Æthereal Gravity: Observational Constraints on

Einstein- Æther Theory

Brendan Foster

University of Maryland

Einstein-æther Theory

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Will & Nordtvedt (1972)Gasperini (1987)Jacobson & Mattingly (2000)

Matter: Assume matter couples universally to gab.

Post-Newtonian Parameters

Eling & TJ (2003), Graesser, Jenkins, Wise (2005), BF & TJ (2005)

non - linearity

space curvature

1,2 preferred frame

3 - body interaction

3,1,2,3,4 4 - momentum not conserved

Describe post-Newtonian-order effects in terms of standard set of potentials and ten “PPN” parameters.

1 1

1 1

0 0

0 0

0 0

GR Æ

Combined PPN, stability, Cerenkov & energy constraints

BF & TJ (2005)

0 c13 1 & 0 < (c1 c3) c13 3(1 c13)

Cerenkov & stability: (spin-2 and spin-0 mode speed)2 ≥1

Also implies (spin-1 mode speed)2 ≥1 AND all energy densities positive.

0by and fix :PPN 2142 cc

Radiation damping

BF (2006)

progress) in (BF, advance... periastron oneffect include alsomust

dipole, vanishing-non :sources

rate.GR to quadrupole equates ( on condition one then , if vanish dipole and monopole :sources

:but generally, radiation quadrupole and dipole Monopole,

1,2

ravitatingStrongly g

ccWeak

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Einstein-Aether WavesMattingly & TJ (2004)spin-2: 2 gravitons

spin-1: 2 transverse aether-metric modes spin-0: 1 longitudinal aether-metric mode

all « massless », speeds all different:

STABILITY constraint: squared speeds > 0

CERENKOV constraint: squared speeds >1Elliott, Moore & Stoica (2005)

Wave Energy

Sign of wave energy densityC.T. Eling (2005)

POSITIVE ENERGY constraint: energy > 0

Spin-2 Spin-1 Spin-0

+ (2c1 - c12

+ c32)/(1-c13) c14(2- c14)

•Found by averaging energy-momentum pseudotensor density over a wave cycle of wave solutions.

•Reduces to result of Lim (2004) in decoupling limit.

Total Energy

C.T. Eling (2005)B.Z.Foster (2005)

E aether c14

8Gd2

S r ut

E total MADM (1 c14

2)

Newtonian limitEling & TJ (2003)Carroll & Lim (2004)Newtonian limit recovered, with

GN G (1 c14 2)

total energy corresponds to M in asymptotic GNM/r term of metric component.

Preferred frame parameters

Graesser, Jenkins, Wise (2005)Foster & TJ (2005)

1 8(c3

2 c1c4 )

2c1 c12 c3

2

2 12cc c1c4 )

2c1 c12 c3

2

Constraints on PPN Parameters

From C.M. Will, gr-qc/0504086

Preferred frame parametersGraesser, Jenkins & Wise (2005); Foster & TJ (2005)

1 0 c4 c32 c1

2 0 c2 ( 2c12 c1c3 c3

2) 3c1 (or c13 c14 0)

...leaves a (c1,c3) parameter space with all PPN parameters

identical to those of GR!

Einstein-Aether Cosmology

TabAether 1

2(c1 3c2 c3) Gab 1

6( 3) R(gab 2uaub )

In RW symmetry the aether field equation is automatic,and the stress tensor is geometric:

Mattingly & TJ (2001), Carroll & Lim (2004)

The first term renormalizes the gravitational constant:

Gcosmo G (1 (c13 3c2) /2)

The second term renormalizes the spatial curvature termin the Friedman eqn.

Primordial nucleosynthesis

Helium abundance implies

Carroll & Lim (2004)

When preferred frame parameters vanish we find

| Gcosmo GN 1 | 1 8

Gcosmo GN

Einstein-Aether Cosmology II

Primordial fluctuations:• Spin-1 perturbations decay exponentially (not sourced).• Inflaton sources spin-0 aether perturbation which mixes with metric mode• scalar and tensor mode speeds differ, and G’s differ • upshot: power spectra differently rescaled. Upsets “inflationary consistency relation”:

tensor/scalar power = - 9/2 tensor spectral index (1+O(ci))

(Lim, 2004)

Spherically symmetric solutionsChris Eling & TJ (2006)

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Static aether solutionChris Eling & TJ (2006)

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Static aether solution – II Chris Eling & TJ (2006)

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Static aether solution – III Chris Eling & TJ (2006)

gravity. surface zeroth horizon wi Killing a

parameter; affine null finiteat

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violatedcondition energy Null

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Aethereal StarsChris Eling & TJ (2006)

• Fluid star interior can be matched to static aether exterior.• Solution determined by eqn. of state and central pressure.• Maximum mass of constant density stars less than in GR.

)18( G

Aethereal Black HolesC. Eling & TJ (2006)

• æther not aligned with Killing vector – flows into hole • analytic solution not possible• to solve can shoot out from horizon or in from infinity• metric horizon generically regular – 2 parameter family??

What is a black hole?• must trap all modes; metric, spin-0,1,2 horizons generally differ.• only spin-0 has spherically symmetric modes• regularity of spin-0 horizon reduces to 1 parameter family

Aethereal Black Holes – cont’d

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C. Eling & TJ (2006)

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Curvature divergence

Aether flow vs. free-fall

Aether flow vs. free-fall inside

Aethereal Black Holes – issues

• other values of c3 ?

• evidence for negative mass black holes?

• æther singular at bifurcation surface – time asymmetry

• 1st law and entropy? (Foster, ‘05)

• numerical collapse? (e.g. imploding aether wave)

• rotating black holes?