Quantum corrections to the spacetime metric from geometric phase space quantization

We consider the possibility that the physical spacetime of a quantum particle may be regarded as a four-dimensional hypersurface locally embedded in eightdimensional phase space. We show that, as a consequence, accelerated particles are seen to live in a curved spacetime, and, in the particular case of uniform acceleration, we are led to a generalization of the Rindler metric which implies, for a uniformly accelerated particle, a discrete energy spectrum.     

Aspects of electrostatics in a weak gravitational field

Several features of electrostatics of point charged particles in a weak, homogeneous, gravitational field are discussed using the Rindler metric to model the gravitational field. Some previously known results are obtained by simpler and more transparent procedures and are interpreted in an intuitive manner. Specifically: (a) We discuss possible definitions of the electric field in curved spacetime (and noninertial frames), argue in favour of a specific definition for the electric field and discuss its properties. (b) We show that the electrostatic potential of a charge at rest in the Rindler frame (which is known and is usually expressed as a complicated function of the coordinates) is expressible as A ₀ = q/λ where λ is the affine parameter distance along the null geodesic from the charge to the field point. (c) This relates well with the result that the electric field lines of a charge coincide with the null geodesics; that is, both light and the electric field lines ‘bend’ in the same manner in a weak gravitational field. We provide a simple proof for this result as well as for the fact that the null geodesics (and field lines) are circles in space. (d) We obtain the sum of the electrostatic forces exerted by one charge on another in the Rindler frame and discuss its interpretation. In particular, we compare the results in the Rindler frame and in the inertial frame and discuss their consistency. (e) We show how a purely electrostatic term in the Rindler frame appears as a radiation term in the inertial frame. (In part, this arises because charges at rest in a weak gravitational field possess additional weight due to their electrostatic energy. This weight is proportional to the acceleration and falls inversely with distance—which are the usual characteristics of a radiation field.) (f) We also interpret the origin of the radiation reaction term by extending our approach to include a slowly varying acceleration. Many of these results might have possible extensions for the case of electrostatics in an arbitrary static geometry.     

Equivalence Principle and Radiation by a Uniformly Accelerated Charge

We address the old question of whether or not a uniformly accelerated charged particle radiates, and consequently, if weak equivalence principle is violated by electrodynamics. We show that radiation has different meanings; some absolute, some relative. Detecting photons or electromagnetic waves is not absolute, it depends both on the electromagnetic field and on the state of motion of the antenna. An antenna used by a Rindler observer does not detect any radiation from a uniformly accelerated co-moving charged particle. Therefore, a Rindler observer cannot decide whether or not he is in an accelerated lab or in a gravitational field. We also discuss the general case.     

Vacuum polarization induced by a uniformly accelerated charge

We consider a point charge fixed in the Rindler coordinates which describe a uniformly accelerated frame. We determine an integral expression of the induced charge density due to the vacuum polarization at the first order in the fine structure constant. In the case where the acceleration is weak, we give explicitly the induced electrostatic potential.     

The equivalence principle and an electric charge in a gravitational field

It is shown that there is no violation of the strong principle of equivalence in the case of an electric charge either falling freely or supported in a static uniform gravitational field. For a freely falling charge, the global electromagnetic field distribution at any instant is found to be the same as that of a charge which is moving uniformly with respect to an inertial frame with a velocity equal to the instantaneous velocity of the freely falling charge. In the case of a charge supported in the gravitational field, the total electromagnetic field energy, as measured by freely falling observers instantaneously at rest with respect to the charge, is shown to be equal to the Coulomb field energy of a charge permanently stationary in an inertial frame. The conclusion here, that in neither of the two cases does the charge emit electromagnetic radiation, is independent of our choice of the observer's frame of reference.     

Contrasting Classical and Quantum Vacuum States in Non-inertial Frames

Classical electron theory with classical electromagnetic zero-point radiation (stochastic electrodynamics) is the classical theory which most closely approximates quantum electrodynamics. Indeed, in inertial frames, there is a general connection between classical field theories with classical zero-point radiation and quantum field theories. However, this connection does not extend to noninertial frames where the time parameter is not a geodesic coordinate. Quantum field theory applies the canonical quantization procedure (depending on the local time coordinate) to a mirror-walled box, and, in general, each non-inertial coordinate frame has its own vacuum state. In particular, there is a distinction between the “Minkowski vacuum” for a box at rest in an inertial frame and a “Rindler vacuum” for an accelerating box which has fixed spatial coordinates in an (accelerating) Rindler frame. In complete contrast, the spectrum of random classical zero-point radiation is based upon symmetry principles of relativistic spacetime; in empty space, the correlation functions depend upon only the geodesic separations (and their coordinate derivatives) between the spacetime points. The behavior of classical zero-point radiation in a noninertial frame is found by tensor transformations and still depends only upon the geodesic separations, now expressed in the non-inertial coordinates. It makes no difference whether a box of classical zero-point radiation is gradually or suddenly set into uniform acceleration; the radiation in the interior retains the same correlation function except for small end-point (Casimir) corrections. Thus in classical theory where zero-point radiation is defined in terms of geodesic separations, there is nothing physically comparable to the quantum distinction between the Minkowski and Rindler vacuum states. It is also noted that relativistic classical systems with internal potential energy must be spatially extended and can not be point systems. The classical analysis gives no grounds for the “heating effects of acceleration through the vacuum” which appear in the literature of quantum field theory. Thus this distinction provides (in principle) an experimental test to distinguish the two theories.     

Fermions’ Tunnelling from the Reissner- Nordström-Anti-de Sitter Black Hole

Very recent work of Kerner and Mann involving fermions tunnelling from the Rindler space-time and a general non-rotating black hole is extended to the case of Reissner-Nordström-anti-de Sitter black hole. Due to the couple between the gravity field and electromagnetic field, we introduce the Dirac equation of the charged particles to determine the action of the radiation. We further consider the correction of the thermal spectrum in the unfixed background space time. It is shown that when the energy and charge conservations are considered, the tunnelling rate of fermions is also related to the change of Bekenstein-Hawking entropy, implying the underlying unitary theory is satisfied.     

Behavior of a charged particle in a schwarzschild field

The influence of the De Witt self-action force on the motion of and electromagnetic emission from a charged particle in a Schwarzschild field is considered. It is shown that a charged particle in a Schwarzschild field is equivalent to a neutral particle of the same mass in a certain Reissner-Nordstrom field. A relationship is found between the power of the electromagnetic emission from an accelerated charge and the power of the thermal emission generated in a reference frame with the same acceleration at the event horizon. The quantum-mechanical problem of the motion of and emission from a charge in the field of a minihole is considered. Wave functions, the energy spectrum, and the widths of quasi-stationary levels are found with allowance for the De Witt self-action force. It is shown that the latter is important for large charges, when the solution becomes oscillatory. "Brainstorm" Little Science and Technology Enterprise. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 75–82, July, 1998.     

The Equivalence Principle and an Electric Charge in a Gravitational Field II. A Uniformly Accelerated Charge Does Not Radiate

The electromagnetic field of a charge supported in a uniform gravitational field is examined from the viewpoint of an observer falling freely in the gravitational field. It is argued that such a charge, which from the principle of equivalence is moving with a uniform acceleration with respect to the (inertial) observer, could not be undergoing radiation losses at a rate implied by Larmor's formula. It is explicitly shown that the total energy in electromagnetic fields, including both velocity and acceleration fields, of a uniformly accelerated charge, at any given instant of the inertial observer's time, is just equal to the self-energy of a non-accelerated charge moving with a velocity equal to the instantaneous present velocity of the accelerated charge. At any given instant of time, and as seen with respect to the present position of the uniformly accelerated charge, although during the acceleration phase there is a radially outward component of the Poynting vector, there is throughout a radially inward Poynting flux component during the deceleration phase, and a null Poynting vector at the instant of the turn around. From Poynting's theorem, defined for any region of space strictly in terms of fixed instants of time, it is shown that a uniformly accelerated charge does not emit electromagnetic radiation, in contrast to what is generally believed. Contrary to some earlier suggestions in the literature, there is no continuous passing of electromagnetic radiation from a uniformly accelerated charge into the region inaccessible to a co-accelerating observer.     

Radiation from a Uniformly Accelerated Charge

The emission of radiation by a uniformly accelerated charge is analyzed. According to the standard approach, a radiation is observed whenever there is a relative acceleration between the charge and the observer. Analyzing difficulties that arose in the standard approach, we propose that a radiation is created whenever a relative acceleration between the charge and its own electric field exists. The electric field induced by a charge accelerated by an external (nongravitational) force is not accelerated with the charge. Hence the electric field is curved in the instantaneous rest frame of the accelerated charge. This curvature gives rise to a stress force, and the work done to overcome the stress force is the source of the energy carried by the radiation. In this way, the energy balance paradox finds its solution.     

Electrodynamics in accelerated frames revisited

Maxwell's equations are formulated in arbitrary moving frames by means of tetrad fields, which are interpreted as reference frames adapted to observers in space‐time. We assume the existence of a general distribution of charges and currents in an inertial frame. Tetrad fields are used to project the electromagnetic fields and sources on accelerated frames. The purpose is to study several configurations of fields and observers that in the literature are understood as paradoxes. For instance, are the two situations, (i) an accelerated charge in an inertial frame, and (ii) a charge at rest in an inertial frame described from the perspective of an accelerated frame, physically equivalent? Is the electromagnetic radiation the same in both frames? Normally in the analysis of these paradoxes the electromagnetic fields are transformed to (uniformly) accelerated frames by means of a coordinate transformation of the Faraday tensor. In the present approach coordinate and frame transformations are disentangled, and the electromagnetic field in the accelerated frame is obtained through a frame (local Lorentz) transformation. Consequently the fields in the inertial and accelerated frames are described in the same coordinate system. This feature allows the investigation of paradoxes such as the one mentioned above.     

Generalized Uncertainty Relation of One-Dimensional Rindler Oscillator

General Minkowski vacuum state is seen to be equivalent to a thermal bath for a Rindler uniformly accelerated observer. This paper calculates the generalized uncertainty relation of one-dimensional Rindler oscillator in the coordinate representation. The calculations show that for a Rindler uniformly accelerated observer there is not only general quantum fluctuation but also thermal fluctuation related to his acceleration.     

The Blackbody Radiation Spectrum Follows from Zero-Point Radiation and the Structure of Relativistic Spacetime in Classical Physics

The analysis of this article is entirely within classical physics. Any attempt to describe nature within classical physics requires the presence of Lorentz-invariant classical electromagnetic zero-point radiation so as to account for the Casimir forces between parallel conducting plates at low temperatures. Furthermore, conformal symmetry carries solutions of Maxwell’s equations into solutions. In an inertial frame, conformal symmetry leaves zero-point radiation invariant and does not connect it to non-zero-temperature; time-dilating conformal transformations carry the Lorentz-invariant zero-point radiation spectrum into zero-point radiation and carry the thermal radiation spectrum at non-zero temperature into thermal radiation at a different non-zero temperature. However, in a non-inertial frame, a time-dilating conformal transformation carries classical zero-point radiation into thermal radiation at a finite non-zero-temperature. By taking the no-acceleration limit, one can obtain the Planck radiation spectrum for blackbody radiation in an inertial frame from the thermal radiation spectrum in an accelerating frame. Here this connection between zero-point radiation and thermal radiation is illustrated for a scalar radiation field in a Rindler frame undergoing relativistic uniform proper acceleration through flat spacetime in two spacetime dimensions. The analysis indicates that the Planck radiation spectrum for thermal radiation follows from zero-point radiation and the structure of relativistic spacetime in classical physics.     

Asymptotic analysis of ultra-relativistic charge

This article offers a new approach for analysing the dynamic behaviour of distributions of charged particles in an electromagnetic field. After discussing the limitations inherent in the Lorentz-Dirac equation for a single point particle a simple model is proposed for a charged continuum interacting self-consistently with the Maxwell field in vacuo. The model is developed using intrinsic tensor field theory and exploits to the full the symmetry and light-cone structure of Minkowski spacetime. This permits the construction of a regular stress-energy tensor whose vanishing divergence determines a system of non-linear partial differential equations for the velocity and self-fields of accelerated charge. Within this covariant framework a particular perturbation scheme is motivated by an exact class of solutions to this system describing the evolution of a charged fluid under the combined effects of both self and external electromagnetic fields. The scheme yields an asymptotic approximation in terms of inhomogeneous linear equations for the self-consistent Maxwell field, charge current and time-like velocity field of the charged fluid and is defined as an ultra-relativistic configuration. To facilitate comparisons with existing accounts of beam dynamics an appendix translates the tensor formulation of the perturbation scheme into the language involving electric and magnetic fields observed in a laboratory (inertial) frame.     

Accelerating electromagnetic magic field from the C-metric

Various aspects of the C-metric representing two rotating charged black holes accelerated in opposite directions are summarized and its limits are considered. A particular attention is paid to the special-relativistic limit in which the electromagnetic field becomes the “magic field” of two oppositely accelerated rotating charged relativistic discs. When the acceleration vanishes the usual electromagnetic magic field of the Kerr–Newman black hole with gravitational constant set to zero arises. Properties of the accelerated discs and the fields produced are studied and illustrated graphically. The charges at the rim of the accelerated discs move along spiral trajectories with the speed of light. If the magic field has some deeper connection with the field of the Dirac electron, as is sometimes conjectured because of the same gyromagnetic ratio, the “accelerating magic field” represents the electromagnetic field of a uniformly accelerated spinning electron. It generalizes the classical Born’s solution for two uniformly accelerated monopole charges.     

Ion acceleration by the space charge electric force arising from the radiation pressure in a magnetized electron–positron plasma

It is shown that ions can be accelerated by the space charge electric force arising from the separation of electrons and positrons due to the ponderomotive force of the magnetic field-aligned circularly polarized electromagnetic (CPEM) wave in a magnetized electron–positron–ion plasma. The ion acceleration critically depends on the external magnetic field strength. The result is useful in understanding differential ion acceleration in magnetized electron–positron–ion plasmas, such as those in magnetars and in some laboratory experiments that aim to mimic astrophysical environments.     

Inertial frame of reference in kerr space

In the gravitational field produced by a uniformly rotating charged ring, a frame of reference that is the analog of an inertial frame in flat space is constructed. Expressions are obtained for the energy in a sphere of radius r and the total energy in such a frame of reference. The results are compared with the results of other papers.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 57–62, January, 1974.     

Space-time structure and bound-charge fields

Together with a “postulate of equivalent situations,” the exact solution for the field of a charge in a uniformly accelerated noninertial frame of reference (NFR) makes it possible to find the space-time structure and fields of charged conductors of arbitrary shape without using the Einstein equations. The energy of the electric field outside of a charged plane, which is equal to the rest energy of the masses of the charges creating the field, is determined. The space-time metric outside of the charged plane is established; it could also have been found from the exact solution of the Einstein-Maxwell equations. This solution describes the equilibrium of charged dust in parallel electric and gravitational fields. The field and metric are found outside of a charged conducting sphere. While it eliminates the self-energy divergence, the proposed method renders the classical electrodynamics internally consistent on transition to any short distance. All-Russian Scientific Research Institute of Opticophysical Measurements. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 63–74, October, 1997.