Ümit V C ¸atalyürek
email: catalyurek.1@osu.edu

Bora Uçar

Cevdet Aykanat
email: aykanat@cs.bilkent.edu.tr

Hypergraph Partitioning

Hypergraphs are generalization of graphs where each edge (hyperedge) can connect more than two vertices. In simple terms, the hypergraph partitioning problem can be defined as the task of dividing the vertices of hypergraph into two or more roughly equal sized parts such that a cost function on the hyperedges connecting vertices in different parts is minimized.

DISCUSSION

Introduction

During the last decade, hypergraph-based models gained wide acceptance in the parallel computing community for modeling various problems. By providing natural way to represent multi-way interactions and unsymmetric dependencies, hypergraph can be used to elegantly model complex computational structures in parallel computing. Here, some concrete applications will be presented to show how hypergraph models can be used to cast a suitable scientific problem as an hypergraph partitioning problem. Some insights and general guidelines for using hypergraph partitioning methods in some general classes of problems are also given.

Formal Definition of Hypergraph Partitioning

A hypergraph H = (V, N ) is defined as a set of vertices (cells) V and a set of nets (hyperedges) N among those vertices. Every net n ∈ N is a subset of vertices, i.e., n ⊆ V. The vertices in a net n are called its pins. The size of a net is equal to the number of its pins. The degree of a vertex is equal to the number of nets it is connected to. Graph is a special instance of hypergraph such that each net has exactly two pins. Vertices can be associated with weights, denoted with w[•], and nets can be associated with costs, denoted with c[•]. Π = {V 1 , V 2 , . . . , V K } is a K-way partition of H if the following conditions hold:

• each part V k is a nonempty subset of V, i.e., V k ⊆ V and V k = ∅ for 1 ≤ k ≤ K,

• parts are pairwise disjoint, i.e., V k ∩ V = ∅ for all 1 ≤ k < ≤ K

• union of K parts is equal to V, i.e., K k=1 V k = V.

In a partition Π of H, a net that has at least one pin (vertex) in a part is said to connect that part. Connectivity λ n of a net n denotes the number of parts connected by n. A net n is said to be cut (external) if it connects more than one part (i.e., λ n > 1), and uncut (internal) otherwise (i.e., λ n = 1). A partition is said to be balanced if each part V k satisfies the balance criterion:

W k ≤ W avg (1 + ε), for k = 1, 2, . . . , K. (1)
In [START_REF] Ababei | Multi-objective circuit partitioning for cutsize and path-based delay minimization[END_REF], weight W k of a part V k is defined as the sum of the weights of the vertices in that part (i.e., W k = v∈V k w[v]), W avg denotes the weight of each part under the perfect load balance condition (i.e., W avg = ( v∈V w[v])/K), and ε represents the predetermined maximum imbalance ratio allowed. The set of external nets of a partition Π is denoted as N E . There are various [START_REF] Lengauer | Combinatorial Algorithms for Integrated Circuit Layout[END_REF] cutsize definitions for representing the cost χ(Π) of a partition Π. Two relevant definitions are:

χ(Π) = n∈N E c[n] (2)
χ(Π) = n∈N E c[n](λ n -1). (3)
In [START_REF] Aykanat | Multi-level direct k-way hypergraph partitioning with multiple constraints and fixed vertices[END_REF], the cutsize is equal to the sum of the costs of the cut nets. In [START_REF] Aykanat | Permuting sparse rectangular matrices into block-diagonal form[END_REF], each cut net n contributes c[n](λ n -1) to the cutsize. The cutsize metrics given in [START_REF] Aykanat | Multi-level direct k-way hypergraph partitioning with multiple constraints and fixed vertices[END_REF] and (3) will be referred to here as cut-net and connectivity metrics, respectively. The hypergraph partitioning problem can be defined as the task of dividing a hypergraph into two or more parts such that the cutsize is minimized, while a given balance criterion (1) among part weights is maintained. A recent variant of the above problem is the multi-constraint hypergraph partitioning [START_REF] ¸atalyürek | Hypergraph Models for Sparse Matrix Partitioning and Reordering[END_REF][START_REF] Karypis | Multilevel algorithms for multi-constraint hypergraph partitioning[END_REF] in which each vertex has a vector of weights associated with it. The partitioning objective is the same as above, and the partitioning constraint is to satisfy a balancing constraint associated with each weight. Here, w[v, i] denotes the C weights of a vertex v for i = 1, . . . , C. Hence, the balance criterion (1) can be rewritten as

W k,i ≤ W avg,i (1 + ε) for k = 1, . . . , K and i = 1, . . . , C . (4)
where the ith weight W k,i of a part V k is defined as the sum of the ith weights of the vertices in that part (i.e., W k,i = v∈V k w[v, i]), and W avg,i is the average part weight for the ith weight (i.e., W avg,i = ( v∈V w[v, i])/K), and ε again represents allowed imbalance ratio. Another variant is the hypergraph partitioning with fixed vertices, in which some of the vertices are fixed in some parts before partitioning. In other words, in this problem, a fixed-part function is provided as an input to the problem. A vertex is said to be free if it is allowed to be in any part in the final partition, and it is said to be fixed in part k if it is required to be in V k in the final partition Π.

Yet another variant is multi-objective hypergraph partitioning in which there are several objectives to be minimized [START_REF] Ababei | Multi-objective circuit partitioning for cutsize and path-based delay minimization[END_REF][START_REF] Selvakkumaran | Multi-objective hypergraph partitioning algorithms for cut and maximum subdomain degree minimization[END_REF]. Specifically, a given net contributes different costs to different objectives.

Sparse Matrix Partitioning

One of the most elaborated applications of hypergraph partitioning (HP) method in the parallel scientific computing domain is parallelization of sparse matrix-vector multiply (SpMxV) operation. Repeated matrix-vector and matrix-transpose-vector multiplies that involve the same large, sparse matrix are the kernel operations in various iterative algorithms involving sparse linear systems. Such iterative algorithms include solvers for linear systems, eigenvalues, and linear programs. The pervasive use of such solvers motivates the development of HP models and methods for efficient parallelization of SpMxV operations.

Before discussing the HP models and methods for parallelizing SpMxV operations, it is favorable to discuss parallel algorithms for SpMxV. Consider the matrix-vector multiply of the form y ← A x, where the nonzeros of the sparse matrix A as well as the entries of the input and output vectors x and y are partitioned arbitrarily among the processors. Let map(•) denote the nonzero-to-processor and vector-entry-to-processor assignments induced by this partitioning. A parallel algorithm would execute the following steps at each processor P k .

1. Send the local input-vector entries x j , for all j with map(x j ) = P k , to those processors that have at least one nonzero in column j.

2.

Compute the scalar products a ij x j for the local nonzeros, i.e., the nonzeros for which map(a ij ) = P k and accumulate the results y k i for the same row index i.

3.

Send local nonzero partial results y k i to the processor map(y i ) = P k , for all nonzero y k i .

4. Add the partial y i results received to compute the final results y i = y i for each i with map(y i ) = P k .

As seen in the algorithm, it is necessary to have partitions on the matrix A and the inputand output-vectors x and y of the matrix-vector multiply operation. Finding a partition on the vectors x and y is referred to as the vector partitioning operation, and it can be performed in three different ways: by decoding the partition given on A; in a post-processing step using the partition on the matrix; or explicitly partitioning the vectors during partitioning the matrix. In any of these cases, the vector partitioning for matrix-vector operations is called symmetric if x and y have the same partition, and non-symmetric otherwise. A vector partitioning is said to be consistent, if each vector entry is assigned to a processor that has at least one nonzero in the respective row or column of the matrix. The consistency is easy to achieve for the nonsymmetric vector partitioning; x j can be assigned to any of the processors that has a nonzero in the column j, and y i can be assigned to any of the processors that has a nonzero in the row i. If a symmetric vector partitioning is sought, then special care must be taken to assign a pair of matching input-and output-vector entries, e.g., x i and y i , to a processor having nonzeros in both row and column i. In order to have such a processor for all vector entry pairs, the sparsity pattern of the matrix A can be modified to have a zero-free diagonal. In such cases, a consistent vector partition is guaranteed to exist, because the processors that own the diagonal entries can also own the corresponding input-and output-vector entries; x i and y i can be assigned to the processor that holds the diagonal entry a ii .

In order to achieve an efficient parallelism, the processors should have balanced computational load and the inter-processor communication cost should have been minimized. In order to have balanced computational load, it suffices to have almost equal number of nonzeros per processor so that each processor will perform almost equal number of scalar products, e.g., a ij x j , in any given parallel system. The communication cost, however, has many components (the total volume of messages, the total number of messages, maximum volume/number of messages in a single processor, either in terms of sends or receives or both) each of which can be of utmost importance for a given matrix in a given parallel system. Although there are alternatives and more elaborate proposals, the most common communication cost metric addressed in hypergraph partitioningbased methods is the total volume of communication.

Loosely speaking, hypergraph partitioning-based methods for efficient parallelization of SpMxV model the data of the SpMxV (i.e., matrix and vector entries) with the vertices of a hypergraph. A partition on the vertices of the hypergraph is then interpreted in such a way that the data corresponding to a set of vertices in a part are assigned to a single processor. More accurately, there are two classes of hypergraph partitioning-based methods to parallelizing SpMxV. The methods in the first class build a hypergraph model representing the data and invoke a partitioning heuristic on the so-built hypergraph. The methods in this class can be said to be models rather than being algorithms. There are currently three main hypergraph models for representing sparse matrices, and hence there are three methods in this first class. These three main models are described below in the next section. Essential property of these models is that the cutsize (3) of any given partition is equal to the total communication volume to be incurred under a consistent vector partitioning when the matrix elements are distributed according to the vertex partition. The methods in the second class follow a mix-and-match approach and use the three main models, perhaps, along with multi-constraint and fixed-vertex variations in an algorithmic form. There are a number of methods in this second class, and one can develop many others according to application needs and matrix characteristics. Three common methods belonging to this class are described later, after the three main models. The main property of these algorithms is that the sum of the cutsizes of each application of hypergraph partitioning amounts to the total communication volume to be incurred under a consistent vector partitioning (currently these methods compute a vector partitioning after having found a matrix partitioning) when the matrix elements are distributed according to the vertex partitions found at the end.

Three main models for matrix partitioning

In the column-net hypergraph model [START_REF] Aykanat | Hypergraph-partitioning-based decomposition for parallel sparse-matrix vector multiplication[END_REF] used for 1D rowwise partitioning, an

M × N matrix A with Z nonzeros is represented as a unit-cost hypergraph H R = (V R , N C ) with |V R | = M vertices, |N C | = N nets, and Z pins. In H R , there exists one vertex v i ∈ V R for each row i of matrix A. Weight w[v i ] of a vertex v i is equal to the number of nonzeros in row i.
The name of the model comes from the fact that columns are represented as nets. That is, there exists one unit-cost net n j ∈ N C for each column j of matrix A. Net n j connects the vertices corresponding to the rows that have a nonzero in column j. That is, v i ∈ n j if and only if a ij = 0.

In the row-net hypergraph model [START_REF] Aykanat | Hypergraph-partitioning-based decomposition for parallel sparse-matrix vector multiplication[END_REF] used for 1D columnwise partitioning, an M ×N matrix A with Z nonzeros is represented as a unit-cost hypergraph

H C = (V C , N R ) with |V C | = N vertices, |N R | = M
nets, and Z pins. In H C , there exists one vertex v j ∈ V C for each column j of matrix A. Weight w[v j ] of a vertex v j ∈ V R is equal to the number of nonzeros in column j. The name of the model comes from the fact that rows are represented as nets. That is, there exists one unit-cost Jagged-Like-Partitioning(A, K = P × Q, ε 1 , ε 2 ) Input: a matrix A, the number of processors K = P × Q, and the imbalance ratios ε 1 , ε 2 . Output: map(a ij ) for all a ij = 0 and totalVolume.

1:

H R = (V R , N C ) ← columnNet(A) 2: Π R = {R 1 , . . . , R P } ← partition(H R , P, ε 1 ) rowwise partitioning of A 3: expandVolume← cutsize(Π R ) 4: foldVolume← 0 5: for p = 1 to P do 6: R p = {r i : v i ∈ R p } 7: A p ← A(R p , :)
submatrix indexed by rows R p 8:

H p = (V p , N p ) ← rowNet(A p ) 9: Π C p = {C 1 p , . . . , C Q p } ← partition(H p , Q, ε 2 ) columnwise partitioning of A p 10:
foldVolume←foldVolume +cutsize(Π C p )

11:

for all a ij = 0 of A p do 12:

map(a ij ) = P p,q ⇔ c j ∈ C q p 13: return totalVolume←expandVolume+foldVolume net n i ∈ N R for each row i of matrix A. Net n i ⊆ V C connects the vertices corresponding to the columns that have a nonzero in row i. That is, v j ∈ n i if and only if a ij = 0.

In the column-row-net hypergraph model, otherwise known as the fine-grain model [START_REF] Aykanat | A fine-grain hypergraph model for 2D decomposition of sparse matrices[END_REF], used for 2D nonzero-based fine-grain partitioning, an M ×N matrix A with Z nonzeros is represented as a unit-weight and unit-cost hypergraph

H Z = (V Z , N RC ) with |V Z | = Z vertices, |N RC | = M +N
nets and 2Z pins. In V Z , there exists one unit-weight vertex v ij for each nonzero a ij of matrix A. The name of the model comes from the fact that both rows and columns are represented as nets. That is, in N RC , there exist one unit-cost row-net r i for each row i of matrix A and one unit-cost column-net c j for each column j of matrix A. The row-net r i connects the vertices corresponding to the nonzeros in row i of matrix A, and the column-net c j connects the vertices corresponding to the nonzeros in column j of matrix A. That is, v ij ∈ r i and v ij ∈ c j if and only if a ij = 0. Note that each vertex v ij is in exactly two nets.

Some other methods for matrix partitioning

The jagged-like partitioning method [START_REF] Aykanat | On two-dimensional sparse matrix partitioning: Models, methods, and a recipe[END_REF] uses the row-net and column-net hypergraph models. It is an algorithm with two steps, in which each step models either the expand phase or the fold phase of the parallel SpMxV algorithm. Therefore, there are two alternative schemes for this partitioning method. The one which models the expands in the first step and the folds in the second step is described below.

Given an M × N matrix A and the number K of processors organized as a P × Q mesh, the jagged-like partitioning model proceeds as shown in Fig. 1. The algorithm has two main steps. First, A is partitioned rowwise into P parts using the column-net hypergraph model H R (lines 1 and 2 of Fig. 1). Consider a P -way partition Π R of H R . From the partition Π R , one obtains P submatrices A p , for p = 1, . . . , P each having roughly equal number of nonzeros. For each p, the rows of the submatrix A p correspond to the vertices in R p (lines 6 and 7 of Fig. 1). The submatrix A p is assigned to the pth row of the processor mesh. Second, each submatrix A p for 1 ≤ p ≤ P is independently partitioned columnwise into Q parts using the row-net hypergraph H p (lines 8 and 9 of Fig. 1). The nonzeros in the ith row of A are partitioned among the Q processors in a row of the processor mesh. In particular, if v i ∈ R p at the end of line 2 of the algorithm, then the nonzeros in the ith row of A are partitioned among the processors in the pth row of the processor mesh. After partitioning the submatrix A p columnwise, the map array contains the partition information for the nonzeros residing in A p .

For each i, the volume of communication required to fold the vector entry y i is accurately represented as a part of "foldVolume" in the algorithm. For each j, the volume of communication regarding the vector entry x j is accurately represented as a part of "expandVolume" in the algorithm.

Figure 2(a) illustrates the column-net representation of a sample matrix to be partitioned among the processors of a 2 × 2 mesh. For simplicity of the presentation, the vertices and the nets of the hypergraphs are labeled with letters "r" and "c" to denote the rows and columns of the matrix. The matrix is first partitioned rowwise into 2 parts, and each part is assigned to a row of the processor mesh, namely to processors {P 1 , P 2 } and {P 3 , P 4 }. The resulting permuted matrix is displayed in Fig. 2(b). Figure 3(a) displays the two row-net hypergraphs corresponding to each submatrix A p for p = 1, 2. Each hypergraph is partitioned independently; sample partitions of these hypergraphs are also presented in this figure. As seen in the final symmetric permutation in Fig. 3(b), the nonzeros of columns 2 and 5 are assigned different parts, resulting P 3 to communicate with both P 1 and P 2 in the expand phase.

The checkerboard partitioning method [START_REF] Aykanat | A hypergraph-partitioning approach for coarse-grain decomposition[END_REF] is also a two-step method, in which each step models either the expand phase or the fold phase of the parallel SpMxV. Similar to jagged-like partitioning, there are two alternative schemes for this partitioning method. The one which models the expands in the first step and the folds in the second step is presented below.

Given an M ×N matrix A and the number K of processors organized as a P × Q mesh, the checkerboard partitioning method proceeds as shown in Fig. 4. First, A is partitioned rowwise into P parts using the column-net model (lines 1 and 2 of Fig. 4), producing Π R = {R 1 , . . . , R P }. Note that this first step is exactly the same as that of the jagged-like partitioning. In the second step, the matrix A is partitioned columnwise into Q parts by using the multi-constraint partitioning to obtain Π C = {C 1 , . . . , C Q }. In comparison to the jagged-like method, in this second step the whole matrix A is partitioned (lines 4 and 8 of Fig. 4), not the submatrices defined by Π R . The rowwise and columnwise partitions Π R and Π C together define a 2D partition on the matrix A, where map(a ij ) = P p,q ⇔ r i ∈ R p and c j ∈ C q .

In order to achieve a load balance among processors, a multi-constraint partitioning formulation is used (line 8 of the algorithm). Each vertex v i of H C is assigned P weights: w[i, p], for p = 1, . . . , P . Here, w[i, p] is equal to the number of nonzeros of column c i in rows R p (line 7 of Fig. 4). Consider a Q-way partitioning of H C with P constraints using the vertex weight definition above. Maintaining the P balance constraints (4) corresponds to maintaining computational load balance on the processors of each row of the processor mesh.

Establishing the equivalence between the total communication volume and the sum of the cutsizes of the two partitions is fairly straightforward. The volume of communication for the fold operations corresponds exactly to the cutsize(Π C ). The volume of communication for the expand Checkerboard-Partitioning(A, K = P × Q, ε 1 , ε 2 ) Input: a matrix A, the number of processors K = P × Q, and the imbalance ratios ε 1 , ε 2 . Output: map(a ij ) for all a ij = 0 and totalVolume.

1:

H R = (V R , N C ) ← columnNet(A) 2: Π R = {R 1 , . . . , R P } ← partition(H R , P, ε 1 ) rowwise partitioning of A 3: expandVolume← cutsize(Π R ) 4: H C = (V C , N R ) ← rowNet(A) 5: for j = 1 to |V C | do 6:
for p = 1 to P do 7:

w j,p = |{n j ∩ R p }| 8: Π C = {C 1 , . . . , C Q } ← MCPartition(H C , Q, w, ε 2 )
columnwise partitioning of A 9: foldVolume← cutsize(Π C ) 10: for all a ij = 0 of A do 11: map(a ij ) = P p,q ⇔ r i ∈ R p and c j ∈ C q 12: totalVolume←expandVolume+foldVolume ORB-Partitioning(A, dim, Kmin, Kmax, ε) Input: a matrix A, the part numbers Kmin (at initial call, it is equal to 1) and Kmax (at initial call it is equal to K, the desired number of parts), and the imbalance ratio ε. Output: map(a ij ) for all a ij = 0 .

1: if Kmax -Kmin > 0 then totalVolume←totalVolume +cutsize(Π) Recursively partition each submatrix along the orthogonal direction 5: Another common method of matrix partitioning is the orthogonal recursive bisection (ORB) [START_REF] Vastenhouw | A two-dimensional data distribution method for parallel sparse matrix-vector multiplication[END_REF]. In this approach, the matrix is first partitioned rowwise into two submatrices using the column-net hypergraph model, and then each part is further partitioned columnwise into two parts using the row-net hypergraph model. The process is continued recursively until the desired number of parts is obtained. The algorithm is shown in Fig. 6. In this algorithm, dim represents either rowwise or columnwise partitioning, where -dim switches the partitioning dimension.

map1(A 1 ) ← ORB-Partitioning(A 1 , -dim, Kmin, Kmin + mid -1, ε) 6: map2(A 2 ) ← ORB-Partitioning(A 2 -dim, Kmin + mid, Kmax, ε) 7: map(A) ← map1(A 1 ) ∪ map2(A 2 ) 8: else 9: map(A) ← Kmin
In the ORB method shown above, the step bisect(A, dim, ε) corresponds to partitioning the given matrix either along the rows or columns with, respectively, the column-net or the rownet hypergraph models into two. The total sum of the cutsizes (3) of each each bisection step corresponds to the total communication volume. It is possible to dynamically adjust the ε at each recursive call by allowing larger imbalance ratio for the recursive call on the submatrix A 1 or A 2 .

Some other applications of hypergraph partitioning

As said before, the initial motivations for hypergraph models were accurate modeling of the nonzero structure of unsymmetric and rectangular sparse matrices to minimize communication volume for iterative solvers. There are other applications that can make use of hypergraph partitioning formulation. Here, a brief overview of general classes of applications is given along with the names of some specific problems. Further application classes are given in bibliographic notes.

Parallel reduction or aggregation operations form a significant class of such applications, including the MapReduce model. The reduction operation consists of computing M output elements from N input elements. An output element may depend on multiple input elements, and an input element may contribute to multiple output elements. Assume that the operation on which reduc-tion is performed is commutative and associative. Then, the inherent computational structure can be represented with an M × N dependency matrix, where each row and column of the matrix represents an output element and an input element, respectively. For an input element x j and an output element y i , if y i depends on x j , a ij is set to 1 (otherwise zero). Using this representation, the problem of partitioning the workload for the reduction operation is equivalent to the problem of partitioning the dependency matrix for efficient SpMxV.

In some other reduction problems, the input and output elements may be pre-assigned to parts. The proposed hypergraph model can be accommodated to those problems by adding K part vertices and connecting those vertices to the nets which correspond to the pre-assigned input and output elements. Obviously, those part vertices must be fixed to the corresponding parts during the partitioning. Since the required property is already included in the existing hypergraph partitioners [START_REF] Boman | Zoltan 3.0: Parallel Partitioning, Load Balancing, and Data-Management Services; User's Guide[END_REF][START_REF] Aykanat | PaToH: A Multilevel Hypergraph Partitioning Tool[END_REF][START_REF] Karypis | hMeTiS A Hypergraph Partitioning Package Version 1.0.1[END_REF], this does not add extra complexity to the partitioning methods.

Iterative methods for solving linear systems usually employ preconditioning techniques. Roughly speaking, preconditioning techniques modify the given linear system to accelerate convergence. Applications of explicit preconditioners in the form of approximate inverses or factored approximate inverses are amenable to parallelization. Because, these techniques require SpMxV operations with the approximate inverse or factors of the approximate inverse at each step. In other words, preconditioned iterative methods perform SpMxV operations with both coefficient and preconditioner matrices in a step. Therefore, parallelizing a full step of these methods requires the coefficient and preconditioner matrices to be well partitioned, e.g., processors' loads are balanced and communication costs are low in both multiply operations. To meet this requirement, the coefficient and preconditioner matrices should be partitioned simultaneously. One can accomplish such a simultaneous partitioning by building a single hypergraph and then partitioning that hypergraph. Roughly speaking, one follows a four-step approach: (i) build a hypergraph for each matrix, (ii) determine which vertices of the two hypergraphs need to be in the same part (according to the computations forming the iterative method), (iii) amalgamate those vertices coming from different hypergraphs, (iv) if the computations represented by the two hypergraphs of the first step are separated by synchronization points then assign multiple weights to vertices (the weights of the vertices of the hypergraphs of the first step are kept), otherwise assign a single weight to vertices (the weights of the vertices of the hypergraphs of the first step are summed up for each amalgamation).

The computational structure of the preconditioned iterative methods is similar to that of a more general class of scientific computations including multi-phase, multi-physics, and multi-mesh simulations.

In multi-phase simulations, there are a number of computational phases separated by global synchronization points. The existence of the global synchronizations necessitates each phase to be load balanced individually. Multi-constraint formulation of hypergraph partitioning can be used to achieve this goal.

In multi-physics simulations, a variety of materials and processes are analyzed using different physics procedures. In these types of simulations, computational as well as the memory requirements are not uniform across the mesh. For scalability issues, processor loads should be balanced in terms of these two components. The multi-constraint partitioning framework also addresses these problems.

In multi-mesh simulations, a number of grids with different discretization schemes and with arbitrary overlaps are used. The existence of overlapping grid points necessitates a simultaneous partitioning of the grids. Such a simultaneous partitioning scheme should balance the computational loads of the processors and minimize the communication cost due to interactions within a grid as well as the interactions among different grids. With a particular transformation (the vertex amalgamation operation, also mentioned above), hypergraphs can be used to model the interactions between different grids. With the use of multi-constraint formulation, the partitioning problem in the multi-mesh computations can also be formulated as a hypergraph partitioning problem.

In obtaining partitions for two or more computation phases interleaved with synchronization points, the hypergraph models lead to the minimization of the overall sum of the total volume of communication in all phases (assuming that a single hypergraph is built as suggested in the previous paragraphs). In some sophisticated simulations, the magnitude of the interactions in one phase may be different than that of the interactions in another one. In such settings, minimizing the total volume of communication in each phase separately may be advantageous. This problem can be formulated as a multi-objective hypergraph partitioning problem on the so-built hypergraphs.

There are certain limitations in applying hypergraph partitioning to the multi-phase, multiphysics, and multi-mesh-like computations. The dependencies must remain the same throughout the computations, otherwise the cutsize may not represent the communication volume requirements as precisely as before. The weights assigned to the vertices, for load balancing issues, should be static and available prior to the partitioning; the hypergraph models cannot be used as naturally for applications whose computational requirements vary drastically in time. If, however, the computational requirements change gradually in time, then the models can be used to re-partition the load at certain time intervals (while also minimizing the redistribution or migration costs associated with the new partition).

Ordering methods are quite common techniques to permute matrices in special forms in order to reduce the memory and running time requirements, as well as to achieve increased parallelism in direct methods (such as LU and Cholesky decompositions) used for solving systems of linear equations. Nested-dissection is a well-known ordering method that has been used quite efficiently and successfully. In the current state-of-the-art variations of the nested-dissection approach, a matrix is symmetrically permuted with a permutation matrix P into doubly bordered block diagonal form

A DB = PAP T        A 11 A 1S A 22 A 2S . . . . . . A KK A KS A S1 A S2 • • • A SK A SS       
, where the nonzeros are only in the marked blocks (the blocks on the diagonal and the row and column borders). The aim in such a permutation is to have reduced numbers of rows/columns in the borders and to have equal sized square blocks in the diagonal. One way to achieve such a permutation when A has symmetric pattern is as follows. Suppose a matrix B is given (if not, it is possible to find one) where the sparsity pattern of B T B equals to that of A (here arithmetic cancellations are ignored). Then, one can permute B nonsymmetrically into the singly bordered form

B SB = QBP T      B 11 B 1S B 22 B 2S . . . . . . B KK B KS     
, so that B T SB B SB = PAP T ; that is one can use the column permutation of B resulting in B SB to obtain a symmetric permutation for A which results in A DB . Clearly, the column dimension of B kk will be the size of the square matrix A kk and the number of rows and columns in the border will be equal to the number of columns in the column border of B SB . One can achieve such a permutation of B by partitioning the column-net model of B while reducing the cutsize according to the cut-net metric (2), with unit net costs, to obtain the permutation P as follows. First, the permutation Q is defined to be able to define P. Permute all rows corresponding to the vertices in part k before those in a part , for 1 ≤ k < ≤ K. Then, permute all columns corresponding to the nets that are internal to a part k before those that are internal to a part , for 1 ≤ k < ≤ K, yielding the diagonal blocks, and then permute all columns corresponding to the cut nets to the end, yielding the column border (the order of column defining a diagonal block). Clearly the correspondence between the size of the column border of B SB and the doubly border of A DB is exact, and hence the cutsize according to the cut-net metric is an exact measure. The requirement to have almost equal sized square blocks A kk decoded as the requirement that each part should have an almost equal number of internal nets in the partition of the column-net model of B. Although such a requirement is neither the objective nor the constraint of the hypergraph partitioning problem, the common hypergraph-partitioning heuristics easily accommodate such requirements.

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BIBLIOGRAPHIC NOTES AND FURTHER READING

The first use of the hypergraph partitioning methods for efficient parallel sparse matrix-vector multiply operations is discussed by C ¸atalyürek and Aykanat [START_REF] Aykanat | A hypergraph model for mapping repeated sparse matrixvector product computations onto multicomputers[END_REF]. A more comprehensive study [START_REF] Aykanat | Hypergraph-partitioning-based decomposition for parallel sparse-matrix vector multiplication[END_REF] describes the use of the row-net and column-net hypergraph models in 1D sparse matrix partitioning. There are different views and alternatives on vector partitioning [START_REF] Bisseling | Communication balancing in parallel sparse matrix-vector multiplication[END_REF][START_REF] Uçar | Encapsulating multiple communication-cost metrics in partitioning sparse rectangular matrices for parallel matrix-vector multiplies[END_REF][START_REF] Uçar | Revisiting hypergraph models for sparse matrix partitioning[END_REF].

A fair treatment of parallel sparse matrix-vector multiplication, analysis and investigations on certain matrix types along with the use of hypergraph partitioning is given by Bisseling [4, Chapter 4]. Further analysis of hypergraph partitioning on some model problems is given by Uçar and C ¸atalyürek [START_REF] Uçar | On the scalability of hypergraph models for sparse matrix partitioning[END_REF].

Hypergraph partitioning schemes for preconditioned iterative methods are given by Uçar and Aykanat [START_REF] Uçar | Partitioning sparse matrices for parallel preconditioned iterative methods[END_REF], where vertex amalgamation and multi-constraint weighting to represent different phases of computations are given. Discussions on application of such methodology for multi-phase, multi-physics, and multi-mesh simulations are also included in the same paper. Some different methods for sparse matrix partitioning using hypergraphs are presented by Uçar et al. [START_REF] Uçar | A matrix partitioning interface to PaToH in MATLAB[END_REF], including jagged-like and checkerboard partitioning methods, and by Vastenhouw and Bisseling [START_REF] Vastenhouw | A two-dimensional data distribution method for parallel sparse matrix-vector multiplication[END_REF], the orthogonal recursive bisection approach. A recipe to choose a partitioning method for the objective of the communication volume reduction for a given matrix is given by C ¸atalyürek et al. [START_REF] Aykanat | On two-dimensional sparse matrix partitioning: Models, methods, and a recipe[END_REF].

The use of hypergraph models for permuting matrices into special forms such as singly bordered block-diagonal form is presented by Aykanat et al. [START_REF] Aykanat | Permuting sparse rectangular matrices into block-diagonal form[END_REF]. This permutation can be leveraged to develop hypergraph partitioning-based symmetric [START_REF] ¸atalyürek | Hypergraph Models for Sparse Matrix Partitioning and Reordering[END_REF][START_REF] Aykanat | Hypergraph partitioning-based fill-reducing ordering[END_REF] and nonsymmetric [START_REF] Grigori | Hypergraph unsymmetric nested dissection ordering for sparse LU factorization[END_REF] nested-dissection orderings.

The standard hypergraph partitioning and the hypergraph partitioning with fixed vertices formulation, respectively, is used for static and dynamic load balancing for some scientific applications [START_REF] Cambazoglu | Hypergraph-partitioning-based remapping models for image-space-parallel direct volume rendering of unstructured grids[END_REF][START_REF] Catalyurek | A repartitioning hypergraph model for dynamic load balancing[END_REF].

Some other applications of hypergraph partitioning are briefly summarized by Aykanat et al. [START_REF] Aykanat | Multi-level direct k-way hypergraph partitioning with multiple constraints and fixed vertices[END_REF]. These include image-space parallel direct volume rendering, parallel mixed integer linear programming, data declustering for multi-disk databases, scheduling file-sharing tasks in heterogeneous master-slave computing environments, and work-stealing scheduling, road network clustering methods for efficient query processing, pattern-based data clustering, reducing software development and maintenance costs, processing spatial join operations, and improving locality in memory or cache performance.

Figure 1 :

 1 Figure 1: Jagged-like partitioning.

  vol = 3 imbal = [-2.1%, 2.1%] (b)

Figure 2 :

 2 Figure 2: First step of 4-way jagged-like partitioning of a matrix; (a) 2-way partitioning Π R of column-net hypergraph representation H R of A, (b) 2-way rowwise partitioning of matrix A Π obtained by permuting A according to the partitioning induced by Π; the nonzeros in the same partition are shown with the same shape and color; the deviation of the minimum and maximum numbers of nonzeros of a part from the average are displayed as an interval imbal; vol denotes the number of nonzeros and the total communication volume.

Figure 3 :

 3 Figure 3: Second step of 4-way jagged-like partitioning: (a) Row-net representations of submatrices of A and 2-way partitionings, (b) Final permuted matrix; the nonzeros in the same partition are shown with the same shape and color; the deviation of the minimum and maximum numbers of nonzeros of a part from the average are displayed as an interval imbal; nnz and vol denote, respectively, the number of nonzeros and the total communication volume.

Figure 4 :

 4 Figure 4: Checkerboard partitioning.

Figure 5 :

 5 Figure 5: Second step of 4-way checkerboard partitioning: (a) 2-way multi-constraint partitioning Π C of row-net hypergraph representation H C of A, (b) Final checkerboard partitioning of A induced by (Π R , Π C ); the nonzeros in the same partition are shown with the same shape and color; the deviation of the minimum and maximum numbers of nonzeros of a part from the average are displayed as an interval imbal; nnz and vol denote, respectively, the number of nonzeros and the total communication volume.

Π = A 1 ,

 1 A 2 ←bisect(A, dim, ε) Partition A along dim into two, producing two submatrices 4:

Figure 6 :

 6 Figure 6: Orthogonal recursive bisection (ORB)

Figure 5 (

 5 b) displays the 2 × 2 checkerboard partition induced by (Π R , Π C ). Here, Π R is a rowwise 2-way partition giving the same figure as shown in Fig. 2, and Π C is a 2-way multiconstraint partition Π C of the row-net hypergraph model H C of A shown in Fig. 5(a). In Fig. 5(a), w[9, 1] = 0 and w[9, 2] = 4 for internal column c 9 of row stripe R 2 , whereas w[5, 1] = 2 and w[5, 2] = 4 for external column c 5 .


Sergey G Fedosin
email: fedosin@hotmail.com

Equations of motion in the theory of relativistic vector fields

Keywords: vector field, equation of motion, field equations

Within the framework of the theory of relativistic vector fields, the covariant expressions are presented for the equations of motion of the matter and the field. These expressions can be written either in terms of the field tensors, that is, the fields' strengths and solenoidal vectors, or in terms the four-potentials, that is, the fields' scalar and vector potentials. This state of things is due to the fact that the Lagrange function initially implied the complementarity of description in terms of the strengths and the field potentials. It is shown that the equation for the fields, obtained by taking the covariant derivative in the equation for the metric, has a deeper meaning than the ordinary equation of motion of the matter, found with the help of the principle of least action. In particular, the above-mentioned equation for the fields leads to the generalized Poynting theorem, and after integration over the volume it allows us to introduce for consideration the integral vector as a measure of the energy and the fields' energy fluxes, associated with a system of particles and fields.

Introduction

One of the most effective ways of obtaining the equations of motion is to apply the principle of least action [1][START_REF] José | Classical dynamics: a contemporary approach[END_REF][START_REF] Goldstein | Classical mechanics, 2-nd edition[END_REF]. In this case, as a rule, either the motion of charged particles in an electromagnetic field or the motion of particles in a potential field or in a scalar pressure field are described. As for the gravitational field, it usually appears as a tensor field within the framework of the general theory of relativity, including the use of modified versions of this theory to describe the dynamic properties of matter [START_REF] Yousaf | Spherical relativistic vacuum core models in a Λ-dominated era[END_REF][START_REF] Yousaf | Stellar filaments with Minkowskian core in the Einstein-Λ gravity[END_REF][START_REF] Yousaf | Structure scalars of spherically symmetric dissipative fluids with f(G,T) gravity[END_REF].

The concept of the electromagnetic field, which is a vector field, is successfully used in science and technology for a comprehensive description of electromagnetic phenomena. It is also possible to represent all the fields acting in macroscopic systems as the corresponding vector fields [START_REF] Fedosin | Estimation of the physical parameters of planets and stars in the gravitational equilibrium model[END_REF][START_REF] Fedosin | Two components of the macroscopic general field[END_REF][START_REF] Fedosin | The virial theorem and the kinetic energy of particles of a macroscopic system in the general field concept[END_REF]. This allows us to apply the well-developed mathematical apparatus of the theory of relativistic vector fields and to describe with its help both the equation of the matter's motion under the action of the fields and the propagation of the fields during the matter's motion.

The main purpose of this paper is to present the equations of motion of the matter and field in a covariant form, which is suitable for use, in particular, in the curved spacetime and in any reference frames. We will systematize the results presented in the previous papers and will supplement them with regard to the equations for the fields' scalar and vector potentials.

Among a great number of vector fields, only the four fields will be considered, including the electromagnetic field, gravitation within the covariant theory of gravitation [START_REF] Fedosin | The physical theories and infinite hierarchical nesting of matter[END_REF], the acceleration field and the vector pressure field [START_REF] Fedosin | The procedure of finding the stress-energy tensor and vector field equations of any form[END_REF]. Other macroscopic vector fields can be easily added to the presented equations, as has already been done for the dissipation field [START_REF] Fedosin | Four-Dimensional Equation of Motion for Viscous Compressible and Charged Fluid with Regard to the Acceleration Field, Pressure Field and Dissipation Field[END_REF] and the fields of strong and weak interactions [START_REF] Fedosin | The Concept of the General Force Vector Field[END_REF].

In our calculations, we will use everywhere the metric signature of the form (+,-,-,-).

The general forms of the equation of motion

The application of the principle of least action, taking into account the energy gauge by means of the cosmological constant within the framework of the covariant theory of gravitation and the four vector fields, leads to the equation for finding the metric tensor components [START_REF] Fedosin | About the cosmological constant, acceleration field, pressure field and energy[END_REF]:

( )

1 1 4 2 R R g U W B P ck α β α β α β α β α β α β - = - + + + , (1)
where R α β is the Ricci tensor; R is the scalar curvature; g α β is the metric tensor; c is the speed of light,

3 16 c k G π β = -
, where G is the gravitational constant; β is a certain coefficient of the order of unity to be determined; U α β , W α β , B α β and P α β are the stress-energy tensors of the gravitational and electromagnetic fields, the acceleration field and the pressure field, respectively.

With the help of the covariant derivative β ∇ we can find the four-divergence of both parts of (1). The divergence of the left-hand side is equal to zero due to the equality to zero of the Einstein tensor's divergence,

1 0 2 R R g α β α β β   ∇ - =    
, and also as a consequence of the fact that outside the body the scalar curvature vanishes, 0 R = , and inside the body it is considered to be a constant. The latter follows from the gauge condition of the system's energy [START_REF] Fedosin | About the cosmological constant, acceleration field, pressure field and energy[END_REF]. The divergence of the right-hand side of (1) is also equal to zero:

( ) 0 U W B P α β α β α β α β β ∇ + + + = . (2)
Expression (2) for the space components of the tensors with the index 1, 2, 3 α = shows that the change over time of the fields' energy fluxes leads to the so-called fields' tensions. As for the time components of the tensors with the index 0 α = , for them (2) is the expression of the generalized Poynting theorem for all the fields, both inside and outside the matter, that determines the energy balance in the system.

The stress-energy tensors of the gravitational field [START_REF] Fedosin | The physical theories and infinite hierarchical nesting of matter[END_REF], [START_REF] Fedosin | The Principle of Least Action in Covariant Theory of Gravitation[END_REF], the electromagnetic field, the acceleration field and the pressure field [START_REF] Fedosin | The procedure of finding the stress-energy tensor and vector field equations of any form[END_REF], [START_REF] Fedosin | About the cosmological constant, acceleration field, pressure field and energy[END_REF], presented in (2), are derived from the principle of least action:

2 1 4 4 c U g g g g g Φ Φ G α β µκ α λ σ β α β σ λ µ λ κ σ π   = - - +     , 2 0 1 4 W c g g g g g F F α β µκ α λ σ β α β σ λ µ λ κ σ ε   = - +     . 2 1 4 4 c B g g g g g u u α β µκ α λ σ β α β σ λ µ λ κ σ πη   = - +     , 2 1 4 4 c P g g g g g f f α β µκ α λ σ β α β σ λ µ λ κ σ πσ   = - +     . (3)
Here Φ µ λ , F µ λ , u µ λ and f µ λ represent the gravitational tensor, electromagnetic tensor, acceleration tensor and pressure field tensor, respectively; 0 ε is the electric constant, η is the acceleration field coefficient; σ is the pressure field coefficient.

All the tensors in (2) and in (3) are expressed in terms of the corresponding fields' strengths and solenoidal vectors. For example, the electromagnetic tensor F µ λ and the stress-energy tensor of the electromagnetic field W α β are expressed in terms of the field strength E and the magnetic field B .

Equation ( 2) inside the matter can be transformed in such a way that the field tensors themselves would appear in it instead of the stress-energy tensors. In particular, we can prove the following equalities [START_REF] Fedosin | The Principle of Least Action in Covariant Theory of Gravitation[END_REF][START_REF] Fock | The Theory of Space, Time and Gravitation[END_REF]:

U g Φ J α β α µ ν β µν ∇ = - , W g F j α β α µ ν β µν ∇ = - , B g u J α β α µ ν β µν ∇ = - , P g f J α β α µ ν β µν ∇ = - . (4)
In order to prove the validity of (4), it is necessary to substitute (3) into (4), to apply the covariant derivatives β ∇ to the products of the field tensors, and then to use the equations of the respective field, so that the mass four-current J ν and the charge four-current j ν would appear in (4).

In view of (4), equation ( 2) inside the matter turns into the equation of motion and takes the following form:

0 Φ J F j u J f J ν ν ν ν µν µν µν µν + + + = .
(

) 5
The equation of motion in the form of (5) can also be obtained directly from the principle of least action [START_REF] Fedosin | The Principle of Least Action in Covariant Theory of Gravitation[END_REF], regardless of (1) and [START_REF] José | Classical dynamics: a contemporary approach[END_REF].

In order to reduce (5) to a more convenient form, we should take into account the expression for the mass current where U ν is the four-potential of the acceleration field. With this in mind, we have the following:

0 0 0 ( ) DU u J u U U u U D µ ν ν ν µν µ ν ν µ µ ν ρ ρ ρ τ = ∇ -∇ = ∇ - . (6)
In this case, the operator of proper-time-derivative D u D ν ν τ ∇ = was used, where D is the symbol of the four-differential in the curved spacetime, τ is the proper time [START_REF] Fedosin | The physical theories and infinite hierarchical nesting of matter[END_REF]. The substitution of (6) into [START_REF] Yousaf | Stellar filaments with Minkowskian core in the Einstein-Λ gravity[END_REF] gives the following:

0 0 DU u U Φ J F j f J D µ ν ν ν ν µ ν µν µν µν ρ ρ τ = ∇ + + + . ( 7
)
If other vector fields are taken into account in the system, such as the dissipation field [START_REF] Fedosin | Four-Dimensional Equation of Motion for Viscous Compressible and Charged Fluid with Regard to the Acceleration Field, Pressure Field and Dissipation Field[END_REF] or the macroscopic vector fields of strong and weak interactions [START_REF] Fedosin | The Concept of the General Force Vector Field[END_REF], then the respective additional terms with the tensors of these fields appear on the right-hand side of [START_REF] Fedosin | Estimation of the physical parameters of planets and stars in the gravitational equilibrium model[END_REF].

For an arbitrary vector B µ we can write:

B B B β ν µ ν µ µν β ∇ = ∂ -Γ , DB dB u B u B u B u B D d µ µ ν ν ν β ν β ν µ ν µ µν β µν β τ τ = ∇ = ∂ -Γ = -Γ , (8)
where

β µν Γ is the Christoffel symbol.
Replacement of B µ by the four-potential of the acceleration field U µ in [START_REF] Fedosin | Two components of the macroscopic general field[END_REF] gives the following:

DU dU u U D d µ µ ν β µν β τ τ = -Γ , U U U β µ ν µ ν µν β ∇ = ∂ -Γ . ( 9
)
Let us substitute both equalities (9) into (7):

0 0 dU u U Φ J F j f J d µ ν ν ν ν µ ν µν µν µν ρ ρ τ = ∂ + + + . ( 10
)
We will divide equation [START_REF] Fedosin | The physical theories and infinite hierarchical nesting of matter[END_REF] into two equations, one for the time components and the other for the space vector components, with the index 1, 2,3 i = :

0 0 0 0 0 0 dU U u Φ J F j f J d c t ν ν ν ν ν ν ν ν ρ ρ τ ∂ = + + + ∂ . 0 0 i i i i i dU u U Φ J F j f J d ν ν ν ν ν ν ν ν ρ ρ τ = ∂ + + + . (11)
In [START_REF] Fedosin | The procedure of finding the stress-energy tensor and vector field equations of any form[END_REF], differentiation with respect to the proper time τ can be replaced by differentiation with respect to the coordinate time t as follows:

d dt d d d dt τ τ = .
In this case, for the mass four-current J ν and for the charge four-current j ν , we can write:

0 0 0 0 ( , ) dx dt dx dt J u c d d dt d ν ν ν ν ρ ρ ρ ρ τ τ τ = = = = v , 0 0 ( , ) q q dt j u c d ν ν ρ ρ τ = = v , (12)
where 0q ρ is the invariant charge density, v is the velocity of the matter particles' motion. In the Cartesian space coordinates, the four-dimensional quantity

1 2 3
( , ) ( , , , ) ( , , , )

x y z c c v v v c v v v = = v
is expressed in terms of the speed of light and the components of the particles' velocity.

Also taking into account that the field tensors in [START_REF] Fedosin | The procedure of finding the stress-energy tensor and vector field equations of any form[END_REF] are expressed in terms of the strengths and solenoidal vectors, and the four-potential of the acceleration field , U c

µ ϑ   = -   
 U is defined by the scalar potential ϑ and the vector potential U , in view of (12), we find the following:

0 0 q U d dx dt dt t ν ν ρ ϑ ρ ∂ = + ⋅ + ⋅ + ⋅ ∂ Γ v E v C v . ( ) 0 0 [ ] [ ] [ ] q i d dx U dt dt ν ν ρ ρ = - ∂ + + × + + × + + × U Γ v Ω E v Β C v I , ( 13
)
where Γ , E and С denote the strengths of the gravitational field, electromagnetic field and pressure field, respectively; and, similarly, Ω is the gravitational torsion field; Β is the magnetic induction, and I is the solenoidal vector of the pressure field.

According to the definition of the acceleration field tensor, its components include the strength S and the solenoidal vector N , expressed in terms of the acceleration field potentials:

t ϑ ∂ = -∇ - ∂ U S , = ∇ × N U.
The vectors S and N can be included in equations [START_REF] Fedosin | The Concept of the General Force Vector Field[END_REF]. To do this, on the left-hand side of ( 13) we must take into account the operator equation

d dt t ∂ = + ⋅∇ ∂ v
, and on the right-hand side of ( 13) we must expand the products of the four-vectors into their components, taking into account the definitions ( , )

x ct ν = r and ( , ) dx c dt ν = v : y x z x y z U U U U dx v v v dt t t t t t t t ν ν ϑ ϑ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = - - - = -⋅ ∂ ∂ ∂ ∂ ∂ ∂ ∂ U v . ( ) [ ] i dx U dt ν ν ϑ ϑ ⋅∇ + ∂ = ∇ -× ∇× = ∇ -× v U v U v N .
With this in mind, it follows from (13):

0 0 0 q ρ ρ ⋅ + ⋅ + ⋅ + ⋅ = S v Γ v E v C v . ( ) 0 0 [ ] [ ] [ ] [ ] 0 q ρ ρ + × + + × + + × + + × = S v N Γ v Ω E v Β C v I . ( 14
)
If in [START_REF] Fedosin | About the cosmological constant, acceleration field, pressure field and energy[END_REF] we multiply scalarly the second equation by the velocity v , we will obtain the first equation in [START_REF] Fedosin | About the cosmological constant, acceleration field, pressure field and energy[END_REF].

Equations ( 10), ( 11), ( 13) and ( 14), written in a covariant way, do not contain Christoffel symbols, which simplifies the solution of these equations. We should note that equations [START_REF] Fedosin | About the cosmological constant, acceleration field, pressure field and energy[END_REF] are completely symmetric with respect to all the four vector fields. This means that, for example, if the field strengths and the velocity v of the system's typical particles are known as functions of time and coordinates, then from the first equation ( 14) we can find the relationship between the field coefficients, that is, between η , G , 0 ε and σ .

Rectilinear motion of an ideal solid body

An ideal solid body can be considered as the limiting case of the relativistic uniform system, in which the system's particles are fixed relative to each other. This means that the Lorentz

factor 2 2 1 1 v c γ ′ = ′ -
of the particles is equal to 1, and the velocity ′ v of their motion is equal to zero in the body's center-of-momentum frame. This fact allows us to substantially simplify the equation of motion of the body.

We will start with the definition of the four-potential of the acceleration field of an arbitrary physical system. According to [START_REF] Fedosin | The procedure of finding the stress-energy tensor and vector field equations of any form[END_REF], the four-potential of any vector field, the vector potential of which is equal to zero in its proper reference frame, that is, in the center-of-momentum frame, in case of rectilinear motion in the laboratory reference frame can be represented by the formula:

2 0 f L k L u c α α ε ρ = , ( 15
)
where

0 0 f q k ρ ρ =
for the electromagnetic field and 1 f k = for the other fields; ε is the invariant energy density of the interaction between the field's four-potential and the corresponding fourcurrent; L u α is the four-velocity with a covariant index that specifies the motion of the center of momentum of the physical system in the laboratory reference frame.

In the proper reference frame , 0,0,0

L dt u c d α τ   =    
, and the vector potential as the space component of L α vanishes according to [START_REF] Fedosin | The Principle of Least Action in Covariant Theory of Gravitation[END_REF]. However, some physical systems, even if they have their own center of momentum fixed, have not only a scalar field potential but also a vector field potential within the system. Therefore, a more general expression for the field's fourpotential in the laboratory reference frame is as follows:

L M L β α α β ′ = , ( 16
)
where M β α is a matrix connecting the coordinates and time of the two reference frames, one of which is laboratory reference frame and the other moves together with the center of momentum of the physical system under consideration, so that there is the field's four-potential L β ′ in it.

In the special case of the system's motion at the constant velocity the matrix M β α represents the Lorentz transformation matrix [START_REF] Fedosin | The physical theories and infinite hierarchical nesting of matter[END_REF].

Let us now assume the simplest case, when the physical system in the form of a sphere has no general rotation, the typical particles of the system move randomly and they also have neither proper rotation nor their own vector potentials in the particles' center-of-momentum frames. In this case, we can use a simpler formula (15) instead of [START_REF] Fock | The Theory of Space, Time and Gravitation[END_REF] to determine the four-potential of the acceleration field in case of rectilinear motion.

In a fixed sphere, the energy density in the volume of each particle

2 0 c ε γ ρ ′ =
, and, according to [START_REF] Fedosin | The Principle of Least Action in Covariant Theory of Gravitation[END_REF], for the acceleration field, when the sphere moves in the laboratory reference frame, the four-potential will equal

L L U u α α γ ′ = .
This means that if for an observer inside the sphere with particles, within the framework of the relativistic uniform model, the quantity γ ′ is an invariantly defined Lorentz factor as a certain function of coordinates and time, then for an observer in the laboratory reference frame, in which the center of the sphere has the fourvelocity L u α , the four-potential of the acceleration field for each point inside the moving sphere will equal

L U α .
By definition, the four-potential of the acceleration field is the four-vector , U c

µ ϑ   = -     U ,
where ϑ and U denote the scalar and vector potentials, respectively. Taking into account [START_REF] Fedosin | The Principle of Least Action in Covariant Theory of Gravitation[END_REF] and the relation In the ideal case, when the system of particles is not a rotating perfectly solid body and the particles inside the system are motionless, it must be 1 γ ′ = , and then the four-potential of the acceleration field would coincide with the four-velocity of the system's center of momentum,

L L U u α α = .
A material point represents a miniature physical system, and if we do not go deeply into the structure of the internal motion of its matter and if we consider this point as a solid body, then the four-potential of the acceleration field of such a point will also be equal to the four-velocity of its motion.

Thus, in case of rectilinear motion of an ideal solid body in the absence of rotation, in equation of motion [START_REF] Fedosin | Estimation of the physical parameters of planets and stars in the gravitational equilibrium model[END_REF] the four-potential of the acceleration field U µ can be replaced by the four-velocity of the body's center of momentum u µ . Taking into account the relation (7) we obtain the following:

0 u u ν µ ν ∇ = , which follows from the equality 2 ( ) 2 0 c u u u u ν ν µ µ ν µ ν ∇ = ∇ =∇ = , instead of
0 0 Du a Φ J F j f J D µ ν ν ν µ µν µν µν ρ ρ τ = = + + . ( 17
)
On the left-hand side of (17) there is the covariant four-acceleration Du a D

µ µ τ =
, and on the right-hand side there is the sum of the densities of the gravitational and electromagnetic forces and the density of the pressure force. In particular, the expression F j ν µν defines the density of the electromagnetic Lorentz force. Equations ( 10) and [START_REF] Fedosin | The procedure of finding the stress-energy tensor and vector field equations of any form[END_REF], in view the relation 0 u u

ν µ ν ∂ = ,
are written as follows:

0 du Φ J F j f J d µ ν ν ν µν µν µν ρ τ = + + . 0 0 0 0 0 du Φ J F j f J d ν ν ν ν ν ν ρ τ = + + , 0 i i i i du Φ J F j f J d ν ν ν ν ν ν ρ τ = + + . ( 18
)
On the left-hand side of (18) we will use the expression for the four-velocity

( , ) dx dx dt dt u c d d dt d µ µ µ τ τ τ = = = -v
, and on the right-hand side we will express the field tensors in terms of the strengths and solenoidal vectors. Then, in view of [START_REF] Fedosin | Four-Dimensional Equation of Motion for Viscous Compressible and Charged Fluid with Regard to the Acceleration Field, Pressure Field and Dissipation Field[END_REF], equation (18) can be transformed as follows:

0 2 0 q d dt c dt d ρ τ ρ   = ⋅ + ⋅ + ⋅     Γ v E v C v , ( ) 0 0 [ ] [ ] [ ] q d dt dt d ρ τ ρ   = + × + + × + + ×     v Γ v Ω E v Β C v I . ( 19
)
In the flat Minkowski spacetime dt d

γ τ = , where 2 2 1 1 v c γ = -
is the Lorentz factor, the Christoffel symbols vanish, and the metric tensor g µν transforms into the metric tensor µν η , which is independent of time and coordinates and has only the diagonal components. As a result, equations (19) are also simplified:

0 2 0 q d c dt ρ γ ρ = ⋅ + ⋅ + ⋅ Γ v E v C v , (20)
( )

0 0 ( ) [ ] [ ] [ ] q d dt ρ γ ρ = + × + + × + + × v Γ v Ω E v Β C v I . ( 21
)
The three-dimensional acceleration vector of a solid body or a particle is

d dt = v a , so that ( ) d d dt dt γ γ γ = + v a v
. Substituting here the expression d dt γ from (20) and comparing with (21), we obtain the equation for the acceleration in the framework of the special theory of relativity, expressed in terms of the fields' strengths and solenoidal vectors:

( ) 0 0 2 0 0 1 [ ] [ ] [ ] q q с ρ ρ γ ρ ρ   = + × + + × + + × - ⋅ + ⋅ + ⋅     a Γ v Ω E v Β C v I v Γ v E v C v . ( 22
)
If we denote the force density as

( ) ( ) ( ) 0 0 0 [ ] [ ] [ ] q ρ ρ ρ = + × + + × + + × f Γ v Ω E v Β C v I ,
then (20-22) can be rewritten as follows:

2 0 d с dt γ ρ = ⋅ f v , 0 ( ) d dt γ ρ = v f , 0 2 с γ ρ ⋅ = - f v a f v.
The relativistic energy and momentum of the solid body with the mass m and charge q are given by the expressions

dW m q m dt = ⋅ + ⋅ + ⋅ = ⋅ Γ v E v C v F v , [ ] [ ] [ ] d m m q q m m dt = + × + + × + + × = p Γ v Ω E v Β C v I F .
In the general case, the fields inside a solid body are non-uniform and differ in value at different points. In this case, the body can be divided into N parts with the masses n m , where the index n ranges from 0 to N . Multiplying equations ( 20) and ( 21) by 1 m , 2 m …, n m , then summing the equations, and passing from the sums to the integrals over the whole body, we arrive at the following:

( ) ( ) ( ) 2 1 1 N N n n n n n n d m c m q m dm dq dt γ = = = ⋅ + ⋅ + ⋅ = ⋅ + ⋅ + ⋅ ∑ ∑ ∫ ∫ Γ v E v C v Γ v C v E v , ( ) ( ) ( ) ( ) 1 1 [ ] [ ] [ ] [ ] [ ] [ ] . N N n n n n n n n n n d m m m q q m m dt dm dq γ = = = + × + + × + + × = = + × + + × + + × ∑ ∑ ∫ ∫ v Γ v Ω E v Β C v I Γ v Ω C v I E v Β
Comparison with the previous equations gives the following:

( )

2 2 1 N n n W mc m c γ γ = = = ∑ , ( )
1 N n n m m γ γ = = = ∑ p v v , 1 N n n m m = = ∑ , ( ) ( ) [ ] [ ] [ ] dm dq = + × + + × + + × ∫ ∫ F Γ v Ω C v I E v Β .
In these relations it is taken into account that in case of rectilinear motion of a solid body without rotation of this body, the velocity v and the Lorentz factor γ are constant for all the points of the body.

The case is possible when the body does not rotate in the center-of-momentum frame, but it does not move along a straight line. This leads to the fact that the body rotates relative to the point located at the instantaneous radius of curvature of the motion trajectory, and then the velocities of the points inside the body become different. In this case, the motion of the body's points should be described by more general equations ( 14).

Expression of the equation of motion in terms of the fields' four-potentials

By definition, the field tensors are expressed in terms of the four-curl applied to the fields' four-potentials. For the acceleration field, gravitational and electromagnetic fields and pressure field, this gives the following:

u U U µν µ ν ν µ = ∇ -∇ , Φ D D µν µ ν ν µ = ∇ -∇ , F A A µν µ ν ν µ = ∇ -∇ , f µν µ ν ν µ π π = ∇ -∇ , (23) where
, U c µ ϑ   = -    
U is the four-potential of the acceleration field, ϑ and U denote the scalar and vector potentials, respectively, , and also taking into account rule [START_REF] Fedosin | Two components of the macroscopic general field[END_REF] for the operator of the proper-time-derivative. We will also use the expression for the covariant derivative for the arbitrary four-vector B ν :

B B B β µ ν µ ν µν β ∇ = ∂ -Γ .
All this gives the following equation:

( ) 0 0 0 0 q q d U D dA u U u D u A u d d µ µ µ µ ν ν ν ν µ ν µ ν µ ν µ ν π ρ ρ π τ ρ τ ρ + + + = ∂ + ∂ + ∂ + ∂ . ( 24
)
Equation of motion (24) falls into two equations, since it can be written separately for the time and space vector components of the fields' four-potentials. Using the expression 0 1 с t

∂ ∂ = ∂
for the time component of the partial derivative, we have:

( ) 0 0 0 0 q q d U D A d u u u u d d t t t t ν ν ν ν ν ν ν ν ρ ρ ϑ ψ π ϕ τ ρ τ ρ + +℘ ∂ ∂ ∂ ∂ + = + + + ∂ ∂ ∂ ∂ , ( ) 0 0 0 0 q q i i i i d d u U u D u A u d d ν ν ν ν ν ν ν ν ρ ρ π τ ρ τ ρ + + + = -∂ -∂ - ∂ -∂ U D Π A . ( 25
)
In ( 25) the three-dimensional spatial gradient operator in Cartesian coordinates has the components , , The peculiarity of the covariant equations of motion ( 24) and ( 25) is the fact that they are fully expressed in terms of the fields' four-potentials or in terms of their components in the form of the fields' scalar and vector potentials. It is to be recalled that, in contrast, the equations of motion in the form of ( 7), ( 10) and ( 11) are expressed in terms of tensors, that is, in terms of the fields' strengths and solenoidal vectors, as was shown in ( 13) and ( 14).

i x y z   ∂ ∂ ∂ ∂ =   ∂ ∂ ∂   ,
We will use the relation 25). This gives the following:

d dt d d d dt τ τ = in equations (
( ) 0 0 0 0 q q d U D A d dx dx dx dx dt dt dt t dt t dt t dt t ν ν ν ν ν ν ν ν ρ ρ ϑ ψ π ϕ ρ ρ + +℘ ∂ ∂ ∂ ∂ + = + + + ∂ ∂ ∂ ∂ , ( 26
) ( ) 0 0 0 0 q q i i i i d d dx dx dx dx U D A dt dt dt dt dt dt ν ν ν ν ν ν ν ν ρ ρ π ρ ρ + + + = - ∂ - ∂ - ∂ - ∂ U D Π A . ( 27
)
For the case of rectilinear motion of an ideal solid body without rotation, the four-potential U µ of the acceleration field at each point of the body coincides with the four-velocity u µ of motion of the body's center of momentum. In this case in (25), ( 26) and ( 27) we will have 2 dt c d

ϑ τ = , dt dτ = U v.
In the Minkowski spacetime dt d γ τ = and the equations of motion of a solid body are further simplified, since then in ( 26) and ( 27)

2 c ϑ γ = , γ = U v, ( , ) U u c ν ν γ γ = = -v . In addition, since 2 2 2 1 1 v c γ = -
, the following relations hold true:

( ) ( ) ( ) 2 2 2 2 2 2 2 1 1 0 2 2 v c v u dx c dt t t t t t ν ν γ γ γ γ γ γ   ∂ ∂ ∂ ∂ ∂   = - = - = ∂ ∂ ∂ ∂ ∂     , ( ) ( ) ( ) 2 2 2 2 2 2 2 1 1 0 2 2 i i i i i dx u c v с v dt ν ν γ γ γ γ γ γ   - ∂ = -∂ + ∂ = - ∂ -∂ =   .
Taking this into account, instead of ( 26) and ( 27) we can write the following:

0 0 2 0 0 ( ) q q D A d d d dx dx dx c dt dt dt dt t dt t dt t ν ν ν ν ν ν ρ ρ π γ ψ ϕ ρ ρ ∂ ∂ ∂ +℘ = - - + + + ∂ ∂ ∂ , ( 28
) 0 0 0 0 ( ) ( ) q q i i i d d d dx dx dx D A dt dt dt dt dt dt ν ν ν ν ν ν ρ ρ γ π ρ ρ + = - - - ∂ - ∂ - ∂ v D Π A . ( 29
)

Complementarity of description in terms of the potentials and field strengths

We will demonstrate the equivalence of equations of motion (26), (27) and equations [START_REF] Fedosin | About the cosmological constant, acceleration field, pressure field and energy[END_REF].

For this we will need expressions (23) written in a covariant vector form. The field tensors' components according to (23) are the strengths and solenoidal vectors, which by definition depend on the time and space derivatives of the corresponding scalar and vector potentials:

t ψ ∂ = -∇ - ∂ D Γ , = ∇ × Ω D, t ϕ ∂ = -∇ - ∂ A E , = ∇ × B A, t ϑ ∂ = -∇ - ∂ U S , = ∇ × N U, t ∂ = -∇℘- ∂ Π C , = ∇ × I Π . (30)
We will use the operator equation

d dt t ∂ = + ⋅∇ ∂ v
on the left-hand side of equation ( 26), and on the right-hand side of (26), in view of the definitions ( , )

x ct ν =
r and ( , ) dx c dt ν = v , we will expand the products of the four-vectors into their components, so we will obtain, for example, the following:

y x z x y z D D D D dx v v v dt t t t t t t t ν ν ψ ψ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = - - - = -⋅ ∂ ∂ ∂ ∂ ∂ ∂ ∂ D v .
Transforming also the remaining terms on the right-hand side of (26), we arrive at the following:

( ) 0 0 0 0 q q t t t t ρ ρ ϑ ψ ϕ ρ ρ ∂ ∂ ∂ ∂ ⋅∇ + +℘ + ⋅∇ = -⋅ -⋅ - ⋅ -⋅ ∂ ∂ ∂ ∂ U D A Π v v v v v v .
Now we can take into account (30):

0 0 0 q ρ ρ ⋅ + ⋅ + ⋅ + ⋅ = v S v Γ v E v C .
This equality coincides with the first equality in [START_REF] Fedosin | About the cosmological constant, acceleration field, pressure field and energy[END_REF].

Taking now the Cartesian coordinates for simplifying, we will project all the terms in (27) onto the axis OX , so that in the equation we will have

1 i = , 1 x x ∂ ∂ =∂ = ∂ : 0 0 0 0 ( ) q q x x x x d U D d A U D A dx dx dx dx dt dt dt x dt x dt x dt x ν ν ν ν ν ν ν ν ρ ρ Π π ρ ρ + + ∂ ∂ ∂ ∂ + = - - - - ∂ ∂ ∂ ∂ . (31)
In view of (30), we will expand the products of the four-vectors into their components on the right-hand side of (31):

[ ] [ ] ( ) [ ] . y x x x x x z x y z x x x y z x x x x x x x D D D D D D D D dx v v v v v v dt x x x x x t x y z D dD D t dt ν ν ψ Γ Γ Γ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ - = - + + + = + + + × + + = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = + + × + ⋅∇ = + + × ∂ v Ω v Ω v v Ω 0 0 0 0 0 0 0 0 [ ] q q q q x x x A dA dx dt x dt ν ν ρ ρ ρ ρ Ε ρ ρ ρ ρ ∂ - = + + × ∂ v B , [ ] x x x d dx C dt x dt ν ν π Π ∂ - = + + × ∂ v I , [ ] x x x U dU dx S dt x dt ν ν ∂ - = + + × ∂ v N .
Substitution of all this into (31) gives the following:

0 0 0 0 [ ] [ ] [ ] [ ] 0 q q x x x x x x x x S C ρ ρ Γ Ε ρ ρ + × + + × + + × + + × = v N v Ω v B v I .
This equation coincides with the projection of the second equation in [START_REF] Fedosin | About the cosmological constant, acceleration field, pressure field and energy[END_REF] onto the axis OX .

Thus, equations of motion (26) and ( 27) are equivalent to equations [START_REF] Fedosin | About the cosmological constant, acceleration field, pressure field and energy[END_REF], which is a consequence of the relation between the strengths and the field potentials in (30).

In addition to the presence of proper equations of motion, the complementarity of the potentials and the field strengths is emphasized by the fact that for them there are proper field equations. Thus, for the four fields used here, the field equations for determining the strengths and the solenoidal vectors have the following form [START_REF] Fedosin | About the cosmological constant, acceleration field, pressure field and energy[END_REF]:

2 4 G Φ J c α β α β π ∇ = , 0 Φ Φ Φ σ µν ν σ µ µ ν σ ∇ + ∇ + ∇ = , 0 F j α β α β µ ∇ = - , 0 F F F σ µν ν σ µ µ ν σ ∇ + ∇ + ∇ = , 2 4 u J c α β α β π η ∇ = - , 0 u u u σ µν ν σ µ µ ν σ ∇ + ∇ + ∇ = , 4 f J c α β α β π σ ∇ = - , 0 f f f σ µν ν σ µ µ ν σ ∇ + ∇ + ∇ = . ( 32
)
In the first equations in (32), the field tensors can be replaced by their expressions in terms of the four-potentials according to (23). After this, we should use the rule for the difference of the second-order covariant derivatives of the arbitrary four-vector B β :

B B B R α β α β µ α β β µ ∇ ∇ =∇ ∇ - .
where R α µ is the Ricci tensor with the mixed indices.

With this in mind, we arrive at the field equations for the four-potentials of the fields:

2 4 G D J D D R c β α α α β µ α β β µ π ∇ ∇ = - + ∇ ∇ - , 0 A j A A R β α α α β µ α β β µ µ ∇ ∇ = + ∇ ∇ - . 2 4 U J U U R c β α α α β µ α β β µ π η ∇ ∇ = + ∇ ∇ - , 2 4 J R c β α α α β µ α β β µ π σ π π π ∇ ∇ = + ∇ ∇ - . ( 33
)
As a rule, the four-potentials of the fields are gauged in such a way that their covariant divergences are equal to zero:

0 D β β ∇ = , 0 A β β ∇ = , 0 U β β ∇ = , 0 β β π ∇ = . ( 34
)
The use of (34) leads to simplification of (33):

2 4 G D J D R c β α α µ α β µ π ∇ ∇ =- - , 0 A j A R β α α µ α β µ µ ∇ ∇ = - . 2 4 U J U R c β α α µ α β µ π η ∇ ∇ = - , 2 4 J R c β α α µ α β µ π σ π π ∇ ∇ = - . ( 35
) 4 4 f J J f R f R R f c c σ ρ ρ η λ σ µν µ ν ν µ ν ρ µ µ ρ ν µν λη π σ π σ ∇ ∇ = ∇ - ∇ + - + . (36)
Equations (36) represent the wave equations for the field tensors in the curved spacetime.

In the Minkowski spacetime, the covariant derivative becomes a four-gradient, the covariant d'Alembertian

β σ β σ ∇ ∇ =∇ ∇ is transformed into the ordinary d'Alembertian 2 2 2 2 2 2 2 2 2 2 2 2 1 1 c t c t x y z β β ∂ ∂ ∂ ∂ ∂ ∂ ∂ = -∆ = - - - ∂ ∂ ∂ ∂ ∂ , the Ricci tensor R α µ
and the curvature tensor , R µν λη vanish. In this case, equations (32) can be presented as the equations for determining the fields' strengths and the solenoidal vectors:

0 4 G π γ ρ ∇ ⋅ = - Γ , 2 2 1 4 G c t c π ∂ ∇ × = - ∂ Γ J Ω , 0 ∇ ⋅ = Ω , t ∂ ∇ × = - ∂ Ω Γ . 0 0q µ γ ρ ∇ ⋅ = E , 0 2 2 1 c t c µ ∂ ∇ × = + ∂ j E B , 0 ∇ ⋅ = B , t ∂ ∇ × = - ∂ B E . 0 4π ηγ ρ ∇ ⋅ = S , 2 2 1 4 c t c πη ∂ ∇ × = + ∂ S J N , 0 ∇ ⋅ = N , t ∂ ∇ × = - ∂ N S . 0 4π σ γ ρ ∇ ⋅ = C , 2 2 1 4 c t c πσ ∂ ∇ × = + ∂ C J I , 0 ∇ ⋅ = I , t ∂ ∇ × = - ∂ I C . (37)
In (37) 0q γ ρ = j v is the electric current density, γ is the Lorentz factor, 0q ρ is the invariant charge density, v is the velocity of motion of the matter's particles, and 0 γ ρ = J v is the mass current density.

In this case, equations (35) turn into the wave equations for the four-potentials in the framework of the special theory of relativity. We can expand these equations and write them separately for the scalar and vector potentials:

0 4 G β β ψ π γ ρ ∂ ∂ =- , 2 4 G c β β π ∂ ∂ =- D J, 0 0 q β β γ ρ ϕ ε ∂ ∂ = , 0 β β µ ∂ ∂ = A j, 0 4 β β ϑ π ηγ ρ ∂ ∂ = , 2 4 c β β πη ∂ ∂ = U J, 0 4 β β π σ γ ρ ∂ ∂ ℘= , 2 4 c β β πσ ∂ ∂ = Π J. (38)
The inhomogeneous wave equations (36) for the field tensors are also simplified, because the terms with curvature vanish. Since the tensor components are expressed in terms of the fields' strength vectors and the fields' solenoidal vectors, the wave equations can be written for each such vector:

0 0 0 1 ( ) q t β β γ ρ µ ε ∂ ∂ ∂ =-∇ - ∂ j E , 0 β β µ ∂ ∂ = ∇× B j, 0 2 4 4 ( ) G G c t β β π π γ ρ ∂ ∂ ∂ = ∇ + ∂ J Γ , 2 4 G c β β π ∂ ∂ = - ∇ × Ω J, 0 2 4 4 ( ) c t β β π η π η γ ρ ∂ ∂ ∂ =- ∇ - ∂ J S , 2 4 c β β π η ∂ ∂ = ∇ × N J, 0 2 4 4 ( ) c t β β π σ π σ γ ρ ∂ ∂ ∂ =- ∇ - ∂ J C , 2 4 c β β π σ ∂ ∂ = ∇ × I J. (39)
It follows from ( 38) and (39) that in the empty space, where there are no charges and currents and therefore 0 0 q ρ = , 0 = j , the following relations must hold for the field potentials, the electric field strength and the magnetic field of the electromagnetic wave: 0

β β ϕ ∂ ∂ = , 0 β β ∂ ∂ = A , 0 β β ∂ ∂ = E , 0 β β ∂ ∂ = B
. The similar relations in the empty space, in the absence of matter and its fluxes, must also be valid for the gravitational field in the framework of the covariant theory of gravitation: 0

β β ψ ∂ ∂ = , 0 β β ∂ ∂ = D , 0 β β ∂ ∂ = Γ , 0 β β ∂ ∂ = Ω .
Although equations (37) are of the first order, they are vector equations and for each field they contain four separate coupled equations for the corresponding strengths and solenoidal vectors, which makes it difficult to solve them. In contrast, the solutions of equations (38) for the potentials have a standard form and are called solutions with retarded potentials. As a result, we can substitute the potentials found from (38) into the equations of rectilinear motion of the solid body (28) and (29), and we can determine the Lorentz factor γ and the velocity v as functions of the time and coordinates in the Minkowski spacetime. In practice, however, it usually happens in a different way: first, using the potentials found, with the help of (30) the fields' strengths and solenoidal vectors are calculated, and then they are substituted into the equations of rectilinear motion of the solid body (20), (21), and (22).

Since the components of the field tensors become known, they can be used to calculate the stress-energy tensors of the fields in (3).

Conclusion

In Section (2) we presented equation ( 2), which follows from the equation for the metric and is expressed in terms of the stress-energy tensors of the gravitational and electromagnetic fields, the acceleration field and the pressure field. Inside the matter, equation ( 2) is equivalent to the equation of motion of the matter, written in the form of ( 5), ( 7), ( 13), [START_REF] Fedosin | About the cosmological constant, acceleration field, pressure field and energy[END_REF]. This becomes possible due to the fact that the fields' energy fluxes transfer the fields' momenta, which are transmitted to the matter's particles during the interaction.

However, the meaning of equation ( 2) is much broader and it does not reduce only to the equation of motion. Indeed, equation ( 2) is valid for the space outside the matter's limits, defining there the balance of the fields' energy in accordance with the generalized Poynting theorem, as well as the corresponding field strengths in the space created by the energy fluxes.

Moreover, equation (2) in the limit of the weak field and low velocities, where the effects of spacetime curvature can be neglected, can be integrated over the four-volume. The subsequent application of the divergence theorem reduces integration over the four-volume to integration of the time components of the fields' total stress-energy tensor over the three-volume, if the instantaneous situation is considered at a certain given time point. This leads to the determination of the integral vector, which is conserved in closed systems. In contrast, the equation of motion of the matter in the form of ( 5) cannot be integrated over the volume and does not result in the integral vector.

The equations of motion are simplified for the ideal solid body that does not rotate in its proper center-of-momentum frame and moves rectilinearly, parallel to itself. In order to describe such a motion, we can apply equations ( 17) and ( 18), where the field tensors are used, and equations ( 19) with the fields' strengths and solenoidal vectors. In the flat Minkowski spacetime, the equations of motion are even more simplified and can be represented in the form of equations ( 20), ( 21) and ( 22), which show the rate of change of the energy density and the momentum density, respectively, depending on the value of the strengths and solenoidal vectors inside the matter.

In Section 4, the equation of motion in the form of (24-27) was fully expressed in terms of the derivatives of the fields' scalar and vector potentials taken with respect to time and coordinates. For the rectilinear motion of the solid body in the flat Minkowski spacetime, the equation of motion can be written in the form of two equations ( 28) and (29), equivalent to equations (20) and (21).

The fact that equations of motion ( 26) and ( 27) coincide with equations ( 14) can be proved using the definitions of the strengths and solenoidal vectors in terms of the fields' scalar and vector potentials indicated in (30). We provided this proof in Section 5.

Due to the relationship between the strengths and solenoidal vectors on the one hand, and the fields' scalar and vector potentials on the other hand, field equations (32) for the strengths and solenoidal vectors can be transformed into wave equations (35) for the fields' scalar and vector potentials. For the field tensors, we also obtained wave equations (36) in the curved spacetime. For all these equations, we provided their expressions in the framework of the special theory of relativity.

  the invariant mass density 0 ρ and the four-velocity u ν of the particles, as well as the definition of the acceleration tensor u U U

  out that in the relativistic uniform system in the form of a fixed sphere the scalar potential will be 2 c ϑ γ′=. As for the global vector potential of the acceleration field U , it will differ from zero only in the systems with directed fluxes of particles.

  Let us assume that the mass and charge of the body are constant in motion. Then after multiplying equations (20) and (21) by the mass m , we can introduce the force F and represent the equations as follows:

  four-potential of the gravitational field, described in terms of the scalar potential ψ and the vector potential D of this field, four-potential of the electromagnetic field, given by the scalar potential ϕ and the vector potential A of this field, four-potential of the pressure field, containing the scalar potential ℘ and the vector potential Π .Let us substitute (23) into[START_REF] Fedosin | Estimation of the physical parameters of planets and stars in the gravitational equilibrium model[END_REF], taking into account the expressions for the mass current 0

It is interesting that the equations with the d'Alembertian β β ∇ ∇ can be obtained not only for the four-potentials, but also for the field tensors themselves. The derivation of one such equation is shown in Appendix A using the example of the electromagnetic field equations in (25).

According to the relation (A10), we obtain the following:

.

Similarly, we can write for the tensors of the gravitational field, the acceleration field and the pressure field:

Thus, the mutual complementarity of the field strengths and field potentials is manifested not only in case of their presence in the Lagrange function and their participation in the equation of motion, but also in the corresponding equations for the fields themselves.

If we know the dependences of the mass density 0 ρ and the charge density 0q ρ , as well as the four-velocity u µ of the matter's motion at each point of the system at any time, this is enough to determine all the characteristics of the system. Indeed, first we find the mass four- , and then, using the equations of the corresponding fields, we calculate the field tensors, as well as the four-potentials of the fields.

After that, we can find the stress-energy tensors of the fields, the system's metric, the densities of the four-forces, we can solve the equation of motion and also calculate the energy, momentum, angular momentum, and the integral vector of the system's fields, which is associated with the fields' energy fluxes and is found by integrating equation ( 2) over the fourvolume. In addition, we can also determine the four-acceleration of the matter as the derivative of the four-velocity with respect to the proper time in the curved spacetime:

The solution of inverse problems, in which, for example, the dependence of the fourvelocity of the matter is unknown, but the fields' distribution as well as the distribution of mass and charge are specified, can be more difficult. In this case, first it is necessary to reestablish the dependence of the four-velocity of the matter with the help of the field equations, and then to determine the remaining characteristics of the system, including the four-acceleration.

Appendix A. Derivation of the covariant wave equation for the electromagnetic field tensor

The covariant equations of the electromagnetic field for the electromagnetic tensor F µν have the following form:

where 0 µ is the magnetic constant, j α is the charge four-current.

Let us take in (A1) the second equation without the field sources, and apply the covariant derivative σ ∇ to it:

Next, we will use the rule for the difference of the second-order covariant derivatives:

where , R ρ µ α β is the curvature tensor.

In view of this rule, we find:

, ,

Now we will take into account the first equation of the electromagnetic field from (A1) and the antisymmetry of the electromagnetic tensor:

Consequently, in (A3) and (A4) we will have the following:

For the curvature tensor, there is a rule: , ,

After multiplying this equality by g νη we obtain the following: , , , ,

Here if we assume that index ρ α = , then the following relation will be valid:

With this in mind, the second term on the right-hand side of (A3) and the third term on the right-hand side of (A4) are transformed as follows:

Let us transform the third term on the right-hand side of (A3) and the second term on the righthand side of (A4), in the latter case we will apply the operation of permutation of the indices

.

In view of (A5), (A7) and (A8), for the sum of (A3) and (A4) in (A2) we find the following:

The last term on the right-hand side of (A9) can also be transformed: