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the that http://pos.sissa.it reAlike Licence. find find which energy-momentum tensor corresponds iro - RJ,- iro20550-013, Brazil RJ,- iro20550-013, Brazil use the energy conditions. This work presents the haracterize a dark energy fluid in the presence of of these two cases produces some bounds for the olutions sometimes have not any physical meaning. he he electromagnetic field in the vacuum and in the ation, between it and the electromagnetic field present in de Janeiro, de Janeiro, mological constant need to have a value bigger than
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1 [email protected] [email protected] The Einstein’s equations can be solved in order to Poster Session
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5th International Schoolon Field Theory and Gravit Cuiabá city, Brazil April 20-24, 2009 Rua São Francisco Xavier 524, Maracanã, Riode Jane da Silva M.F.A. Instituto de Física, Universidade do Estado do Rio Rua São Francisco Xavier 524, Maracanã, Riode Jane O. O. Goldoni Instituto de Física, Universidade do Estado do Rio Energy Conditions for Electromagnetic Field in in Field Electromagnetic for Conditions Energy Constant Cosmological of presence E-mail: E-mail: To choose which solutions have physical meaning, we space-time. In particular, it is shown that the cos
to a given geometry of space-time. However, these s explicit expressions of the energy conditions for presence t of the cosmological constant. The Analysis electromagnetic fields. value value of cosmological constant, showing a relation one considered in cosmological models in order to c
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is the Λ O. Goldoni is the energy- µν T ime ime geometry, , (1.1) ured ured for any observer need to be ervable ervable fluid is always less than the . . (1.8) anisotropic anisotropic fluid , with energydensity (1.5) (1.7) and and the energy are always related with (1.6) have have any physical meaning. To choose (1.4) 0, behaved” matter exercises an effect of energy energy in this space time, and 0. 0. (1.3) 0 0. (1.2) direction need to be positive too, too, positive to be need direction i WEC 0. i i p i i i i is is assured if: if: assured is is ≥ energy conditions. conditions. energy ield Equations. Equations. ield p ≥ p 1 s equation can be represented by: by: represented be can s equation 3 0 = pressure pressure of the fluid do not exceed the energy 0 − ≥ + ≥ i ∑ ρ + ≥ − ≥
T 0. 0 0 + ≥ 0 0 ρ ρ = + Λ 2 2 T T T T ρ µν µν µν G kT g
, respectively. , respectively. i p and ρ a fluid need to obey also the the also obey to fluid need a DEC is the Einstein tensor, which describe the space-t µν G Following Following are present the energy conditions for an This condition imposes that the energy density meas The General Relativity Theory says that the matter This condition says that the local speed of any obs This This condition guarantes that all physically “well Weak Energy Condition (WEC) (WEC) Condition Weak Energy Strong Energy Condition (SEC) (SEC) Condition Strong Energy Dominant Energy Condition (DEC) (DEC) Condition Energy Dominant
always positive. positive. always cosmological constant. constant. cosmological Some solutions of the Einstein’s Equations do not which solutions have physical meaning we use we the meaning physical have which solutions
1.1. and pressures given by by given and pressures or or in any pressure the plus density the energy Also local speed of light. This is assured if the local density, momentum momentum tensor, which describe the matter and the
1.Introduction 1.Introduction Energy conditions for electromagnetic Field in presence of cosmological constant tensor: of terms the energy-momentum In 1.3. or in terms of the energy-momentum tensor:or ofterms in the energy-momentum where convergence in the time-like and null geodesics. Th geodesics. and null time-like the in convergence 1.2. the To obey In terms of the energy-momentum tensorthi of energy-momentum terms the In the geometry of the space-time, by the Einstein’s F by the Einstein’s of space-time, the the geometry
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O. Goldoni need to be µ α Π (2.1) ,
) odels recently provide that (2.3) . , µ α ( (2.2) Π (3.3) momentum momentum is written as (2.3), the , : ed by: ed by: . (3.4) -diagonal, -diagonal, the tensor ic ic field will be purely electric and (3.2) (3.1) . (1.9) space-time without andspace-time rotation δ 0. conditions we can rewrite the energy 0. 2 1 = = == 0. , , which exercises in all bodies just the 2 1 0 0 0 0 i i 4 1 es. The dark energy must repel matter. matter. repel must energy The dark es. ; ; 0 0 the tensor the λδ + ≥
0 0 i i i i 0 = − 0 µ µ 3 3 α α i i Π ≤ Π Π −Π ≥ T T Π −Π ≥ Π π 1 1 , . () 4 4 Π − = Π +Π ≤ Π µν ν µ µ ν ∑ F A A = −Π+ Π λρ λρ Dark Dark Energy
α α α µ µ µ T FF FF π 1 1 4 4 µ µλ λρ µ = − + α αλ λρµ Π= Π =Π µν µλ νρ µν λρ T gFF gFF (SEC) (SEC) (DEC) (DEC) (WEC) (WEC) a fluid need to satisfy also the equation (1.4). (1.4). equation the also to satisfy need fluid a SEC is particularly interesting because cosmological m is the electromagnetic tensor and it is defined by defined is it and tensor electromagnetic the is SEC µν is the four-potential.electromagnetic F in equation (1.1). Once the electromagnetic energy- µ For tensor diagonal energy-momentum we have: In In the case of the energy-momentum tensor to be non In In order to simplify the calculations of the energy In In a space-time without rotation, the electromagnet The electromagnetic energy-momentum tensoris defin energy-momentum The electromagnetic A 0 with constant cosmological Λ = where
energy conditionenergy canwrittenfunction also in be of
where: where: made diagonal,made so: 3.Energy conditions for electromagnetic field in a electromagnetic 3.Energy conditionsfor 2.Energy-momentum tensor of electromagnetic field of electromagnetic 2.Energy-momentum tensor Energy conditions for electromagnetic Field in presence of cosmological constant the satisfy To and The momentum tensormomentum as:
must must to exist a type of energy, called counter effect of the “well behaved” energy exercis energy behaved” “well the of effect counter PoS(ISFTG)072
O. Goldoni ession can be found: essioncan found: be and then, applying this µ (3.8) α (3.10) Π (3.12) . 2 nents of the electromagnetic (4.2) i , i 1 φ 2 2 q (3.7) ∂ e tensor ∂ . (4.1) (3.6) α β (3.9) e presence of cosmological presence e (3.5) , . (3.11) rotation can rotation be can by: given ns can are written as. as. can written are ns ) and (3.7), all of the energy conditions k 0. i
2 1 () , i 0, 1 i ef 3 = 0 2 ( ) 1 by: by: 1 i ∑ , , 2 2 tion tion in an universe without cosmological 0 4 1 0 () α β 0 1 “well behaved”,“well expected. as φ s are found: s ≤ Π = 3 = − ≥ − ≥ Π = = i ∑
= 0 i + ≤ Π i 1 2 λ λ λ 0 µν µν 4 4 λ λ i i iii λ λ = + Λ A t q , . λ λ G kT , , . i i ∑ φ φ q q ef ()() ( ) ∂ ∂ ∂ ∂ µν µν µν
= = − T T g α β 3 0 = − = λ λ = − i 22 2 2 2 0 F F ds tqdt tqdq (DEC) (DEC) (SEC) (WEC) (WEC) analogue to the equation (2.3), the following expr analoguethe equationtothefollowing (2.3),
ef ( ) µν
. T k Λ π 4 k () sing sing the equation (2.2), (3.8) and (3.9), the compo Λ = The general withoutmore metric for the space-time Applying Applying these eigenvalues in equations (3.5), (3.6 Using these components to find the components of th Writing Writing the The equationcan (1.1) redefined by: be Once the eigenvalues are found, the energy conditio energy the found, are eigenvalues Once the Considering the general more as: electric potential constant
results equation in eigenvaluethe following (3.4),
4.Energy conditions for electromagnetic field in th electromagnetic 4.Energy conditionsfor Energy conditions for electromagnetic Field in presence of cosmological constant with and u where the effective energy-momentum tensorwheretheeffective energy-momentum is given
are satisfied. So for a general metric constant, thefieldelectromagnetic always be will without rota tensor are by: given are tensor
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O. Goldoni y, y, for this the (4.3) . (4.8) k (4.7) ic field. We can see that the c (4.6) 2 (4.5) i k φ (4.4) q () k ∂ 2 , ∂ () otential (3.9), and considering thatotential and(3.9), considering the s (3.5), (3.6) and (3.7) in to: to: in(3.7) and(3.6) (3.5), s k 2 . i () φ are found. found. are q 2 . i of electromagnetic energy-momentum ∂ n exotic fluid that exercises a repulsive ∂ ervals are found: 1 2 2 δ nt nt would exercise a repulsive gravitational 0. A B i 2 1 onstant onstant once the purely electric field always he eigenvalues given by equations (3.11) and re interesting to use this energy conditions to nferior limit on the onlimit constant. nferior cosmological 1 2 1 0 1 3 = 4 1 2 2 i ∑
0 A B , that depends on the electromagnetic field, the ≤ Π+Λ − ≥ π k 0 5 5 1 c 8 0 i 3 = − ≥ Π+Λ i ∑ Λ λ λ λ i + ≤ Π+Λ π 1 1 π λ λ 4 4 i k 8 λ λ
= −Π+ Π+Λ ≤ Λ ≤ ∑ 0 α α α µ µ µ Λ> =Λ T ef ( ) is a candidate to represent a model for dark energ Λ (DEC) (DEC) (SEC) (WEC) (WEC) values higher than Λ needs to be positive, but limited for electromagnet Λ Today Today is believed that Analyzing theseAnalyzing resultsthe following expressions, conditions,To allsatisfy energy intthefollowing Using Using the by (3.8) same metric and the given same p Therefore, for So, The cosmological term will transform the expression transform will term The cosmological The interval to brake the SEC SEC the brake to interval The The intervals wich saisfy all energy conditions conditions energy all wich saisfy intervals The
SEC SEC needs to be violated, once the dark energy gravitational force, so: is a 4.2. 4.1.
Energy conditions for electromagnetic Field in presence of cosmological constant
presencetheofelectromagneticimposesi anfield combination f this one with the cosmological consta force. force. cosmological cosmological constant is just added in tensor, the energy conditions can be analyzed for t the diagonal (3.12) plus the cosmological constant, but it is mo estimate intervals at values for the cosmological c conditions. the energy obey
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Λ i i O. Goldoni e of the universe, , Cambridge Monographs on constitute a way to find the a presençade constante , International Student Edition, 2009) , Washington University (1996) , Graduation Monograph, Instituto de onsider onsider all the fields of energy when all universe ages and to analyze each value value can change, in order to represent a omagnetic omagnetic field. Once this limits of ted to the accelerated phase. phase. the to accelerated ted Λ , DidáticosCadernos – UFRJ; f dark energy. To study this more carefully it ifferent type of matter across the time. time. the across matter of type ifferent space-time, itspace-time, is reasonable to conclude that the
, Oxford University Press, (1992); , Laboratoire Physique de Théorique, Institute Henr 6 6
Introduction to General Relativity The Large ScaleStructure of Space-time , based on the energy conditions, depends on the ag Λ Condições de energia paraum fluidogeral Lecturs on General Relativity Introducing relativity Einstein's Graduation Monograph, Instituto de Física UERJ,- ( Condições de energia para o campo eletromagnético n FundamentosRelatividade da Especial Lorentzian Wormholes: Form Einstein to Hawking where the energy conditions are violated. This can Λ This work is only an example of the importance to c In In this work we show how the limits for the M. Visser, R. R. D'Inverno, Mathematical Physics (1973). S.W. Hawking, G.F.R. Ellis, Poincaré (1974); Física Física UERJ,- (2007) T.S. Magalhães, McGraw-Hill Kogakusha LTD. A. Papapetrou,A. J. Barcelos, R. R. Adler, M. Bazin, M. Schiffer, O. Goldoni, cosmológica,
[5] [8] [4] [3] [7] [6] [2] [1] time when occurred transitions between the decelera the between transitions occurred when time depends depends on the type of the matter that fulfill the limit for the parameter
References References Energy conditions for electromagnetic Field 5.Conclusion in presence of cosmological constant dark dark energy fluid, due to the presence of an electr since different ages of universe are dominated by d by dominated are ofages universe since different the constant cosmological is taken as a candidate o is necessary to take the energy-momentum tensor for interval of