Abstract
We state three different familiar definitions of electromagnetic field tensor in curvilinear coordinates and curved spacetimes, and we show the relation between these definitions. As two explicit examples, we give expressions for the electromagnetic field tensor and electromagnetic field equations for a general comoving observer in the Schwarzschild and Kerr backgrounds in terms of noncoordinate components of fields. We show that ignoring differentiation between coordinate and noncoordinate bases could lead to inconsistent and confusing results. Finally, because of its practical aspects, we obtain the electromagnetic field tensor and field equations in terms of noncoordinate components in the spacetime of a Galilean rotating observer.
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1 Introduction
In the presence of gravitational fields, as well as in non-inertial reference frames, one should find and employ the electromagnetic field tensor, the Maxwell equations, and constitutive equations in a curved spacetime or a curved 3-space with the corresponding line element expressed in a curvilinear coordinate system adapted to its symmetries [1,2,3]. An example that clearly demonstrates such a necessity is to obtain the electromagnetic field tensor and the Maxwell equations in a rotating frame. [4,5,6,7,8,9,10,11]. To encounter this problem, one needs to specify what is meant by space and spatial metric in a curved spacetime or a spatially non-Euclidean flat spacetime. When dealing with stationary spacetimes such as the spacetime of a rotating source, the decomposition of the line element into spatial and temporal sections is not a trivial task, and one has to choose a decomposition formalism. Here, we show how the electromagnetic field equations defined by this decomposition are consistent with those based on coordinate and non-coordinate bases in a 4-dimensional treatment
In this paper, we will express our final results in terms of the noncoordinate components of the fields. We remind that noncoordinate (physical) components of tensors are the component values that can be measured by experiment with standard physical instruments. But coordinate components are used in tensor analysis in a coordinate system and vary with the coordinate chosen. For the particular case where our metric is Minkowski metric in Cartesian coordinates, these components are the same, i.e. coordinate components equal the physical components we measure. But in other cases, we need to know how to find physical components from coordinate components. Hence, calculating physical components from coordinate components is essential for comparing experiments with theory. Of course, we know invariant quantities are the same for any coordinate system. They are not components of tensors, and they are equivalent to physically measured values [12,13,14].
The familiar covariant forms of electromagnetic field equations in curved spacetime are given by
These equations are usable in a coordinate basis (“Appendix A”). To have the explicit form of electromagnetic field equations, we should write the electromagnetic field tensor in a coordinate system and then insert the result in(1). But we usually encounter electromagnetic field tensor in flat spacetime in Cartesian coordinate, which is given by
and the explicit forms of electromagnetic tensor and field equations in curved spacetimes or even in curvilinear coordinates in flat spacetimes are less familiar. For example, if we want to obtain electromagnetic field equations in cylindrical coordinates in flat spacetime or Schwarzschild spacetime, a more general definition than (2) is needed. In these coordinates and spacetimes, unlike Cartesian coordinate in flat spacetime, the coordinate and non-coordinate components of electromagnetic field tensor are not the same. We need to have a general definition in the coordinate basis to use (1). But there are different definitions of electromagnetic field tensor in curved spacetime in the literature. The relationship between these definitions is usually unclear, and each leads to different field equations. Moreover, the distinction between coordinate and non-coordinate components of the field is essential, and ignoring it leads to some confusion. Sometimes we encounter different field tensor and field equations for an observer. For example, in the case of the Galilean rotating observer, which is more interested, electromagnetic field tensor and equations are given in different forms in [15, 16, 18], without mentioning the relation between them or explaining differences. Here, we will show that these definitions can have the same explicit form for general comoving observers if we state them in terms of non-coordinate basis components of fields. In addition, in some cases, there are errors in the final form of the equations relating to the Galilean rotating observer. We will present the explicit form of these equations in Sect. 4.
In summary, this article is written for the following purposes:
1. In this paper, three different definitions of field tensors are expressed together, and the relationship between them is explained. Usually, only one of these definitions is used in the literature, and the relationship between them is unclear.
2. The relationship between the noncoordinate (physical) and coordinate components of electromagnetic fields is considered, and all final results are expressed in terms of physical components. In some papers, the relationship between coordinate and noncoordinate components of electromagnetic field is unclear. Sometimes the noncoordinate form of electromagnetic field tensor is used in a coordinate analysis, which is ambiguous. Some articles in electrical engineering use Landau formalism for electromagnetic field tensor and field equations. Although the determination of noncoordinate components is vital in this area, the relationship between field components in Landau formalism and noncoordinate components is not specified in these articles.
3. In this paper, the electromagnetic field equations in Schwarzschild and Kerr spacetimes, as well as in the rotating observer spacetime, are explicitly expressed in terms of noncoordinate components of electromagnetic field. In previous papers, these results are usually expressed in terms of coordinate components.
4. It is a straightforward translation among different conventions. It would be helpful for a student trying to use sources with different conventions.
The plan of the paper is as follows. In the next section, we review three definitions of electromagnetic field tensor in curvilinear coordinates and curved spacetimes in the literature. In Sect. 3, we show the relation between these definitions. As explicit examples, we give expressions for the electromagnetic field tensor and Maxwell equations for the general comoving observer in the Schwarzschild and Kerr backgrounds in terms of noncoordinate components of fields. We show that these results are the same as those obtained using coordinate tetrad or vielbein. We also explain the relationship between the components of the field in Landau formalism and the noncoordinate components of the field. In Sect. 4, we obtain the electromagnetic field tensor and field equations in terms of non-coordinate components in the spacetime of a Galilean rotating observer. In the final section, we will present a summary of the discussion.
2 Three definitions of electromagnetic field tensor
We can find three definitions for electromagnetic field tensor in the literature. Here we outline them:
Definition 1
Using the relation between the tensor components in the coordinate and noncoordinate bases via the properties of coordinate tetrad [15, 19];
First, the noncoordinate field tensor is defined as
then the coordinate field tensor is obtained using the relation between the components of tensor in the coordinate and noncoordinate bases
in which \(h_{\mu }^{\,(a)}\) (with mixed indices) are called coordinate tetrad or vielbein and their inverse denoted by \(h_{(a)}^{\,\mu }\). These tetrads can be obtained from the metric (see “Appendix A”). The advantage of this definition is that its final form is expressed in terms of non-coordinate (physical [20]) components of fields.
Definition 2
The electromagnetic field tensor in 1+3 formalism [21, 22];
From theories of relativity, both special and general, we know that physical phenomena happen in an arena called spacetime, a new entity that was introduced by fusing the concepts of space and time. But when it comes to the measurements of these phenomena, it is their spatial and temporal characters which are being analyzed. So a decomposition of spacetime into space and time is a necessity if we are going to measure 3-dimensional quantities such as spatial distances and velocities. There are two well known formalisms called \(3+1\) or foliation decomposition and \(1+3\) or threading decomposition. Here, we are interested in the second approach to obtain electromagnetic field tensor and field equations. Ellis [22] formulate the electromagnetic field tensor in curved spacetimes based on the 4-velocity of a timelike observer. It is also called field tensor relative to a fundamental observer [23] and is given by
in which \(u^\alpha\) is a timelike tangent vector not necessarily orthogonal to the surface of constant time, and the 1+3 splitting is made relative to it. The projection tensor h, projecting into the three-dimensional tangent plane orthogonal to \(u^\alpha\), is given by
and E and B are defined to be orthogonal to \(u^\alpha\).
Definition 3
Landau definition;
Landau and Lifshitz [24] in their very physical approach to spacetime decomposition, which among other things enables one to express Einstein field equations in the so-called quasi-Maxwell form in the context of the so-called gravitoelectromagnetism [25], introduce a three-dimensional form of Maxwell equations and an alternative form for the electromagnetic field tensor in stationary spacetimes. Their definitions, which are seemingly ad hoc, are as follows,
in which \(E_{L}\) and \(B_L\) are vectors in three-dimensional space, and the index L refers to Landau definition. Although we sometimes refer to this definition as 1+3 definition, there are some differences between this definition and definition 2 that we will mention later. The electromagnetic field tensor in Landau definition can be written as
Substituting this definition into (1), we will have usual three-dimensional form of Maxwell equations
in which div and curl are three-dimensional operators and \(\gamma\) is space metric were defined in “Appendix B”. For example \(div\mathbf{D }=0\) and (81) give
and substituting \(D^i=-\sqrt{g_{00}}F^{0i}\) and (82) in (10), one ends up with the same equation obtained from(1) and (8) for \(\nu =0\). It seems that obtaining this three-dimensional form provided the motivation for Landau to define such a different definition of electromagnetic tensor.
These definitions have been used and have led to different explicit forms for electromagnetic tensor and electromagnetic field equations. For example, in [15] (using the first definition) and [16] (using third definition), we find two different forms for the electromagnetic tensor in the spacetime correspond to a rotating observer who uses Galilean rotational transformation. But the relationship between these two forms is unclear. We will show the relationship between them as well as the final corrected forms in Sect. 4.
As mentioned in [26], the covariant form of Maxwell’s equations does not uniquely determine the three-vector explicit forms of electromagnetic tensor and field equations in curved spacetimes or accelerated frames. So there are few applications of these three-vector formulations. However, some applications of this formalism have been discussed by some authors. In [27], using definition 1, the electrostatic field of a point charge at rest in the Schwarzschild metric is given in algebraic form, and the magnetostatic field of a current loop surrounding a black hole is given in the integral form. In [28], a procedure for calculating the Debye potential corresponding to a stationary axisymmetric distribution of charges and currents in the Kerr metric is given. In [29], using definition 2, the authors derive the equations of motion of charged particles around a magnetized rotating star. In [30], using definition 2, Rezzolla et al. present analytic solutions of Maxwell equations in the internal and external background spacetime of a slowly rotating magnetized neutron star.
The most obvious application of these definitions is in electrodynamics of accelerated media. Electrical engineers are often confronted with configurations involving rotating bodies, and the solution of electromagnetic problems involving rotating bodies is of fundamental importance for those concerned with rotating machinery [17]. For example in [16], using definition 3, Bladel illustrates the solution of Maxwell’s equations by evaluating the fields scattered by a rotating cylinder. In [15], using definition 1, a plane wave scattered by a rotating sphere is solved in the laboratory frame.
3 The relation between different definitions of electromagnetic field tensor
First, to distinguish physical quantities from coordinate quantities, it is necessary to express the final form of the field tensor and field equations in terms of noncoordinate components of fields. So the noncoordinate definition (3) is our basic definition, and all other definitions are expressed in terms of its components. In the first definition, the final form is expressed in terms of noncoordinate components of fields. But in the second and third definitions, the components of fields are not noncoordinate components. So to be able to compare the final forms, we have to express all the components of the field in terms of non-coordinate components.
3.1 The relation between definitions 1 and 2
Definition 2 depends on the timelike tangent vector \(u^\mu\). In general, if we choose the timelike tangent vector \(u_\mu\) in the second definition as the timelike basis vector \(h_{\mu }^{\,(0)}\) of tetrad in the first definition (Or equivalently \(u^\mu\) as inverse timelike basis vector \(h_{(0)}^{\,\mu }\)), then the electromagnetic field tensor obtained from the two methods will have the same explicit form. In other words, in the first definition we set \(h_{(0)}^{\,\mu }=u^\mu\) and then we define the three spacelike unit vectors upon it. We will then see that the results obtained from the two definitions are exactly the same. This is because in Ellis formalism, the velocity \(u^\mu\) on which the decomposition is performed acts as a timelike tetrad vector. If we use the general comoving observer velocity in definition 2, the tetrad based on which it is made is the same as the coordinate tetrad (vielbein) obtained directly from the metric according to the relations (46). Therefore, if we use the general comoving observer velocity in definition 2 and the coordinate tetrad in definition 1, the results of the two approaches will be the same. This similarity arises from the properties of the general comoving observer. It can be said that the metric of a curved spactime (such as Schwarzschild spacetime) is written from the point of view of the general comoving observer. Therefore, the results obtained for field tensor from the point of view of a general comoving observer based on Ellis’s formalism are the same as those obtained by the first definition with the help of coordinate tetrad.
As important examples, we obtain the electromagnetic field tensor for the general comoving observer in the Schwarzschild and Kerr backgrounds. In [30] another example for zero angular momentum observers (ZAMO) in a slow rotation metric is given by some approximations.
The velocity of the general comoving observer is given by
and the Schwarzschild metric in spherical coordinates \((t,r,\theta ,\phi )\) is given by
Using equations (51) and choosing the general comoving observer velocity (11) as \(h_{(0)}^{\,\mu }\), the coordinate tetrad for this metric is obtained as follows (As we mentioned before, these tetrads can be obtained directly from the metric (12) according to the relations (46))
The electromagnetic tensor in a general noncoordinate basis is given by equation (3). To transform it into spherical coordinates, we choose
so that
then using (4a) and (4b), the electromagnetic field tensor in a Schwarzschild background is given by
On the other hand for general comoving observe by (5), (11) and (12) we have
in which \(g_{00}\gamma =g\) and
But the field components in (5) are not noncoordinate components, and we can write them in terms of noncoordinate components as below
Then we can see that the field tensor is the same as (17). As an example
Now we obtain the electromagnetic field equations. By working on a coordinate basis, that is, using equations (1) and (16), we have the following vacuum electromagnetic field equations in the Schwarzschild metric
for \(\nu =0,1,2,3\), respectively. For the source free equations by using (1) and (17) we have
It can easily be seen that by setting \(m=0\), all the equations reduce to those expected in flat spacetime in spherical coordinates. Ignoring the difference between coordinate and noncoordinate bases has led to confusing expressions in the literature. As an example, in [27], without mentioning that equations (1) are expressed in a coordinate basis, the definition of the field tensor in a non-coordinate basis has been used to obtain the electromagnetic field equations in the Schwarzschild background. Therefore, the results presented for the field equations there (relations 7 and 11 in [27]) are not the same as Equations (22) and (23) here.
Now we check our idea for another example. The weak field, slow rotation limit of the Kerr metric representing the spacetime of a spherically symmetric rotating mass m with angular momentum per unit mass a in spherical (Schwarzschild-type) coordinates is given by,
Again we choose the general comoving observer velocity (11) as \(h_{(0)}^{\,\mu }\) and so we have
and the electromagnetic field tensor in this background, using \(F^{\mu \nu }={{h^{-1}}^T}F^{(ab)}{h^{-1}}\), is given by
and
On the other hand, for general comoving observe in this stationary spacetime, we have
by substituting this in (5) and using (24), we repeat what was done for the Schwarzschild metric, and we can again see that definition 2 leads to the same result as definition 1 for the electromagnetic field tensor in Kerr background. Here we obtain two components of field tensor by definition 2:
Electromagnetic field equations in Kerr background can also be obtained using (1), (26) and (27). For example setting \(\nu =0\) in the left hand side of equation (1) gives
in the vacuum. And setting \(\mu ,\nu ,\lambda =1,2,3\) in the right-hand side of equation (1), we have
Again it can be easily seen that if we set \(m=0\), all the equations reduce to those expected in flat spacetime in spherical coordinates.
3.2 The relation between definitions 2 and 3
Landau and Lifshitz introduce the definitions of the electromagnetic field in a curved background in terms of a field tensor without referring to a specified frame. In this subsection, we show how their definitions are related to those given by Ellis [22] in terms of the 4-velocity of a timelike observer in its comoving frame. Definitions 2 and 3 are both based on 1+3 decomposition. In [22] an approach to this decomposition is given that focuses on applications of the 1 + 3 decomposition in relativistic cosmology. We refer to it as Ellis formalism. Another approach that we refer to as Landau formalism has been discussed in “Appendix B”. It is easy to show that if we use Ellis formalism for general comoving observers, then we will have Landau formalism. If we substitute the velocity of the general comoving observer (11) into projecting tensor (6), we obtain the spatial tensor introduced by Landau
So we can say that Landau formalism is a special case of a general definition in Ellis formalism. But the interesting point is that by substituting the general comoving observer velocity in definition(5), the Landau definition for electromagnetic field tensor (8) is not obtained. In other words, although inserting the velocity of the general comoving observer into (6) gives the Landau projecting tensor, we can not obtain Landau definition for electromagnetic tensor(8) from Ellis definition (5) in the same way. So Landau definition is a particular case and we should classify it as a different definition. Using this definition, Landau and Lifshitz introduce a three-dimensional form of Maxwell equations and an alternative form for the electromagnetic field tensor in stationary spacetimes [24]. It seems that the purpose of Landau definition is to introduce electromagnetic field equations in a way that is similar to the usual three-dimensional form of Maxwell equations, and so the start point of Landau is (9).
Now the question arises as to what is the relation between field components in Landau definition and the corresponding components in a noncoordinate basis. For example, what is the relation between \({E_L}_i\) and \(E_r\)? To answer this question, we take equation (8) as the electromagnetic field tensor in the coordinate basis. As an example, in the case of Schwarzschild metric, we set equations (8) and (17) equal to each other and find the following relations
along with using equations (7) to get,
in which for the Schwarzschild metric we have used:
Now setting from (33) and (34) into (9) and using (81), one can obtain the electromagnetic field equations in Schwarzschild metric, which are the same as the equations we obtained before. For example, by using these equations, one gets
which is the same as (22d), i.e. the two formalisms are equivalent, leading to the same field equations. This is a consistency check for the applicability of \(1+3\) formulation and its definition of 3-space as the arena where the 3-objects \(\mathbf{E}\) and \(\mathbf{B}\) play their physical role.
4 Electromagnetic field tensor and field equations for Galilean rotating observer
The electrodynamics of rotating frames has been studied because of its practical aspects, especially in electrical engineering [1, 10, 15,16,17,18, 31, 32]. In these works, the Galilean rotational transformation (GRT) and the three-dimensional form of the field equations are usually used. In [33, 34] Nouri-Zonoz et al. have shown the properties, limitations and problems of GRT. It has been demonstrated that GRT can only be used for the centric rotating observers. However, here we obtain electromagnetic field tensor and field equations in the Galilean rotating observer’s frame, as a particular case of a stationary spacetime. We use definition 1 introduced in 2. It should be noted that these equations are obtained in [16, 17] based on definition 3, but like most works in this field, the relationship between their field components and the non-coordinate field components is not clear.
Due to the axial symmetry of the problem, one naturally employs cylindrical coordinates, assigning events in the non-rotating frame with coordinates\((t^\prime , \rho ^\prime , \phi ^\prime , z^\prime )\) and in the one rotating with constant angular velocity \(\Omega\) around their common axis with \((t, \rho , \phi , z)\). In the simplest rotational transformation, these are related through the Galilean rotational transformation [24],
or in its differential form
So that the line element from the spatially Euclidean flat spacetime in cylindrical coordinates, i.e.
transforms into
which is the spatially non-Euclidean flat spacetime in a rotating frame. The corresponding coordinate tetrad is given by
so that the electromagnetic field tensor in this background, using \(F^{\mu \nu }={{h^{-1}}^T}F^{(ab)}{h^{-1}}\), is given by
and
in the coordinate basis, by using left-hand side equation of (1) and (42), we have the following vacuum electromagnetic field equations
for \(\nu =0,1,2,3\), respectively. For the source free equations by using right-hand side equation of (1) and (43) we have
It can easily be seen that by setting \(m=0\) all the equations reduce to those expected in flat spacetime in cylindrical coordinates.
5 Conclusions
Clarifying the roles of coordinate (holonomic) and noncoordinate (anholonomic) bases in the presentation of the electromagnetic field tensor and Maxwell equations, we stated that three different often-used definitions of electromagnetic field tensor in curvilinear coordinates and curved spacetimes can have the same explicit final forms. If we choose tetrad in definition 1 ( the definition uses tetrad formalism) so that its timelike basis vector \(h_{\mu }^{\,(0)}\) be equal to the timelike tangent vector \(u_\mu\) in definition 2 (via Ellis formalism), and then state them in terms of noncoordinate basis components of fields, the final form of the equations will be the same. Moreover, the results obtained for field tensor from the point of view of general comoving observer based on Ellis’s formalism are the same as those obtained by the first definition with the help of coordinate tetrad. Definition 3 (via Landau formalism) is a particular definition that its components can be obtained in terms of noncoordinate components by comparison with other definitions. We checked our idea for the general comoving observer in two backgrounds: Schwarzschild and weak Kerr spacetimes. Finally, we obtained the electromagnetic field tensor and field equations in terms of noncoordinate components in the spacetime of a Galilean rotating observer.
It should be noted that from the physical point of view, the components of the fields that are measured in an experiment are those components introduced in the noncoordinate basis. For example in a cylindrically symmetric distribution of charges, the azimuthal electromagnetic fields in the noncoordinate basis, e.g. \(E_\phi \equiv E^{(2)} = h^{(2)}_2 E^2\) and \(B_\phi \equiv B^{(2)} = h^{(2)}_2 B^2\), are the so-called physical or ordinary components [20, 35]. Some of the results of this article could be found in different books and papers, and one of our goals is to gather them in a systematic way in one place so that they could be easily accessible to those interested. Also, it is a straightforward translation among different conventions.
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Appendices
Appendix A: Coordinate and noncoordinate bases
In this appendix, we give a brief account of coordinate and noncoordinate bases and the relation between the components of a tensor in the two bases.
1.1 A.1: Coordinate or holonomic basis
In a coordinate basis (also called holonomic basis) [35,36,37] the basis vectors and their duals are denoted by \({\hat{e}}_\mu\) and \({\hat{e}}^{\mu }\), respectively. They are unit vectors and satisfy the following inner product relations,
where \(g_{\mu \nu }\) and \(g^{\mu \nu }\) are the metric components. In this basis, the vector V and the second rank tensor T, in terms of their covariant and contravariant components, are given by
raising and lowering of the indices are done with the help of the general metric components \(g^{\mu \nu }\) and \(g_{\mu \nu }\), respectively.
1.2 A.2: Non-coordinate or anholonomic basis
In a noncoordinate basis (also called orthonormal or anholonomic basis) [35,36,37], the basis vectors and their duals, denoted by \({\hat{\epsilon }}_{(a)}\) and \({\hat{\epsilon }}^{(a)}\), respectively, are given by tangent and cotangent vectors of unit length satisfying the following inner product relations
where \(\eta _{(ab)}\) is the flat spacetime metric in Cartesian coordinates. For example, in such a basis, a vector V and a second rank tensor T are written as as follows
Raising and lowering of the indices are done with the Minkowski metric elements \(\eta ^{(ab)}\) and \(\eta _{(ab)}\), respectively. A very important point that should be emphasized here is that the components of T and V in a noncoordinate basis ( i.e \(V_{(a)}\), \(V^{(a)}\), \(T^{(ab)}\) and \(T_{(ab)}\) ) all are scalars under coordinate transformation. Later we will obtain the components of the same tensors T and V in other bases in terms of these components.
One can express coordinate basis vectors in terms of the noncoordinate basis vectors as follows
where \(h_{\mu }^{\,(a)}\) (with mixed indices) are called tetrads and with their inverses denoted by \(h_{(a)}^{\,\mu }\) satisfy the following relations
1.3 A.3: Covariant and contravariant components of tensors in the two bases
According to the definitions given in the previous subsections, we have the following relations between contravariant and covariant components in the two bases
The inverse relations are
The divergence of a vector in the coordinate basis with generalized coordinates \(q^\mu\) can be written as follows
where (46c) was used and \({\Gamma ^k}_{ij}\) is also defined by
Using the relation \(\frac{\partial g}{\partial {q^k}}= 2g\Gamma ^i_{ik}\) in (54) we end up with the well known relation for the divergence of a vector
In the same way, for the gradient operator we have
And for the covariant derivative of a second rank tensor \(\mathbf{T}\), we have
The expression in parenthesis gives the components of the covariant derivative of \(\mathbf{T}\). Finally for the divergence of a tensor \(\mathbf{T}\) denoted by \(\nabla \cdot {\mathbf{T }}\) we have:
If \(T^{\mu \nu }=-T^{\nu \mu }\) then
as expected. In other words, the familiar relations that are employed for covariant derivative or divergence of a tensor are obtained in a coordinate basis.
Example: Cylindrical coordinate and noncoordinate bases in flat 3-space
As an example of the above discussions in a curvilinear coordinate, we introduce cylindrical coordinate and noncoordinate bases in flat three-dimensional Euclidean space. Using \(\rho\), \(\phi\) and z as the coordinates, the line element of such a space is given by
Then by using equation (51a) we obtain the tetrad components
and their inverses
Now the holonomic basis vectors are chosen to be
and a vector V is represented by
On the other hand, the noncoordinate basis vectors are given by,
and the contravariant vector components are
Now using the above relations and employing (53a), (63) and (67) we have,
so that according to (56) the divergence of a vector is given by
Appendix B: 1+3 (threading) formulation of spacetime decomposition (Landau formalism)
To define spatial metric and spatial distances in a given spacetime (metric), we choose the \(1+3\) (threading) formulation of spacetime decomposition. Unlike the \(3+1\) (or foliation) formulation of spacetime decomposition in which spacetime is foliated into constant-time hypersurfaces, in the \(1+3\) formulation, it is decomposed into threads tracking history of each spatial point. This formulation of spacetime decomposition starts from the following general form for the spacetime metric [24]
in which \(d\tau _{syn}=\sqrt{g_{00}}(dx^0 - {{A_g}_i} dx^i)\) is the synchronized proper time, \({A_g}_{i}=-\frac{g_{0i}}{g_{00}}\) is the so-called gravitomagnetic potential and
is the spatial line element (also called the radar distance element) of the 3-space in terms of its three-dimensional spatial metric \(\gamma _{ij}\). It is the integral of this line element between two spatial points which gives the spatial distance between two events,
The 3-velocity is defined in terms of the synchronized proper time as follows
where now using (70) and (73) the spacetime line element could be written as follows
Now the components of the 4-velocity \(u^\alpha = \frac{dx^\alpha }{ds}\), in terms of the components of the 3-velocity are given by
where in the comoving frame, \(v^i =0\), it reduces to \(u^\alpha = (\frac{1}{\sqrt{g_{00}}}, 0, 0, 0)\) as expected. It is this same formulation of spacetime decomposition which allows one to use the analogy with electromagnetism and define the so called gravitoelectric and gravitomagnetic fields as follows;
In terms of the above fields and in the context of the so-called Gravitoelectromagnetism, vacuum Einstein field equations could be rewritten in the following quasi-Maxwell form [25]
where \({^{(3)}}R^{ij}\) is the 3-dimensional Ricci tensor of the 3-space constructed from the three-dimensional metric \(\gamma _{\alpha \beta }\) in the same way that the usual 4-dimensional Ricci tensor \(R^{\mu \nu }\) is made out of \(g_{\mu \nu }\).
It should also be noted that in the above equations, all the differential operations are defined in the 3-space with metric \(\gamma _{ij}\) [24, 25], in particular, divergence and curl are defined as follows:
in which \(\gamma =det\,\gamma _{ij}\) and one can show that
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Ramezani-Aval, H. The relation between different definitions of electromagnetic field tensor and Maxwell’s equations in stationary spacetimes. Indian J Phys 96, 4347–4359 (2022). https://doi.org/10.1007/s12648-022-02330-2
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DOI: https://doi.org/10.1007/s12648-022-02330-2