arXiv subjects

Dan N. Vollick

Publications and source records attributed to Dan N. Vollick.

At least 19 recordsLinked to original sources

arXiv2021

On the Meaning of Various Mass Definitions for Asymptotically Flat Spacetimes

The mass contained in an arbitrary spacetime in general relativity is not well defined. However, for asymptotically flat spacetimes various definitions of mass have been proposed. In this paper I consider eight masses and show that some of them correspond to the active gravitational mass while the others correspond to the inertial mass. For example, the ADM mass corresponds to the inertial mass while the M$ø$ller mass corresponds to the active gravitational mass. In general the inertial and active gravitational masses are not equal. If the spacetime is vacuum at large $r$ the Einstein equations force the inertial and active gravitational masses to be the same. The Einstein equations also force the masses to be the same if any matter that extends out to large $r$ satisfies the weak, strong or dominant energy condition. I also examine the contributions of the inertial and active gravitational masses to the gravitational redshift, the deflection of light, the Shapiro time delay, the precession of perihelia and to the motion of test bodies in the spacetime.

gr-qc↗

arXiv2021

Lagrangian and Hamiltonian Formulation of Classical Electrodynamics without Potentials

In the standard Lagrangian and Hamiltonian approach to Maxwell's theory the potentials $A^μ$ are taken as the dynamical variables. In this paper I take the electric field $\vec{E}$ and the magnetic field $\vec{B}$ as the the dynamical variables. I find a Lagrangian that gives the dynamical Maxwell equations and include the constraint equations by using Lagrange multipliers. In passing to the Hamiltonian one finds that the canonical momenta $\vecΠ_E$ and $\vecΠ_B$ are constrained giving 6 second class constraints at each point in space. Gauss's law and $\vec{\nabla}\cdot\vec{B}=0$ can than be added in as additional constraints. There are now 8 second class constraints, leaving 4 phase space degrees of freedom. The Dirac bracket is then introduced and is calculated for the field variables and their conjugate momenta.

physics.class-ph↗

arXiv2021

Hamiltonian Formulation of the Pilot-Wave Theory

In the pilot-wave theory of quantum mechanics particles have definite positions and velocities and the system evolves deterministically. The velocity of a particle is determined by the wave function of the system (the guidance equation) and the wave function evolves according to Schrodinger's equation. In this paper I first construct a Hamiltonian that gives Schrodinger's equation and the guidance equation for the particle. I then find the Hamiltonian for a relativistic particle in Dirac's theory and for a quantum scalar field.

quant-ph↗

arXiv2016

Modified Palatini action that gives the Einstein-Maxwell theory

The actions for bosonic fields typically contain terms quadratic in the derivatives of the fields. This is not the case in the Palatini approach to general relativity. The action does not contain any derivatives of the metric and it only contains terms linear in the derivatives of the connection. In general relativity the covariant derivative of the metric vanishes, so it is not possible to include such terms in the action. However, in more general theories this is not the case. In this paper I consider an action which is quadratic in the derivatives of the metric and connection and show that it leads to the coupled Einstein-Maxwell theory or the coupled Einstein-Proca theory with the antisymmetric part of the Ricci tensor playing the role of the electromagnetic field strength.

gr-qc↗

arXiv2016

Null Surfaces in Static Space-times

In this paper I consider surfaces in a space-time with a Killing vector $ξ^α$ that is time-like and hypersurface orthogonal on one side of the surface. The Killing vector may be either time-like or space-like on the other side of the surface. It has been argued that the surface is null if $ξ_αξ^α\rightarrow 0$ as the surface is approached from the static region. This implies that, in a coordinate system adapted to $ξ$, surfaces with $g_{tt}=0$ are null. In spherically symmetric space-times the condition $g^{rr}=0$ instead of $g_{tt}=0$ is sometimes used to locate null surfaces. In this paper I examine the arguments that lead to these two different criteria and show that both arguments are incorrect. A surface $ξ=$ constant has a normal vector whose norm is proportional to $ξ_αξ^α$. This lead to the conclusion that surfaces with $ξ_αξ^α=0$ are null. However, the proportionality factor generally diverges when $g_{tt}=0$, leading to a different condition for the norm to be null. In static spherically symmetric space-times this condition gives $g^{rr}=0$, not $g_{tt}=0$. The problem with the condition $g^{rr}=0$ is that the coordinate system is singular on the surface. One can either use a nonsingular coordinate system or examine the induced metric on the surface to determine if it is null. By using these approaches it is shown that the correct criteria is $g_{tt}=0$. I also examine the condition required for the surface to be nonsingular.

gr-qc↗

arXiv2016

Lorentz gauge quantization in synchronous coordinates

It has been shown that the Gupta-Bleuler method of quantization can be used to impose the Lorentz gauge condition in static space-times but not in cosmological space-times. This implies that the Gupta-Bleuler approach fails in general in non-static space-times. More recently, however, the Dirac method of quantizing constrained dynamical systems has been successfully employed to impose the Lorentz gauge in conformally flat space-times. In this paper we generalize this result by using Dirac's method to impose the Lorentz gauge in a general space-time region where the metric is expressed in synchronous coordinates.

gr-qc↗

arXiv2015

Lorenz gauge quantization in conformally flat spacetimes

Recently it was shown that Dirac's method of quantizing constrained dynamical systems can be used to impose the Lorenz gauge condition in a four-dimensional cosmological spacetime. In this paper we use Dirac's method to impose the Lorenz gauge condition in a general four-dimensional conformally flat spacetime and find that there is no particle production. We show that in cosmological spacetimes with dimension $D\neq 4$ there will be particle production when the scale factor changes, and we calculate the particle production due to a sudden change.

gr-qc↗

arXiv2013

Lorentz Gauge Quantization in a Cosmological Space-time

It has recently been shown that it is not possible to impose the Lorentz gauge condition in a cosmological space-time using the Gutpa-Bleuler method of quantization. It was also shown that it is possible to add $\nabla_μA^μ$ as a new degree of freedom to the electromagnetic field and that this new degree of freedom might be the dark energy which is producing the accelerated expansion of the Universe. In this paper I show that the Lorentz gauge condition can be imposed using Dirac's method of quantizing constrained dynamical systems. I also compute the vacuum expectation value of the energy-momentum tensor and show that it vanishes. Thus, in Dirac's approach, the electromagnetic field does not make a contribution to the dark energy.

gr-qc↗

arXiv2010

Addendum to `Gravitational Geons in 1+1 Dimensions'

In a recent paper [arXiv:0807.0611] I found gravitational geons in two classes of 1+1 dimensional theories of gravity. In this paper I examine these theories, with the possibility of a cosmological constant, and find strong field gravitational geons. In the spacetimes in [arXiv:0807.0611] a test particle that is reflected from the origin suffers a discontinuity in $d^2t/dτ^2$. The geons found in this paper do not suffer from this problem.

gr-qc↗

arXiv2008

Homogeneous and isotropic cosmologies with nonlinear electromagnetic radiation

In this paper I examine cosmological models that contain a stochastic background of nonlinear electromagnetic radiation. I show that for Born-Infeld electrodynamics the equation of state parameter, $w=P/ρ$, remains close to 1/3 throughout the evolution of the universe if $E^2=B^2$ in the late universe to a high degree of accuracy. Theories with electromagnetic Lagrangians of the form $L=-{1/4}F^2+αF^4$ have recently been studied in magnetic universes, where the electric field vanishes. It was shown that the $F^4$ term can produce a bounce in the early universe, avoiding an initial singularity. Here I show that the inclusion of an electric field, with $E^2\simeq B^2$ in the late universe, eliminates the bounce and the universe "begins" in an initial singularity. I also examine theories with Lagrangians of the form $L=-{1/4}F^2-μ^8/F^2$, which have been shown to produce a period of late time accelerated expansion in magnetic universes. I show that, if an electric field is introduced, the accelerated phase will only occur if $E^2<3B^2$.

gr-qc↗

arXiv2008

Gravitational Geons in 1+1 Dimensions

It is well known that general relativity does not admit gravitational geons that are stationary, asymptotically flat, singularity free and topologically trivial. However, it is likely that general relativity will receive corrections at large curvatures and the modified field equations may admit solutions corresponding to this type of geon. If geons are produced in the early universe and survive until today they could account for some of the dark matter that has been "observed" in galaxies and galactic clusters. In this paper I consider gravitational geons in 1+1 dimensional theories of gravity. I show that the Jackiw-Teitelboim theory with corrections proportional to $R^2$ and $\Box R$ admits gravitational geons. I also show that gravitational geons exist in a class of theories that includes Lagrangians proportional to $R^{2/3}$.

gr-qc↗

arXiv2007

Noether Charge and Black Hole Entropy in Modified Theories of Gravity

The entropy of black holes in modified theories of gravity is examined in the Palatini formalism using the Noether Charge approach. It is shown that, if the gravitational coupling constant is properly identified, the entropy of a black hole is one-quarter of the horizon area in f(R) theories coupled to conformally invariant matter. If matter is present that is not conformally invariant the entropy is still proportional to the area of the black hole, but the coefficient is generally not one-quarter. The entropy of black holes in generalized dilaton theories and in theories with Lagrangians that depend on an arbitrary function of the Ricci tensor are also examined.

gr-qc↗

arXiv2006

Black hole and cosmological space-times in Born-Infeld-Einstein theory

In this paper I examine black hole and cosmological space-times in Born-Infeld-Einstein theory with electric and magnetic charges. The field equations are derived and written in the form $G_{μν}=-κT_{μν}$ for spherically symmetric space-times. The energy-momentum tensor is not the Born-Infeld energy-momentum tensor, but can be obtained from Born-Infeld theory by letting $a\to ia$, where $a$ is the Born-Infeld parameter. It is shown that there is a curvature singularity in spherically symmetric space-times at a nonzero radial coordinate and that, as in Reissner-Nordstrom space-times, there are zero, one or two horizons. Charged black holes have either two horizons and a timelike singularity or one horizon with a spacelike, timelike, or null singularity. Anisotropic cosmological solutions with electric and magnetic fields are obtained from the spherically symmetric solutions.

gr-qc↗

arXiv2006

Einstein-Maxwell and Einstein-Proca theory from a modified gravitational action

A modified gravitational action is considered which involves the quantity $F_{μν}=\partial_μΓ_ν-\partial_νΓ_μ$, where $Γ_μ=Γ^α_{μα}$. Since $Γ_μ$ transforms like a U(1) gauge field under coordinate transformations terms such as $F^{μν}F_{μν}$ are invariant under coordinate transformations. If such a term is added to the usual gravitational action the resulting field equations, obtained from a Palatini variation, are the Einstein-Proca equations. The vector field can be coupled to point charges or to a complex scalar density of weight $ie$, where $e$ is the charge of the field. If this scalar density is taken to be $g^{-ie/2}$ and the overall factor of the scalar density Lagrangian takes on a particular value the resulting field equations are the Einstein-Maxwell equations.

gr-qc↗

arXiv2005

Born-Infeld-Einstein theory with matter

The field equations associated with the Born-Infeld-Einstein action including matter are derived using a Palatini variational principle. Scalar, electromagnetic, and Dirac fields are considered. It is shown that an action can be chosen for the scalar field that produces field equations identical to the usual Einstein field equations minimally coupled to a scalar field. In the electromagnetic and Dirac cases the field equations reproduce the standard equations only to lowest order. The spherically symmetric electrovac equations are studied in detail. It is shown that the resulting Einstein equations correspond to gravity coupled to a modified Born-Infeld theory. It is also shown that point charges are not allowed. All particles must have a finite size. Mass terms for the fields are also considered.

gr-qc↗

arXiv2004

On the Dirac field in the Palatini form of 1/R gravity

In recent papers (astro-ph/0306630, gr-qc/0312041) I have argued that the observed cosmological acceleration can be accounted for by the inclusion of a 1/R term in the gravitational action in the Palatini formalism. Subsequently, Flanagan (astro-ph/0308111, gr-qc/0403063) argued that this theory is equivalent to a scalar-tensor theory which produces corrections to the standard model that are ruled out experimentally. In this article I examine the Dirac field coupled to 1/R gravity. The Dirac action contains the connection which was taken to be the Christoffel symbol, not an independent quantity, in the papers by Flanagan. Since the metric and connection are taken to be independent in the Palatini approach it is natural to allow the connection that appears in the Dirac action to be an independent quantity. This is the approach that is taken in this paper. The resulting theory is very different and much more complicated than the one discussed in Flanagan's papers.

gr-qc↗

arXiv2003

On the viability of the Palatini form of 1/R gravity

Recently Flanagan [astro-ph/0308111] has argued that the Palatini form of 1/R gravity is ruled out by experiments such as electron-electron scattering. His argument involves adding minimally coupled fermions in the Jordan frame and transforming to the Einstein frame. This produces additional terms that are ruled out experimentally. Here I argue that this conclusion is false. It is well known that conformally related theories are mathematically equivalent but not physically equivalent. As discussed by Magnano and Sokolowski [2] one must decide, in the vacuum theory, which frame is the physical frame and add the minimally coupled Lagrangian in this frame. If this procedure is followed the resulting theory is not ruled out experimentally. The discussions in this paper also show that the equivalence between the generalized gravitational theories and scalar tensor theories discussed by Flanagan [gr-qc/0309015] is only mathematical, not physical.

gr-qc↗

arXiv2003

Palatini approach to Born-Infeld-Einstein theory and a geometric description of electrodynamics

The field equations associated with the Born-Infeld-Einstein action are derived using the Palatini variational technique. In this approach the metric and connection are varied independently and the Ricci tensor is generally not symmetric. For sufficiently small curvatures the resulting field equations can be divided into two sets. One set, involving the antisymmetric part of the Ricci tensor $R_{\stackrel{μν}{\vee}}$, consists of the field equation for a massive vector field. The other set consists of the Einstein field equations with an energy momentum tensor for the vector field plus additional corrections. In a vacuum with $R_{\stackrel{μν}{\vee}}=0$ the field equations are shown to be the usual Einstein vacuum equations. This extends the universality of the vacuum Einstein equations, discussed by Ferraris et al. \cite{Fe1,Fe2}, to the Born-Infeld-Einstein action. In the simplest version of the theory there is a single coupling constant and by requiring that the Einstein field equations hold to a good approximation in neutron stars it is shown that mass of the vector field exceeds the lower bound on the mass of the photon. Thus, in this case the vector field cannot represent the electromagnetic field and would describe a new geometrical field. In a more general version in which the symmetric and antisymmetric parts of the Ricci tensor have different coupling constants it is possible to satisfy all of the observational constraints if the antisymmetric coupling is much larger than the symmetric coupling. In this case the antisymmetric part of the Ricci tensor can describe the electromagnetic field, although gauge invariance will be broken.

gr-qc↗