Acta Gead. Geoph. Hung., Vol. 36(4), pp. 471-476 (2001)

THE GENERAL RELATIVISTIC AND COVARIANT FORM OF THE I. HELMHOLTZ VORTICITY THEOREM

AND GEOPHYSICAL APPLICATION

W SCHRODER1 and H-J TREDER2

[Manuscript received November 8, 1999, revised February 25, 2000]

The general relativistic and covariant differential form of Helmholtz's first vortic­ity theorem is presented. We prove in relation with it an invariant kinematic identity which is the generalisation of the Helmholtz theorem for general continua.

Keywords: geophysical hydrodynamics; Helmholtz vorticity theorem; invariant kinematic identity

The general-relativistic and with it, the general-covariant formulation of the equations of the mechanics of continua has a significant advantage, namely that the so formulated equations are automatically valid in arbitrary systems of co­ordinates, but also in arbitrary systems of reference. The studied continuum may correspondingly be exposed to arbitrary inertial forces; the effect of these inertial forces (especially the centrifugal and the Coriolis forces) on the continuum is through the general relativistic covariance automatically taken into account. Moreover, the general-relativistic form also contains in the case of not disappearing Riemannian tensor R~rii the effect of the gravity field on the continuum. Thus the effect of weak Newtonian gravity fields on the continuum is included for the limiting case of weak static gravity fields. Finally the general-relativistic form contains all special­relativistic effects, too and therefore it generalises the consequences of the classical mechanics of continua on velocities which are comparable to the velocity of light.

Additionally the covariant tensor calculus is especially elegant and simple, and as a consequence, it can describe complicated physical situations of a high generality with few symbolic operations.

We use in the following the tensor calculus with the symbolism used by Einstein (1969). Comma means the usual partial derivative after the corresponding index:

a <I> ,>. = ax>' <I>.

The semicolon means the corresponding covariant derivative, thus e.g. for a vector field Aa:

A~>. = A~>. + r~>.A;3. where r~>., is Christoffel's three-index-symbol:

r a 1 aE( ) ;3>' ="2g -g;3>.,E + g>.e,;3 + gE;3,>.

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472 W SCHRODER and H-J TREDER

and gO!(3 is the metric fundamental tensor and it is

A,,;A - AA;" = A",A - AA,,,.

According to Einstein, one has to sum after the double indices. The Greek indices run from zero to three, the Latin ones from 1 to 3, the sign is fixed so that time-like vectors have a negative square of the absolute value (the inertial index is -1, +1, +1, +1). Latin indices are generally three-dimensional spatial indices.

We define the four-vector of the velocity of a particle of the continuum according to:

(la)

where the following expression is:

(lb)

(where c is the velocity of light in the inertial system) the differential of the eigen­time. The four-velocity uO! is then normed as

0!(3_ O! _ 2 gO!(3U U - U UO! - -c . (2)

The velocity uO! is a contravariant vector to which the co-vector

(3)

belongs. Using this co-vector, we define the antisymmetric tensor of the rotation of the continuum:

(4)

We understand as the total (material) derivative of a covariant quantity <I> after the eigen-time T the following:

(5)

Without any supposition about the dynamics and structure of the continuum we find a simple expression for the total temporal variation grwO!(3 of the rotation wO!(3: it is at first valid that

D A A A -WO!(3 = WO!(3' A U = UO!'(3A U - U(3'O!A U Dr ' , , (6)

and by changing the sequence of differentiation we obtain:

::T wO!(3 = (UO!;A(3 - U(3;AO!)U A - (R",O!(3A - R",(3O!A)U"'U A . (7)

Here R",O!(3A is the great Riemann-tensor. On the base of the symmetry property of this tensor, the second term on the

right side of Eq. (7) disappears. The first term is transformed using the identity:

(Sa)

Acta Geod. Geoph. Hung. 36, 2001

1. HELMHOLTZ VORTICITY THEOREM 473

With the further identities:

(Sb)

we finally obtain by antisymmetrization of Eq. (Sa):

(9a)

The terms on the right side of Eq. (9a),

(9b)

are homogenous in the rotation waf3 and they disappear if the curl of the velocity disappears. The term

( ..E....ua ) - (..E....Uf3) DT ,(3 DT ,a

(9c)

is, however, the curl of the acceleration vectors

According to the Einstein-Planck-Minkowski axioms of the relativistic mechan­ics, it is valid that

Dua __ =pa DT

where pa is the four-vector of the force related of the rest-mass unit;

is then the four-vector of the force related to the rest-volume. Therefore Eq. (9a) says that

(10)

A vortex movement can develop (or disappear) if the curl of the four-vector of force related to rest mass unit,

Pa,f3 - P(3,a does not disappear. Nevertheless, neither gravity, nor inertial forces contribute to pa, as they are both included in the covariant form of the other terms of Eqs (9) and (10).

Equations (9) and (10) are, respectively, general cinematic identities which are valid irrespective of the dynamics and structure of the continuum.

If the continuum is such that the term Eq. (9c), namely the curl of the accel­eration also disappears, then Helmholtz's 1. vorticity theorem is clearly valid: the total temporal variation of the curl is proportional with the curl itself, it disappears

Acta Geod. Geoph. Hung. 36, 2001

474 W SCHRODER and H-J TREDER

consequently if the curl disappears (Sommerfeld 1966). This is valid both for the covariant total derivative (D / DT)Wa(3 as well as due to

corresponding to the usual total derivative (d/dT)Wa(3. Generally the total temporal variation of the curl contains an inhomogeneity, Eq. (9c).

The general-relativistic dynamics of continua follows then from Einstein's dy­namic equation for the material tensor Ta(3 of the continuum. This dynamic equa­tion is simply the four-dimensional covariant formation of the differential energy­impulse law (Einstein 1969):

(11)

Therefore the dynamics depends only on the form of the material tensors Ta(3. In the case of an incoherent, pressure-free the material tensor is:

(12a)

Here g is the rest mass density and one obtains from Eqs (10) and (11) with Eq. (2) the equation of motion:

(12b)

and the equation of continuity:

(12c)

It follows from Eq. (12b) the identical disappearance of Eq. (9c). For an incoherent fluid which moves due to the influence of the general gravity and of arbitrary inertial forces, the 1. vorticity theorem of Helmholtz is valid.

For a general friction-free fluid with the pressure p the material tensor is:

where here one has P

g=0"-2 c

(13a)

(see Einstein 1969) as mass density. The dynamic equation (11) gives now the equation of motion with Eq. (2):

(13b)

and the generalised equation of continuity

2 (a) dp C O"u 'a = -d . , T

(13c)

Acta Geod. Geoph. Hung. 36, 2001

1. HELMHOLTZ VORTICITY THEOREM 475

Equation (13c) contains corrections to the classical equation of continuity which became, however relevant only in case of high velocities.

Helmholtz's theorem has to be valid now for incompressible fluids. The notion incompressibility is in the framework of relativity a problematic one (see e.g. Synge 1956, 1961), as an exact incompressibility is contrary to the fact that the velocity of light, c, is a limiting velocity, while for a constant Q the velocity of sound increases to infinity. The notion "incompressibility" has a sense therefore only if the velocities are so low, that the velocity of light can be considered in comparison to them as infinitely large.

In this case we define the incompressibility by the four-dimensional generalisa­tion of the classical condition of incompressibility (Synge 1956, 1961):

and then it follows from Eqs (13b) and (13c) with Eq. (2):

dQ = O. dT

It is valid for the four-acceleration:

D dp U a P,a DT U a = dT c2 (J' - ----;;-

and Eq. (9c) has the form:

( dP Ua ) Po:,{3 - P{3,a = -d -2-T c (J' ,{3

(dP;) + (P,{3) (p,a) . dT C (J' (J' ,a (J' ,(3 ,a

(14)

(15a)

(15b)

(16)

With the previously mentioned condition, namely that the velocity in the con­tinuum is small in relation to the velocity of light c, that means exactly that it is valid:

and P does not depend explicitly on time, Eqs (15a) and (15b) can be written as:

D -Uo =0 Dt

(17a)

(17b)

(17c)

These equations are the well-known, non-relativistic expressions. Equation (17) disappears together with Eq. (16). Thus it follows that the vorticity theorem of Helmholtz is valid for incompressible fluids which move under the influence of ar­bitrary inertial and gravitational forces in the approximation in which it is allowed us to consider the velocity of light as an infinitely large quantity, that is neglecting special relativistic effects and the problems with the notion "incompressibility".

Acta Geod. Geoph. Hung. 36, 2001

476 W SCHRODER and H-J TREDER

The transition from four-dimensional formulas to the usual three-dimensional ones follows simply on the basis of the remark that the axial four-vector of the local angular velocity of the vorticity motion w" of he rotatory motion is connected with the tensor of the rotation W KA according to:

(18)

Here c,,(3KA is the Levi-Civita complete obliquely symmetric pseudo-tensor. In the limiting case oflow velocities v 2 « c2 , is the eigen-time T equal to the Newtonian time t and it is valid for the three-dimensional axial vorticity vector wi:

With Eqs (19) and (17) Eq. (10) gets the three-dimensional form:

dw i 1·· - = U w'w' dt ,1

(19)

(20)

where the semicolon refers only to the metrics of the three-dimensional space. Equa­tion (20) is the known differential form of the Helmholtz theorem (see Einstein 1969, Truesdell and Toupin 1960). - The transition from D/Dt to d/dt means in Eq. (20) the restriction to inertial frames of reference.

References

Einstein A 1969: Grundziige der Relativitatstheorie, 5th ed., Berlin/Oxford/Braunschweig Ertel H 1964: Monatsber. Dt. Akad. Wiss. Berlin, 6, 939 Ertel H 1971: Zeitschr. Meteor., 22, 329 Sommerfeld A 1966: Mechanik der deformierbaren Medien, 5. ed., Leipzig Synge J L 1961a: Relativity - the Special Theory, Amsterdam Synge J L 1961b: Relativity - the General Theory, Amsterdam Truesdell C, Toupin R A 1960: The Classical Field Theories, in: Handbuch der Physik,

Vol. 111/1, Berlin/Gottingen/Heidelberg

Acta Geod. Geoph. Hung. 36, 2001