I have recently published, in the journal Science, a paper giving a general model-independent analysis of the way in which current observational cosmology interweaves with the evolution of the galaxies and the abstract energy conditions of general relativity.
In this paper, I trade off precision for robustness: I derive results that are extremely general with an absolute minimum of input assumptions that are explicitly exhibited. (I could add to the precision of the result by adding more technical assumptions-but the fewer technical assumptions one makes about the early universe the more one trusts the results.)
I pointed out that you do not need to know what is going on in the very early universe to realise that there is a serious conflict between various cosmological observations. The discussion of the age of the oldest stars and their conflict with the observed Hubble parameter does not need to worry about the Big Bang itself. Rather, it is only the relatively late stages of the formation of the universe (redshift less than 20, microwave background temperature less than 60 Kelvin) that are important when addressing this problem.
If one believes high values for the Hubble parameter (H0 > 60 km sec-1 Mpc-1), and high values for the age of the oldest stars (tformation > 15 Gyr), then the strong energy condition of general relativity must be violated between galaxy formation and the current epoch.
One way of violating the strong energy condition is with a large cosmological constant (at least 33% of the current energy density of the universe), but other ways of violating the strong energy condition are known.
One of the peculiar features of the strong energy condition is that if violations of the strong energy persist to sufficiently early times then the classical singularity theorem guaranteeing the existence of a Big Bang singularity does not apply.
The initial paper in this project was published in Science 276 (4 April 1997) 88-90. A longer more explicit development of these ideas is available as gr-qc/9705070, this paper has been accepted for publication in Physical Review D15 (15 December 1997).
It is reasonably well known that a gravitational field will polarize the quantum mechanical vacuum in the same way that the electric field surrounding an electron polarizes the QED vacuum. In QED this phenomenon leads to charge screening for the electron. In general relativity this phenomenon implies that the ``vacuum'' outside a star, or planet, or black hole, acquires a non-zero stress energy tensor of order \hbar (plus even higher order corrections). What is not commonly appreciated is that this gravitational vacuum polarization violates all of the classical energy conditions that are commonly used to prove such results as the black hole collapse theorems, the second law of black hole thermodynamics, the positive mass theorem, etcetera.
In a series of papers on this topic I have investigated the extent to which these violations of the energy conditions are actually a problem: I have investigated the vacuum expectation value of the quantum stress-energy tensor in the vicinity of a black hole, and checked to see which energy conditions are violated in which particular regions. I have also been able to relate the behaviour of the stress-energy tensor in black hole spacetimes to that of the stress-energy in the gravitational field surrounding a planet or star. I have checked the various standard vacuum states (the quantum vacuum state is no longer unique in the gravitational field of a black hole), and have shown that the energy condition violations are ubiquitous (albeit small). [Physical Review D54 (1996) 5103-5115; Physical Review D54 (1996) 5116-5122; Physical Review D54 (1996) 5123-5128; Physical Review D56 (1997) 936-952.]
These papers have influenced the important work of Robert Wald and Eanna Flanagan [gr-qc/9602052] in their analysis of how the energy conditions are influenced by perturbative back-reaction effects (around flat spacetime).
Furthermore, this whole question naturally feeds back into questions of the extent to which the positive mass theorem, singularity theorems, and topological mass theorem survive the introduction of quantum physics. This research field is far from being mined out, and I expect this project to continue for several years.
A related project has to do with the reliability horizon that I have developed for semi-classical quantum gravity. With this notion of a reliability horizon I have quantified the otherwise somewhat vague notion of the Planck regime and the onset of Planck scale physics. I show that it is not enough to simply look at the expectation value of the metric and the expectation value of the curvature, but that fluctuations in the spacetime metric are often more important than Planck scale curvatures in signaling the onset of Planck scale physics.
This reliability horizon has direct applications to Hawking's Chronology Protection Conjecture and the Kay-Radzikowski-Wald singularity theorems concerning the breakdown of semi-classical quantum gravity on or near the chronology horizon.
A paper [gr-qc/9702041] has been accepted for publication in Physics Letters B, and a related Brief Report [gr-qc/9702043] is already published as Physical Review D45 (1977) 5212-5214. This Brief Report has been referenced by Stephen Hawking in his most recent paper [hep-th/9709066] on chronology protection.
(Research initiated in collaboration with David Hochberg [LAEFF, Spain])
In this project we have undertaken a general analysis of generic static wormholes without assuming any particular symmetry. We first develop a general definition of what it means to be a wormhole throat, and show that it is geometry [not topology] that is the crucial issue in locating the existence and properties of the throat.
By explicitly calculating the Riemann tensor in Gaussian normal coordinate patch surrounding the throat we are able to show that the results for spherically symmetric wormholes generalise in a suitable fashion to arbitrarily shaped wormhole mouths. Generically, the null energy condition will be violated in some region at or near the throat. This analysis is complementary to the so-called topological censorship theorem in that we are able to give explicit information on where the energy condition violations occur.
A paper [gr-qc/9704082] has been accepted for publication in Physical Review D, and is scheduled to appear on 15 October 97. The survey paper [gr-qc/9710001] we have been invited to provide for the Workshop on Black Hole Interiors and Spacetime Singularities will also deal with these issues.
(Research initiated in collaboration with: Carl Carlson [William and Mary, Virginia], Carmen Molina-París [Los Alamos], and Juan Pérez-Mercader [LAEFF, Spain])
Recently, I have initiated, and am maintaining, an international collaboration that involves a renewed attack on the well-known Casimir effect. Remarkably enough there is still a large amount of confusion in the literature concerning this effect-especially as regards the Casimir effect in dielectrics and conductors. One can find disagreements by factors of a billion or more in the published literature regarding the claimed overall physical scale of the effect.
The Casimir effect is associated with the fact that any attempt at distorting the quantum electrodynamic vacuum state will generically cause a shift in the zero-point energies of the photon modes, and thereby lead to a net change in the total zero-point energy. In many cases of physical interest this shift in the zero-point energy can be shown to be finite. In many other cases, the energy shift is formally infinite. In such cases, in order to make the energy shift finite and meaningful, one requires either (1) a physical regulating mechanism, or (2) an unphysical regulator, followed by renormalization.
In our recent papers, we address the question of whether or not the Casimir energy of a dielectric body includes a contribution proportional to the volume of the dielectric. (There has been some dispute, and considerable confusion, regarding this elementary point in the published literature.) We unambiguously resolve the issue by computing the physical difference in total zero point energy between the two relevant geometries: having the dielectric body present, and removing the dielectric body (replacing it by vacuum). The volume term is indeed present, and the bulk Casimir energy density is explicitly calculable as a function of the photon dispersion relation. [Physics Letters B395 (1997) 76-82; Physical Review D56 (1997) 1262-1281.]
With Carmen Molina-París, I have also developed an extension of the Balian-Bloch asymptotic estimates for the density of states to directly include the case of dielectric junction conditions. This has relevance to the derivation of a simple general formulation that side-steps much of the special function technology currently in vogue. With this technique we are able to derive the existence of a general sub-dominant contribution to the Casimir energy that is proportional to surface area. The coefficient of this term is a specifically calculable function of the refractive indices. [hep-th/9707073; To appear in Physical Review D; 15 November 1997.]
(Research initiated in collaboration with Carmen Molina-París [Los Alamos])
We are also working on resolving some of the conceptual difficulties associated with exactly which ``infinity'' one should subtract in order to get a finite physical result. The conceptual difficulties have to do with the precise nature of the regulator-and whether or not the regulator is good physics or simply an artifice.
We hope to soon be able to turn attention to the issue of when exactly we can define a finite Casimir energy without requiring explicit cutoffs. We can show that the infinite piece of the Casimir energy is always a linear combination of the first few Seeley-DeWitt coefficients. This should help us to elucidate the relationship between regularization and renormalization in this problem, and will hopefully give deeper insight into the unreasonable effectiveness of the zeta function approach.
These research topics have the potential for including a computational component-once a suitable collection of eigenvalues is obtained via either numerical or analytical techniques (or looking them up on the web), the behaviour of Casimir energy sums can be tested numerically as well as being understood analytically. For objects of arbitrary shape, numerical methods are often the only realistic hope for calculating the finite part of the Casimir energy.
There are a number of interesting diseases that occur when you try to give the graviton a small mass. Some of these diseases occur already at the classical level, while other problems first show up at the level of quantum field theory.
If one is dealing with a weak gravitational field, the idea of a graviton mass is perfectly well defined. To then go to strong gravitational fields, even in the classical theory, is not trivial. I have developed a particular way of doing so by setting up a theory that has two metrics. The particular theory I developed has been carefully chosen to be compatible with all current experimental results, but still have non-trivial consequences in strong fields.
The paper I wrote [gr-qc/9705051] has been submitted for publication, and has already encouraged Professor Clifford Will to write a paper developing additional phenomenological constraints on the graviton mass (both using presently available data, and developing analysis techniques suitable for a LIGO based precision limit on the graviton mass.)
I have investigated a class of simple model wormholes
that permit one to relax the requirement of spherical symmetry.
These simple models are extremely benign in that they permit travel
from one universe to another without the traveller being exposed
either to tidal forces or to exotic matter.
[Physical Review D39 (1989) 3182--3184]
I have also investigated the question of the stability of such
traversable wormholes, both at the semi--classical and at the
quantum levels. A class of equations of state for exotic matter
has been found that leads to dynamically stable wormholes. A
detailed field theoretic model leading to such exotic matter is
lacking, but it is clear that at least semiclassical quantum effects
must be involved.
[Nuclear Physics B328 (1989) 203--212]
In a related fully quantum mechanical calculation, I have investigated
the physics of a simple minisuperspace model of a wormhole that
connects two Minkowski spacetimes. This minisuperspace model leads
to a variant of the usual Wheeler--DeWitt equation. This modified
Wheeler--DeWitt equation is explicitly solvable, and, within the
limits of validity imposed by the minisuperspace approximation,
this calculation shows that quantum physics stabilizes this
classically unstable system with a radius of order the Planck
length. It is argued that this stability property is likely not
to be an artifact of the minisuperspace approximation; rather, this
stability property is likely to be a generic feature of a full
theory of quantum gravity.
[Physics Letters B242 (1990) 24--28]
I have also investigated generalizations of this procedure to
wormholes based on surgical modification of Schwarzschild and
Reissner--Nordstrom spacetimes. The results of this analysis
are qualitatively similar to those of the previous calculation and
the model is more realistic in that the wormhole is now permitted
to carry both mass and electric charge. For some of the more
complicated geometries presently under consideration, it does not
appear possible to write down closed form expressions for the
wave--function describing the location of the throat of the wormhole.
For these cases approximation techniques must be employed, such as
WKB analysis and/or numerical estimation of the wave--function.
[Physical Review D43 (1991) 402--409]
The class of simplified models discussed in this series of papers has been adopted by a number of authors and various extensions of these models continue to appear --- for example: (2+1)--dimensional traversable wormholes, traversable wormholes based on surgically modified Schwarzschild--de Sitter spacetime, etc.
This line of enquiry has led to several further publications. For
instance, one paper discusses the singularity structure of the
stress--energy tensor near the onset of time machine formation.
[Nuclear Physics B416 (1994) 895--506]
A further outgrowth of this research is a general study of focussing
and defocussing effects in Lorentzian spacetimes, as embodied in
the behaviour of the van Vleck determinant.
[Physical Review D47 (1993) 2395--2402]
A further paper extends these general considerations to the particular
class of traversable wormhole spacetimes of interest to the chronology
protection conjecture.
[Physical Review D49 (1994) 3963--3980]
Traversable wormholes, if they exist, must violate the averaged null energy condition. Because of this, there is no longer any guarantee that the overall mass of the wormhole system be positive. (The hypotheses of the positive mass theorem no longer apply.) It is possible to present reasonably plausible scenarios which lead to an overall negative mass for the wormhole system.
Such negative masses lead to a distinctive signature in gravitational
micro--lensing events. We have explored this possibility and made
specific suggestions that MACHO searches for gravitational
micro--lensing events be extended to search for these exotic
signatures.
[Physical Review D51 (1995) 3124--3127]
This implies that the averaged null energy condition is generically
violated by quantum effects. Since the averaged null energy condition
is an input hypothesis to many theorems in general relativity
(singularity theorems, topological censorship theorems, positive
mass theorems) this indicates that all of these issues may potentially
be affected by, and be modified by, inclusion of quantum effects.
[Physics Letters B349 (1995) 443--447]
The class of spherically-symmetric thin-shell wormholes provides
a particularly elegant collection of exemplars for the study of
traversable Lorentzian wormholes. We consider
linearized (spherically symmetric) perturbations around some assumed
static solution of the Einstein field equations. This permits us to
relate stability issues to the (linearized) equation of state of
the exotic matter which is located at the wormhole throat.
[Physical Review D52 (1995) 7318--7321]
Specifically, for an arbitrary static spherically symmetric black hole I have established a general formula for the Hawking temperature in terms of the energy density (rho) and radial tension (tau). Adopting Schwarzschild coordinates, I discovered
k T_H = [hbar/(4 pi r_H)] exp(-phi(r_H)) ( 1 - 8 pi G rho_H r_H^2).
Generalizations of this elegant result to axisymmetric spacetimes (for instance, to Kerr--Newman black holes embedded in an axisymmetric cloud of matter) would clearly be of interest. Generalizations to arbitrary event horizons are probably unmanageable. On the one hand, the dominant energy condition (DEC) guarantees the constancy of the surface gravity (and hence the constancy of the Hawking temperature) over the surface of an arbitrary stationary event horizon. For instance, one might conceivably hope to generalize the factor {4 pi r_H} to sqrt{4 pi A_H}. On the other hand, there is no particular reason to believe that rho_H is constant over the event horizon, nor is it clear how to generalize the notion of phi(r_H). If this result is supplemented by the weak energy condition (WEC), I obtain (for static spherically symmetric dirty black holes) the general inequality
k T_H < = hbar/(4 pi r_H).
S = k A_H/(4 l_P^2).
However, in many other interesting cases this simple relationship is false. I have obtained a preliminary result, relating the total entropy of a black hole to the geometry (and other properties)
S = k A_H/(4 l_P^2) + (1/T_H) int_Sigma (rho - L_E) K^mu dSigma_mu.
If no hair is present the validity of the "entropy = (1/4) area" law reduces to the question of whether or not the energy density for the system under consideration is formally equal to the Euclideanized Lagrangian. [Physical Review D48 (1993) 583--591]
With a bit more work, it is possible to show that the term proportional to (rho - L_E) can be converted to a surface term integrated over the event horizon. Thus (ignoring statistical hair), I have obtained
S = k A_H/(4 l_P^2) + int_H rho_S sqrt[g] d^2x.
The integration runs over a spacelike cross-section of the event horizon H. The surface entropy density, rho_S, is related to the behaviour of the matter Lagrangian under time dilations. [Physical Review D48 (1993) 5697--5705]
In certain cases I have made this formula more explicit: consider the specific case of Einstein--Hilbert gravity coupled to an effective Lagrangian that is an arbitrary function of the Riemann tensor (though not of its derivatives). Here
S = k A_H/(4 l_P^2) + 4 pi k/hbar int_H (partial L/partial R_mu_nu_lambda_rho) h_mu_lambda h_nu_rho sqrt[g] d^2x,
where the symbol h_mu_nu denotes the projection onto
the two-dimensional subspace orthogonal to the event horizon.
Though the derivation used to obtain this result proceeds via
Euclidean signature techniques the result can be checked against
certain special cases previously obtained by other techniques,
e.g. (Ricci)^n gravity, R^n gravity, and Lovelock gravity.
[Physical Review D48 (1993) 5697--5705]
The calculation, though elementary, has important conceptual
implications: (1) It seems that the existence of Hawking radiation
from black holes is ultimately dependent only on the fact that
massless quanta (and all other forms of matter) couple to gravity.
(2) The thermal nature of the Hawking spectrum seems to depend only
on the fact that the number of internal states of a large mass
black hole is enormous. (3) Remarkably, the resulting formula for
the decay rate gives meaningful answers even when extrapolated to
low mass black holes.
[Modern Physics Letters A8 (1993) 1661--1670]
This research project addresses the effect on radar propagation of lateral electromagnetic waves.
In order to fully understand low--altitude radar propagation it is essential that one correctly accounts for the complete solution to Maxwell's equations in the vicinity of the air--sea (or air--earth) interface. The study of the electromagnetic field which arises due to a source situated in the presence of an interface separating two electromagnetically--distinct media comprising semi--infinite half--spaces has been fraught with controversy and confusion for eight decades. Progress in this venerable subject has been hampered by the complex formulae which emerge in the analysis, as well as by certain historical lacunae which have clouded the interpretation of the interim results. In spite of the massed efforts of many researchers, the understanding of lateral electromagnetic waves has, until recently, remained relatively poor. The current state of the art is best summarized by the recent tome: "Lateral Electromagnetic Waves" by R. W. P. King, M. Owens, and T. T. Wu.
The present research problem utilizes the King--Owens--Wu analysis
to calculate explicit analytic expressions for the radar propagation
factor.
[Radio Science 29 (1994) 483--494]
Delta psi = 1/sqrt[-g] partial_mu ( sqrt[-g] g^mu^nu partial_nu psi ) = 0.
The propagation of sound is governed by the acoustic metric g_mu_nu(t,x). This acoustic metric describes a Lorentzian (pseudo--Riemannian) geometry and depends on the density, velocity of flow, and local speed of sound in the fluid. Specifically
[ -(c^2-v^2)| - v ]
g_mu_nu(t,x) == (rho/c) [ -----------------]
[ - v | I ].
(Here I denotes the 3x3 identity matrix.) In general, when
the fluid is non--homogeneous and flowing, the acoustic Riemann
tensor associated with this Lorentzian metric will (in general)
be nonzero. It is quite remarkable that even though the underlying
fluid dynamics is Newtonian, nonrelativistic, and takes place in
flat space + time, the fluctuations (sound waves) are governed by
a curved Lorentzian (pseudo--Riemannian) geometry. This connection
between fluid dynamics and techniques more commonly encountered in
the context of general relativity opens up many opportunities for
cross--pollination between the two fields.
A paper based on this research can be found at [gr-qc/9311028; gr-qc@xxx.lanl.gov]