^{*}

Edited by: José Fernando Cariñena, Facultad de Ciencias, Spain

Reviewed by: Douglas Alexander Singleton, California State University, Fresno, United States; Jan Sladkowski, University of Silesia of Katowice, Poland

*Correspondence: David V. Svintradze

This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

We propose dynamic non-linear equations for moving surfaces in an electromagnetic field. The field is induced by a material body with a boundary of the surface. Correspondingly the potential energy, set by the field at the boundary can be written as an addition of four-potential times four-current to a contraction of the electromagnetic tensor. Proper application of the minimal action principle to the system Lagrangian yields dynamic non-linear equations for moving three dimensional manifolds in electromagnetic fields. The equations in different conditions simplify to Maxwell equations for massless three surfaces, to Euler equations for a dynamic fluid, to magneto-hydrodynamic equations and to the Poisson-Boltzmann equation.

Fluid dynamics is one of the most well understood subjects in classical physics [^{1}

An example of fluid dynamics modeling as moving surfaces embedded in Euclidian space is moving two dimensional surfaces of fluid films such as soap films. Another, biologically relevant example is dynamic fluid membranes, vesicles, and micelles where a large body of notable theoretical results has already been produced [

Soap films can be formed by dipping a closed contour wire or by dipping two rings into the soapy solution. Stationary fluid films or films in mechanical equilibrium with the environment form a surface with minimal surface area. Usually surfaces such as soap films are modeled as two dimensional manifolds. Fluid films not in mechanical equilibrium may have large displacements and can undergo big deformations [

The equations of motion for free liquid films were initially proposed in Grinfeld [

where ρ is the two dimensional mass density of the fluid film,

As indicated above, fluid dynamics can be described by motion of fluid surfaces, where the motion can happen in Euclidean ambient space, corresponding to the non-relativistic case or in Minkowski ambient space, corresponding to the fully relativistic case. Minkowskian space-time is more general and we will carry out derivations in Minkowski space that can be trivially simplified for non-relativistic cases. Instead of motion of free fluid films, we discuss motion of charged or partially charged material bodies with the boundary of charged or partially charged surfaces^{2}^{6}^{3}

The theoretical concept of hydrophobicity is already developed [

Hydrophobic and hydrophilic interactions can be described as dispersive interactions between permanent or induced dipoles and ionic interactions throughout the molecules [

In this paper we discuss motion of compact and closed manifolds induced by electromagnetic field, where the field is generated by a continuously distributed charge in the material body. The boundary of the body is a semi-permeable surface (manifold) with a charge (or partial charge) and the charge can flow through the surface. Since, the charge in general is heterogeneously distributed in the body, the charge flow induces a time variable electromagnetic field on the surface of the body, forcing the motion of the manifold. Consequently, the problem is to find an equation of motion of moving manifolds in the electromagnetic field. The problem may be connected to many physics sub-fields, such as fluid dynamics, membrane dynamics or molecular surface dynamics. For instance the surface of macromolecules in aqueous solutions is permeable to some ions and water molecules and the charge on the surface is heterogeneously distributed. Flow of some ions and water molecules through the surface and the uneven distribution of charge in the macromolecules induce the surface dynamics. The same processes occur in biological membranes, vesicles, micelles, etc. Here we deduce general partial differential equations for moving manifolds in an electromagnetic field and demonstrate that the equations, in different conditions, simplify to the Euler equation for fluid dynamics, the Poisson-Boltzmann equation for describing the electric potential distribution on surface and the Maxwell equations for electrodynamics.

The formalism presented in this paper can be easily extended to hypersurfaces of any dimension. The limitation three surfaces embedded in four space-time, which is necessary to describe electromagnetism [

Since Minkowskian space-time does not follow Riemannian geometry, we need a small adjustment of definitions. For Minkowski space-time, which fits to pseudo-Riemannian geometry, we need definitions of arbitrary base pairs of ambient space, even though the definitions look exactly the same as for Riemannian geometry embedded in Euclidean ambient space [

Combination of three ordinary dimensions with the single time dimension forms a four-dimensional manifold and represents Minkowski space-time. In this framework Minkowski four-dimensional space-time is the mathematical model of physical space in which Einsteins general theory is formulated. Minkowski space is independent of the inertial frame of reference and is a consequence of the postulates of special relativity [

Euclidean space is the flat analog of Riemannian geometry while Minkowski space is considered as the flat analog of curved space-time, which is known in mathematics as pseudo-Riemannian geometry. Considerations of four-dimensional space-time make embedded moving manifolds three dimensional, where parametric time

To briefly describe Minkowskian space-time, let us refer to arbitrary coordinates ^{α}, α = 0, …, 3, where the position vector ^{α}). Bold letters throughout the manuscript designate vectors. Latin letters in indexes indicate surface related tensors. Greek letters in indexes show tensors related to the ambient space. All equations are fully tensorial and follow the Einstein summation convention.

Suppose that ^{i} (^{i}, ^{α} are arbitrarily chosen so that sufficient differentiation is achieved in both space and parametric time. The surface equation in ambient coordinates can be written as ^{α} = ^{α}(^{i}) and the position vector can be expressed as

Two dimensional illustration of a curved three dimensional surface embedded in Minkowski space-time. ^{α} represents the analog of Cartesian coordinates of Minkowski space-time. _{i} are base vectors defined in tangent space and ^{α} = ^{α}(^{i}) is the general equation of the surface.

Covariant bases for the ambient space are introduced as _{α} = ∂_{α}

The contravariant metric tensor is defined as the matrix inverse of the covariant metric tensor, so that _{00} = _{0}·_{0} and consequently if for Minkowskian space-time, the space like signature is set (−1, +1, +1, +1), then _{0} = (^{4}

vanish and the equality between partial and curvilinear derivatives follows ∂_{α} = ∇_{α}. In Minkowski space-time (later space) the ∂_{α} partial derivative and ∇_{α} curvilinear derivative are the same. Everywhere in calculations we use ∂ letter for the ambient space derivative and keep in mind that when referring to Minkowski space the derivative has index in Greek letters and, in that case, it is the same as partial derivative. When indexes are mixed Greek and Latin letters the last statement, as is shown below, does not hold in general.

Now let's discuss tensors on the embedded surface with arbitrary coordinates ^{i}, where _{i} is no longer the same as the partial derivative _{i} = ∂_{i}

The definition (4) dictates that the surface is three dimensional pseudo Riemannian manifold, because ambient space is four dimensional Minkowskian space and the surface in four manifold is three manifold.

Analogically to space metric tensor ^{ij} the contravariant surface metric tensor is the matrix inverse of the covariant one _{ij}. The matrix inverse nature of covariant-contravariant metrics gives possibility to raise and lower indexes of tensors defined on the manifold. The surface Christoffel symbols are given by

and along with Christoffel symbols of the ambient space provide all the necessary tools for covariant derivatives to be defined as tensors with mixed space/surface indexes:

where

The metrilinic property ∇_{i}_{mn} = 0 of the surface metric tensor is a direct consequence of (4, 5) definitions, therefore _{m} · ∇_{i}_{n} = 0. The _{m} and ∇_{i}_{n} vectors are orthogonal, so that ∇_{i}_{n} must be parallel to the

where _{ij} is the tensorial coefficient of the (6) relationship and is generally referred to as the symmetric curvature tensor. The trace of the curvature tensor with upper and lower indexes is the mean curvature and its determinant is the Gaussian curvature. It is well-known that a surface with constant Gaussian curvature is a sphere, consequently a sphere can be expressed as:

When the constant becomes null the surface becomes either a plane or a cylinder. Equation (7) is the expression of constant mean curvature (CMC) surfaces in general. Finding the curvature tensor defines the way of finding covariant derivatives of surface base vectors and so (6, 7) provide the way of finding surface base vectors which indirectly leads to the identification of the surface.

After defining the metric tensor for ambient space η_{μν} (3) and the metric tensor for a moving surface _{ij} (4), we now proceed with a brief review of surface velocity,

For the definition of surface velocity we need to define ambient coordinate velocity ^{α} first and to show that the coordinate velocity is the α component of the surface velocity. Indeed, by the velocity definition

taking into account (2), ^{i}. Therefore, by the partial time differentiation of (2) and definition of ambient base vectors, we find that

Consequently ^{α} is the ambient component of the surface velocity. According to (9), the normal component of the surface velocity is the dot product with the surface normal

The normal component ^{α}. Its sign depends on a choice of the normal. The projection of the surface velocity on the tangent space (Figure

2D Graphical illustration of the arbitrary chosen three manifold and it's local tangent space. _{0}, _{1}, _{2}, and _{i}, _{i} directions.

Graphical illustrations of coordinate velocity ^{α}, interface velocity ^{i} are given in Figure

There is a clear geometric interpretation of the interface velocity [_{t}, _{t + Δt}. Suppose that _{t} surface and the corresponding point _{t + Δt}, has the same surface coordinate as _{t} intersects the surface _{t+Δt}. Then for small enough Δ

and can be interpreted as the instantaneous velocity of the surface in the normal direction. It is worth mentioning that the sign of the interface velocity depends on the choice of the normal. Although,

Geometric interpretation of the invariant time derivative _{t} surface so that it lies on the _{t}) curve. _{t+Δt} surface. _{t} surface normal, applied on the point _{t+Δt}. For infinitely small Δ_{i}^{i} on the surface _{t+Δt}.

In this section we briefly explain the concept behind the invariant time derivative for scalar and tensor fields defined on moving manifolds, even though these concepts are already given [_{t}, find the point _{t+Δt} and _{t+Δt} and the straight line orthogonal to _{t} (Figure

Because (13) is entirely geometric, it must be free from choice of a reference frame. Therefore, it is invariant. On the other hand, from the geometric construction it follows that

_{t+Δt} surface _{t + Δt}. Then

since ∇_{i}^{i} indicates the directed distance

Generalization of (16) to any arbitrary tensors with mixed space and surface indexes is given by the formula

where Christoffel symbol

Time differentiation of surface and space integrals have a central role in evaluation of the principle of least action. Dependence of time variation of the potential energy on the geometry becomes rigorously clarified from these theorems. For any scalar field ^{i}) defined on a Minkwoskian domain Ω with boundary

The first term in the integral represents the rate of change of the tensor field, while the second term shows changes in the geometry. It therefore properly takes into account the convective and advective terms due to volume motion. We are not going to reproduce proof of these theorems here^{5}^{6}

In this section we provide several theorems, that will be directly used to deduce equations of motions. The first such theorem is the general Gauss theorem about integration, which gives the rule for the reciprocal transfer of space integral to surface integral. For a domain Ω in Minkowski space with the boundary ^{α}, the generalized Gauss theorem reads

The proof is simple if one uses the Voss-Weyl formula to deduce the theorem. For any sufficiently smooth tensor field in Minkowski space, the Voss-Weyl formula [

Using (21) in the right part of (20) and the designation η = −|η_{..}|, we have

where ^{α} = ^{0}^{1}^{2}^{3}. This term is subject to the Gausss theorem in the arithmetic space. Since arithmetic space and Minkowski space, which is pseudo-Euclidean, can be related to Cartesian coordinates, Minkowski space can be identified as an arithmetic one and the Gauss theorem for the arithmetic space can be used. Thus, using unity of the Minkowski space metric tensor determinant one may prove that^{7}

where _{..}|. This proves that the generalized Gauss's theorem holds for pseudo-Riemannian manifolds embedded in Minkowski space.

The next step is to provide short proofs for Weingarten's and Thomas' formulas by using the relation between the surface derivative and the interface velocity.

Weingarten's formula expresses the surface covariant derivative of the surface normal in the product of the shift and mixed curvature tensors. Proof follows from the definition

If we apply the covariant derivative to (22) and take into account that from (6)

Let's contract both sides of (23) with η^{iβ} and take into account the commonly used relationship in tensor calculus

Since the second term of the last equality vanishes, we get (24), also known as Weingarten's formula.

Now we turn to the Thomas formula which allows calculation of the invariant time derivative of the surface normal. Indeed, using the invariant time derivative formula for the surface base vector [

and dotting both sides of (25) with _{i} = 0, we find

Equation (26) is generally referred to as the Thomas formula.

Since we have all mathematical preliminaries in hand we can proceed with derivation of master equations of motion. To derive the equations we apply the calculus of moving surfaces to the motion of compact and closed manifolds in an electromagnetic field. In this step we only discuss free motion of the single closed surface, where “single” surface means boundary of the single material body and “free” means contact with environment is ignored^{8}^{9}

where the electromagnetic tensor _{αβ} is the combination of the electric and magnetic fields in a covariant antisymmetric tensor [_{·} = (−φ/^{·} = (_{0} is the magnetic permeability of the vacuum. The Minkowski space metric tensor signature is set to be space-like (− + + +) throughout the paper. This formulation is fully relativistic though it can be easily simplified for non-relativistic cases. Raising and lowering the indexes is performed by the Minkowski metric η_{αβ}. The relation between the four potentials and the electromagnetic tensor is given by

As far as the boundary of the material body is a moving three manifold, the surface kinetic energy with variable surface mass density ρ and surface velocity

Subtraction of the potential energy (27) from the kinetic energy (29) leads to the system Lagrangian

where

For proper evaluation of the (31) Lagrangian we start from the simplest term first, the potential energy. Since (27) is the space integral by theorem (18) we have

According to (32) determination of variation of potential energy is calculated from the time differential of the space integrand. Following standard algebraic manipulations for classical electrodynamics, we find

where _{α} and ∂_{β}_{α} and at the boundary condition ∂_{α}/∂

To calculate the last integrand (33), we take into account the definition (28) and note that the covariant electromagnetic tensor can be obtained by lowering indexes in contravariant tensor ^{αβ} = −^{βα}, so that

Taking into account (33–35) in (32) we find the variation of the potential energy

Now we turn to the calculation of the kinetic energy variation. To deduce the variation for the kinetic energy let's define the generalization of conservation of mass law first. The variation of the surface mass density must be so that

is the surface mass with ρ surface mass density. Since we discuss compact closed manifolds the boundary conditions

where

Incidentally, an equation for the surface charge conservation can analogously be deduced and it has exactly the same form. The equation (39) was also reported in Grinfeld [

Here we used the fact that at the end of variations the surface reaches the stationary point and therefore, by the Gauss theorem integral for

Using Weingartens formula (24), the metrilinic property of the Minkowski space base vectors ∇_{i}_{α} = 0 and the definition of the surface normal

Taking into account (12) and its covariant and invariant time derivatives in (42), we find

Continuing algebraic manipulations using the formula for the surface derivative of the interface velocity (25), Thomas formula (26), and the definition of the curvature tensor (6) in (43), yield

Dotting (44) on

where the first part is the normal component and the second part is the tangent component of the dot product. Combination of (36, 45) with (31) reveals

To find the final form of the equations of motion we separate the dot product of the space integrand from (46) into normal and tangential components. Let the vector

where

where

Since equation (46) must hold for every

After applying the Gauss theorem to the surface integrals in (50), the surface integrals are converted to a space integral so that one gets

To summarize (39, 50–52), equations of moving manifolds in an electromagnetic field read

Equations (53) are the master equations of motions.

A case that deserves some attention is the homogeneous symmetrical surface. In that case the only nonzero allowed “force” is

Equations (53) are correct for freely moving manifolds of the body in a vacuum. Generalization can be trivially achieved if instead of the electromagnetic tensor _{αβ} one proposes the electromagnetic stress energy tensor ^{αβ}, which is related to the electromagnetic tensor by the relationship

For objects in matter the electromagnetic tensor ^{μν} in (47, 53) is replaced by the electric displacement tensor ^{μν} and by the magnetization-polarization tensor ^{μν} so that

The charge density

To link the above formulated problem with real physical surfaces it is necessary to do some modeling. To begin let's illustrate macromolecules^{10}^{11}

Top

Let's model a bio-macromolecular surface as a Gaussian map contoured at 2 Å to 8 Å resolution. Figure _{i} base vectors are defined in the tangent space of the Gaussian map. _{ij} is the metric tensor of the map. These are illustrations of surfaces as two-manifolds embedded in Euclidian space and are only true for non-relativistic representations, therefore they do not show the shape of three-manifolds in Minkowski space-time. Figures ^{12}

As we already stated in the introduction, hydrophobic and hydrophilic interactions incorporate dispersive interactions throughout the molecules, mainly related to electrostatics and electrodynamics (Van der Waals forces), induced by permanent (water molecules) or induced dipoles (dipole-dipole interactions) and possibly quadrupole-quadrupole interactions (for instance stacking or London forces) plus ionic interactions (Coulomb forces) [

To demonstrate effectiveness of (53) let's discuss free motion of two manifolds embedded in three dimensional Euclidean space for the stationary surface in an electrostatic field. We have the following conditions: _{·} = (−φ/^{·} = (_{x}, ∂_{y}, ∂_{z}). Then from second equation of (53) with the condition (46), we find

Taking into account the definition of electromagnetic tensor and that we consider the electrostatic field, the partial derivative of the electromagnetic tensor in (56) is

By the definition of the electric field ^{β} = −∂^{β}φ and

The equation (58) is generally known as the Poisson-Boltzmann equation in vacuum and was proposed to describe the distribution of the electric potential in the direction of the normal to a charged surface [

Here we demonstrate that the Poisson-Boltzmann equation is a special case and can be obtained from the equations of motion (53) for stationary surfaces in an electrostatic field. To support this statement we have generated electrostatic field lines using the Adaptive Poisson-Boltzmann Solver (APBS) [

Color-coded electrostatic surface where red indicates negatively charged regions of the surface, white neutrally charged and blue positively charged one. Simulated electrostatic field lines are displayed as hairs on the surface of the protein

In this subsection we demonstrate that, the equations of motion simplify to Maxwell equations for stationary interfaces

Adding (59) to (60) and taking into account (47) and (49) one obtains

(61) must hold for any partial time derivative of the four vector potential, therefore

and the Maxwell equations with the source in the vacuum follow.

This is a somewhat unexpected result: any three manifold with stationary interface ^{13}

In this section we simplify the equations of motion using physical arguments and demonstrate that the equation system (53) yields the Euler equation for a dynamic fluid for some simplified cases. Let's propose that a moving fluid has a planar surface _{ij} = 0 with stationary interface

The first equation of (63) is the continuity equation for the surface mass density and is conservation of mass at the flat space; the second one yields that normal component of the dot product

On the other hand

Taking into consideration (64, 65) in (63) and applying Gauss' theorem to the space integral, we find

According to Weingarten's formula (24) ^{α} is invariant vs the surface derivative for flat manifolds and therefore, can be taken into the surface covariant derivative, so that ^{14}_{i} yields

Taking into account that for flat surfaces Christoffel symbols vanish and ∇_{j} = ∂_{j} one immediately recognizes that the last equation (67) becomes the classical Euler equation of fluid dynamics.

As stated above the equations of motion (53) are formulated for freely moving manifolds; i.e., interaction with the environment is ignored and matter is set to be a vacuum. Though it can be trivially generalized for the matter and then simplified, instead of giving Euler equation, will lead to the more complete Navier-Stokes equation and or magnetohydrodynamic equations. For instance, in matter, according to (55), the electromagnetic tensor becomes the sum of the electric displacement and magnetization tensors. Therefore, in (67), instead of a pure pressure gradient we will have an additive term coming from the magnetic field so that (67) will transform to the ideal magneto hydrodynamic equation.

Analogously, if interaction with an environment is taken into account, then instead of a single surface we have two surfaces at the surface/environment interface and the Lagrangian (30) is split into two kinetic energy terms, one for surface and another one for the environmental interface. All these will occur as additive terms in the third equation of (53) so that equation (67) will transform to the Navier-Stokes equation.

Let's answer the question: what is the shape of micelles formed from lipid molecules when they are in thermodynamic equilibrium with solvent. Lipids have hydrophilic heads and hydrophobic tails, so that in solutions they tend to form a surface with heads on one side and tails on the other. Since the tails disperse the water molecules, the surface made is closed and has some given volume. Such structures are called micelles [

and ^{2} < < ^{2} and the potential energy becomes

Using the first law of thermodynamics, (69) can be modeled as a volume integral from the surface pressure [

On the other hand, taking into account the conditions (68, 69), the total potential energy of the surface can be modeled as

Taking into account (71), the system Lagrangian becomes the same as it is in (1) and its variation leads to the equation

(72) was first reported in Grinfeld [

When the homogeneous surface, such as a micelle, is in equilibrium with the environment then the solution of the (73)^{15}

From equation (74) the generalized Young-Laplace relation which connects the surface pressure to the curvature and the surface tension immediately follows. (74) dictates that the homogeneous surfaces in equilibrium with environment adopt a shape with constant mean curvatures (CMC), which explains the well anticipated lamellar, cylindrical and spherical shapes of micelles. This is another unexpected and surprisingly simple solution to the equations of motion (53).

Equations of motion (53) further simplify for two dimensional surfaces. In non-relativistic framework the space is three dimensional Euclidean (α = 1, 2, 3 and ^{0} = 0 limit), the surface is two-dimensional Riemannian (

where

Taking (76, 77) along with that

Alternative way of deducing (78–80) without using (53) is given in Svintradze [

We have proposed equations of moving surfaces in an electromagnetic field and demonstrated that the equations simplify to: (1) Maxwell equations for massless three manifolds with stationary interfaces; (2) Euler equations for dynamic fluid for planar two manifolds with stationary interface embedded in Euclidean space, which can be generalized to Navier-Stokes equations and to magneto-hydrodynamic equations; (3) Poisson-Boltzmann equation for stationary surfaces in electrostatic field.

We have applied the equation to analyze the motion of hydrophobic-hydrophilic surfaces and explained “equilibrium” shapes of micelles. Analyses were in good qualitative as well as quantitative agreement with known experimental results for micelles [

Also we have shown that hydrophobic-hydrophilic effects are just another expression of well known electromagnetic interactions. In particular, equations of motion for moving surfaces in hydrophobic and hydrophilic interactions, together with the analytic solution, provide an explanation for the nature of the hydrophobic-hydrophilic effect. Hydrophobic and hydrophilic interactions are dispersive interactions throughout the molecules and conform to electromagnetic interaction dependence on surface morphology of the material bodies.

The author confirms being the sole contributor of this work and approved it for publication.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We were partially supported by personal savings accumulated during the visits to Department of Mechanical Engineering, Department of Chemical Engineering, OCMB Philips Institute and Institute for Structural Biology and Drug Discovery of Virginia Commonwealth University in 2007–2012 years. We thank Dr. Alexander Y. Grosberg from New York University for comments on an early draft of the paper and Dr. H. Tonie Wright from Virginia Commonwealth University for editing the English. Limited access to Virginia Commonwealth University's library in 2012–2013 years is also gratefully acknowledged.

^{1}If one proposes to treat particles as classical objects, then the framework fits in Newtonian mechanics. The application of Newton laws and it's stochastic generalizations in simulations is commonly known as molecular dynamics simulations.

^{2}e.g., bio-membranes, macromolecular surfaces, lipid bilayers, micelles, etc.

^{3}The formalism should be relativistic not only because relativistic calculations are more general than classical calculations, or because a proper electrodynamics description requires a relativistic frame work, but also because molecular surface dynamics can be very fast [

^{4}Here the speed of light is set to be unit

^{5}Proofs for the time derivative of integrals can be found in tensor calculus text books. See for instance [

^{6}same explanation, with more details, is given in Svintradze [

^{7}Details about the proof for Euclidean space can be found in tensor calculus text book [

^{8}The environment is set to be vacuum.

^{9}In the case of taking into account interaction with the environment we no longer have single surface. Instead there are double surfaces where one is the boundary of the material body and another one is the surface of the environment at the boundary/environment interface. Having two surfaces raises the terms related to surface-surface interactions and may enter into final equations as a viscoelastic effect incorporated in coefficient of viscosity.

^{10}Or surface made from groups of molecules, for instance lipids.

^{11}Especially for relatively slowly moving surfaces, for instance: cell surface, which is bio-membrane; vesicles; micelles etc.

^{12}The model protein is the peroxide sensitive gene regulator with Protein Data Bank (PDB) ID 3HO7 [

^{13}Here surface mass density is the same as the mass density of the three manifold, because the three manifold is the surface in 4D space.

^{14}More information about how the term

^{15}A shorter alternative way to deduce (73) and it's solution in equlibrium conditions is given in Svintradze [