Gravitational Duality, Topologically Massive Gravity and Holographic Fluids

Abstract

Self-duality in Euclidean gravitational set ups is a tool for finding remarkable four-dimensional geometries. From a holographic perspective, self-duality sets a relationship between two a priori independent boundary data: the boundary energy–momentum tensor and the boundary Cotton tensor. This relationship, which can be viewed as resulting from a topological mass term for gravity boundary dynamics, survives under the Lorentzian signature and provides a tool for generating exact bulk Einstein spaces carrying, among others, nut charge. In turn, the holographic analysis exhibits perfect-fluid-like equilibrium states and the presence of non-trivial vorticity allows to show that infinite number of transport coefficients vanish.

Appendices

Appendix 1: On Vector-Field Congruences

We consider a manifold endowed with a space–time metric of the generic form

$$\displaystyle{ \mathrm{d}s^{2} = g_{\mu \nu }\mathrm{d}\mathrm{x}^{\mu }\mathrm{d}\mathrm{x}^{\nu } =\eta _{\mu \nu }E^{\mu }E^{\nu } }$$

(to avoid inflation of indices we do not distinguish between flat and curved ones). Consider now an arbitrary time-like vector field u, normalised as $u^{\mu }u_{\mu } = -1$, later identified with the fluid velocity. Its integral curves define a congruence which is characterised by its acceleration, shear, expansion and vorticity (see e.g. [81, 82]):

$$\displaystyle{ \nabla _{\mu }u_{\nu } = -u_{\mu }a_{\nu } + \frac{1} {D - 1}\varTheta \varDelta _{\mu \nu } +\sigma _{\mu \nu } +\omega _{\mu \nu } }$$

with³¹

$$\displaystyle\begin{array}{rcl} a_{\mu }& =& u^{\nu }\nabla _{\nu }u_{\mu },\quad \varTheta = \nabla _{\mu }u^{\mu }\;, {}\\ \sigma _{\mu \nu }& =& \frac{1} {2}\varDelta _{\mu }^{\rho }\varDelta _{\nu }^{\sigma }\left (\nabla _{\rho }u_{\sigma } + \nabla _{\sigma }u_{\rho }\right ) - \frac{1} {D - 1}\varDelta _{\mu \nu }\varDelta ^{\rho \sigma }\nabla _{\rho }u_{\sigma } {}\\ & =& \nabla _{(\mu }u_{\nu )} + a_{(\mu }u_{\nu )} - \frac{1} {D - 1}\varDelta _{\mu \nu }\nabla _{\rho }u^{\rho }\;, {}\\ \omega _{\mu \nu }& =& \frac{1} {2}\varDelta _{\mu }^{\rho }\varDelta _{\nu }^{\sigma }\left (\nabla _{\rho }u_{\sigma } -\nabla _{\sigma }u_{\rho }\right ) = \nabla _{[\mu }u_{\nu ]} + u_{[\mu }a_{\nu ]}\;. {}\\ \end{array}$$

The latter allows to define the vorticity form as

$$\displaystyle{ 2\omega =\omega _{\mu \nu }\,\mathrm{d}\mathrm{x}^{\mu } \wedge \mathrm{ d}\mathrm{x}^{\nu } =\mathrm{ d}\mathrm{u} +\mathrm{ u} \wedge \mathrm{ a}\;. }$$

(13.65)

The time-like vector field u has been used to decompose any tensor field on the manifold in transverse and longitudinal components. The decomposition is performed by introducing the longitudinal and transverse projectors:

$$\displaystyle{ U_{\nu }^{\mu } = -u^{\mu }u_{\nu },\quad \varDelta _{\nu }^{\mu } = u^{\mu }u_{\nu } +\delta _{ \nu }^{\mu }\;, }$$

(13.66)

where Δ _{μ ν} is also the induced metric on the surface orthogonal to u. The projectors satisfy the usual identities:

$$\displaystyle{ U_{\rho }^{\mu }U_{\nu }^{\rho } = U_{\nu }^{\mu },\quad U_{\rho }^{\mu }\varDelta _{\nu }^{\rho } = 0,\quad \varDelta _{\rho }^{\mu }\varDelta _{\nu }^{\rho } =\varDelta _{ \nu }^{\mu },\quad U_{\mu }^{\mu } = 1,\quad \varDelta _{\mu }^{\mu } = D - 1\;, }$$

and similarly:

$$\displaystyle{ u^{\mu }a_{\mu } = 0,\quad u^{\mu }\sigma _{\mu \nu } = 0,\quad u^{\mu }\omega _{\mu \nu } = 0,\quad u^{\mu }\nabla _{\nu }u_{\mu } = 0,\quad \varDelta _{\mu }^{\rho }\nabla _{\nu }u_{\rho } = \nabla _{\nu }u_{\mu }\;. }$$

Appendix 2: Papapetrou–Randers Backgrounds and Aligned Fluids

In this appendix, we collect a number of useful expressions for stationary Papapetrou–Randers three-dimensional geometries (13.39) with B = 1, and for fluids in perfect equilibrium on these backgrounds. The latter follow geodesic congruences, aligned with the normalized Killing vector ∂ _t , with velocity one-form given in (13.41).

We introduce the inverse two-dimensional metric a ^ij, and b ⁱ such that

$$\displaystyle{ a^{\mathit{ij}}a_{\mathit{ jk}} =\delta _{ k}^{i}\;,\quad b^{i} = a^{\mathit{ij}}b_{ j}\;. }$$

The three-dimensional metric components read:

$$\displaystyle{ g_{00} = -1\;,\quad g_{0i} = b_{i}\;,\quad g_{\mathit{ij}} = a_{\mathit{ij}} - b_{i}b_{j}\;, }$$

and those of the inverse metric:

$$\displaystyle{ g^{00} = a^{\mathit{ij}}b_{ i}b_{j} - 1\;,\quad g^{0i} = b^{i}\;,\quad g^{\mathit{ij}} = a^{\mathit{ij}}\;. }$$

Finally,

$$\displaystyle{ \sqrt{\vert g\vert } = \sqrt{a}\;, }$$

where a is the determinant of the symmetric matrix with entries a _ij .

Using (13.41) and (13.65) we find that the vorticity of the aligned fluid can be written as the following two-form (the acceleration term is absent here)

$$\displaystyle{ \omega = \frac{1} {2}\omega _{\mu \nu }\mathrm{d}x^{\mu } \wedge \mathrm{ d}x^{\nu } = \frac{1} {2}\mathrm{d}\mathrm{b}\;. }$$

The Hodge-dual of ω _{μ ν} is

$$\displaystyle{ \psi ^{\mu } =\eta ^{\mu \nu \rho }\omega _{\nu \rho } \Leftrightarrow \omega _{\nu \rho } = -\frac{1} {2}\eta _{\nu \rho \mu }\psi ^{\mu }\;. }$$

In 2 + 1 dimensions it is aligned with the velocity field:

$$\displaystyle{ \psi ^{\mu } = qu^{\mu }\;, }$$

where, in our set-up,

$$\displaystyle{ q(x) = -\frac{\epsilon ^{\mathit{ij}}\partial _{i}b_{j}} {\sqrt{a}} \;. }$$

It is a static scalar field that we call the vorticity strength, carrying dimensions of inverse length. Together with $\hat{R}(x)$—the curvature of the two-dimensional metric $\text{d}\ell^{2}$ introduced in (13.42), the above scalar carries all relevant information for the curvature of the Papapetrou–Randers geometry. We quote for latter use the three-dimensional curvature scalar:

$$\displaystyle{ R =\hat{ R} + \frac{q^{2}} {2} \;, }$$

(13.67)

the three-dimensional Ricci tensor

$$\displaystyle{ R_{\mu \nu }\,\mathrm{d}x^{\mu }\mathrm{d}x^{\nu } = \frac{q^{2}} {2} \mathrm{u}^{2} + \frac{\hat{R} + q^{2}} {2} \mathrm{d}\ell^{2} -\mathrm{ u}\,\mathrm{d}x^{\rho }u^{\sigma }\eta _{\rho \sigma \mu }\nabla ^{\mu }q\;, }$$

(13.68)

as well as the three-dimensional Cotton–York tensor:

$$\displaystyle\begin{array}{rcl} C_{\mu \nu }\,\mathrm{d}x^{\mu }\mathrm{d}x^{\nu }& =& \frac{1} {2}\left (\hat{\nabla }^{2}q + \frac{q} {2}(\hat{R} + 2q^{2})\right )\left (2\mathrm{u}^{2} +\mathrm{ d}\ell^{2}\right ) \\ & & -\frac{1} {2}\left (\hat{\nabla }_{i}\hat{\nabla }_{j}q\,\mathrm{d}x^{i}\mathrm{d}x^{j} +\hat{ \nabla }^{2}q\,\mathrm{u}^{2}\right ) \\ & & -\frac{\mathrm{u}} {2}\mathrm{d}x^{\rho }u^{\sigma }\eta _{\rho \sigma \mu }\nabla ^{\mu }(\hat{R} + 3q^{2})\;. {}\end{array}$$

(13.69)