13.30 o1 m visser

Victoria University of WellingtonTe Whare Wananga o te Upoko o te Ika a Maui

VUWHorava gravity

Matt Visser

NZIP 2011Wellington

Monday 17 October 2011

Matt Visser (VUW) Horava gravity NZIP 2011 1 / 34

Abstract:

VUW

Horava gravity is a recent (Jan 2009) idea in theoretical physics fortrying to develop a quantum field theory of gravity, an idea that hasalready generated approximately 575 scientific articles.

It is not a string theory, nor loop quantum gravity, but is instead aquantum field theory that breaks Lorentz invariance at ultra-high(trans-Planckian) energies, while retaining Lorentz invariance at lowand medium energies.

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Abstract:

VUWThe challenge is to keep the Lorentz symmetry breaking controlledand small — small enough to be compatible with experiment.

I will give a very general overview of what is going on in this field,paying particular attention to the disturbing role of the scalargraviton.

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Key questions:

VUWIs Lorentz symmetry truly fundamental?

Or is it just an “accidental” low-momentum emergent symmetry?

Opinions on this issue have undergone a radical mutation over thelast few years.

Historically, Lorentz symmetry was considered absolutely fundamental— not to be trifled with — but for a number of independent reasonsthe modern viewpoint is more nuanced.

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Key questions:

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What are the benefits of Lorentz symmetry breaking?

What can we do with it?

Why should we care?

Where are the bodies buried?

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Horava gravity:

Quantum Gravity at a Lifshitz Point.Petr Horava. Jan 2009.Physical Review D79 (2009) 084008. arXiv: 0901.3775 [hep-th](cited 552 times; 355 published; as of 13 Oct 2011).

Membranes at Quantum Criticality.Petr Horava. Dec 2008.JHEP 0903 (2009) 020. arXiv: 0812.4287 [hep-th](cited 290 times; 199 published; as of 13 Oct 2011).

Spectral Dimension of the Universe in Quantum Gravityat a Lifshitz Point.Petr Horava. Feb 2009.Physical Review Letters 102 (2009) 161301.arXiv: 0902.3657 [hep-th](cited 229 times; 155 published; as of 13 Oct 2011).


Horava gravity:

VUWAs of 13 Oct 2011 the Inspire bibliographic reports that(apart from Horava’s own articles) this topic has generated:

38 published papers with 100 or more citations.

41 published papers with 50–99 citations.



However you look at it, this topic is “highly active”.

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Horava gravity:

VUWWarning:

There has been somewhat of a tendency to charge full steam aheadwith applications, (sometimes somewhat fanciful), without first fullyunderstanding the foundations of the model.

For that matter, there is still, after 212 years, considerable

disagreement as to what precise version of the model is “best”.

When reading the literature, a certain amount of caution isadvisable...

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Horava gravity:

My interpretation of the central idea:

Abandon ultra-high-energy Lorentz invariance as fundamental.

One need “merely” attempt to recover an approximate low-energyLorentz invariance.

Typical dispersion relation:

ω =

√m2 + k2 +

k4

K 2+ . . .

Nicely compatible with the “analogue spacetime” programme...


Quantum Field Theory — QFT:

Every QFT regulator known to mankind either breaks Lorentzinvariance explicitly (e.g. lattice), or does something worse,something outright unphysical.

For example:

Pauli–Villars violates unitarity;Lorentz-invariant higher-derivatives violate unitarity;dimensional regularization is at best a purely formal trick with no directphysical interpretation...(and which requires a Zen approach to gamma matrix algebra).


QFT:

Old viewpoint:

If the main goal is efficient computation in a corner of parameterspace that we experimentally know to be Lorentz invariant to a highlevel of precision, then by all means, go ahead and develop aLorentz-invariant perturbation theory with an unphysical regulator— hopefully the unphysical aspects of the computation can first beisolated in a controlled manner, and then ultimately be banished byrenormalization.

This is exactly what is done, (very efficiently and very effectively),in the “standard model of particle physics”.


QFT:

New viewpoint:

If however one has reason to suspect that Lorentz invariance mightultimately break down at ultra-high (trans–Planckian?) energies,then a different strategy suggests itself.

Maybe one could use the Lorentz symmetry breaking as part of theQFT regularization procedure?

Could we at least keep intermediate parts of the QFT calculation“physical”?

Note that “physical” does not necessarily mean “realistic”,it just means we are not violating fundamental tenets of quantumphysics at intermediate stages of the calculation.


QFT:

Physical but Lorentz-violating regulator:

Specific dispersion relation:

ω2 = m2 + k2 +k4

K 24

+k6

K 46

.

QFT propagator [momentum-space Green function]:

G (ω, k) =1

ω2 −[m2 + k2 + k4

K24

+ k6

K46

] .Note rapid fall-off as spatial momentum k →∞.

This improves the behaviour of the integrals encountered inFeynman diagram calculations (QFT perturbation theory).


QFT:

Key result:

In any (3+1) dimensional scalar QFT, with arbitrary polynomialself-interaction, this is enough (after normal ordering),to keep all Feynman diagrams finite.

For details see:

Lorentz symmetry breaking as a quantum field theory regulator.Matt Visser. Feb 2009.Physical Review D80 (2009) 025011.arXiv: 0902.0590 [hep-th]

Related articles by Anselmi and collaborators.


QFT:

Key results:

Gravity is a little trickier, but you can at least argue forpower-counting renormalizability of the resulting QFT.

This is unexpected, seriously unexpected...

And yes there are still significant technical difficulties...(of which more anon)...

For details see:

Horava’s papers.

Power-counting renormalizability of generalized Horava gravity.Matt Visser. Dec 2009.e-Print: arXiv:0912.4757 [hep-th]


Implications for quantum gravity:

Develop a non-standard extension of the usual ADM decomposition:

Choose a “preferred foliation” of spacetime into space+time.

Implicit return of the aether...

Decompose the Lagrangian

L = (kinetic term)− (potential term)

Now add extra “kinetic” and “potential” terms,beyond the usual textbook Einstein–Hilbert Lagrangian.


Horava gravity:

It is roughly at this stage that Horava makes his two greatsimplifications:

“projectability”;“detailed balance”.

Even after 212 years — it is still somewhat unclear whether these are

just “simplifying ansatze” or whether they are fundamental toHorava’s model.

In particular, Silke Weinfurtner, Thomas Sotiriou, and I have arguedthat “detailed balance” is not fundamental, and we have beencarefully thinking about the issue of “projectability”.


Horava gravity:

Eliminating detailed balance:

Phenomenologically viable Lorentz-violating quantum gravity.Thomas Sotiriou, Matt Visser, Silke Weinfurtner. Apr 2009.Physical Review Letters 102 (2009) 251601.arXiv: 0904.4464 [hep-th]

Quantum gravity without Lorentz invariance.Thomas Sotiriou, Matt Visser, Silke Weinfurtner. May 2009.JHEP 0910 (2009) 033.arXiv: 0905.2798 [hep-th]


Projectability:

What is Horava’s projectability condition?

This is the assertion that the so-called lapse function is always trivial,(or more precisely trivializable).

In standard general relativity the “projectability condition” can alwaysbe enforced locally as a gauge choice;

Furthermore for “physically interesting” solutions of general relativityit seems that this can always be done (more or less) globally.

For this purely pragmatic reason we decided to put “projectability” off toone side for a while, and first deal with “detailed balance”.


Detailed balance:

What is Horava’s detailed balance condition?

It is a way of constraining the potential:

V(g) is a perfect square.

That is, there is a “pre-potential” W (g) such that:

V(g) =

(g ij δW

δgjkgkl δW

δgli

).

This simplifies some steps of Horava’s algebra, it makes other featuresmuch worse.


Detailed balance:

VUWIn particular, if you assume Horava’s detailed balance condition, and try torecover usual Einstein-Hilbert physics in the low energy regime, then:

You are forced to accept a non-zero cosmological constant of the“wrong sign”.

You are forced to accept intrinsic parity violation in the purelygravitational sector.

The second point I could live with, the first point will require somemutilation of detailed balance.

So we might as well go the whole way and discard detailed balance entirely.

VUWMatt Visser (VUW) Horava gravity NZIP 2011 21 / 34

Discarding detailed balance:

The standard Einstein–Hilbert piece of the action is now:

SEH = ζ2

∫ {(K ijKij − K 2) + R − g0 ζ

2}√

g N d3x dt.

The “extra” Lorentz-violating terms are:

SLV = ζ2

∫ {ξ K 2 − g2 ζ

−2 R2 − g3 ζ−2 RijR

ij

−g4 ζ−4 R3 − g5 ζ

−4 R(RijRij )

−g6 ζ−4 R i

jRjkRk

i − g7 ζ−4 R∇2R

−g8 ζ−4∇iRjk ∇iR jk

}√g N d3x dt.

(Some 10 pages of careful calculation required.)


Physical units:

From the normalization of the Einstein–Hilbert term:

(16πGNewton)−1 = ζ2; Λ =g0 ζ

2

2;

so that ζ is identified as the Planck scale.

The cosmological constant is determined by the free parameter g0,and observationally g0 ∼ 10−123.(Renormalized after including vacuum energy contributions.)

In particular, we are free to choose the Newton constant andcosmological constant independently.(And so can be made compatible with observation.)


Physical units:

The Lorentz violating term in the kinetic energy leads to an extrascalar mode for the graviton, with fractional O(ξ) effects at allmomenta.

Phenomenologically, this behaviour is potentially dangerous andshould be carefully investigated.

The various Lorentz-violating terms in the potential becomecomparable to the spatial curvature term in the Einstein–Hilbertaction for physical momenta of order

ζ{2,3} =ζ√|g{2,3}|

; ζ{4,5,6,7,8} =ζ

4

√|g{4,5,6,7,8}|

.


Physical units:

The Planck scale ζ is divorced from the various Lorentz-breakingscales ζ{2,3,4,5,6,7,8}.

We can drive the Lorentz breaking scale arbitrarily high by suitableadjustment of the dimensionless couplings g{2,3} and g{4,5,6,7,8}.

Based on his intuition coming from “analogue spacetimes”,Grisha Volovik has been asserting for many years that theLorentz-breaking scale should be much higher than the Planck scale.

This model naturally implements that idea.


Horava gravity — Problems and pitfalls:

Where are the bodies buried?

Projectability: This yields a spatially integrated Hamiltonianconstraint rather than a super-Hamiltonian constraint.

Prior structure: Is the preferred foliation “prior structure”?Or is it dynamical?

Scalar graviton: As long as ξ 6= 0 there is a spin-0 scalar graviton,in addition to the spin-2 tensor graviton.

Hierarchy problem?(Renormalizability may still require fine tuning.)

Beta functions? Beyond power-counting? RG flow?(Very limited explicit calculations so far.)

There is still the potential for violent conflicts with empirical reality.


Graviton propagator linearized around flat space:

The spin-0 scalar graviton is potentially dangerous:

Binary pulsar?(Extra energy loss mechanism?)

PPN physics?(Detailed solar system tests?)

Eotvos experiments?(Universality of free-fall?)

Negative kinetic energy?(Between the UV conformal and IR general relativistic limits.)

Lots of careful thought still needed...


Spin-zero graviton:

The “Healthy Extension” of Horava-Lifshitz gravity.

Consistent Extension of Horava Gravity.Diego Blas, Oriol Pujolas, Ssergey Sibiryakov. Sep 2009.Physical Review Letters 104 (2010) 181302.

e-Print: arXiv:0909.3525 [hep-th]

Models of non-relativistic quantum gravity:the good, the bad and the healthy.Diego Blas, Oriol Pujolas, Sergey Sibiryakov. Jul 2010.JHEP 1104 (2011) 018e-Print: arXiv:1007.3503 [hep-th]

Key idea: Make the “preferred foliation” dynamical.(Stably causal spacetime with preferred “cosmic time”.)


Spin-zero graviton:

Search for “analogue spacetime” hints:

Emergent gravity at a Lifshitz point from a Bose liquidon the lattice.Cenke Xu, Petr Horava. Mar 2010.Physical Review D81 (2010) 104033.e-Print: arXiv:1003.0009 [hep-th]

Add an extra gauge symmetry:

General covariance in quantum gravity at a Lifshitz point.Petr Horava, Charles M. Melby-Thompson. Jul 2010.Physical Review D82 (2010) 064027.e-Print: arXiv:1007.2410 [hep-th]

Extra symmetry not related to diffeomorphism invariance.


Spin-zero graviton:

Projectability?

Projectable Horava-Lifshitz gravity in a nutshell.Silke Weinfurtner, Thomas P. Sotiriou, Matt Visser. Feb 2010.Journal of Physics Conference Series 222 (2010) 012054.e-Print: arXiv:1002.0308 [gr-qc]

Other?

Status of Horava gravity: A personal perspective.Matt Visser. Mar 2011.Journal of Physics Conference Series 314 (2011) 012002.e-Print: arXiv:1103.5587 [hep-th]


Spin-zero graviton:

VUWTo get rid of the scalar graviton, the choices seem to be:

Approximately decouple the scalar mode?

Kill the scalar mode with more symmetry?

Kill the scalar mode with more a priori restrictions on the metric?(Even more restrictions than projectability?)

While significant progress is being made,none of these approaches is as yet fully satisfying.

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Causal Dynamical Triangulation — CDT:

Comparing CDT with Horava gravity:

Horava’s spectral dimension paper.

CDT meets Horava-Lifshitz gravity.J. Ambjorn, A. Gorlich, S. Jordan, J. Jurkiewicz, R. Loll. Feb 2010.Physics Letters B690 (2010) 413–419.e-Print: arXiv:1002.3298 [hep-th]

Spectral dimension as a probe of the ultraviolet continuum regime ofcausal dynamical triangulations.Thomas P. Sotiriou, Matt Visser, Silke Weinfurtner. May 2011.Physical Review Letters 107 (2011) 131303.e-Print: arXiv:1105.5646 [gr-qc]


Causal Dynamical Triangulation — CDT:

We have now laid out all of the technical tools that allowus to fit the spectral dimension for CDT in the ultravioletcontinuum regime in (2! 1) dimensions using the simu-lations of Ref. [5]. These simulations were repeatedfor 6 different values of the number of simplicesNs " #40; 50; 70; 100; 140; 200$ % 103. For each indepen-dent simulation Ns was held fixed, and one sums over 1000different histories (geometries). On each history the(fictitious) diffusion process was implemented and usedto calculate the corresponding spectral dimension ds.

In Ref. [7] the spectral dimensions of CDT and HLgravity were compared at the two limits of the ultravioletcontinuum regime and they were found to be in goodagreement. Both theories seem to predict that at smallscales ds flows to 2, whereas at length scales large enoughfor the quantum correction to become largely subdomi-nant, but small enough for curvature correction to be un-important, ds flows to 3, the number of topologicaldimensions. Encouraging as it may be, this result is farfrom being conclusive, given that there is an infinity ofcurves joining two points.

Our goal is far more ambitious: we numerically evaluatethe spectral dimensions in Eq. (8) for a dispersion relationof the kind (5), and fit the CDT data using the method ofnonlinear regression. This will allow us to obtain theparameters of the dispersion relation that provide the bestfit for the whole ultraviolet continuum region, and accesshow good such a fit can be. This will provide a conclusiveanswer to whether HL gravity can reproduce the behaviorof ds for CDT in this regime.

This procedure will not uniquely determine all of theparameters of HL gravity. First of all, it is already clear

from Eqs. (6) that g3, g4 and g5 do not enter into thelinearized propagator and, therefore, they cannot be deter-mined without taking into account curvature corrections.Second, there are 7 couplings that do enter in the propa-gator, but only in 4 combinations. This implies that anyobservable that probes the dispersion relation will give ussome but not full insight into the fundamental theory.Finally, without loss of generality, we can choose towork in units where the infrared light speed is one, i.e.

A " 1. This amounts to the rescaling ! ! !=!!!!A

p.

In Fig. 1(a) we plot the spectral dimension ds as result-ing from the CDT simulation with the largest number ofsimplices. As illustrated with the differently shaded areas,the diffusion process can be divided into three physicallydifferent regions. For very small s 2 #1; 15$ the spectraldimension shows discrete behavior as expected. For verylarge s 2 #614; 29 999$, the behavior of the spectral di-mension is expected to be dominated by curvature effects.This is the region were the flow of ds is compatible with adiffusion process on a continuous manifold with the ge-ometry of at stretched sphere [5]. The intermediate region,where a continuous behavior emerges but quantum correc-tions are predominant, is the one we are interested in. Ourmodel-fit for this region (s 2 #15; 614$) is represented by ared solid line. Figure 1(b) zooms into this region.We have chosen to present a fit for the simulation with

the larger number of simplices because the higher thenumber of simplices, the larger the volume of the universewill be in the simulations. If the volume is not largeenough, curvature effects can kick in before quantumeffects become subdominant. There might then be noregime where both effects can be neglected, so no regime

FIG. 1 (color online). (a) The spectral dimension as a function of (fictitious) diffusion time s (semilog scale). Crosses represent datapoints for CDT for simulation with 200 000 simplices, whereas the red solid line represent the fit based of HL gravity. The regionwhere discretium effects are important is lightly shaded. The region where curvature effects dominate is darkly shaded. Theintermediate is the ultraviolet continuum region. (b) Zoom into the latter region. (c) Individual residuals for the model fits for50 000 100 000 and 200 000 simplices. The residuals grow as expected at very small and very large scales under the influence ofdiscretium and curvature effects, respectively. The oscillatory pattern at intermediate scales seems to indicate some systematicdeviation from the model, which however, tends to disappear as the number of simplices is increased. For 200 000 simplices thefractional residual remains below 0.5% almost throughout the entire ultraviolet continuum region.

P HY S I CA L R EV I EW LE T T E R S

3 3

Comparing CDT numerical experiments with predictions from aHorava-type dispersion relation in 2+1 dimensions. 3-parameter fit.


Summary:

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Still a very active field...

Initial feeding frenzy somewhat subsided...

Very real physics challenges remain...

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Technology

13.30 o1 m visser