Electromagnetic wave propagation in general relativistic situations: A general solution of the problem

Authors


Abstract

[1] The exact full-wave modeling and computation of the electromagnetic signals propagating through curved space-time is important from theoretical and practical aspects (e.g., for satellite positioning systems). In this paper a general solution method is presented for these problems. Beside this, it is demonstrated that the Method of Inhomogeneous Basic Modes is a natural consequence of Maxwell's equations in the general relativity.

1. Introduction

[2] The basic formalism of Maxwell's equations in the general relativity is known [see, e.g., Novobátzky, 1962; Stephani, 2004]. This formalism is coherent with the whole structure of the general relativity and it is widely used. In the applications the electromagnetic signals (light of stars, signals of satellites or space probes) propagate through large, curved space-time regions in which the structure is predefined by gravitational forces. In these cases a common tool of the modeling of the propagation is the geometrical optics approximation [e.g., Stephani, 2004]. A lot of successful applications are known.

[3] However, it is not known and earlier it was not necessary to know a general full-wave solution giving a complete and detailed propagating wave form for either a monochromatic or an ultra wide band (UWB) signal. In this paper a general solution method of Maxwell's equations is presented, which is valid in general relativity and which results in detailed, full-wave solutions of the propagating signals. Beside this, it will demonstrate that using this method it is not necessary to predefine the curved space-time structure as a known boundary condition. Finally, the method is a generalization of the Method of Inhomogeneous Basic Modes (MIBM) [Ferencz, 1978, 2004, 2011; Ferencz et al., 2001]. The presented method is not only an accidental choice between more possibilities, but in itself is the natural consequence of the structure of Maxwell's equations in the studies.

2. Formalism and Initial Assumptions

[4] It is known that in an arbitrarily given point it is possible to define a local Minkowski system with the help of suitable transformation matrices Amm or Amm [see, e.g., Stephani, 2004]. Let us use hence forth the formalism which is common and accepted in general relativity related texts [Novobátzky, 1962; Stephani, 2004], where superscripts mean contravariant components and subscripts mean covariant components. (If it is necessary to use indices differently, they will be defined in the text on that place.) Let the coordinates in the Riemann space be denoted by xn and xn, the contravariant and covariant components respectively, and by xn and xn in the local Minkowski system. Then

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i.e.

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It is known [Stephani, 2004] that

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where δmn and δmn are the Kronecker-delta symbols.

[5] In the local Minkowski systems the local metric is given as

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and the Riemann metric is

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In the local Minkowski system the space-time coordinates in contravariant and in covariant form are

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where c is the velocity of light in vacuum.

[6] The electromagnetic field tensor in the local Minkowski system is in covariant and in contravariant form

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respectively and the energy-momentum tensor is

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as it is well known, where

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as it is usual.

[7] Therefore in the Riemann system

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and

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and the energy-momentum tensor remains symmetric, i.e.

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relation will be fulfilled, as well.

[8] Beside these we shall use in the following other symbols:

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and the Ricci (or Einstein) tensor as Rmn and the invariant curvature R, together with the Christoffel symbols [Stephani, 2004], if it is necessary.

3. The Full-Wave Solution of Maxwell's Equations

[9] It is known [Novobátzky, 1962; Stephani, 2004] that in a Riemann space-time system the form of Maxwell's equations is

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where jm is the four-current density and jm ≡ 0 is possible. The definition of the four-current density in the general relativity is [Stephani, 2004]

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where j1, j2 and j3 are the common current density components in the given space-time system and ρ is the charge density there. (It is practical to start from the local Minkowski system and to derive the actual form of jm from jm.)

[10] In the first equation of (10) we can reformulate the members by applying (1) and (7):

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Therefore a new form of the first equation of (10) is

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[11] In the same manner we can reformulate the second equation of (10) too:

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or

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The second term of the left side of (12) is

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In this relation

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and the second and third terms are

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Therefore the new form of the second equation of (10) is

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[12] If we see the members in {…} brackets in (11) and (13), we find there the members of Maxwell's equations valid in (local) Minkowski systems. Let us search the solution of Maxwell's equations as a sum of “basic modes” in every point, i.e.

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where the index i is the marker of independent basic modes, i = 1, … N and N is the number of the modes in a given point, and we use in the case of index i the normal summation formalism as in the work of Ferencz [2011] too. With this choice Maxwell's equations are

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3.1. The Definition of the Basic Modes

[13] Because only the resultant of the modes, i.e., the resultant in (14) is the solution of Maxwell's equations we have some degree of freedom choosing from basic modes. (More about this can be seen in [Ferencz, 2011] or in earlier publications of the MIBM.) The structure of (15) and the fact that Amm , Ann are non-degenerative in every case suggest that the “basic modes” will be the solutions of the parts of (15), in which the {…} brackets are present. With this choice to find basic modes is possible to derive these modes from the following equations:

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Because Amm , Ann are non-degenerative,

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These are Maxwell's equations in the local Minkowski systems. Let us choose the “basic modes” in such a way that every mode fulfill these (16) equations alone. Then (16) is also fulfilled. Therefore let the definition of the “basic modes” be the possible solutions of the equations

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where ijm = 0 is also a possible case.

[14] With this choice the complete Maxwell's equation-system is fulfilled if the remaining part of the equation-system is fulfilled by fixing the free parameters of the iFm′ n modes.

[15] Because this solution procedure is exactly the Method of Inhomogeneous Basic Modes (MIBM), which is in this case a natural consequence of the general relativistic structure of Maxwell's equations, let us use the MIBM nomenclature, i.e., let the remaining part of (15) be the “coupling equations.”

3.2. The Coupling Equations

[16] After some simple reformulations, in which we apply (7) and (1), it is possible to write, that

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[17] After fixing of the free parameters of the basic modes, the full-wave solution of Maxwell's equations in the form (14) is known.

4. A Way to Find New Type of Solutions

[18] By the application of the presented solution method it is possible to search interesting class or classes of the solutions. In this case we can add the Einstein's gravitational equation [Novobátzky, 1962; Stephani, 2004], which is

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where κ is a known constant, to the equation-system (17) and (18), and to search by self-consistent-field method such kind of solutions, in which the space-time structure (curvature) is defined by the electromagnetic field itself.

[19] To find the actual form of (19) we can use the relations (3) and the definition of the Christoffel-symbols which is

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and

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In (20) let us apply (3) and consider that the relations (1) and (2) are also valid. During the derivations of the Christoffel-symbols Γmna we can assume that the space-time functions and the AmmAnn functions are non-degenerative and continuously derivable functions. Therefore it is valid to the second order derivatives that

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The Christoffel-symbols in covariant form, see (20), are

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and

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After applying (22) in Γmna the new form of these symbols become

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and

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[20] As it is known, the Ricci (or Einstein) tensor is [Novobátzky, 1962; Stephani, 2004]

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Applying (22), (23) and (24) in (25) it is possible to derive that

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Applying these quantities in (25), the Ricci (Einstein) tensor is

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From (26)

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and by applying this relation the second term in (19)

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[21] With (26) and (27) it is possible to write the Einstein-equation (19) in this case. After a few trivial reformulations the form of (19) is:

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[22] Equations (17), (18) and (28) form a complete differential equation system which uses the MIBM and has a Self Consistent Field (SCF) character. By solving of these equations assuming that ijm = 0, it is possible to derive new type of solutions of the Maxwell's equations and simultaneously the space-time structure too. (The SCF character of the Maxwell's equations is well known.) The work of deriving such kind of solutions is currently under way.

5. Conclusions

[23] Based on the experiences of this derivation, mainly based on the structure of equation-system (15) it is possible to say that the MIBM is an inherent solution method of Maxwell's equations, in general relativistic electrodynamics.

[24] If the space-time structure, the Amm , Ann transformation derivatives are known, e.g., these are predetermined (for example) by the gravitational field, than it is possible to derive a full-wave solution of the propagating signal applying the method presented in section 3.

[25] Because ijm have an important role in equation (17), it is clear, that the constitutive relations valid in the traversed media will influence the determination of the basic modes.

[26] Finally, it is necessary to remark that it is possible to search for an interesting class of solutions by the application of the presented solution method, i.e., by solving the equation-systems (17), (18) and (28) together if ijm = 0, in which the space-time structure (curvature) is defined by the electromagnetic field itself.

Acknowledgment

[27] The author sincerely thanks János Lichtenberger for the discussions of the problems.

Ancillary

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