The following are the (incomplete, possibly erroneous) notes I took during Thomas Sotiriou’s talk at the YETI 2017 conference in Durham.

The old problem of quantum gravity can be stated as: “can we make gravity renormalizable?”. The immediate answer is “no”, but possible solutions have been devised over the years. One might be tempted to add higher-order curvature invariants, but these come with higher order time derivatives which in turn introduce ghosts.

Another simple solution could be to simply give up on Lorentz invariance: one is then free to introduce higher order space derivatives without pairing them with suitable time derivatives, circumventing the problem of creation of ghosts entirely. The theory is then renormalizable at the power counting level. But is it really that simple?

## Lifshitz gravity

Consider for instance the action

with the natural scaling dimensions $[\mathrm{dt}]=[\kappa]^{-z}$ and $[\mathrm{dx}]=[\kappa]^{-1}$, which imply $[\phi]=[\kappa]^{(d-z)/2}$, $[a_m]=[\kappa]^{2(z-m)}$ and $[g_n]=[\kappa]^{d+z-n(d-z)/2}$. Then the theory is indeed power-counting renormalizable for $z\ge d$.

Furthermore, we note that standard QFT results can be recovered for, e.g. $d=4$ or $d=1+1$. More interestingly, the special case $m=d=z$ introduces twice as many spatial derivatives than time derivatives, and leads to $[g_n]=[g]>0$, i.e. the theory is renormalizable for any $n$!

### Foliation

We introduce a preferred foliation of spacetime as:

where $N^i$ are shift-vectors. We are left with foliation preserving diffeormorphisms: time reparametrizations of the form $t\to\tilde{t}(t)$, and spacetime-dependent 3-diffeomorphisms of the form $x^i\to\tilde{x}^i(t,x^i)$. The most general action we can write down is

where we defined the extrinsic curvature

A general feature of the theory is that the potential should be at least $6^{\text{th}}$ order in spatial derivatives:

The theory propagates a scalar mode, and comes with more than 60 couplings and 2 types of Lorentz-violating terms (lower and higher order terms). Generally speaking, trying to reduce the number of couplings imposes constraints on $\xi$ and $\eta$, driving the theory out of the GR observational space.

## Einstein-Aether theory

We consider the action

where

and the aether satisfies $u^\mu u_\mu=1$. This is in fact the most general theory with a unit timelike vector field that is also second order in derivatives. We now impose hypersurface orthogonality by assuming

and choosing $T$ as the time coordinate that satisfies

Essentially, we’re killing off the vector mode and making $T$ the fundamental degree of freedom. The action is then

with

### Constraints

One could try constraining the Einstein-Aether theory with stability/Cherenkov tests: light travelling in a vacuum could decay via the Cherenkov channel to new subliminal modes. This depletion could then be observable in cosmic rays; precise measurements of the latter therefore enforce strong constraints on such theories.

Pertinent questions to be asked at this point are: do couplings get the correct values under the renormalization group flow? how is the relevant phenomenology modified? what about the vacuum energy? how do we make sense of causality without Lorentz invariance? do black holes and singularities still exist?

## Lorentz violation…

### … and causal structure

In a Lorentz violating theory, linear dispersion relations now follow $\omega\propto k$: different modes have different speeds and different light cones. Yet light cones still exist; causality takes a local definition…

### … and black holes

They now have multiple horizons !

### … and non-linear dispersion relations

What if we consider instead the case where $\omega^2\propto k^2+ak^4+\ldots$? In that case we lose light cones entirely! The causal structure is reduced to a foliation of simultaneous spacetime slices. One might think this does away completely with singularities and black holes, but Sotiriou and others recently showed that this isn’t quite right!

In fact, the theory naturally comes with the concepts of constant preferred time and universal horizons (where the foliation wraps around the black hole). Essentially, the metric horizon itself disappears, but a universal horizon had always been hiding behind it and so black holes are conserved.