The Electromagnetic Radiation and Gravity

Transcription

1 International Journal of Theoretial and Mathematial Physis 016, 6(3): DOI: /j.ijtmp The Eletromagneti Radiation and Gravity Bratianu Daniel Str. Teiului Nr. 16, Ploiesti, Romania Abstrat Is already known that a non-inertial referene frame is equivalent to a ertain gravitational field. In this paper, a non-uniformly aelerated linear motion of the referene frame is analyzed. We will try to prove that this motion generates a non-uniform and variable gravitational field, desribed by a Finsler metri. The attahing to the non-inertial referene frame of an eletri harge, leads to hanges in the struture of this gravitational field. The hanges are produed by the EM radiation emitted, and an be easily reognized in the mathematial expression of the metri of spae-time. The motion of the eletri harge is analyzed also from a quantum perspetive. The onnetion of the Shrodinger equation with the metri of spae-time an be realized by means of a funtion of oordinates, whih defines some Lorentz non-linear transformations. At the basis of these theoretial approahes is lay a variational priniple in whih the veloity of the partile is onsidered as a funtion of oordinates Keywords Damped quantum osillator, Lorentz non-linear transformations, a different type of variational priniple, Finsler spae 1. Introdution Let us onsider that we have a positive harge q loated in the enter of a inertial referene frame (O; x, y, z, t), and a negative harge q loated in the enter of a non-inertial referene frame (O ; x, y, z, t ). Then, let us imagine that the non-inertial referene frame is moving with the aeleration (a x ) in the (x) diretion with respet to the inertial referene frame. Also, let us assume that the aelerated harge is moving suh that the bilinear form x t (1.1) is an invariant with respet to the following non-linear transformations x = a 11 x + a 1 t t = a 1 x + a t (1.) where, a ij = a ij x, t (i,j=1,). So, we try to find the funtions a ij x, t suh that x t = x t (1.3) First we replae the oeffiients a ij = a ij x, t as follows a 11 = osh β x, t a = osh θ x, t a 1 = sinh β x, t a 1 = sinh θ x, t (1.4) Then, substituting the transformations (1.) into equation (1.3), we get sinh β θ = 0 (1.5) * Corresponding author: daniel.bratianu@yahoo.om (Bratianu Daniel) Published online at Copyright 016 Sientifi & Aademi Publishing. All Rights Reserved So, we must have β x, t = θ x, t, and the transformations (1.) beome x = x osh β + t sinh β t = x sinh β + t osh β (1.6) Also, we an admit the following inverse transformations x = x osh α t sinh α t (1.7) = t osh α x sinh α where α = α x, t. Differentiating now the diret transformations (1.6), we dx = osh β dx + sinh β + = osh β + sinh β dx + t dβ (1.8) dβ (1.9) Differentiating also the inverse transformations (1.7), we dx = osh α dx sinh α + dα (1.10) = osh α 1 t sinh α dx + dα (1.11) α In order to find the physial semnifiation of the funtion x, t, we observe that, for the origin x = 0 of the non-inertial referene frame, the equation (1.10) beomes osh α dx sinh α = 0 (1.1) From this, we an dedue the derivative of the position x with respet to time t x = dx x =0 = tanh α x, t (1.13) whih signifies the veloity of non-inertial referene frame with respet to the inertial referene frame. Also, in order to find the physial semnifiation of the funtion β x, t, we observe that, for the origin x = 0 of the inertial referene α

2 94 Bratianu Daniel: The Eletromagneti Radiation and Gravity frame, the equation (1.8) beomes osh β dx + sinh β = 0 (1.14) From this, we an dedue also the derivative of the position x with respet to time t x = dx x=0 = tanh β x, t (1.15) whih signifies the veloity of the inertial referene frame with respet to the non-inertial referene frame. But, aording to the motion relativity priniple, we must have x = x (1.16) Therefore, we an admit the following identity. The Wave Funtion β x, t = α x, t (1.17) Further on, we an assoiate a de Broglie wave with the aelerated harge. The wave must be stationary with respet to the non-inertial frame. Therefore, we an introdue the following wave funtion for the aelerated harge Ψ t = Ae iω t (.1) where t is given by the equation (1.7). Thus, we an rewrite this funtion with respet to the inertial referene frame as follows Ψ x, t = Ae iω t osh α x,t 1 x sinh α x,t (.) Taking the partial time derivative of this funtion, we get t Ψ x, t = iω osh α 1 x α t Ψ x, t (.3) Now, we an introdue by definition the Hamiltonian operator HΨ x, t = i Ψ x, t (.4) t x =0 Aording to the equation (.3), for the point x = 0 we get HΨ x, t = ω osh α x, t Ψ x, t (.5) Therefore, we an dedue from here the following Hamiltonian funtion H x, t = H 0 osh α x, t (.6) where H 0 is the rest energy of the aelerated harge H 0 = ω = m 0 (.7) and m 0 is the harge rest mass. Taking now the partial x derivative of the funtion Ψ x, t, we get Ψ x, t = iω 1 sinh α 1 x α Ψ x, t (.8) Further on, we an introdue by definition the linear momentum p x Ψ x, t = i Ψ x, t (.9) x =0 So, aording to the equation (.8), for the point x = 0 we an write p x Ψ x, t = 1 ω sinh α x, t Ψ x, t (.10) Thus, we the following expression for the linear momentum of the aelerated harge p x = 1 H 0 sinh α x, t (.11) Substituting now the equation (1.13) into the equation (.6), we get the well known formula H p x = m = p x + m 0 (.1) where m is the harge moving mass m = m 0 osh α = m 0 1 x (.13) Also, substituting the equation (1.13) into the equation (.11), we get the well known formula 3. The Hamilton s Equations p x = mx (.14) Aording to the priniples of analytial mehanis, we an introdue now the Lagrange funtion L x, x, t = x p x H p x, x, t (3.1) and the linear momentum p x as the derivative of lagrange funtion with respet to x p x = L (3.) where, aording to the equation (1.13), we an onsider x = v x x, t (3.3) Let us onsider also the ation funtional S x = t L t, x t, x x, t (3.4) where, t are onstants. We assume that the ation S x attains a loal minimum at f t, and g t is an arbitrary funtion that has at least one derivative and vanishes at the endpoints, t. So, for any number ε lose to zero, we an write S f S f + εg = G ε (3.5) Thus, if the funtional S x has a minimum for x = f, the funtion G ε has a minimum at ε = 0 G 0 dg dε ε=0 = t dl = 0 (3.6) dε ε=0 where, aording to the equation (3.3), x t and x x, t are funtions of ε, of the form x t = f t + εg t (3.7) x x, t = v x f t + εg t, t + εg t (3.8) Taking the total derivative of L t, x t, x x, t, we get dl = L dx + L dx (3.9) dε dε dε But, aording to the equations (3.7) and (3.8), we have dx dε = g t (3.10) dx = v x g t + g t (3.11) dε

3 International Journal of Theoretial and Mathematial Physis 016, 6(3): G Therefore, we get 0 t t1 dl d 0 t L L vx L t g t g t 1 x x x x = t L + L v x d L g t L + g t t (3.1) Aording to the fundamental lemma of alulus of variations, we get d L L v x L = 0 (3.13) In onsequene, we an introdue now the Hamiltonian funtion as H = x p x L x, x, t = H p x, x, t (3.14) Differentiating now this funtion, aording to the equation (3.13), we get the following equation dh = x dp x p x v x p x dx L (3.15) t Comparing this expression with the mathematial expression dh = H dp p x + H x We get the following Hamilton equations H dx + (3.16) t p x v x x = H (3.17) H = x = v p x x, t x (3.18) 4. The Shrodinger Equation Further on, we onsider a funtion F p x, x, t whih desribes a physial quantity of the aelerated harge. Taking the total time derivative of the funtion F p x, x, t, we = F F x + p p x + F x t (4.1) Aording to the equations (3.17) and (3.18), we an rewrite this formula as follows = F, H + v x F p p x + F x t where F, H is the lassial Poisson braket F, H = F H F H p x p x (4.) (4.3) Let us write the equation (4.) under the following form = DF + F, H (4.4) where D t is an operator whose expression is given by D t = D = + v x p t x (4.5) p x Taking into onsideration the analogy between quantum mehanis and analytial mehanis, we an introdue an operator F orresponding to the physial quantity F. Aording to the equation (4.4), the orresponding equation for this operator must be = DF + F, H (4.6) where F, H is the quantum Poisson braket F, H = i H F F H (4.7) Aording to the equation (4.6), we an define the time derivative of the expetation value of the observable F as follows = D t F (4.8) where F is the mean value (expetation value) of the observable F. Also, the law that desribes the time evolution of a partile must be of the form HΦ p x, t = ihd t Φ p x, t (4.9) In this equation, Φ p x, t is the wave funtion of a quantum system, in the p x representation. Aording to this representation, the operators x and p x are given by the expressions x = i p x p x = p x (4.10) Therefore, in the expression (4.5), we an make the replaing = i x (4.11) p x whih leads us to the following expression of the operator D t in the p x representation D t = D = i v x p t xx (4.1) But we seek the Shrodinger equation (4.9) in the x representation. For this, we must introdue the operators p x = i x = x and the anonial ommutation relation (4.13) p x x = x p x i (4.14) Aording to this relation, in the expression (4.1) we an make the replaing p x x = i x i (4.15) whih leads us to the following expression of the operator D t in the x representation D t = t v x x v x (4.16) Therefore, we an write now the following Shrodinger equation HΨ x, t = id t Ψ x, t (4.17) where the new Hamiltonian operator for the aelerated harge is given by H = id t = i t i v x x i v x (4.18) The additional term is of omplex nature and its real part

4 96 Bratianu Daniel: The Eletromagneti Radiation and Gravity signify the potential energy of the field of fore in whih the partile is moving. 5. The Damped Osillator Let us onsider that the aelerated harge ( q) is bound to the fixed harge (q) by an elasti fore of the form F el. = Kx (K = m 0 ω 0 ) (5.1) Taking the total time derivative of the equation (.6), we an introdue, aording to the speial relativity theory, the following equation dh = mx dα = Fv = x F x (5.) where F x is the resultant fore whih ats on the aelerating harge F x = dp x (5.3) This fore must be the sum between the elasti fore and the radiative reation fore F rad. = μ 0q x (5.4) Therefore, for non-relativisti veloities, we get the equation of motion x μ 0q 6π x + ω 0 x = 0 (5.5) We know that this equation is appliable only to the extent that the radiative reation fore is small ompared with the elasti fore μ 0 q 6π x ω 0 x This ondition leads us to the following equation of motion where 6π x + γx + ω 0 x = 0 (5.6) γ = μ 0q ω 0 1πm 0 (5.7) Thus, the approximate solution of the equation (5.6) an be written as x t = Ae γt os ωt + φ 0 (5.8) where ω is the angular frequeny of the damped osillator ω = ω 0 γ (5.9) Also, from the equation (5.), for non-relativisti veloities, we an now dedue a new differential equation dα = ω 0 γ x x (5.10) The integration of this equation leads us to the solution α x, t = μ x φ(t) μ = γ/ where φ(t) is given by the expression φ t = ω 0 (5.11) x (5.1) But, aording to the equation (1.13), we must have the ondition dx = tanh μx ω 0 x (5.13) Taking the total time derivative of this equation, we get d x = 1 os μx ω 0 Now, we an replae 1 os μx ω 0 x Therefore, we get the equation x μ dx ω 0 x = 1 1 dx 1 d x dx = μ ω 0 x whih is the same equation (5.6). Substituting this result into the equation (5.10), we an write the approximate equation dα 1 x (5.14) The solution of this equation is given by the expression α x, t = argtan x (5.15) whih results from the equation (1.13). Taking now the total time derivative of the displaement (5.8), we get x = γae γt os ωt + φ 0 ωae γt sin ωt + φ 0 (5.16) This expression an be written as a funtion of the displaement funtion x t, under the following form x = v x x, t = γx u(t) (5.17) where the funtion u(t) is given by the expression u t = ω Ae γt sin ωt + φ 0 (5.18) In this way, the veloity beomes a funtion of the oordinate x, beause here the displaement x is onsidered as a variable whih no longer depends on time. Consequently, we may admit for α x, t a solution of the form α x, t = argtan μx u(t) (5.19) 6. The Damped Quantum Osillator Substituting now the solution (5.17) into the equation (4.18), we get, for non-relativisti veloities H = i + iγx + iγ (6.1) t For the Hamiltonian operator of the osillator, we take the expression H = + 1 m m 0 0ω 0 x (6.) Admitting for the wave funtion a solution of the form Ψ x, t = R x T t, after the separation of variables, we get the following two equations

5 International Journal of Theoretial and Mathematial Physis 016, 6(3): m 0 R x + 1 m 0ω 0 x R x iγ xr x + Rx=ERx (6.3) i t T t = E T t (6.4) where the onstant E is the separation onstant, and signifies the total energy of the quantum osillator. Thus, after a rearrangement of the terms, the first equation an be rewritten as follows '' m0 R x i xr x h m0 1 E i h m 00 x R x 0 h Further on, we an introdue a dimensionless variable ρ = m 0 ω 0 suh that the equation beomes x = νx (6.5) R ρ + ibρr ρ + a ρ R = 0 (6.6) in whih we have introdued the notations a = a n + ib (6.7) a n = E ω 0 (6.8) b = γ ω 0 (6.9) Now, we an take for R ρ a solution of the form R ρ = w ρ e kρ (6.10) whih generates the differential equation for the unknown funtion w ρ '' w k ib w k a w 4k 4ibk 1 w 0 The onstant k an be determined from the equation (6.11) 4k + 4ibk 1 = 0 (6.1) In order to get a finite solution for the funtion R ρ, we hoose the root where we have introdued the notation k = ib d / (6.13) d = 1 b (6.14) Therefore, the equation (6.11) beomes w ρ dρw ρ + a n d + ib w ρ = 0 (6.15) We an introdue now a power series expansion as solution w ρ = j =0 A j ρ j (6.16) Substituting this expansion into the differential equation (6.15), we get j j + 1 j + A j + jd a j d + ib A j y j = 0 (6.17) whih leads us to the reursion relation A j + = jd a j d+ib j +1 j + A j (6.18) Beause the series must be finite, there exists some n suh that when j = n, the numerator will be zero. Therefore, we get a n d + ib = nd (6.19) Substituting here the formula (6.8), we the following omplex expression for the total energy of the quantum osillator E = E n i γ (6.0) where E n are the quantized energies of the quantum osillator E n = ω n + 1 (6.1) Substituting now the onstant k into the expression (6.10), we get R ρ = w ρ e 1 b ρ e ib ρ (6.) Also, substituting the equation (6.19) into the equation (6.15), we get for the unknown funtion w ρ the equation w ρ dρw ρ + ndw ρ = 0 (6.3) If we introdue now a new dimensionless variable r = d ρ = m 0ω the equation (6.3) beomes x = τx (6.4) H n r rh n r + nh n r = 0 (6.5) where H n r are the Hermite polynomials. So, w r = H n r, and the solution R x an be written as R x = N n H n τx e 1 b ν x e ib ν x (6.6) where N n are the multipliation fators, whose expression an be determined by the normalization ondition of the wave funtion. For the seond equation (6.4), we have the solution T t = e i E t (6.7) Substituting here the solution (6.0), we get the expression T t = e γ t e i E n t (6.8) whih desribe the time evolution of the wave funtion for the aelerated harge. 7. The Metri of Spae-Time Let us write now the distane between two infinitesimally lose events, into the inertial referene frame (O; x, y, z, t) ds = dx (7.1) Substituting here the equations (1.8) and (1.9), we get ds = g 00 + g 01 dx + g 11 dx (7.)

6 98 Bratianu Daniel: The Eletromagneti Radiation and Gravity where the omponents of the metri tensor are g 00 = t g 11 = 1 x t + x t (7.3) x t + t (7.4) g 01 = x t + 1 t t x t (7.5) But, aording to the equations (1.17) and (5.11), we have β x, t = α x, t = μx φ(t) (7.6) Then, aording to the transformations (1.6), we an impose where x = Λ v x x + v x t (7.7) t = Λ v x t + v x x (7.8) Λ v x = 1 1 v x (7.9) Substituting now these expressions into the equation (7.6), we get where β x, t = μ Λ v x x + v x t φ x, t (7.10) φ x, t = φ t t=λ vx t + v x (7.11) x Taking now the partial x derivative of β x, t, we = μλ v x + dφ t (7.1) Also, taking the partial t derivative of β x, t, we = μv t x Λ v x + dφ t (7.13) t But, aording to the transformations (7.7) and (7.8), we an write dφ t = v x Λ v x (7.14) t t = Λ v x (7.15) = ω 0 x = ω 0 Λ v x x + v x t (7.16) Substituting these expressions into the equations (7.1) and (7.13), we get = μλ v x ω 0 v xλ v 3 x x + v x t (7.17) = μv t x Λ v x ω 0 Λ v x x + v x t (7.18) where the veloity v x is given by the expression (5.17), whih no longer an be written as a funtion of the oordinates x, t. So, in this partiular ase, the metri of spae-time (7.) beomes a Finsler metri, and the spae beomes a Finsler spae. 8. Conlusions Intuitively, the non-uniformly aelerated linear motion of a osillator must be equivalent to a uniform but variable gravitational field. However, the gravitational field desribed by the metri of spae-time (7.), is a non-uniform field. This is due to the partiular mode of writing of the veloity of a partile as a funtion of the oordinates x, t, and the funtion α x, t, depending on the same oordinates, by means of the veloity. Also, from the equation (3.17), we an observe that the term v x p x orresponds at the half of the radiative reation fore v x p x = γm 0 x 1 F rad. Therefore, the funtion α x, t whih determines the Lorentz non-linear transformations, intervenes in the determination of the radiative reation fore of the EM field and, also, in the determination of the gravitational field whih appears in the non-inertial referene frame. Indeed, if we impose now α = onst., we get F rad. = 0 ds = dx so, both, the radiative reation fore and the gravitational field, vanish. On the other hand, the radiative reation fore vanishes again when the osillator is a neutral partile, but the gravitational field does not vanish, beause ω 0 is different from zero. This suggests that the EM energy arried by the EM radiation ontributes to the energy of the gravitational field. Also we an onlude that the field whih appears as an EM field into the inertial referene frame, appears as a gravitational field into the non-inertial referene frame. REFERENCES [1] The Classial Theory of Fields, Third Revised English Edition, Course of Theoretial Physis, Volume, L. D. Landau and E. M. Lifshitz, Pergamon Press, [] General Relativity and Cosmology, by Niholas Ionesu-Pallas, Sientifi and Enylopedi Publishing house, Buharest, Romania [3] From Albert Einstein to R. G. Beil, by Valentin Gartu, Fair Partners Publishing house, Buharest, Romania [4] General Physis, by Traian Cretu, Tehnial Publishing house, Buhares984. [5] The Meaning of Relativity, by Albert Einstein, Prineton University Press, Prineton, New Jersey 1955.