A time varying space-time metric is shown to be a source of electromagnetic
radiation. The post-Newtonian approximation is used as a realistic model of the
connection between the space-time metric and a time varying gravitational
potential. Large temporal variations in the metric from the coalescence of
colliding black holes and neutron stars are shown to be possible progenitors of
gamma ray burst and millisecond pulsars.Comment: Accept by International Journal of Modern Physics 

Jones, Preston

English

arXiv

A time varying space–time metric is shown to be a source of electromagnetic radiation even in the absence of charge sources. The post-Newtonian approximation is used as a realistic model of the connection between the space–time metric and a time-varying gravitational potential. Rapid temporal variations in the metric from the coalescence of relativistic stars are shown to be likely progenitors of gamma ray burst and millisecond pulsars

Embry-Riddle Aeronautical University

Publications 11-2007 Electromagnetic Radiation From Temporal Variations in Space-Time and Progenitors of Gamma Ray Burst and Millisecond Pulsars Preston Jones University of Louisiana at Monroe, jonesp13@erau.edu Follow this and additional works at: https://commons.erau.edu/publication  Part of the Astrophysics and Astronomy Commons Scholarly Commons Citation Jones, P. (2007). Electromagnetic Radiation From Temporal Variations in Space-Time and Progenitors of Gamma Ray Burst and Millisecond Pulsars. International Journal of Modern Physics D, 16(1). https://doi.org/10.1142/S0218271807011164 This Article is brought to you for free and open access by Scholarly Commons. It has been accepted for inclusion in Publications by an authorized administrator of Scholarly Commons. For more information, please contact commons@erau.edu. arXiv:gr-qc/0606102v4  31 Aug 2007Electromagnetic radiation from temporalvariations in space-time and progenitors ofgamma ray burst and millisecond pulsarsPreston JonesDepartment of Mathematics and PhysicsUniversity of Louisiana at MonroeMonroe, LA 71209August 16, 2007AbstractA time varying space-time metric is shown to be a source of electromag-netic radiation even in the absence of charge sources. The post-Newtonianapproximation is used as a realistic model of the connection between thespace-time metric and a time varying gravitational potential. Rapid tem-poral variations in the metric from the coalescence of relativistic starsare shown to be likely progenitors of gamma ray burst and millisecondpulsars.1 IntroductionWith the observation of the bending of light around the sun by Eddington in1919 it has been generally accepted that gravity and electromagnetism are inter-related. In the intervening years this relation between gravity and electromag-netism has been established beyond the association of gravity with curvatureof space-time and the path of a beam of light. For example recent research [1]has established a theoretical connection between the gravity wave predictions ofgeneral relativity and stimulation of electromagnetic radiation in charged plas-mas. A more fundamental relationship between gravity and electromagneticradiation will be established here by solving the covariant Maxwell’s equationsin the presence of a rapidly time varying gravitational potential and in the ab-sence of any charge sources. In particular rapid variations in space-time, as aconsequence of the collision of astronomical objects, will be shown to be a sourceof observed electromagnetic emissions.It is well established that the homogeneous and inhomogeneous 4-space equa-tions of electrodynamics reduce to the familiar four Maxwell equations for thespecial case of a Lorentz inertial frame and flat space-time metric. Where the1space-time is not flat these equations are complicated by space-time variationsin the metric. This complication follows from the appearance of the metricin the relation between the electric and magnetic fields and the differential 4-space operator in the 4-space equations. Because of this explicit dependence onthe metric in the 4-space equations of electrodynamics variations in space-timewould be expected to contribute to the electrodynamic fields. It would also beexpected that temporal variations in the metric will contribute to the electricand magnetic fields independent of the local charge distribution.The 4-space equations of electrodynamics, in order to insure the invarianceof the equations, must relate the 4-space invariant electric and magnetic fieldsand differential operator to the invariant charge sources. This is most commonlyachieved by first relating a 2-rank tensor, the Electromagnetic Tensor, to theelectric and magnetic fields [2]. The association between the ElectromagneticTensor, the 4-space metric, and the electromagnetic fields in the equations ofelectrodynamics is then somewhat obscure since the fields only appear indirectlyin the equations. It is much easier to identify the relation between the electricand magnetic fields and the 4-space metric in an alternative representation ofthe equations of electrodynamics that includes the fields directly in the invariantequations. The explicit appearance of the fields in the 4-space equations wasestablished by Ellis [3][4] where the electromagnetic tensor is replaced with thedirect products of the fields and the local 4-velocity.In order to examine the relation between a time varying space-time metricand the electromagnetic fields the 4-space electrodynamic equations are ex-panded here under the conditions of a rapidly time varying metric in a space-time region that is free of charge sources. Solving for the divergence of the elec-tromagnetic power the time variations in space-time are shown to be a source ofelectromagnetic radiation. These rapid time variations in the space-time metriccan be related to realistic astronomical phenomena by expressing the metric interms of the gravitational potential using the post-Newtonian approximation.From this approximation the rapid changes in the gravitational potential, asso-ciated with collisions of black-holes, are shown to be a probable source for thepower spectra from gamma ray burst. As a second example of this phenomenon,temporal changes in the gravitational potential during ring-down following neu-tron star collisions are shown to be the possible progenitor of millisecond pulsars.2 Covariant Equations of ElectrodynamicsThe equations of electrodynamics developed by Ellis [3] exhibit an implicit con-nection between the electric and magnetic fields and the 4-space metric throughthe covariant derivatives. Following Ellis [3] [5], the homogeneous Maxwell equa-tions can be expressed in covariant form as the direct product of the fields andthe local 4-velocity,[Bγuδ];γ− [uγBδ];γ+[eγδαβuαEβ];γ= 0. (1)The 4-velocity uα is the velocity of the differential volume element associated2with the fields and Bγ and Eβ are the components of the 4-vector magnetic andelectric fields respectively. The Levi-Civita tensor eαβγδ is defined in terms ofthe permutation symbol Eαβγδ, which is zero if any indices are repeated, onefor even permutations of 0, 1, 2, 3, and negative one for odd permutations,eαβγδ = − 1√−gEαβγδ, (2)where g = det (eˆα · eˆβ) is the determinant of the metric components. Theinhomogeneous Maxwell equations are written in a similar fashion [5],[Eγuδ];γ− [uγEδ];γ− [eγδαβuαBβ];γ = 4piJδ, (3)where Jδ is the 4-vector electric current density. Assuming a Lorentz inertialframe (LIF) the covariant derivatives can be replaced with partial derivatives.Adopting the convention of a negative time part for the metric the 4-velocity isuα = u0 = −1. Expanding the homogeneous equation with δ = 0 leads toBk,k = 0 (4)and in the inhomogeneous equation toEk,k = 4piJ0, (5)where the Roman indices range over 1, 2, 3. Taking δ = i in the homogeneousand inhomogeneous equations produces the remaining Maxwell equations,Bi,0 + EijkEk,j = 0 (6)andEijkBk,j − Ei,0 = 4piJ i. (7)This expansion demonstrates that the covariant equations 1 and 3 are the correctform invariant equations of electrodynamics.3 Time Dependent Field in Post Newtonian Ap-proximationIn order to examine the relation between the covariant equations of electrody-namics and variations in the gravitational potential the 4-space metric will beexpanded in terms of the potential. The Post-Newtonian approximation as-sumes that only the lowest order terms in this expansion contribute appreciablyto the metric. Making this assumption and requiring that the gravitationalpotential be time dependent, the time component of the metric isg00 = − (1 + 2φ) , g00 = − (1− 2φ) . (8)3In this post-Newtonian approximation the spacial components are,gii = 1− 2φ, gii = 1 + 2φ, (9)with all other components equal to zero. The Christoffel symbols will be neededin the expansion of the covariant derivatives and the nonzero terms areΓ000 = −Γ0ii = φ˙− 2φφ˙, Γii0 = −φ˙− 2φφ˙. (10)where the dots represent time derivatives and there is no sum on i. It will behelpful later to define a sum of two Christoffel symbols,Γ000 + Γii0 = −2∂∂tφ2 = −Σ. (11)In a comoving frame, ui = 0, the covariant derivatives in the equations ofelectrodynamics can be expanded in terms of this sum. The time derivatives interms of some arbitrary vector Ak and the direct product with the 4-velocityare[u0Ak];0= Ak,0 − ΣAk. (12)The covariant space derivatives are simply the partial derivatives,[u0Ak];k= Ak,k. (13)The dual term in the equations requires a bit more attention. Since the metricis not position dependent the Christoffel symbols will vanish and the covariantderivatives are again partial derivatives,[eij0ku0Ak];i=[(− 1√−gEij0k)(−1)Ak],i. (14)With our assumptions of a time dependent metric in the post-Newtonian ap-proximation the spacial derivatives of the determinate make no contribution andto first order the covariant derivatives of the dual terms,[eij0ku0Ak];i= −EijkAk,j , (15)are simply the negative of the curl.4 Electromagnetic Power SpectraThe post-Newtonian approximation for a time varying 4-space metric will beused to demonstrate the connection between the gravitational potential andthe electromagnetic fields in the covariant equations of electrodynamics. Theequations of electrodynamics will be expanded assuming a comoving referenceframe and small second order terms in the metric, φ2 ≺≺ 1 and φ2 ≺≺ ∂∂tφ2. To4demonstrate that a time varying gravitational potential generates electromag-netic radiation assume also that there are no charge sources, Jα = 0. Takingthe equations for δ = i and no charge sources,[u0Bi];0+ EijkEk,j = 0 (16)and[u0Ei];0− EijkBk,j = 0. (17)Making use of the expansions of the covariant derivatives from the previoussection and contracting each equation with the appropriate field,BiBi,0 +BiEijkEk,j = ΣB2 (18)andEiEi,0 − EiEijkBk,j = ΣE2. (19)Adding these two equations and rearranging the terms,BiEijkEk,j − EiEijkBk,j + 12∂∂t(B2 + E2)= Σ(B2 + E2). (20)The terms on the left hand side can be simplified using the triple productidentity, ∇ · (a× b) = b · (∇×a)− a · (∇×b), dividing by 14pi, and substitutingthe energy density U = 18pi(E2 +B2),∇ · S+ ∂U∂t=4c4U∂∂tφ2, (21)where use has been made of the Poynting vector, S = c4piE×B, and the factorof c has been reinstated to illustrate the order of magnitude of this effect. Thisdemonstrates that in a region of nonzero internal energy density U and rapidlytime varying gravitational potential, φ, electromagnetic radiation will be gen-erated. Due to the square of the gravitational potential in the time derivativethe period of the electromagnetic power spectra would be half the period of thevariation in the gravitational potential. This relationship between electromag-netic radiation and a time varying metric is formally similar to the expressionfor the interaction between gravity waves and charged plasmas [1]. The currentdevelopment is a more general consequence of the covariant derivatives in theMaxwell equations and in particular does not assume a charged plasma or anyother charge sources.5 Gamma Ray Burst and Millisecond PulsarsLet us consider the electromagnetic radiation that will be generated from thecollision of relativistic stars. Relativistically the internal energy density U wouldbe associated with the total mass-energy density of the binary system. Duringcoalescence a fraction of the rest mass energy will be radiated from the system5as electromagnetic radiation. Since the gravitational potential increases mostrapidly at the end of the collision process this is a likely progenitor of theradiation spike seen in gamma ray burst. Collisions of neutron star binarieshave been previously identified as a potential source of gamma ray burst [1][6][7].However, the magnitude of the radiated power calculated from the collisions ofneutron stars is much smaller than the observed radiated power.Since the typical energy radiated from gamma ray burst is much greaterthan the internal energy density of neutron star binaries the progenitor of thethese burst are much more likely to be black hole collisions. Hawking [8] hasestimated that as much as 50 percent of the original rest mass energy escapesduring the coalescence of black hole binaries. Hawking’s estimate would placea lower bound on the radiated energy of the order of one solar mass. However,the internal energy associated with a black hole binary does not have an upperbound and explains the magnitude of the energy radiated in even the largestgamma ray burst. The period of the coalescence of black hole binaries should beassociated with the interval of the transition from two separate event horizonsto a single event horizon which is consistent with the short duration of the peakenergy of gamma ray burst. This association between black holes and gammaray burst provides a method of determining the order of magnitude of the massesof black holes and could also provide a method for determining the distributionof black holes in the universe and the associated mass distribution.Gamma ray bursts are one example of an electromagnetic power spectra thatwould be associated with an astronomical event involving a rapidly varyinggravitational potential. However, the generation of electromagnetic radiationby time variations in a gravitational potential should be ubiquitous. In theexample of the collision of a neutron star binary there is also the possibilityof a ring-down of the coalesced object, provided that the combined mass doesnot produce a black hole. During this ring-down the gravitational potentialwould be expected to generate an electromagnetic power spectra due to thetime variation in the square gravitational potential. The pulsation modes of themetric and associated gravitational potential, in the ring down of a coalescedneutron star, have been calculated by Allen et al [7][9]. The calculated modespredict millisecond time variations in the gravitational potential which wouldexplain the mechanism for electromagnetic radiation from millisecond pulsars.6 ConclusionExpanding the covariant equations of electrodynamics, in terms of the directproducts of the electromagnetic fields and the local 4-velocity and assumingthe post-Newtonian approximation, a time varying metric was shown to bea source of electromagnetic radiation even in the absence of charge sources.Electromagnetic radiation produced by rapid time variations in the gravitationalpotential of coalescing black holes is the likely progenitor of gamma ray burst.This gravity induced electromagnetic radiation is also the likely mechanism forproducing the radiation from millisecond pulsars. The phenomena of gravity6induced electromagnetic radiation would not be expected to be limited to thesetwo examples. However, due to the factor of c4 in the relation between theradiated power and the time rate of change of the potential this phenomenonwill be observable only in astronomical events where space-time variation in thepotential and the internal energy are great.References[1] J. Moortgat and J. Kuijpers, “Gravitational and magnetosonic waves ingamma-ray burst,” Astronomy and Astrophysics aa3292 (2003) 1-7.[2] J. D. Jackson, Classical Electrodynamics (Wiley and Sons, 2nd Edition,1975) 236.[3] G. F. R. Ellis, Cargese Lectures in Physics (Gordon and Breach, New York,1973) ed. Evry Schatzman, 1-60.[4] P. Jones, A dyad theory of hydrodynamics and electrodynamics,arXiv:physics/0502072.[5] S. Sonego, M. A. Abramowicz, “Maxwell equations and the optical geome-try,” Journal of Mathematical Physics 39 (1998) 3158-3166.[6] P. Me´sza´ros and M. J. Rees, “Relativistic fireballs and their impact on exter-nal matter: model for gamma-ray burst,” Astrophysical Journal 405 (1993)278-284.[7] G. D. Allen, N. Anderson, P. Laguan, J. A. Pullin, and J. Ruoff, “Close-limitapproximation to neutron star collisions,” Physical Review D 60 (1999)104021.[8] S. W. Hawking, “Gravitational radiation: the theoretical aspect,” Contem-porary Physics 13 (1972) 272-282.[9] G. D. Allen, N. Anderson, K. D. Kokkotas, and B. F. Schutz, “Gravita-tional waves from pulsating stars: evolving the perturbation equations forrelativistic stars,” Physical Review D 58 (1998) 124012.7

Electromagnetic Radiation From Temporal Variations in Space-Time and Progenitors of Gamma Ray Burst and Millisecond Pulsars

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Temporal variations in space-time and progenitors of gamma ray burst and millisecond pulsars

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