View source for Fary–Milnor theorem

In [[theoretical physics]], particularly [[fringe physics]], '''polarizable vacuum''' (PV) and its associated theory refers to proposals by [[Harold Puthoff]], [[Robert H. Dicke]], and others to develop an analogue of [[general relativity]] to describe [[gravity]] <ref>{{Cite arxiv | eprint=0902.1305v1 | author1=Xing-Hao Ye | title=Polarizable vacuum analysis of electromagnetic fields | class=physics.gen-ph | year=2009 }}</ref> and its relationship to [[electromagnetism]].

== Description ==
{{Infobox Pseudoscience 
|topics=* [[physics]] 
|claims=* Gravitation can be described via a ''scalar theory of gravitation'', using a ''stratified conformally flat metric'', in which the field equation arises from the notion that the vacuum behaves like an optical polarizable medium.
|origyear=* 1957 (R. H. Dicke), 1998 (H. E. Puthoff)
|origprop=* Robert Dicke, Harold Puthoff
|currentprop=* Harold Puthoff (see also [[Bernard Haisch]] and [[Stochastic electrodynamics|SED]])}}

In essence, Dicke and Puthoff proposed that the presence of mass alters the [[electric permittivity]] and the [[magnetic permeability]] of flat [[spacetime]], ε<sub>o</sub> and μ<sub>o</sub> respectively by multiplying them by a scalar function, '''K''':
<blockquote>ε<sub>o</sub>→ε = Kε<sub>o</sub>, μ<sub>o</sub>→μ = Kμ<sub>o</sub></blockquote>
arguing that this will affect the lengths of rulers made of ordinary matter, so that in the presence of a gravitational field the spacetime metric of [[Minkowski spacetime]] is replaced by
:<math> ds^2 = -\frac{1}{\kappa^2} dt^2 + \kappa^2 \, ( dx^2 + dy^2 + dz^2) </math>
where <math>\kappa^2 = K</math> is the so-called ''"[[Vacuum permittivity|dielectric constant of the vacuum]]"''.  This is a "diagonal" metric given in terms of a Cartesian chart and having the same ''stratified conformally flat'' form in the [[Scalar theories of gravitation#Additional variations|Watt-Misner theory of gravitation]].  However, according to Dicke and Puthoff, κ must satisfy a field equation which differs from the field equation of the Watt-Misner theory.  In the case of a static spherically symmetric vacuum, this yields the asymptotically flat solution
:<math> \kappa = \exp(m/r) = 1 + m/r + O\left( \frac{1}{r^2} \right) </math>
The resulting Lorentzian spacetime happens to agree with the analogous solution in the Watt-Misner theory, and it has the same weak-field limit, and the same far-field, as the [[Schwarzschild metric|Schwarzschild vacuum]] solution in general relativity, and it satisfies three of the four [[classical tests of general relativity|classical tests of relativistic gravitation]] (redshift, deflection of light, precession of the perihelion of Mercury) to within the limit of observational accuracy.  However, as shown by Ibison (2003), it yields a different prediction for the inspiral of test particles due to gravitational radiation.

However, requiring stratified-conformally flat metrics rules out the possibility of recovering the weak-field [[Kerr metric]], and is certainly inconsistent with the claim that PV can give a ''general'' "approximation" of the [[general theory of relativity]].  In particular, this theory exhibits no [[frame-dragging]] effects.  Also, the effect of gravitational radiation on test particles differs profoundly between scalar theories and tensor theories of gravitation such as general relativity.  [[LIGO]] is not intended primarily as a test ruling out scalar theories, but is widely expected to do so as a side benefit once it detects unambiguous gravitational wave signals exhibiting the characteristics expected in general relativity.

Ibison has considered a "cosmological solution" of PV, analogous to the Friedmann dust solution with flat orthogonal hyperslices in general relativity, and argues that this model is inconsistent with various observational and theoretical constraints.  He also finds a rate of inspiral disagreeing with observation. The latter result disagrees with that of Watt and Misner, whose Lorentzian manifold differs from PV in the case of cosmology.

It is widely accepted that no scalar theory of gravitation can reproduce all of general relativity's successes, contrary to Puthoff's claims. It might be noted that De Felice uses [[constitutive relation]]s to obtain a ''susceptibility tensor'' which lives in spatial hyperslices; this provides extra degrees of freedom which help make up for the degree of freedom lacking in PV and other scalar theories.

==Modified PV Model==
In 2005 Depp proposed a modification to the original PV work of Dicke that addresses many of the objections above. The revised model is not a replacement for general relativity but is meant to provide insight into the possible underlying physics of general relativity.{{Citation needed|date=June 2009}}  

Depp presents a PV Lagrangian field from which can be derived the exact Schwarzschild solution and the exact Reissner-Nordstrom solution.{{Citation needed|date=June 2009}} In addition it is also shown that electron charge renormalization can be obtained from the model that is in good agreement with QED.
<ref>http://www.webcitation.org/query?url=http://www.geocities.com/psistar%40sbcglobal.net/PV.pdf&date=2009-10-26+01:04:55 Polarizable Vacuum and the Schwarzschild Solution</ref> 
<ref>http://www.webcitation.org/query?url=http://www.geocities.com/psistar%40sbcglobal.net/PV_and_RN.pdf&date=2009-10-26+01:04:58 Polarizable Vacuum and the Reissner-Nordstrom Solution</ref> 
<ref>http://www.webcitation.org/query?url=http://www.geocities.com/psistar%40sbcglobal.net/Renormalization.pdf&date=2009-10-26+01:05:01 Electron Charge Renormalization and the PV-RN Model</ref>

==Criticism==

===Generalization===
Puthoff himself has apparently offered various characterizations of his proposal, which has been variously characterized as
*an attempt to ''reformulate'' general relativity in terms of a purely formal analogy with the propagation of light through an optical medium,
*an attempt to ''replace'' general relativity with a scalar theory of gravitation featuring formal analogies with Maxwell's theory of electromagnetism,
*an attempt to ''unify'' gravitation and electromagnetism in a theory of ''electrogravity'',
*an attempt to provide a ''physical mechanism'' for how spacetime gets curved in general relativity, which suggests (to Puthoff) the possibility of ''"metric engineering"'' for such purposes as spacecraft propulsion (see [[Breakthrough Propulsion Physics Program]]).

PV has origins in more mainstream work by such physicists as Robert Dicke{{Citation needed|date=June 2009}}, but in current parlance the term does appear to be most closely associated with the speculations of Puthoff. The claims have not been accepted in mainstream physics.{{Citation needed|date=June 2009}}

===Not a unified field theory?===
Mainstream physicists agree that PV is {{Citation needed|date=June 2009}}
#not viable as a unification of gravitation and electromagnetism
#not a "reformulation" of general relativity,
#not a viable theory of gravitation, since it violates observational and theoretical requirements.

== Related work ==
Antecedents of PV and more recent related proposals include the following:
#A proposal in 1921 by H. A. Wilson to reduce gravitation to electromagnetism by pursuing the formal analogy between "light bending" in [[classical theories of gravitation|metric theories of gravitation]] and propagation of light through an optical medium having a spatially varying [[refractive index]].  Wilson's approach to a [[unified field theory]] is not considered viable today.
#An attempt (roughly 1960-1970) by [[Robert Dicke]] and [[Fernando de Felice]] to resurrect and improve Wilson's idea of an optical analogue of gravitational effects. If interpreted conservatively as an attempt to provide an alternative approach to gtr, rather than as work toward a theory unifying electromagnetism and gravitation, this is not an unreasonable approach, although most likely of rather limited utility.
#The 1967 proposal of [[Andrei Sakharov]] that gravitation might arise from underlying [[quantum field theory]] effects, in a manner somewhat analogous to the way that the (simple) classical theory of elasticity arises from (complicated) particle physics.  This work is generally regarded as mainstream and not entirely implausible, but highly speculative, and most physicists seem to feel that little progress has been made.
#In a series of papers, [[Bernard Haisch]] and Alfonso Rueda have proposed that the [[inertia]] of massive objects arises as a ''"electromagnetic reaction force"'', due to interaction with the so-called [[zero point field]].  According to mainstream physics, their claims rest upon incorrect computations using quantum field theory.
#Recent work, motivated in large part by the discoveries of the [[Unruh effect]], [[Hawking radiation]], and [[black hole thermodynamics]], to work out a complete theory of physical analogues such as ''optical black holes''.  This is not work toward a unified field theory, but in another sense can be regarded as work towards an even more ambitious unification, in which some of the most famous effects usually ascribed to general relativity (but actually common to many metric theories of gravitation) would be seen as essentially ''thermodynamical effects'', not specifically ''gravitational effects''.  This work has excited great interest because it might enable experimental verification of the basic concept of Hawking radiation, which is widely regarded as one of the most revolutionary proposals in twentieth century physics, but which in its gravitational incarnation seems to be impossible to verify in experiments in earthly laboratories.
#The 1999 proposal by Keith Watt and [[Charles W. Misner]] of a [[scalar theories of gravitation|scalar theory of gravitation]] which postulates a ''stratified conformally flat'' metric of the form <math>ds^2 = -\exp(2 \, \phi) + \exp(-2 \phi) (dx^2+dy^2+dz^2</math>, given with respect to a [[Cartesian coordinates|Cartesian chart]], where φ satisfies a certain [[partial differential equation]] which reduces in a vacuum region to the [[flat spacetime]] [[wave equation]] <math>\Box \phi = 0</math>.  This is a "toy theory", not a fully fledged theory of gravitation, since as Watt and Misner pointed out, while this theory does have the correct [[Newtonian limit]], it disagrees with the result of certain observations.

==See also==
*[[Induced gravity]] (for Sakharov's proposal)
*[[Maxwell's equations in curved spacetime]]
*[[Electromagnetic stress-energy tensor]]

== References ==
{{More footnotes|date=June 2009}}
*{{cite web | author=Visser, Matt | title=Analog Gravity | work=Living Reviews in Relativity | url=http://relativity.livingreviews.org/Articles/lrr-2005-12/index.html | year=2005 | accessdate=2006-06-02}}
*{{cite arXiv | author=Ibison, M. | title=Investigation of the polarizable vacuum cosmology | year = 2003 | version=2003| eprint=astro-ph/0302273}}
*{{cite arXiv | author=Watt, Keith; and Misner, Charles | title=Relativistic Scalar Gravity: A Laboratory for Numerical Relativity | year = 1999| eprint=gr-qc/9910032 | version=10 Oct 1999}}
*{{cite journal|author = Puthoff, H. E. | title=Polarizable-Vacuum (PV) representation of general relativity | journal=Found. Of Phys. | volume=32 | year=2002 | pages=927–943 | doi=10.1023/A:1016011413407|issue = 6}} [http://www.arxiv.org/abs/gr-qc/9909037 arXiv eprint]
*{{cite journal | author=de Felice, F. | title = On the gravitational field acting as an optical medium | journal = General Relativity and Gravitation | year = 1971 | volume = 2 | pages = 347– | doi = 10.1007/BF00758153 | bibcode=1971GReGr...2..347D | issue=4}}
*{{cite journal | author=Dicke, R. H. | title=Gravitation without a principle of equivalence |journal = Reviews of Modern Physics | year = 1957 | volume = 29 | pages = 363–376 | doi=10.1103/RevModPhys.29.363 | bibcode=1957RvMP...29..363D | issue=3}}
*{{cite journal | author=Wilson, H. A. | title=An electromagnetic theory of gravitation |journal = Physical Review | year = 1921 | volume = 17 | pages = 54–59 | doi=10.1103/PhysRev.17.54|bibcode = 1921PhRv...17...54W }}

==External links==
*H. E. Puthoff, M. Ibison, [http://earthtech.org/publications/Mitre%20Conference.pdf Polarizable Vacuum "Metric Engineering" Approach to GR-Type Effects], MITRE Conference, McLean, VA, May 8, 2003.  From the website of [http://earthtech.org/ EarthTech], a company founded by Puthoff.
*H. E. Puthoff, S. R. Little, M. Ibison, [http://stardrive.org/Jack/puthoff1.pdf Engineering the Zero-Point Field and Polarizable Vacuum For Interstellar Flight], Institute for Advanced Studies, Austin, Texas, January 2001.

==Notes==
{{reflist}}

[[Category:Theories of gravitation]]
[[Category:Fringe physics]]