General Covariance in Gravity at a Lifshitz Point
ABSTRACT This paper is based on the invited talks delivered by the author at GR 19:
the 19th International Conference on General Relativity and Gravitation, Ciudad
de M\'exico, M\'exico, July 2010. In Part 1, we briefly review some of the main
features of quantum gravity with anisotropic scaling, and comment on its
possible relation to the causal dynamical triangulations (CDT) approach to
lattice quantum gravity. Part 2 explains the construction of gravity with
anisotropic scaling with an extended gauge symmetry -- essentially a
nonrelativistic version of general covariance. This extra symmetry eliminates
the scalar graviton polarization, and thus brings the theory closer to general
relativity at long distances.
[show abstract] [hide abstract]
ABSTRACT: A new set of projection operators is constructed to suitably handle non-relativistic theories of gravity with anisotropic scaling, including the ones with parity-violating terms. This alternative procedure allows us to discuss unitarity and spectral properties for different formulations of the Horava-Lifshitz gravity. This task, that generally involves lengthy algebraic steps, becomes more systematic and greatly simplified in terms of the projectors we work out. Moreover, this procedure allows us to fix the number of propagating degrees of freedom and the structure of gauge symmetries is readily determined. In order to test the efficacy of the technique at hand, the unitarity and low-energy regime of a general Horava-Lifshitz Gravity model are investigated.11/2011;
arXiv:1101.1081v1 [hep-th] 5 Jan 2011
General Covariance in Gravity at a Lifshitz Point
Petr Hoˇ rava
Berkeley Center for Theoretical Physics and Department of Physics
University of California, Berkeley, CA, 94720-7300, USA
Theoretical Physics Group, Lawrence Berkeley National Laboratory
Berkeley, CA 94720-8162, USA
Abstract: This paper is based on the invited talks delivered by the author at GR 19:
the 19th International Conference on General Relativity and Gravitation, Ciudad de M´ exico,
M´ exico, July 2010.
1 Gravity with Anisotropic Scaling
1.1The minimal theory
1.2The nonprojectable case
1.3 Entropic origin of gravity?
1.4Causal dynamical triangulations and the spectral dimension of spacetime
1.5 Phases of gravity
2 General Covariance in Gravity with Anisotropic Scaling
2.1 Fields and symmetries
2.2The Lagrangian and Hamiltonian formulations
2.3Comparing to general relativity in the infrared
Gravity may be the one force of nature we are intuitively most familiar with, but its theoretical
understanding – despite the beauty of general relativity and string theory – is still shrouded
in surprisingly many layers of mystery. Perhaps we already have all the pieces of the puzzle,
and just need to find the correct way of putting them together; or perhaps new ideas are
needed. In this context, the idea of gravity with Lifshitz-type anisotropic scaling [1–3] has
attracted a lot of attention recently.
In Part 1 of this paper, we briefly review some of the main features of quantum gravity
with anisotropic scaling, in its original formulation initiated in [1–3], and comment on its
possible relation to the causal dynamical triangulations (CDT) approach to lattice quantum
gravity. Part 2 explains the construction of gravity with anisotropic scaling with an extended
gauge symmetry – essentially a nonrelativistic version of general covariance – presented in .
This extra symmetry eliminates the scalar graviton polarization, and thus brings the theory
closer to general relativity at long distances.
1Gravity with Anisotropic Scaling
The central idea of [1, 2] is a minimalistic one: to formulate quantum gravity as a quantum
field theory, with the spacetime metric as the elementary field, in the standard path-integral
language. Quantum field theory (QFT) has emerged from the 20th century as the universal
language for understanding systems with many degrees of freedom, ranging from high-energy
particle physics to condensed matter, statistical physics and more. Before giving up on QFT
for quantum gravity, it makes sense to apply its full machinery to this problem, without prior
restrictions such as microscopic relativistic invariance. The novelty of [1, 2] is that gravity
is combined with the idea of anisotropic scaling of spacetime, more familiar from condensed
matter, and characterized by
x → bx,
Here z is an important observable, the “dynamical critical exponent,” associated with a given
fixed point of the renormalization group (RG). Systems with many different values of z are
known, for example in dynamical critical phenomena or quantum criticality. It is natural
to ask whether one can construct theories with anisotropic scaling and with propagating
gravitons. Why? A consistent theory of gravity with anisotropic scaling can be potentially
useful for a number of possible applications:
t → bzt.(1.1)
(i) Phenomenology of gravity in our Universe of 3 + 1 macroscopic dimensions.
(ii) New gravity duals for field theories in the context of the AdS/CFT correspondence; in
particular, duals for a broader class of nonrelativistic QFTs.
(iii) Gravity on worldsheets of strings and worldvolumes of branes.
(iv) Mathematical applications to the theory of the Ricci flow on Riemannian manifolds .
(v) IR fixed points in condensed matter systems, with emergent gravitons (new phases of
algebraic bose liquids) .
(vi) Relativistic gravity and string theory in asymptotically anisotropic spacetimes ;
and possibly others. Note that only application (i) is subjected to the standard observational
tests of gravity, while the others are only constrained by their mathematical consistency. And
of course, applications (i–vi) aside, this system can serve as a useful theoretical playground
for exploring field-theory and path-integral methods for quantum gravity.
This approach shares some philosophical background with the idea of asymptotic safety,
initiated in  and experiencing a resurgence of recent interest. Both approaches are equally
minimalistic, suggesting that gravity can find its UV completion as a quantum field theory of
– 2 –
the fluctuating spacetime metric, without additional degrees of freedom or a radical departure
from standard QFT. While both approaches look for a UV fixed point, they differ in the
nature of the proposed fixed point: In asymptotic safety, one benefits from maintaining
manifest relativistic invariance, and pays the price of having to look for a nontrivial, strongly
coupled fixed point. In gravity with anisotropic scaling, one gives up Lorentz invariance as
a fundamental symmetry at short distances, and looks for much simpler, perhaps Gaussian
or at least weakly coupled fixed points in the UV. The price to pay, if one is interested in
application (i), is the need to explain how the experimentally extremely well-tested Lorentz
symmetry emerges at long distances.
Such Gaussian fixed points of gravity with z > 1 can also serve as IR fixed point in con-
densed matter systems, as shown for z = 2 and z = 3 in . This may be important because it
leads to new phases of algebraic bose liquids, and gives a new mechanism for making gapless
excitations technically natural in condensed matter. Implications for quantum gravity are less
dramatic: The gapless excitations at the IR fixed points of  are linearized gravitons, only
allowed to interact in a way which respects linearized diffeomorphism invariance. Hence, this
lattice model is not a theory of emergent gravity with nonlinear diffeomorphism symmetries.
1.1 The minimal theory
In our construction of Lifshitz-type gravity, we assume that the spacetime manifold M carries
the additional structure of a codimension-one foliation F, by D-dimensional leaves Σ of
constant time. We will use coordinate systems (t,x ≡ xi), i = 1,...D, adapted to F.
Perhaps the simplest relevant example of systems with Lifshitz-type anisotropic scaling
is the Lifshitz scalar theory with z = 2,
with ∆ the spatial Laplacian. Compared to the relativistic scalar in the same spacetime
dimension, the Lifshitz scalar has an improved UV behavior. The scaling dimension of φ
changes to [φ] = (D − 2)/2, and conseqently the (lower) critical dimension also shifts, from
the relativistic 1 + 1, to 2 + 1 when z = 2.
The most “primitive” theory of gravity similar to (1.2) would describe the dynamics of the
spatial metric gij(x,t), invariant under time-independent spatial diffeomorphisms. Because
of the lack of (time-dependent) gauge invariance, this model would propagate not only the
tensor polarizations of the graviton, but also the vector and the scalar. This “primitive”
theory becomes more interesting when we make it gauge invariant under foliation-preserving
diffeomorphisms Diff(M,F), generated by
δt = f(t), δxi= ξi(t,x). (1.3)
The minimal multiplet of fields now contains, besides gij, also the lapse function N and the
shift vector Ni. Since the lapse and shift play the role of gauge fields of Diff(M,F), we can
assume that they inherit the same dependence on spacetime as the corresponding generators
(1.3): While Ni(t,x) is a spacetime field, N(t) is only a function of time, constant along Σ.
Making this assumption about the lapse function gives to the minimal theory of gravity with
– 3 –
anisotropic scaling, sometimes referred to as the “projectable” theory . (For its brief review
and some phenomenological applications, see .)
The dynamics of the projectable theory is described by the most general action which
respects the Diff(M,F) symmetry. At the lowest orders in time derivatives, the action is
dtdDx√g N?KijKij− λK2− V?, (1.4)
2N(˙ gij− ∇iNj− ∇jNi)(1.5)
is the extrinsic curvature of Σ, K = gijKij, λ is a dimensionless coupling, and the potential
term V is an arbitrary Diff(Σ)-invariant local scalar functional built out of gij, its Riemann
tensor and the spatial covariant derivatives, but no time derivatives.
Which terms in V are relevant will depend on our choice of z at short distances. Terms
with 2z spatial derivatives have the same classical scaling dimension as the kinetic term, and
their quadratic part defines the Gaussian fixed point. Terms with fewer derivatives represent
relevant deformations of the theory. They induce a classical RG flow, which can lead to an IR
fixed point, with the isotropic z = 1 scaling in the deep infrared regime. As usual in effective
field theory, terms of higher order in derivatives, or involving additional time derivatives, are
of higher dimension and therefore superficially irrelevant around the UV fixed point.
Compared to general relativity, the minimal model is different in three interconnected
ways: It has one fewer gauge symmetry per spacetime point, its field multiplet has one fewer
field component per spacetime point (since N is independent of xi), and it propagates an
additional scalar graviton polarization in addition to the standard tensor polarizations, at
least around flat spacetime. While the number of gauge symmetries and field components
may not be observable, the number of propagating graviton polarizations is.
1.2The nonprojectable case
Another possibility is to insist on matching the field content of general relativity, and promote
the lapse N to a spacetime field. This is the “nonprojectable” theory [1, 2]. If we postulate the
same Diff(M,F) gauge symmetry as in the projectable case, the generic action will contain
new terms, constructed from the new ingredient ai ≡ ∂iN/N.
such new terms is sometimes referred to in the literature as the “healthy extension” of the
projectable theory. This is a misnomer – indeed, the basic rules of effective field theory clearly
instruct us to include all terms compatible with the postulated symmetries, since such terms
would otherwise be generated by quantum corrections. Hence, including all terms compatible
with the gauge symmetry should not be called a “healthy extension” of the nonprojectable
theory; it is just the correct implementation of the assumptions of the nonprojectable theory.
(For a recent review of the nonprojectable theory, see .)
In contrast, leaving the ai-dependent terms artificially out deserves to be called an “un-
healthy reduction” of the projectable theory.
justified if it is protected by additional symmetries. However, a closer analysis of the un-
healthy reduction indeed reveals difficulties with the closure of the constraint algebra and no
The general theory with
Such an unhealty reduction could only be
– 4 –
new gauge symmetry [10, 11], possibly with the interesting exception of the deep infrared
The nonprojectable model may be described in terms of the same field content as general
relativity, but the scalar graviton polarization is still present in its physical spectrum. In
Part 2, we discuss a mechanism proposed in , which eliminates the scalar graviton, by
enlarging the gauge symmetry to “nonrelativistic general covariance.”
1.3 Entropic origin of gravity?
There is another concept originally introduced in [1, 2] which has caused some level of
confusion in the literature: the “detailed balance” condition. This concept has its roots
in nonequlibrium statistical mechanics and dynamical critical phenomena. Oversimplifying
slightly, the theory is said to be in detailed balance if the potential in (1.4) is of a special
form, effectively a square of the equations of motion associated with a (Euclidean-signature)
theory in D dimensions with some action W. For example, the Lifshitz scalar (1.2) is in
detailed balance, with W =1
In [1, 2], this condition was suggested simply as a technical trick, which can possibly
reduce the number of independent couplings in V, if one can show that detailed balance is
preserved under renormalization (which is the case in many nongravitational examples in
condensed matter). If one is interested in getting close to general relativity with a small cos-
mological constant at long distances, detailed balance would clearly have to be broken, at least
in the minimal theory. If that breaking happens only at the level of relevant deformations,
the restrictive power of the detailed balance condition can still be useful for constraining the
terms whose dimension equals that of the kinetic term.
Is it possible that the detailed balance condition could play a more physical role in our un-
derstanding of gravity? While this question remains open, one intriguing analogy seems worth
pointing out: When gravity with anisotropic scaling satisfies the detailed balance condition,
its path integral in imaginary time is formally analogous to the Onsager-Machlup theory of
nonequilibrium thermodynamics [13, 14]. In this path-integral formulation of nonequilibrium
systems, a collection of thermodynamic variables Φais governed by the Onsager-Machlup
action, given – up to surface terms – by
Here the Onsager kinetic coefficients Labrepresent a metric on the Φaspace,
preted as entropic forces, and the action W itself plays the role of entropy!
This analogy leads to a natural speculation, implicit in [1, 2], that the nature of gravity
with anisotropic scaling is somehow entropic. It would be interesting to see whether this
analogy can be turned into a coherent framework in which some of the intriguing recent ideas
about the entropic origin of gravity  (also ) and cosmology  can be made more
δΦa are inter-
1.4 Causal dynamical triangulations and the spectral dimension of spacetime
In the study of quantum field theory, it is often useful to construct the system by a lattice
regularization, and study the approach to the continuum limit using computer simulations.
– 5 –