1 4

pdetau

=0 =Vµ³_L +¹₃M²µ²_LµU+ 4MVµ_Lµ²_U−₂₇⁴M³µ³_U+ 32V²µ³_U= 0.

(13.29) Since the matrixA⁻¹adAis diagonal, it must lie in the Cartan subalgebra ofsl(3,R).

Diagonalizing simultaneouslyL₀ andU₀ one finds

A⁻¹adA=diag 0,0, f_L+2f_U, f_L−2f_U,−f_L+2f_U,−f_L−2f_U,2f_L,−2f_L, (13.30) with some functions f_L, f_U of the charges and chemical potentials that can be determined by explicitly calculating the characteristic polynomial of the matrixiβLau

for the eigenvaluesλas derived from the solution (13.3) (with constant charges and chemical potentials) and comparing it with the characteristic polynomial that follows from (13.30). The first set of relations (13.27) yields the conditions

fL= mπ βL

, fU = (n−^m₂)π βL

, n, m∈Z. (13.31)

Thus, the first half of the holonomy conditions leads to a discrete family of solutions parametrized by two integers n and m. For the choicem = 2 and n = 1 these conditions reproduce precisely the guess (13.28) and (13.29). This choice is unique by requiring that in the absence of spin-3 chemical potentials and spin-3 charges the holonomy conditions reduce to the ones for flat space cosmologies. I will therefore always make this choice in the following.

So far only half of the holonomy conditions have been solved. The other half emerges from imposing the second set of relations (13.27). After a straightforward calculation⁵ one finds that one of these conditions is linear in the charges and

4Since the even part in this representation corresponds tosl(3,R).

5There are numerous different ways to obtain these results, but it is not always easy to extract the simple conditions (13.32) and (13.33). For instance, one can contract the AdS holonomy conditions using the map (13.5), but this leads naturally to nonlinear relations between charges and chemical potentials. Two combinations of these relations immediately provide the holonomy conditions (13.28) and (13.29), but it takes a bit of work to extract the other two conditions in their simplest form. Alternatively, one can explicitly construct the matrixAin (13.25) that diagonalizes the sl(3,R)part of the generators and then determine the two eigenvectors associated with the two

chemical potentials, while the other is quadratic in the charges and linear in the chemical potentials

M(1 +µM) +Lµ_L+ 12VµV+ 16UµU = 0, (13.32) 9V(1 +µM) + 6UµL+M²µV+ 2LMµ_U = 0. (13.33) These results are considerably simpler than the corresponding holonomy conditions in AdS, which are at least quadratic in chemical potentials and charges.

The linear holonomy condition (13.32) simplifies the entropy (13.23) to

S=β_L µLQ_L+µUQ_U. (13.34) For the special caseµU = 0, the entropy (13.34) depends only on spin-2 charges and chemical potentials. Moreover, the solution to the four holonomy conditions (13.28), (13.29), (13.32), (13.33) is given by

M= 4π² For that case entropy is given by the Bekenstein–Hawking area law withk= _4G¹

N

S

µU=0=kβ_L|µ_LL|=k2π|L|

√M. (13.36)

The absolute values used above ensure that entropy is positive regardless of the sign of the chargeL. The inverse temperature

β =− ∂S

then coincides with the spin-2 result (see e.g. [63]; note that in their conventions M=r²₊and|L|=|r₀r₊|)

T = 1 2π

M^3/2

|L| . (13.38)

The minus sign in the definition (13.37) can be seen as a remnant of the inner horizon first law of black hole mechanics [148, 149, 165–167] as explained in [63] when thinking about a flat space cosmology as a flat space limit of a non-extremal BTZ black hole as elaborated on at the beginning of Chapter10. From the corresponding first law

−dQ_M=T dS+ Ω dQL, (13.39) one can deduce the angular potential

Ω =−T ∂S

zero eigenvalues. This approach makes it clear from the start that the remaining two holonomy conditions must be linear in the chemical potentials.

which again coincides with the spin-2 result [63].

In the general caseµU6= 0not all holonomy conditions are linear. Instead, one has to solve one quadratic and one cubic equation, similar to the AdS case. Defining µ = µLµU and η = µL/µU + ¹₉M²/V the holonomy conditions (13.28), (13.29) simplify to

η³+η4M − M⁴ 27V²

+ 32V −16M³

27V + 2M⁶

729V³ = 0, (13.41) µ= 4π²

β²_L µL

µU

M+ 24V+4µU

3µL

M²⁻¹. (13.42)

Solving the cubic equation (13.41) yields a result for the ratio µL/µU, which can then be plugged into the linear equation (13.42) to determine the product of the chemical potentials. The sign of the discriminantDof the cubic equation (13.41) is given by

signD=sign M³−108V². (13.43) IfDis negative there is exactly one real solution. This happens only if the spin-3 chargeV is sufficiently large or if the massMis negative. For a critical tuning of the charges,

criticality: 108V²=M³, (13.44)

the discriminant vanishes,D= 0, and there is a unique real solutionη= 0. However, the linear equation (13.43) has no finite solution for µ in this case. Therefore, starting from finite and positiveMit is not possible to smoothly increase the spin-3 chargeV beyond the critical value (13.44).

Henceforth, the following inequality will be assumed to hold

M> 108V^21/3 ≥0. (13.45)

In other words, from now on exclusively the case of positive discriminantD >0will be considered. In this case there are three real solutions forη. The resulting entropy is real for all three branches. However, only one branch recovers the same entropy (13.34) as for the spin-2 case in the limitV →0. It is exactly this branch that will be of interest for the following discussion.

On this particular branch, there is a neat way to express all results in terms of the charges M,L,U and a new parameterRthat depends on the ratio of spin-3 and spin-2 charges _M^V23, just like in the AdS case [159]

R −1

4R^3/2 = |V|

M^3/2, R>3. (13.46)

The restriction toR>3guarantees that indeed the correct branch is chosen.

The chemical potentials then read while the entropy is given by

S(M,L,R,P) = 2πk |L|

The expression for entropy (13.51) is the main result of this section. The pre-factor containing the spin-2 chargesM,Lcoincides with the spin-2 result (13.36). The spin-3 correction depends nonlinearly on one of the combinations of spin-3 charges, R, and linearly on the other,P.

For some purposes it can be useful to have a simpler perturbative result for entropy in the limit of small spin-3 chargeV(largeR) given by

S(M,L,V,U) = 2πk |L| I will close this discussion on entropy of higher-spin charged flat space cosmologies by addressing sign issues. The mass should be positive,M>0, motivated by the necessity of this condition in the spin-2 case. The sign ofLdoes not matter, which is why absolute values in the final result for entropy (13.51) have been used. Suppose thatL>0(L<0). Then one can exploit the sign ambiguity in the definitions ofµL, µUby choosingµL >0 (µL <0) so that the first term in (13.34) is always positive and thus entropy is positive in the limit of vanishing spin-3 fields. The sign ofV is taken care of by the definition (13.46), which ensures positiveRregardless of the sign ofV. Thus, the only remaining signs of potential relevance are the signs of the spin-3 chargeU and the corresponding chemical potentialµU. The latter is fixed through the sign choice ofµLexplained above, but the former is free to change, and this change is physically relevant. This implies that the quantityP defined in (13.52) can have either sign, so that the last term in the entropy (13.51) can have either sign. Demanding positivity of entropy then establishes an upper bound on U.

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