Elastodynamics In A Continuum Of Infinite Extension

ELASTODYNAMICS IN A CONTINUUM OF INFINITE EXTENTION BJØRN URSIN KARLSEN To Elizabeth Abstract. Already in the Nineteenth century William Thomson (Lord Kelvin) (1824–1907) pointed out the resemblance between elastodynamic and electrodynamic equations [2, page 279-280]. In this paper I will follow up some of his thoughts in order to see exactly how far this resemblance stretches. I will start by recapitulating some topics from the Linear Theory of Elasticity in an elastic continuum of infinite extension – which I will call a spatial continuum – and show that they can be reformulated to terms that exactly match those applied in Electrodynamics. With the additional assumption that there may be free moving sinks and sources in the spatial continuum, I will show that they will behave exactly like electric charges do. Finally I will show that if disturbance energy can be confined in small areas of the spatial continuum, the energy packets will behave in the same way as matter except for gravitation, which goes beyond the scope of this paper.

1. The Linear Theory of Elasticity The Linear Theory of Elasticity is a discipline in its own right, and this section is only meant as an introduction to the topic. The theory was probably originally intended to describe ordinary elastic bodies consisting of particles bound together by molecular forces, but it is through the centuries refined to be a theory that can describe deformations in a true elastic continuum. Here, I will focus on the part of the theory that describes deformations of an elastic continuum of infinite extension or nearly so. In its undeformed state it is supposed to be homogeneous and isotropic if anything else isn’t stated explicitly. For a more thorough investigation I refer to [1]. Only a couple of the equations, namely (1.7), (1.16), and (1.18) are used in later developments so if those equations are familiar, this section may be skipped. 1.1. Displacement fields. The space B under consideration, also called a Kelvin space, is all filled up with an elastic continuum which in its undeformed state is homogeneous and isotropic with mass density ρs that obeys the deformation laws of The Linear Theory of Elasticity. Notice that I already now put the index s on the mass density in order to distinguish it from the charge density ρ of electrodynamics. Date: 20:08:08. Thanks to a friend who wants to be anonymous, because he has supplied me with important books, and given me the term ”The spatial continuum”. 1

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BJØRN URSIN KARLSEN

The displacement in this space is described by the displacement field ; its value u(x) at a point x is the infinitesimal displacement of x. The symmetric part ¢ ¡ (1.1) ² = 12 ∇u + ∇uT , 1 2

²ij =

(u i, j + uj, i ),

of the displacement gradient, ∇u, is the infinitesimal strain field, and the above equation, relating ² to u, is called the strain-displacement relation. In this context the (local) space is to be understood as the whole of the deformed area, or at least an area through which border no significant forces due to the inside deformation are conveyed. In addition u has got to be continuous and sufficiently smooth. We call div u = tr ², the dilatation. The infinitesimal volume change δv(P ) of a part P of space due to a continuous displacement of the field u is defined by I δv (P ) = u · n da, P

where n is the unit vector normal to the surface element da of the surface of P , and we say that u is solenoidal if δv(P ) = 0 for every P . By the divergence theorem we have Z Z δv (P ) = div u dv = tr ² dv. P

P

Thus u is solenoidal if and only if tr ² ≡ 0, or equivalently, div u ≡ 0, all over space. Then there exist a vector field Ψ such that u = curl Ψ. A deformation field is said to be irrotational if it satisfies the condition curl u ≡ 0, all over the field. Then there exist a scalar field φ in space such that u = ∇φ. Let u be a vector field where [u]∞ = 0, then Helmholtz’s theorem states that there exist a smooth scalar field φ and a vector field Ψ on B such that u = ∇φ + curl Ψ,

where div Ψ = 0,

(see e.g. http://mathworld.wolfram.com/HelmholtzsTheorem.html). In plain words this theorem states that an arbitrary deformation field, u, can be decomposed into two fields; an irrotational field u1 = ∇φ and an solenoidal field u2 = curl Ψ, such that ½ u1 = grad φ, curl u1 = 0 u = u1 + u2 , where (1.2) u2 = curl Ψ, div u2 = 0. which implies that the superposition of u1 and u2 gives a complete description of any local deformation field in the spatial continuum.

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1.2. System of forces. In an elastic continuum there may be a system of forces acting on a part, S, of space basically consisting of a surface force sn and a body force b. By the same right as u can be divided in an irrotational and an solenoidal component, so can also the body force b, making b = b1 + b2 with b1 and b2 belonging to the irrotational and solenoidal field respectively. ½ b1 = grad ϕ (1.3) b = b1 + b2 b2 = curl A, div A = 0. Since ϕ initially can be set to any level, it might as well be associated with a possible uniform pressure in the continuum, so an initial uniform pressure will not alter the equations in the least. We assume that there all over space is a strictly positive function ρs called the density such that the mass of any part P of space is given by Z ρs dv P

The motion of the body is described by the (infinitesimal) displacement field u(x, t) such that ∂u ∂2u and u ¨= 2 ∂t ∂ t are the velocity and acceleration respectively. The linear momentum l of P is Z l(P ) = ρs u˙ dv, u˙ =

P 0

and the body counterforce b caused by acceleration is Z 0 ˙ b (P ) = −l(P ) = − ρs u ¨ dv. P

In addition to this initial body force, I will keep the possibility open that there might be another hypothetical body force b caused by the external world, just in order to see how such a force would change the spatial continuum. The total force f (P ) on a part P of space is the total surface force from the stress vector sn exerted across the surface ∂P plus the total body force exerted on P by the external world Z Z f (P ) = sn da + b dv. ∂P

P

The Cauchy-Poisson theorem [1, page 44] states that if u is an admissible motion and f is a system of forces, then [u, f ] is a dynamic process if and only if the following two conditions are satisfied: (1) there exists a symmetric tensor field σ called the stress field, such that for each unit vector n, σn = σ n; (2) u, σ, and b satisfy the equation of motion (1.4)

div σ + b = ρs u ¨.

This theorem is one of the major results of continuum mechanics.

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1.3. The stress-strain relation. In a linearly elastic continuum there exists a relation between strain and the stress it causes, which can be expressed by the relation £ ¤ σ(x) = C ²(x) , where C is a fourth-order symmetric tensor that maps the space of strain onto the space of stress according to σij = Cijkl ²kl . C is called the elasticity tensor, and since the continuum under consideration is assumed to be homogeneous over the actual space, the 36 components of C is independent of the position vector x. Since ² is the symmetric part of the deformation gradient, ∇u, it may look like this relation rules out the possibility that there might be a residual pressure in the continuum, but that is not so if ² is interpreted as the actual stress minus the residual stress, provided that we interpret the corresponding surface trajectory accordingly [1, footnote 1 on page 68]. The spatial continuum under consideration is not only homogeneous, but also isotropic. This property immediately reduces the 36 components of C such that C may be described by only two different scalar constants. The stress-strain relation in a homogeneous and isotropic continuum thus takes the relatively simple form (1.5)

σ = 2µs ² + λs (tr²)I, σij = 2µs ²ij + λs ²kk δij = µs (u i, j + uj, i ) + λs uk, k δij ,

where µs and λs are Lam´e’s elastic moduli 1, which are constants in a homogeneous elastic continuum, and δij is the Kronecker delta ½ 1 if i = j , δij = 0 if i 6= j . 1.4. The Navier-Cauchy equation. From the strain field (1.1), the Stress-strain relation (1.5) and the Equation of motion (1.4) one can derive the Navier-Cauchy equation [1, page 213] µs ui, jj + µs uj,ij + (λs uk,k δij ), j + bi = ρs u ¨i µs ui, jj + µs uj, ji + λs uk,ki + bi = ρs u ¨i , (1.6)

µs ∇2 u + (λs + µs )∇divu + b = ρs u ¨,

or equivalently by the mathematical identity curl curl u = ∇ div u − ∇2 u (1.7)

(λs + 2µs )∇divu − µs curl curl u + b = ρs u ¨.

At this point it may be appropriate to stress the point that the Navier-Cauchy equation only treats the limit where deformations can be considered infinitesimal, and it must not be mixed up with Navier-Stokes equation, which also incorporates viscosity and takes into account the hydrodynamic property that v˙ may be different from ∂v/∂t [i.e. v˙ = ∂v/∂t + (v · ∇)v]. 1Note that I have put on the indices s to avoid mixing them up with other properties in electrodynamics.

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According to Helmholtz’s Theorem any vector field satisfying [∇ · v]∞ = 0, [∇ × v]∞ = 0, (no velocities at infinite distance from considered area) may be written as the sum of an irrotational part and a solenoidal part, v = −∇φ + ∇ × A, where

Z

∇·v d3 r0 , 0 − r| 4π|r ZV ∇×v 3 0 A= d r, 0 V 4π|r − r|

φ=−

(see http://mathworld.wolfram.com/HelmholtzsTheorem.html). As the spatial continuum is of infinite extension, or nearly so, any deformations have to be confined to a finite part of space, so this theorem will be applicable on all deformations. Hence the displacement field can be decomposed into two properties u = u1 + u2 , where u1 = −∇φ = − grad φ, u2 = ∇ × Ψ = curl Ψ. Since curl grad φ ≡ 0, and div curl Ψ ≡ 0, the Navier-Cauchy equation (1.7) can be divided into two independent equations, one for an irrotational field ρs b1 (1.8) ∇divu1 = u ¨1 − , (λs + 2µs ) λs + 2µs and the other for a solenoidal field (1.9)

−curl curl u2 =

ρs b2 u ¨2 − . µs µs

By defining two new constants s (1.10)

c1 =

λs + 2µs , ρs

r c2 =

µs , ρs

the N-C equation takes the form (1.11)

c12 ∇divu − c22 curl curl u +

b =u ¨. ρs

Operating on Equation (1.7) with the div operator and on Equation (1.6) with the curl operator yields respectively (1.12) (1.13)

∇2 (div u) −

1 ∂ 2 (div u) div b =− 2 2 c1 ∂t λs + 2µs

∇2 (curl u) −

1 ∂ 2 (curl u) curl b =− c22 ∂t2 µs

With b = 0 we have two wave equations where the dilatation, divu, satisfies a wave moving with the speed c1 , while the rotational component curlu, satisfies

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BJØRN URSIN KARLSEN

a wave moving with the speed c2 . In fact the Propagation theorem for isotropic bodies states that if a body is isotropic, then a wave is either longitudinal, in which case c = c1 , or transversal, in which case c = c2 [1, page 256]. This diversion of the Navier-Cauchy equation into one irrotational and one solenoidal part, allows us to examine these two parts separately and thereby simplifies the strain-stress relation immensely by reducing the elastic constants to only one single constant (the wave speed) in each equation (c1 6= c2 ). We see from Equation (1.10) that the two wave p speeds are related to each other with a fixed constant given by the relation c1 = 2 + λs /µs · c2 . We notice that c1 is about the double of c2 . Finally we notice that all information of curlu is lost in (1.12) and all information of divu in (1.13).

1.5. Field energy and energy transport. From the Navier-Cauchy equation one can find the internal field energy in an admissible field in B by performing the following thought experiment: Introduce a hypothetical body force, −b (negative because b is a breaking force), from the outside world such that it eradicates the entire field in B; i.e. u and all functions of u become constant like zero all over B. In addition I will assume that the entire field is confined inside B such that u is zero on the surface of B and beyond. The energy released by this operation, E, would then be like the total field energy in B.

Z E=−

Z0 dv

B

Z =

h

f (u) fZ(u)

¡

dv B

bdu ¢ i ρs u ¨ − (λs + 2µ)grad divu + µcurl curlu du

0

E can be separated into three integrals, i.e. E = E1 + E2 + E3 . The first of these integrals is simply the kinetic energy of the system Z E1 =

Zu˙

³ dv ρs

0

B

Z =

Zu˙

³ dv ρs

´ du˙ · u˙ ,

0

B

Z

1 2

E1 =

du˙ ´ du dt

2

ρs u˙ dv.

B

The next part can be integrated by using the mathematical identity (1.17) and inserting φ = div u and A = du

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Z

div Z u

E2 = (λs + 2µ)

h i div u · div(du) − div(du · div u)

dv 0

B

div Z u

Z = (λs + 2µ)

£ ¤ div u · d(divu) −

dv 0

B

Z (λs + 2µ)

¡ dv · div

div Z u

¢ du · divu .

0

B

The first part of the integral can readily be integrated, and the last part can be transformed into a surface integral over ∂B by the Divergence theorem2 and disappear because u is constant like zero on the border of B and beyond. Thus Z 2 1 E2 = 2 (λs + 2µ)(div u) dv. B

We can find E3 in much the same way by using the identity (1.14)

div (A × B) = curl A · B − curl B · A,

and inserting B = curl u and A = du Z E3 = −µ

curl Z u

h

dv 0

B

Z =µ

i div(du × curl u) − curl u · curl(du)

curl Z u

curl u · d(curl u)−

dv B

0

Z µ B

Z u ³ curl ´ dv · div du × curl u . 0

Again the first part can be integrated and the last part disappear by the same reason as above, and we get Z 2 1 E3 = 2 µ(curl u) dv. B

Finally we can write the total energy in the deformed area Z h i 2 2 2 1 1 1 E= (1.15) ρ u ˙ + (λ + 2µ)(div u) + µ(curl u) dv. s 2 s 2 2 B

The development may be a bit unorthodox, but the result is already known as Kelvin’s theorem [1, page 208], and is a proven theorem in the Linear Theory of Elasticity. The result can be interpreted as the local energy density even if this development does not prove where in the field the energy is to be found; only that there to a curl u and a div u always corresponds an energy given by the 2 H (A · n) df = R div A dv ∂B

B

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BJØRN URSIN KARLSEN

equation above, and no other energy is present as long as we deal with infinitesimal deformations restricted to a limited area of a homogeneous and isotropic continuum covered by the Linear Theory of Elasticity.3 With this restriction in mind the local energy density, e, in the spatial continuum is given by (1.16)

e=

1 2

ρs u˙ 2 +

1 2

(λs + 2µs )( div u)2 +

1 2

µs ( curl u)2

leaving the possibility open that there may be a residual pressure and a corresponding homogeneous residual energy density in addition to this field energy. It is noteworthy that the energy density in any field of strain and motion is nonnegative even if the space itself should happen to contain a huge amount of uniformly distributed energy due to an initial pressure. The energy transport in the deformation field can be found by deriving the equation above with respect on time. We acquire ∂e ˙ u + (λs + 2µs ) div u div u˙ + µs curl u curl u. ˙ = ρs u¨ ∂t We substitute ρs u ¨ from Equation (1.7) and get ¤ £ ∂e =u˙ (λs + 2µs ) grad div u − µs curl curl u + b ∂t ˙ + (λs + 2µs ) div u div u˙ + µs curl u curl u, £ ¤ ∂e ˙ curl u) bu˙ = + µs curl ( curl u) · u˙ − curl u( ∂t £ ¤ − (λs + 2µs ) ( div u) div u˙ + u˙ grad ( div u) . By the mathematical identity (1.14) and the identity (1.17)

div (φA) = φ div A + A grad φ,

this equation develops into £ ¤ ∂e + div (µs curl u × u) ˙ − div (λs + 2µs ) div u · u˙ = bu. ˙ ∂t We define a new vector (1.18)

S = µs curl u × u˙ − (λs + 2µs ) div u · u, ˙

and in the absence of external forces we acquire the compact equation: ∂e + div S = 0. ∂t Since the increase in energy density has got to be equal to the inflow of energy per unit volume, S can be interpreted as the energy flow vector. 2. Solenoidal deformations and Electrodynamics In this section I will redefine some of the terms used in elastodynamics to terms that can be directly compared to those in electrodynamics. I will stress that these redefinitions only are intended to make the comparison simpler and will not change the physics behind the original terms in any way. With the additional assumption that there might be true sinks and sources in the elastic continuum, I will also show that they will influence the elastodynamic fields in exactly the same way as electric charges influence the electromagnetic fields. How sinks and sources can be 3The corresponding expression for the energy density in an electromagnetic field has the same

limitation, but nonetheless it is usually interpreted as the local energy density.

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more than pure mathematical entities will be discussed in another paper4. Some terms are used quite differently in mechanics and electrodynamics. For example the Greek letter ρ is used for mass density in mechanics, but as charge density in electrodynamics. To avoid confusion I will use an index s on the mechanical terms whenever necessary. Hence ρs means spatial mass density while ρ means the density of sinks – the spatial counterpart to charge density. 2.1. Reformulation of the Navier-Cauchy’s Equation. First we define some new properties def

1 , µs

(2.1)

µ0 =

(2.2)

ε0 = ρs ,

(2.3)

E = −

(2.4)

B = curl u2 ,

(2.5) (2.6)

[L2 F −1 ] = [LT 2 M −1 ],

def

def

[F T 2 L−4 ] = [M L−3 ],

∂u2 , ∂t

[LT −1 ],

def

[ ],

def

[F L−3 ] = [M L−2 T −2 ],

j = b, 1 , c2 = ε0 µ0

[L2 T −2 ].

Note that in this section all the defined properties refer exclusively to elastic properties. The notations inside the square brackets are the dimensions of the properties in front, but in this context I have found it convenient to alternatively replace mass with force as a fundamental unit. Hence the mass unit is converted to the force unit by the relation [F ] = [M LT −2 ]. By the identities curl (∂(·)/∂t) = ∂ curl (·)/∂t and div curl (·) = 0 we immediately get the relation between B and E (2.7)

˙ = 0, curl E + B

and by the mathematical identity div curl a = 0 we have (2.8)

div B = 0.

Navier’s Equation (1.9) takes the form (2.9)

curl B −

1 ˙ E = µ0 j. c2

Now I will make the assumption that there may be real sinks and sources in the spatial continuum. How this is possible will be discussed elsewhere, but here I take entities like that for granted. I will take sinks as positive entities and sources as negative sinks, and assume that they can only be created by pair production; one sink for for one equally strong source. If there are more sinks than sources in an area, the sink density is positive, and if there are more sources than sinks, then the 4See a proposition on how it might work in http://www.pdfcoke.com/doc/3014850/The-GreatPuzzle.

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sink density is negative. The strength of a spatial sink, Qs , can be defined as the inflow of spatial mass through a closed surface around the sink I def Q = −ρs undf, ˙ [F T L−1 ] = [M T −1 ], V

where n is an outwards pointing unit vector normal to df . Now let sinks and sources with strength Q1 , Q2 , Q3 , · · · , Qn be sufficiently smoothly distributed in space. Then the density of sinks is given by m 1 X Qsn V →² V n=1 I 1 undf ˙ = −ρs lim V →² V V = −ρs div u, ˙ def

ρ = lim

or with the new terms incerted ρ = ε0 div E,

[F T L−4 ] = [M L−3 T −1 ],

where m is the number of sinks in a volume V of space, and ² is a small volume, but still great enough to contain many sinks. (Note the difference between the spatial mass density ρs and the density of sinks ρ.) There is a hidden dependency between the sink density ρ and and the volume force j. By taking the divergence of Equation (2.9): ³E ˙ ´ div curl B − div 2 = div µ0 j, c ˙ = 1 div j, (I) − div E ε0 and the partial derivative with respect on time of Equation (??): ∂ 1 ∂ρ div E = ∂t ε0 ∂t ˙ = 1 ∂ρ , div E ε0 ∂t

(II)

and evaluating the combination I + II, we acquire (2.10)

ρ˙ + div j = 0.

Since sinks and sources by definition can only be created or disappear in pairs, the only way the density can change in a volume is by out- or inflow, hence the vector j can be interpreted as a flow of sinks or sources, i.e. a flow of sinks or sources will create a force field (a drag) in the spatial continuum. 2.2. The stress energy tensor. According to (1.16) and the newly defined properties the elastodynamic field energy in a divergence-free field is (2.11)

e=

ε0 2 1 2 E + B , 2 2µ0

[F L−2 ] = [M L−1 T −2 ].

Since this field may contain energy, we must also expect that it can move around in space as the field changes. To examine this property we can start by deriving

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the field energy (2.11) with respect on time and get ∂e ˙ + 1 B · B. ˙ = ε0 E · E ∂t µ0 From this expression we can eliminate the time derivatives of E and B by applying (2.9) and (2.7) ³ ∂e 1 1 ´ = ε0 E · c2 curl B − j − B · curl E, ∂t ε0 µ0 ¢ ∂e 1¡ = curl B · E − curl E · B − j · E, ∂t µ0 and further by the mathematical identity (1.14) it develops into ³ ´ ∂e 1 + div E × B = −j · E, ∂t µ0

(2.12)

{= u˙ · b}.

The right side of this equation is the rate of work done by external forces per unit volume on the continuum, and the left side can be interpreted as the rate of increase in energy density plus the rate at which the energy is leaving per unit volume. Thus the energy flow vector is (2.13)

def

S =

1 (E × B), µ0

[F L−1 T −1 ] = [M T −3 ].

With this property inserted, Equation (2.12) takes the form (2.14)

1 1 ∂e 1 + div S = − j · E. c ∂t c c

To examine the forces involved by a change of momentum, we can derive the momentum vector with respect on t and again eliminate the time derivatives of E and B by applying (2.9) and (2.7) ¢ S˙ 1 ¡ ˙ +E ˙ ×B , = 2 E×B c2 c µ0 £ ¤ S˙ = ε0 E × (− curl E) + (c2 curl B − c2 µ0 j) × B , 2 c S˙ 1 = −ε0 E × curl E + curl B × B + B × j. c2 µ0 By applying the mathematical identity (2.15)

grad (A · A) = 2[A × curl A + (A · ∇)A],

we obtain

(2.16)

¡ ε0 ¢ ¡ 1 ¢ S˙ + grad E · E − ε0 (E · ∇)E + grad B·B 2 c 2 2µ0 1 − (B · ∇)B = (B × j). µ0

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BJØRN URSIN KARLSEN

We then write out the above equation in component form5: S˙ i ε0 1 1 +( E·E+ B · B),i − ε0 Ej Ei, j − Bj Bi, j 2 c 2 2µ0 µ0 = ²ijk Bj jk , and expand it by adding ±ε0 Ej,j Ei and ± µ10 Bj,j Bi S˙ i ¡ ε0 2 1 2¢ + E + B , i − ε0 Ej Ei, j − ε0 Ej, j Ei + ε0 Ej, j Ei 2 c 2 2µ0 1 1 1 Bj, j Bi + Bj, j Bi = ²ijk Bj jk . − Bj Bi, j − µ0 µ0 µ0 The term Bj, j is like zero by (2.8), Ej,j = ρ/ε0 by Equation (??), and the rest can be manipulated into ∂Si + σij,j = ρEi − ²ijk Bj jk . c2 ∂t where the new tensor σij is given by 1 1¡ 1 2¢ def (2.18) σij = ε0 Ei Ej + Bi Bj − ε 0 E2 + B δij . µ0 2 µ0 (2.17)

−

Now we write out (2.14) and (2.17) in component form and obtain the set of equations (the zeroes are inserted for clarity) ∂e ∂Sx + c∂t c∂x ∂Sx ∂σxx − c2 ∂t ∂x ∂σyx ∂Sy − c2 ∂t ∂x ∂Sz ∂σzx − c2 ∂t ∂x

∂Sy c∂y ∂σxy − ∂y ∂σyy − ∂y ∂σxy − ∂y

+

∂Sz c∂z ∂σxz − ∂z ∂σyz − ∂z ∂σzz − ∂z +

=0 −

Ex jx Ey jy Ez jz − − , c c c

=Ex ρ + 0 − Bz jy + By jz , =Ey ρ + Bz jx + 0 − Bx jz , =Ez ρ − By jx + Bx jy + 0.

The four differential equations can be written as one matrix equation   ∂   e Sx /c Sy /c Sz /c c∂t  ∂   Sx /c −σxx −σxy −σxz   ∂x     ∂  ·  Sy /c −σyx −σyy −σyz  ∂y ∂ Sz /c −σzx −σzy −σzz ∂z     0 −Ex /c −Ey /c −Ez /c cρ  Ex /c   0 −Bz By   ·  jx  = (2.19)  Ey /c Bz 0 −Bx   jy  Ez /c −By Bx 0 jz which formally can be written Tαβ ,β = Fαβ Jβ , or in frame independent notation ∇ · T = F · J. 5Notice that Latin indices go from 1 to 3.

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Here the second order tensor T, the  e  αβ def  Sx /c (2.20) T =  Sy /c Sz /c

stress energy tensor, is given by  Sx /c Sy /c Sz /c −σxx −σxy −σxz  , −σyx −σyy −σyz  −σzx −σzy −σzz

the second order tensor F by  (2.21)

Fαβ

 0 −Ex /c −Ey /c −Ez /c Ex /c 0 −Bz By  def  , =   Ey /c Bz 0 −Bx  Ez /c −By Bx 0

and finally J by def

(2.22)

Jα = (cρ, jx , jy , jz ).

In four-space we need some definitions: First the Minkowski metric  −1  0 ηαβ =   0 0

0 0 1 0 0 1 0 0

 0 0  . 0  1

Coordinates in 4-space xα =(x0 , x1 , x2 , x3 ) = (ct, x, y, z), xβ =xα ηαβ = (−ct, x, y, z). The del operator in four space

³1 ∂ ∂ ∂ ∂ ´ , , , , c ∂t ∂x ∂y ∂z ³ 1 ∂ ∂ ∂ ∂ ´ , , , , ∇α = ∂ α = − c ∂t ∂x ∂y ∂z ³ 1 ∂2 ∂2 ∂2 ∂2 ´ ∇ 2 = ∂ α ∂α = − 2 2 , 2 , 2 , 2 . c ∂t ∂x ∂y ∂z ∇α = ∂α =

2.3. The vector potential in the elastic continuum. Equation (2.8) means that the field B can be derived from some vector potential A (2.23)

B = curl A,

where div A is temporarily arbitrary, but can be given a fixed meaning later without changing the term curl A. By inserting (2.23) into (2.7) we get curl (E + A,t ) = 0. Therefore E + A,t may be represented as some gradient E + A,t = −c grad φ, hence (2.24)

E = −(c grad φ + A,t ).

Thus both E and B can be represented by some potentials A and φ. For the choice of A and φ the Equations (2.7) and (2.8) are fulfilled.

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BJØRN URSIN KARLSEN

By adding and subtracting the same term c grad ψ,t into (2.24), we acquire E = −[c grad (φ − ψ,t ) + (A + c grad ψ),t ]. We also have that adding c grad ψ to A leaves B unchanged. Hence the substitutions (2.25)

φ → φ0 = φ + ψ,t ,

A → A0 = A + c grad ψ

leave the properties E, B, j, and ρ unchanged for arbitrary functions ψ. The substitutions (2.25) are called Gauge transformations (see http://www.mathematik.tudarmstadt.de/ bruhn/Maxwell-Theory.html). The most obvious gauge is to set div A = div u which means to infer that the spatial continuum is uncompressed. It would work equally well to set div u = const. This picture is complicated by the assumption that there are true point-like sinks and sources around, hence − div u˙ = ρ/ε0 (see Equation (??)), so we can introduce a potential φ such that ρ −∇2 φ = . ε0 This leads to the Coulomb gauge which works well if we consider a fixed frame in the spatial continuum. What we need, however, is a gauge that works equally well in a moving frame. This requirement leads to the Lorenz gauge after the Danish physicist Ludvig Valentin Lorenz (1829-1891): 1 (2.26) div A + φ, t = 0. c Inserting (2.23) and (2.24) into (2.9) yields 1 curl curl A + 2 (A,tt + c grad φ) = µ0 j, c and by applying the mathematical identity (2.27)

curl curl A = grad div A − ∇2 A,

we obtain 1 1 A,tt − ∇2 A + grad ( div A + φ, t ) = µ0 j. c2 c Analogously by inserting the same properties into (??) we obtain ρ −(c∇2 φ + div A,t ) = , ε0 or by adding and subtracting 1/cφ,tt we acquire 1 1 ρ φ,tt − c∇2 φ − ( div A + φ, t ),t = , c c ε0 1 1 cρ 1 2 φ,tt − ∇ φ − ( div A + φ, t ),t = µ0 · 2 . c2 c c c ε0 µ0 By the Lorenz gauge and (2.6) the two potentials reduce to 1 (2.28) − 2 φ,tt + ∇2 φ = −µ0 · cρ, c 1 (2.29) − 2 A,tt + ∇2 A = −µ0 · j. c These two equations can be expressed as one vector potential in four-space ∂α ∂ α Aβ = −µ0 · Jβ ,

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where Aβ = (φ, Ax , Ay , Az ), Jβ = (cρ, jx , jy , jz ), or in frame independent notation (2.30)

∇2 A = −µ0 · J.

2.4. Energy flow and momentum. Think of disturbance energy as a substance that flows through space with velocity c. Then the energy flow vector alternatively can be expressed as S = e · c, and the energy density as e=

|S| . c

Next if we only consider disturbance energy without the presence of any sinks or sources, Equation (2.17) reduces to ∂Si = σij,j , c2 ∂t ∂ ³S´ = ∇ · σ. ∂t c2 Now, if σ is a stress tensor, then the divergence of it represents a force, hence ∂ ³S´ = f, ∂t c2 and we can define a new vector (2.31)

def

p = S/c2 ,

to obtain ∂p = f. ∂t Let us imagine some elastodynamic radiation trapped inside an imaginary box with reflecting walls, and let the box be subdivided into m small cells containing small parts of the radiation energy En . At a given time the sum of the energy flow is given by (2.32)

E·v =

m X

Sn ,

n=1

hence the box is moving with some velocity v in the direction of S and it contains an amount of energy given by E=

m X n=1

En .

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BJØRN URSIN KARLSEN

We divide Equation (2.32) by c2 and take the time derivative of it. We obtain m ∂ ³ Ev ´ X ∂ ³ Sn ´ = , ∂t c2 ∂t c2 n=1 m ∂ ³ Ev ´ X = fn , ∂t c2 n=1 ∂ ³ Ev ´ = f. ∂t c2

Finally we define a new property m given by def

m =

E , c2

or E = mc2 ,

(2.33) and acquire

∂ (mv) = f . ∂t

(2.34)

We can interpret this equation such that if we have a box containing a disturbance energy similar to m, then a force f is needed to give it an acceleration a = v, ˙ provided that the property m, which we could call the mass of radiation, is kept constant. When the box is accelerated from zero velocity6, however, we have got to add energy to it given by dE = f · ds, = f ds, if ds is in the direction of f . By the equations above we can develop this equation further into d(mv) ds dt ³ d E ´ = v ds dt c2 ¢ 1¡ = 2 v · dE + E · dv dv, c 1 v · dv dE = 2 , E c 1 − v 2 /c2 dE =

6The situation is considerably more complicated if the box and the observer have an initial velocity, say v0 . To address that question, one first has got to assume that the phenomenon is observed in a Lorentz frame that makes the equations above invariant for the change of the observer’s coordinate system, as Lorentz showed already in the fall of the nineteenth century. That would make the observation fully relativistic, and v0 could be set to zero from where the deduction could proceed as shown.

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which can be solved 1

ln E = ln p

1 − v 2 /c2 C

= ln p

(2.35)

1 − v 2 /c2 E0 E=p , 1 − v 2 /c2 m0 m= p . 1 − v 2 /c2

+ ln C ,

Equations (2.34) and (2.35) bring the dynamics of confined disturbance energy in line with Newton’s second law of motion and the relativistic mass increases with velocity. Equation (2.33) is of course like the famous Einstein energy/mass relation. 3. Summing up In this paper we have seen that the four equations (2.7) through (??) correspond to James Clerk Maxwell’s (1831–1879) electrodynamic equations. Provided that there are free moving sinks and sources in the spatial continuum, Equation (2.10) demonstrates that they will generate a ”drag” just like Lord Kelvin postulated for moving electrons in 1890 [2, page 247]. The energy flow vector in Equation (2.13) is formally like Poynting’s vector after John Henry Poynting (1852-1914). In a notation introduced by Hermann Minkowski (1864–1909), the field tensor Fαβ in Equation (2.21) is like the Electromagnetic tensor, and the Elastodynamic stress-energy tensor, Tαβ , corresponds exactly to the Electromagnetic stress-energy tensor. Note also that the spatial stresses, σxy , correspond exactly to Maxwell’s stress tensor that represent the mechanical stresses caused by electromagnetic fields in space. Finally it is possible to describe deformation fields as a vector potential in the spatial continuum. In this notation the fields, like electromagnetic fields, are invariant by transformations between different Lorentz frames, after Hendrik Antoon Lorentz (1853–1928), in rectilinear motion relative to each other. A moving weightless box containing an amount of disturbance energy will have a momentum corresponding to the energy in a material body with the same energy content. The force needed to change its velocity corresponds to Newton’s second Law of motion, and moreover, to increase the velocity of such a box towards the propagating speed c of transversal waves will increase its energy towards infinity. References 1. S. Fl˝ ugge (ed.), Mechanics of solids ii, Encyclopedia of Physics, vol. VIa/2, Springer, 1972. 2. Sir Edmund Whittaker, A history of the theories of aether and electricity, vol. I and II, Philosophical Library, 1951. E-mail address: [email protected]