13.30 o1 m visser

VUW
Victoria University of Wellington
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Te Whare W¯nanga o te Upoko o te Ika a Maui
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Hoˇava gravity
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Matt Visser

NZIP 2011
Wellington

Monday 17 October 2011
Matt Visser (VUW) Hoˇava gravity
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Abstract:

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Hoˇava gravity is a recent (Jan 2009) idea in theoretical physics for
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trying to develop a quantum field theory of gravity, an idea that has
already generated approximately 575 scientific articles.

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

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and medium energies.

r NZIP 2011 2 / 34

Abstract:

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The challenge is to keep the Lorentz symmetry breaking controlled
and 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 scalar

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graviton.

r NZIP 2011 3 / 34

Key questions:

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Is Lorentz symmetry truly fundamental?
Or is it just an “accidental” low-momentum emergent symmetry?
Opinions on this issue have undergone a radical mutation over the
last few years.
Historically, Lorentz symmetry was considered absolutely fundamental
— not to be triﬂed with — but for a number of independent reasons

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the modern viewpoint is more nuanced.

r NZIP 2011 4 / 34

Key questions:

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What are the beneﬁts of Lorentz symmetry breaking?
What can we do with it?
Why should we care?

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Where are the bodies buried?

r NZIP 2011 5 / 34

Hoˇava gravity:
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Quantum Gravity at a Lifshitz Point.
Petr Hoˇava. Jan 2009.
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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 Hoˇava. Dec 2008.
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JHEP 0903 (2009) 020. arXiv: 0812.4287 [hep-th]

Spectral Dimension of the Universe in Quantum Gravity
at a Lifshitz Point.
Petr Hoˇava. Feb 2009.
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Physical Review Letters 102 (2009) 161301.
arXiv: 0902.3657 [hep-th]

r NZIP 2011 6 / 34

Hoˇava gravity:
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As of 13 Oct 2011 the Inspire bibliographic reports that
(apart from Hoˇava’s own articles) this topic has generated:
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38 published papers with 100 or more citations.
41 published papers with 50–99 citations.

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

r NZIP 2011 7 / 34

Hoˇava gravity:
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Warning:
There has been somewhat of a tendency to charge full steam ahead
with applications, (sometimes somewhat fanciful), without ﬁrst fully
understanding the foundations of the model.
For that matter, there is still, after 2 1 years, considerable
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disagreement as to what precise version of the model is “best”.
When reading the literature, a certain amount of caution is

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advisable...

r NZIP 2011 8 / 34

Hoˇava gravity:
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My interpretation of the central idea:
Abandon ultra-high-energy Lorentz invariance as fundamental.
One need “merely” attempt to recover an approximate low-energy
Lorentz invariance.
Typical dispersion relation:

k4
ω= m2 + k 2 + + ...
K2
Nicely compatible with the “analogue spacetime” programme...

r NZIP 2011 9 / 34

Quantum Field Theory — QFT:

Every QFT regulator known to mankind either breaks Lorentz
invariance 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 direct
physical interpretation...
(and which requires a Zen approach to gamma matrix algebra).

r NZIP 2011 10 / 34

QFT:

Old viewpoint:

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

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

r NZIP 2011 11 / 34

QFT:

New viewpoint:
If however one has reason to suspect that Lorentz invariance might
ultimately 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 the
QFT 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 quantum
physics at intermediate stages of the calculation.

r NZIP 2011 12 / 34

QFT:

Physical but Lorentz-violating regulator:
Speciﬁc dispersion relation:

k4 k6
ω 2 = m2 + k 2 + 2
+ 4.
K4 K6

QFT propagator [momentum-space Green function]:

1
G (ω, k) = .
k4 k6
ω2 − m2 + k2 + 2
K4
+ 4
K6

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

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

r NZIP 2011 13 / 34

QFT:

Key result:
In any (3+1) dimensional scalar QFT, with arbitrary polynomial
self-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.
Related articles by Anselmi and collaborators.

r NZIP 2011 14 / 34

QFT:

Key results:
Gravity is a little trickier, but you can at least argue for
power-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:
Hoˇava’s papers.
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Power-counting renormalizability of generalized Horava gravity.
Matt Visser. Dec 2009.
e-Print: arXiv:0912.4757 [hep-th]

r NZIP 2011 15 / 34

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.

r NZIP 2011 16 / 34

Hoˇava gravity:
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It is roughly at this stage that Hoˇava makes his two great
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simpliﬁcations:
“projectability”;
“detailed balance”.

Even after 2 1 years — it is still somewhat unclear whether these are
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just “simplifying ansatze” or whether they are fundamental to
Hoˇava’s model.
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In particular, Silke Weinfurtner, Thomas Sotiriou, and I have argued
that “detailed balance” is not fundamental, and we have been
carefully thinking about the issue of “projectability”.

r NZIP 2011 17 / 34

Hoˇava gravity:
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Eliminating detailed balance:

Phenomenologically viable Lorentz-violating quantum gravity.
Thomas Sotiriou, Matt Visser, Silke Weinfurtner. Apr 2009.

Quantum gravity without Lorentz invariance.
Thomas Sotiriou, Matt Visser, Silke Weinfurtner. May 2009.
JHEP 0910 (2009) 033.

r NZIP 2011 18 / 34

Projectability:

What is Hoˇava’s projectability condition?
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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 always
be enforced locally as a gauge choice;
Furthermore for “physically interesting” solutions of general relativity
it seems that this can always be done (more or less) globally.

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

r NZIP 2011 19 / 34

Detailed balance:

What is Hoˇava’s detailed balance condition?
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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:

δW kl δW
V(g ) = g ij g .
δgjk δgli

This simpliﬁes some steps of Hoˇava’s algebra, it makes other features
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much worse.

r NZIP 2011 20 / 34

Detailed balance:

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In particular, if you assume Hoˇava’s detailed balance condition, and try to
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recover 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 purely
gravitational sector.
The second point I could live with, the ﬁrst point will require some
mutilation of detailed balance.

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

r NZIP 2011 21 / 34

Discarding detailed balance:

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

√
SEH = ζ 2 (K ij Kij − K 2 ) + R − g0 ζ 2 g N d3 x dt.

The “extra” Lorentz-violating terms are:

SLV = ζ 2 ξ K 2 − g2 ζ −2 R 2 − g3 ζ −2 Rij R ij

−g4 ζ −4 R 3 − g5 ζ −4 R(Rij R ij )
−g6 ζ −4 R i j R j k R k i − g7 ζ −4 R 2 R
√
−g8 ζ −4 i Rjk i R jk g N d3 x dt.

(Some 10 pages of careful calculation required.)

r NZIP 2011 22 / 34

Physical units:

From the normalization of the Einstein–Hilbert term:
g0 ζ 2
(16πGNewton )−1 = ζ 2 ; Λ= ;
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 and
cosmological constant independently.
(And so can be made compatible with observation.)

r NZIP 2011 23 / 34

Physical units:

The Lorentz violating term in the kinetic energy leads to an extra
scalar mode for the graviton, with fractional O(ξ) eﬀects at all
momenta.
Phenomenologically, this behaviour is potentially dangerous and
should be carefully investigated.
The various Lorentz-violating terms in the potential become
comparable to the spatial curvature term in the Einstein–Hilbert
action for physical momenta of order
ζ ζ
ζ{2,3} = ; ζ{4,5,6,7,8} = .
|g{2,3} | 4
|g{4,5,6,7,8} |

r NZIP 2011 24 / 34

Physical units:

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

We can drive the Lorentz breaking scale arbitrarily high by suitable
adjustment 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 the
Lorentz-breaking scale should be much higher than the Planck scale.

This model naturally implements that idea.

r NZIP 2011 25 / 34

Hoˇava gravity — Problems and pitfalls:
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Where are the bodies buried?
Projectability: This yields a spatially integrated Hamiltonian
constraint rather than a super-Hamiltonian constraint.

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

Scalar graviton: As long as ξ = 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.

r NZIP 2011 26 / 34

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?)

E¨tv¨s experiments?
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(Universality of free-fall?)

Negative kinetic energy?
(Between the UV conformal and IR general relativistic limits.)
Lots of careful thought still needed...

r NZIP 2011 27 / 34

Spin-zero graviton:

The “Healthy Extension” of Horava-Lifshitz gravity.
Consistent Extension of Horava Gravity.
Diego Blas, Oriol Pujolas, Ssergey Sibiryakov. Sep 2009.
Models of non-relativistic quantum gravity:
the good, the bad and the healthy.
Diego Blas, Oriol Pujolas, Sergey Sibiryakov. Jul 2010.
JHEP 1104 (2011) 018
Key idea: Make the “preferred foliation” dynamical.
(Stably causal spacetime with preferred “cosmic time”.)

r NZIP 2011 28 / 34

Spin-zero graviton:

Search for “analogue spacetime” hints:
Emergent gravity at a Lifshitz point from a Bose liquid
on the lattice.
Cenke Xu, Petr Hoˇava. Mar 2010.
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Add an extra gauge symmetry:
General covariance in quantum gravity at a Lifshitz point.
Petr Horava, Charles M. Melby-Thompson. Jul 2010.
Extra symmetry not related to diﬀeomorphism invariance.

r NZIP 2011 29 / 34

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.

r NZIP 2011 30 / 34

Spin-zero graviton:

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To 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,

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none of these approaches is as yet fully satisfying.

r NZIP 2011 31 / 34

Causal Dynamical Triangulation — CDT:

Comparing CDT with Hoˇava gravity:
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Hoˇava’s spectral dimension paper.
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CDT meets Horava-Lifshitz gravity.
J. Ambjorn, A. Gorlich, S. Jordan, J. Jurkiewicz, R. Loll. Feb 2010.
Physics Letters B690 (2010) 413–419.

Spectral dimension as a probe of the ultraviolet continuum regime of
causal dynamical triangulations.
Thomas P. Sotiriou, Matt Visser, Silke Weinfurtner. May 2011.
e-Print: arXiv:1105.5646 [gr-qc]

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s for CDT in this regime. enough, curvature effects
his procedure will not uniquely determine all — the
Causal Dynamical Triangulation of CDT:effects become subdomina
ameters of HL gravity. First of all, it is already clear regime where both effects c

1 (color online).CDT The spectral dimension as a function of (fictitious) diffusion time s (semi
Comparing (a) numerical experiments with predictions from a
ts for CDT for simulation withrelation simplices, whereas the red solid line represent the fit
Hoˇava-type dispersion 200 000 in 2+1 dimensions. 3-parameter fit.
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re discretium effects are important is lightly shaded. The region where curvature effects d
mediate Mattthe ultraviolet continuum region. (b) Zoom into the latter region. 2011 Individual
is Visser (VUW) Hoˇava gravity
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(c) 33 / 34

Summary:

—— VUW ——

Still a very active ﬁeld...

Initial feeding frenzy somewhat subsided...

Very real physics challenges remain...

—— VUW ——

r NZIP 2011 34 / 34

13.30 o1 m visser

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