Gravitational Poynting theorem: interaction of gravitation and electromagnetism

Transcription

1 Graitational Poynting theorem 433 Journal of Foundations of Physics and Chemistry, 0, ol. (4) Graitational Poynting theorem: interaction of graitation and electromagnetism M.W. Eans Alpha Institute for Adanced Studies ( The geometrical basis of ECE theory is used to deduce the existence of a graitational equialent of the Poynting Theorem and four graitational fields: g, d, h and b. These are the equialents of E, D, H and B in electromagnetism, the field equations of graitation haing the same structure as those of electromagnetism, two homogeneous and two inhomogeneous. The interaction of graitation and electromagnetism is deeloped on the principle that all forms of energy are interconertible, so the mechanism of conersion of electromagnetic to graitational energy is elucidated ia the respectie Poynting Theorems. Keywords: ECE theory, graitational Poynting Theorem, graitational fields, interconersion of electromagnetic and graitational energy.. Introduction. The geometrical structure of Einstein Cartan Eans (ECE) theory [ 0] shows that the field equations of graitation hae a richer structure than thought hitherto, and that that structure is the same as for electromagnetism. In this paper the graitational field equations are deeloped further and in parallel with electromagnetism. There are two homogeneous equations of electromagnetism, with the fields E and B, respectiely the electric field strength and magnetic flux density. So it is shown in Section that there are two homogeneous field equations of graitation with the same structure and describing the interaction of the acceleration due to graity g and the graitomagnetic flux density b. There are two inhomogeneous field equations of electromagnetism, with the electric displacement D and magnetic field strength H, and in Section it is shown that there are two inhomogeneous field equations of graitation, inoling the graitational displacement d, and the field strength of magnetograitation h. In Section the units of these fields are defined and also the way in which they interact with the mass density ρ m and current of mass density J m. The tensor structure of the equations is defined by the Cartan geometry [] of ECE theory. EMyrone@aol.com

2 434 M.W. Eans In Section 3 the graitational Poynting Theorem is deeloped in parallel with the well known Poynting Theorem of electromagnetism, the law of conseration of energy. In ECE theory both laws of conseration of energy are deried from geometry within the context of general relatiity and again, both hae the same structure, defined in this Section. Since all forms of energy are interconertible, the structure of the graitational Poynting theorem may be used to inestigate the way in which electromagnetism affects graitation. This inestigation may lead towards a practical counter graitational deice in which the electromagnetic field decreases g, the acceleration due to graity.. The field equations of graitation and magnetograitation As shown in the preceding paper, UFT 67 of this series [ 0] the geometrical structure of the homogeneous field equation is: a a T = j H. () a Here T is the Hodge dual of the torsion tensor, R a the Hodge dual of the curature tensor, and Ω a b the releant spin connection. In general the homogeneous four current j R T a a a a H = Ω b =0, () is non-zero, but from experimental results in electromagnetism, it is assumed to be zero. The basic geometrical structure of the inhomogeneous field equation is: a a a a b T ji R bt = = ω (3) in which the current j a originates in mass density and the current of mass I density. In ECE theory the basic geometrical structures () and () are the same for graitation and electromagnetism. The geometrical structure (3) gies the inhomogeneous field equation of graitation in tensor format. For each sense of polarization a the field equation is: K = J M. (4) The field tensor is defined as: ρ σ K = g g T k ρσ (5) where k is the Einstein constant:

3 Graitational Poynting theorem 435 8πG k = = m kg c 6 (6) where G is the Newton constant, and where the inerse metrics hae been used to raise indices as usual. Note that these are the inerse metrics of a four dimensional spacetime with both torsion and curature [-]. The field tensor is a 4 4 matrix defined as follows: 0 d d d K X Y Z = dx 0 h Z/c h Y/c dy h Z/c 0 h X/c dz h Y/c h X/c 0 (7) so the tensor equation (4) becomes two ector equations: d = ρ, m (8) d h = J. m t (9) Equation (8) is the direct analogy of the Coulomb law in electromagnetism, and Eq. (9) the direct analogy of the Ampere Maxwell law. In direct analogy with electromagnetism the graitational displacement is defined by: d = g (0) 8πG where g is the acceleration due to graity. The units of d are kg m in direct analogy with the units of electric displacement D in electromagnetism (C m ). The analogue of Eq. (0) in electromagnetism is: D= 0 E, () where E is the electric field strength (V m ), the analogue of g in graitation, and where 0 is the acuum permittiity. Here ρ m is the mass density in units of kg m 3, in direct analogy with the charge density in units of C m 3. The current of mass density J m has units of C ρ m or kg s m in analogy with the electric current density J with units of C s m. The graitomagnetic field strength h has units of kg m s in analogy with magnetic field strength H in units of C s m or A m. The homogeneous field tensor is defined within the factor c as the torsion of spacetime:

4 436 M.W. Eans g ct 0 gx /c gy /c gz /c g /c 0 b b gy / c bz 0 -b X gz /c by bx 0 = = X Z Y () The metric of the homogeneous field equation is defined as: ( metric ) diag (,,, ) g = (3) so that the homogeneous field equation is: g = 0 (4) where: 0 -bx -by -b Z g = bx 0 gz /c gy /c b Y gz /c 0 gx /c b Z gy / c gx/c 0 (5) Equation (4) may be deeloped in terms of two ector equations: b =0, (6) and b g = 0. (7) in which the units of the graitomagnetic flux density b are s and in which g is the acceleration due to graity in m s. Equation (6) is the direct analogue of the Gauss law of magnetism: B =0 (8) and Eq. (7) is the direct analogue of the Faraday law of induction:

5 Graitational Poynting theorem 437 B E + = 0. (9) The metric (3) is defined in terms of the conjugate product of tetrads [ 0]: ( ) a b g metric = q q +η ab (0) and should not be confused with the Minkowski metric of flat spacetime in which there is no torsion. It is based on experimental data. The two equations (8) and (9) of electromagnetism are thought to be well erified experimentally, i.e. there is no magnetic monopole or magnetic current. In analogy it is assumed that there is no magneto-graitational monopole or current. Note carefully that in general the metrics of the homogeneous and inhomogeneous structures are different. As shown in UFT 67 the inhomogeneous metric elements in electromagnetism define the permittiity and permeability of a gien material. The analogous concept is present in the inhomogeneous structure of graitation. In summary of this section, the ECE field equations of graitation are, for each index of polarization: b = 0, b + g = 0, d =ρm, d h = Jm. () The ECE field equations of electromagnetism, in direct analogy, are, for each index : B = 0, B + E = 0, D =ρ, D H = J. () 3. Graitational and electromagnetic Poynting theorems The graitational Poynting Theorem is deduced in direct analogy with the well known electromagnetic Poynting Theorem [] - the law of conseration of energy. Therefore the usual Poynting Theorem is reiewed first as follows.

6 438 M.W. Eans The theory of this section is deeloped with the understanding that it is alid for each polarization index a, both for electromagnetism and graitation. The electromagnetic energy density in units of joules per cubic metre is: U= + ( E D B H) (3) and the total rate of doing work by the electromagnetic field in a olume V is: P 3 = V J E dx (4) in joules per second, the units of power. This power is the conersion of electromagnetic energy to other forms of energy, notably graitational energy. The Poynting theorem is D J E = H E (5) where Eq. (.d) has been used to eliminate J. Using Eq. (9) the theorem can be expressed as: U + S= J E (6) where the Poynting ector is defined as: S : =E H. (7) Similarly, the total rate of doing work by the graitational field in a olume V is: P gra = V J m g 3 dx (8) where J m is the mass current density. The graitational energy density is: U gra = + ( g d b h) (9) which has the correct units of kg m - s -, or kg m s - m -3, or Jm -3. The graitational Poynting theorem is therefore:

7 Graitational Poynting theorem 439 U gra + S = J g (30) gra m where the graitational Poynting ector is: S = g h gra. (3) When considering the interaction of electromagnetism and graitation J. E is the work done per unit time per unit olume by the electromagnetic field on the graitational field, and j m. g is the work done per unit time per unit olume by the graitational field on the electromagnetic field. For practical applications we wish to consider the effect of an electromagnetic deice on graitation, notably on g. The oerall aim is to make g smaller in magnitude and to counter graitation. In the presence of electromagnetism the graitational Poynting theorem becomes: d g h = g Jm + E J. (3) For simplicity, consider situations where: h =0 (33) then: d g = g Jm + E J (34) where: d = g (35) ck g ( ) g = ck g Jm + E J (36) If the mass is a fixed mass such as that of the Earth (the object responsible for g), then:

8 440 M.W. Eans J m = 0 (37) so: g g = c k E J (38) where the Einstein constant is: 8πG k =. c (39) In the Z axis: g Z c k = E Z J Z =.7 0 gz E J 0 Z Z (40) so g z changes with time in the opposite direction to E z and J z. The effect is ery small, but could be amplified by a resonance mechanism, which could be found within the structure of the Poynting theorem itself. References. [] M.W. Eans, Generally Coariant Unified Field Theory (Abramis, 005 onwards), in seen olumes to date. [] M.W. Eans, S. Crothers, H. Eckardt and K. Pendergast, Criticisms of the Einstein Field Equation (Cambridge International Science Publishing, 0). [3] L. Felker, The Eans Equations of Unified Field Theory (Abramis 007). [4] K. Pendergast, The Life of Myron Eans(Cambridge International Science Publishing, 0). [5] The ECE websites, (National Library of Wales), [6] M.W. Eans and L. B. Crowell, Classical and Quantum Electrodynamics and the B(3) Field (World Scientific, 00). [7] M.W. Eans and J. - P. Vigier, The Enigmatic Photon (Kluwer, Dordrecht, 994 to 00), in fie olumes. [8] M.W. Eans, ed., Modern Nonlinear Optics (Wiley, 00, second edition). [9] M. W. Eans and S. Kielich (eds.), ibid., first edition (99, 993, 997). [0] M.W. Eans and A. A. Hasanein, The Photomagneton in Quantum Field Theory (World Scientific, 994). [] S.P. Carroll, Spacetime and Geometry: an Introduction to General Relatiity (Addison Wesley, New York, 004). [] J.D. Jackson, Classical Electrodynamics (Wiley, 999, third edition).