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Strain energy density in the Elastodynamics of the Spacetime Continuum and the electromagnetic field.

1 Introduction

The Elastodynamics of the Spacetime Continuum (STCED) is based on the application of a continuum mechanical approach to the analysis of the spacetime continuum [1-3]. The applied stresses from the energy-momentum stress tensor result in strains in, and the deformation of, the spacetime continuum (STC). In this paper, we explore the resulting strain energy per unit volume, that is the strain energy density, resulting from the Elastodynamics of the Spacetime Continuum. We then calculate the strain energy density of the electromagnetic field from the electromagnetic energy-momentum stress tensor.

2 Strain energy density of the spacetime continuum

The strain energy density of the spacetime continuum is a scalar given by [4, see p. 51]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [[epsilon].sub.[alpha][beta]] is the strain tensor and [T.sup.[alpha][beta]] is the energy-momentum stress tensor. Introducing the strain and stress deviators from (12) and (15) respectively from Millette [2], this equation becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Multiplying and using relations [[epsilon].sup.[alpha].sub.[alpha] = 0 and [t.sup.[alpha].sub.[alpha]] = 0 from the definition of the strain and stress deviators, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Using (11) from [2] to express the stresses in terms of the strains, this expression becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where the Lame elastic constant of the spacetime continuum [[mu].sub.0] is the shear modulus (the resistance of the continuum to distortions) and [K.sub.0] is the bulk modulus (the resistance of the continuum to dilatations). Alternatively, again using (11) from [2] to express the strains in terms of the stresses, this expression can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

3 Physical interpretation of the strain energy density

The strain energy density is separated into two terms: the first one expresses the dilatation energy density (the "mass" longitudinal term) while the second one expresses the distortion energy density (the "massless" transverse term):

[epsilon] = [[epsilon].sub.[parallel]] + [[epsilon].sub.[perpendicular to]] (6)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Using (10) from [2] into (7), we obtain

[[epsilon].sub.[parallel]] = 1/32[K.sub.0][[rho][c.sup.2]].sup.2] (9)

The rest-mass energy density divided by the bulk modulus [K.sub.0], and the transverse energy density divided by the shear modulus [[mu].sub.0], have dimensions of energy density as expected. Multiplying (5) by [32.sub.[K.sub.0]] and using (9), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

Noting that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is quadratic in structure, we see that this equation is similar to the energy relation of Special Relativity [5, see p. 51] for energy density

[[??].sup.2]= [[rho].sup.2][c.sup.4]+ [[??].sup.2][c.sup.2] (11)

where [??] is the total energy density and [??] the momentum density.

The quadratic structure of the energy relation of Special Relativity is thus found to be present in the Elastodynamics of the Spacetime Continuum. Equations (10) and (11) also imply that the kinetic energy pc is carried by the distortion part of the deformation, while the dilatation part carries only the rest mass energy.

This observation is in agreement with photons which are massless ([[epsilon].sub.[parallel]] = 0), as will be shown in the next section, but still carry kinetic energy in the transverse electromagnetic wave distortions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]).

4 Electromagnetic strain energy density

The strain energy density of the electromagnetic energy-momentum stress tensor is calculated. Note that Rationalized MKSA or SI (Systeme International) units are used in this paper as noted previously in [3]. In addition, the electromagnetic permittivity of free space [[member of].sub.[epsilon]m] and the electromagnetic permeability of free space [[mu].sub.[epsilon]m] are written with "em" subscripts as the "0" subscripts are used in the spacetime constants. This allows us to differentiate between [[mu].sub.[epsilon]m] and [[mu].sub.0].

Starting from the symmetric electromagnetic stress tensor [6, see pp. 64-66]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

with [g.sup.uv] = [[eta].sup.uv] of signature (+ -), and the field-strength tensor components [6, see p. 43]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

we obtain [6, see p. 66] [7, see p. 141],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

where [S.sup.j] is the Poynting vector, and where we use the notation [[sigma].sup.[mu]v] = [[THETA].sup.[mu]v] as a generalization of the [[sigma].sup.ij] Maxwell stress tensor notation. Hence the electromagnetic stress tensor is given by [6, see p. 66]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

where [[sigma].sup.ij] is the Maxwell stress tensor. Using the relation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to lower the indices of [[sigma].sup.[mu]j], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

4.1 Calculation of the longitudinal (mass) term

The mass term is calculated from (7) and (17) of [2]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

The term [[sigma].sup.[alpha].sub.[alpha]] is calculated from:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

Substituting from (16) and the metric [[eta].sup.[mu]v] of signature (+ -), we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

Substituting from (15), this expands to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

and further,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

Hence

[[sigma].sup.[alpha].sub.[alpha]] = 0 (23)

and, substituting into (18),

[[epsilon].sub.[parallel]] = 0 (24)

as expected [6, see pp. 64-66]. This derivation thus shows that the rest-mass energy density of the photon is 0.

4.2 Calculation of the transverse (massless) term

The transverse term is calculated from (8), viz.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

Given that t = 1/4 [[sigma].sup.[alpha].sub.[alpha]] = 0, then [t.sup.[alpha][beta]] = [t.sup.[alpha][beta]] and the terms [[sigma].sup.[alpha][beta]] [t.sub.[alpha][beta]] are calculated from the components of the electromagnetic stress tensors of (16) and (17). Substituting for the diagonal elements and making use of the symmetry of the Poynting component terms and of the Maxwell stress tensor terms from (16) and (17), this expands to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

The E-B terms expand to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

Simplifying,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

which gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

and finally

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

Including the E-B terms in (26), substituting from (15), expanding the Poynting vector and rearranging, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

Expanding the quadratic expressions,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

Grouping the terms in powers of c together,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

Simplifying,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

which is further simplified to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

Making use of the definition of the Poynting vector from (15), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

and finally

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

Substituting in (25), the transverse term becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)

where [U.sub.em] = 1/2 [[member of].sub.[member of]m] ([E.sup.2] + [c.sup.2][B.sup.2]) is the electromagnetic field energy density.

4.3 Electromagnetic field strain energy density and the photon

S is the electromagnetic energy flux along the direction of propagation [6, see p. 62]. As noted by Feynman [8, see pp. 27-1-2], local conservation of the electromagnetic field energy can be written as

-[partial derivative][U.sub.em]/[partial derivative]t = [nabla] x S, (40)

where the term E * j representing the work done on the matter inside the volume is 0 in the absence of charges (due to the absence of mass [3]). By analogy with the current density four-vector [j.sup.v] = (c[??], j), where g is the charge density, and j is the current density vector, which obeys a similar conservation relation, we define the Poynting four-vector

[S.sup.v] = ([cU.sub.em],S) (41)

where [U.sub.em] is the electromagnetic field energy density, and S is the Poynting vector. Furthermore, as per (40), [S.sup.v] satisfies

[[partial derivative].sub.v][S.sup.v]= 0 (42)

Using definition (41) in (39), that equation becomes

[[epsilon].sub.[perpendicular to]] = 1/[[mu].sub.0][c.sup.2] [S.sub.v][S.sup.v] (43)

The indefiniteness of the location of the field energy referred to by Feynman [8, see p. 27-6] is thus resolved: the electromagnetic field energy resides in the distortions (transverse displacements) of the spacetime continuum.

Hence the invariant electromagnetic strain energy density is given by

[epsilon] = 1/[[mu].sub.0][c.sup.2] [S.sub.v][S.sup.v] (44)

where we have used p = 0 as per (23). This confirms that [S.sup.v] as defined in (41) is a four-vector.

It is surprising that a longitudinal energy flow term is part of the transverse strain energy density i.e. [S.sup.2]/[[mu].sub.0][c.sup.2] in (39). We note that this term arises from the time-space components of (16) and (17) and can be seen to correspond to the transverse displacements along the time-space planes which are folded along the direction of propagation in 3-space as the Poynting vector. The electromagnetic field energy density term [U.sub.em.sup.2]/[[mu].sub.0] and the electromagnetic field energy flux term [S.sup.2]/[[mu].sub.0][c.sup.2] are thus combined into the transverse strain energy density. The negative sign arises from the signature (+ -) of the metric tensor [[eta].sup.[mu]v].

This longitudinal electromagnetic energy flux is massless as it is due to distortion, not dilatation, of the spacetime continuum. However, because this energy flux is along the direction of propagation (i.e. longitudinal), it gives rise to the particle aspect of the electromagnetic field, the photon. As shown in [9, see pp. 174-5] [10, see p. 58], in the quantum theory of electromagnetic radiation, an intensity operator derived from the Poynting vector has, as expectation value, photons in the direction of propagation.

This implies that the [(pc).sup.2] term of the energy relation of Special Relativity needs to be separated into transverse and longitudinal massless terms as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (45)

where [[??].sub.[parallel]] is the massless longitudinal momentum density. Equation (39) shows that the electromagnetic field energy density term [U.sub.em.sup.2]/[[mu].sub.0] is reduced by the electromagnetic field energy flux term [S.sup.2]/[[mu].sub.0][c.sup.2] in the transverse strain energy density, due to photons propagating in the longitudinal direction. Thus the kinetic energy is carried by the distortion part of the deformation, while the dilatation part carries only the rest-mass energy, which in this case is 0.

As shown in (9), (10) and (11), the constant of proportionality to transform energy density squared ([[??].sup.2]) into strain energy density ([epsilon]) is 1/([32.sub.K.sub.0]):

[[epsilon].sub.[parallel]] = 1/32[K.sub.0][[rho][c.sup.2]].sup.2] (46)

[epsilon] = 1/32[K.sub.0][[??].sup.2] (47)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (48)

Substituting (39) into (48), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (49)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (50)

This suggests that

[[mu].sub.0]= 32[K.sub.0], (51)

to obtain the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (52)

5 Discussion and conclusion

In this paper, we have analyzed the strain energy density of the spacetime continuum in STCED and evaluated it for the electromagnetic stress tensor. We have found that the strain energy density is separated into two terms: the first one expresses the dilatation energy density (the "mass" longitudinal term) while the second one expresses the distortion energy density (the "massless" transverse term). We have found that the quadratic structure of the energy relation of Special Relativity is present in the strain energy density of the Elastodynamics of the Spacetime Continuum. We have also found that the kinetic energy pc is carried by the distortion part of the deformation, while the dilatation part carries only the rest mass energy.

We have calculated the strain energy density of the electromagnetic energy-momentum stress tensor. We have found that the dilatation longitudinal (mass) term of the strain energy density and hence the rest-mass energy density of the photon is 0. We have found that the distortion transverse (massless) term of the strain energy density is a combination of the electromagnetic field energy density term [U.sub.em]2/[[mu].sub.0] and the electromagnetic field energy flux term [S.sup.2]/[[mu].sub.0][c.sup.2], calculated from the Poynting vector. This longitudinal electromagnetic energy flux is massless as it is due to distortion, not dilatation, of the spacetime continuum. However, because this energy flux is along the direction of propagation (i.e. longitudinal), it gives rise to the particle aspect of the electromagnetic field, the photon.

Submitted on January 7, 2013/Accepted on January 11, 2013

References

[1.] Millette P. A. On the Decomposition of the Spacetime Metric Tensor and of Tensor Fields in Strained Spacetime. Progress in Physics, 2012, v. 4, 5-8.

[2.] Millette P. A. The Elastodynamics of the Spacetime Continuum as a Framework for Strained Spacetime. Progress in Physics, 2013, v. 1, 55-59.

[3.] Millette PA. Derivation of Electromagnetism from the Elastodynamics of the Spacetime Continuum. Progress inPhysics, 2013, v. 2, 12-15.

[4.] Flugge W. Tensor Analysis and Continuum Mechanics. Springer-Verlag, New York, 1972.

[5.] Lawden D.F. Tensor Calculus and Relativity. Methuen & Co, London, 1971.

[6.] Charap J.M. Covariant Electrodynamics, A Concise Guide. The John Hopkins University Press, Baltimore, 2011.

[7.] Misner C.W., Thorne K.S., Wheeler J.A. Gravitation. W.H. Freeman and Company, San Francisco, 1973.

[8.] Feynman R.P, Leighton R.B., Sands M. Lectures on Physics, Volume II, Mainly Electromagnetism and Matter. Addison-Wesley Publishing Company, Reading, Massachusetts, 1975.

[9.] Loudon R. The Quantum Theory of Light, Third Edition. Oxford University Press, Oxford, 2000.

[10.] Heitler W. The Quantum Theory of Radiation, Third Edition. Dover Publications, Inc, New York, 1984.

Pierre A. Millette

University of Ottawa (alumnus), Ottawa, Canada. E-mail: PierreAMillette@alumni.uottawa.ca
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Author:Millette, Pierre A.
Publication:Progress in Physics
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Date:Mar 1, 2013
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