This paper provides derivation of some basic identities for complex four-component vectors defined in a complex four-dimensional spacetime frame specified by an imaginary temporal axis. The resulting four-vector identities take exactly the same forms of the standard vector identities established in the familiar three-dimensional space, thereby confirming the consistency of the definition of the complex four-vectors and their mathematical operations in the general complex spacetime frame. Contravariant and covariant forms have been defined, providing appropriate definitions of complex tensors, which point to the possibility of reformulating differential geometry within a spacetime frame.
Complex Spacetime Frame Four-Vector Identities Contravariant and Covariant Forms Complex Tensors1. Introduction
In a recent derivation [1] , the present author identified the unit wave vector to be the temporal unit vector within four-dimensional spacetime frame. The temporal direction is specified as an imaginary axis with unit vector, where is the imaginary number. General spacetime frame is then defined as a complex four-dimensional coordinate system spanned by the temporal unit vector and the three mutually perpendicular spatial unit vectors specifying the x, y, z axes, respectively. We take the temporal unit vector to have general orientation relative to the spatial unit vectors according to
The usual assumption, implicit in conventional four-vector mathematics that the temporal axis is perpendicular to all the three mutually perpendicular spatial axes, with, , , may occur only as a special case to be specified. The basic elements of the complex spacetime frame are complex four-component vectors, which we generally call four-vectors. We define a general four-vector in the form
with the imaginary temporal component defined by
where is a scalar quantity specifying the nature of the temporal component of the four-vector. As usual, denotes time. In general, the temporal component of each four-vector occurs along the imaginary temporal axis, , multiplied by a factor, where is the speed of light. We follow standard convention denoting four- vectors by uppercase letters, e.g., , while the usual three-component spatial vectors are denoted with arrows over letter symbols, e.g.,.
According to the general definition in Equations (2a)-(2b), the spacetime displacement four-vector X takes the form
while the corresponding event interval between two neighboring spacetime points, takes the form
The spacetime derivative four-vector, is defined as
where and is the usual three-component spatial gradient vector defined by
Interpreting in Equation (4a) in terms of the general four-vector in Equations (2a)-(2b) gives
We observe that the concept of imaginary temporal axis developed here, represents a rediscovery of the idea of imaginary time first introduced independently by Poincare [2] , Lorentz [3] and Einstein [4] in their original theories of electrodynamics or special relativity in a four-dimensional spacetime frame. These authors did not identify the temporal unit vector and therefore could not completely specify the complex spacetime frame and develop the full mathematical operations using complex four-vectors in the manner presented in this paper.
2. Mathematical Operations with Four-Vectors
The general four-component vector form in Equation (2a) with all unit vectors specified allows us to carry out four-vector mathematical operations in the complex spacetime frame in exactly the same manner as the standard mathematical operations with the familiar three-component vectors in three-dimensional space.
In developing the mathematical operations in general form, we shall take the temporal unit vector to be of general orientation relative to the spatial unit vectors, satisfying the conditions in Equation (1). We denote four-vectors with arrows, while the three-component spatial vectors are written in boldface as stated above. We use two general four-vectors and defined by
to develop the mathematical operations with four-vectors. The basic mathematical operations are essentially addition, subtraction, dot product, cross product, divergence and curl.
2.1. Addition and Subtraction
Four-vector addition and subtraction is straightforward, taking the form
2.2. The Dot Product
The dot product of the four-vectors and is obtained as
which we expand term by term, maintaining the order of components in the products and then substitute, from Equation (5), to obtain the dot product in the final form
2.3. The Cross Product
The cross product of the four-vectors and is obtained as
which we expand term by term, using, and then substitute, from Equation (5) to obtain the cross product in the final form
2.4. Divergence of a Four-Vector
Setting equal to the spacetime derivative four-vector according to
with
in the general four-vector dot product obtained in Equation (7b), we obtain the divergence of a general four- vector in the final form
2.5. Curl of a Four-Vector
Setting equal to the spacetime derivative four-vector according to Equations (9a), (9b) in the general four-vector cross product obtained in Equation (8b), we obtain the curl of a general four-vector in the final form
3. Four-Vector Identities
We now derive some basic four-vector identities in complex four-dimensional spacetime frame, which generalize standard vector identities in three-dimensional Euclidean space [5] [6] .
3.1. Curl of Gradient Four-Vector
A gradient four-vector generated through application of the spacetime derivative four-vector on a scalar function is obtained as
Setting the general four-vector equal to the gradient four-vector according to
with
in the general curl of a four-vector obtained in Equation (10), we obtain the curl of a gradient four-vector in the form
which on using the standard three-dimensional space vector analysis results
gives the final result
This shows that the curl of a gradient four-vector vanishes. This four-vector identity generalizes the corresponding vector identity in standard three-dimensional Euclidean space [5] [6] given in the first part of Equation (11e).
3.2. Divergence of Curl of a Four-Vector
Taking the divergence of the curl of the general four-vector in Equation (10), we obtain
which on expanding term by term becomes
Applying standard three-dimensional space vector analysis results
we express Equation (13b) in the form
Application of standard three-dimensional space vector identity
gives
which on using
takes the final form
Substituting Equation (13h) into Equation (13d) gives the final result
This shows that the divergence of curl of a four-vector vanishes. This four-vector identity generalizes the corresponding vector identity in standard three-dimensional Euclidean space [5] [6] given in the first part of Equation (13c).
3.3. General Vanishing Four-Vector Dot Product: <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/1-5300737x89.png" xlink:type="simple"/></inline-formula>
The important identity on the vanishing of the divergence of curl of a four-vector in Equation (14) can be generalized by taking the dot product of the four-vector and the cross product of the four-vectors and which on using the general result in Equation (8b) is obtained as
which we expand term by term and use standard three-dimensional space vector identities
to obtain
Applying a three-dimensional space vector identity
and using
gives
which we substitute into Equation (15c) to obtain the final result
This result generalizes the divergence of curl of a four-vector obtained in Equation (14). It is a generalization of the corresponding vector identity in standard three-dimensional Euclidean space [5] [6] given in the first part of Equation (15b).
3.4. Divergence and Curl of <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/1-5300737x100.png" xlink:type="simple"/></inline-formula>
For a scalar function, we use the definitions of and from Equations (2a) and (4a) to obtain
Expanding these term by term gives
which we reorganize to obtain the four-vector identities
These four-vector identities generalize the corresponding vector identities in standard three-dimensional Euclidean space [5] [6] .
3.5. Divergence of Four-Vector Cross Product: <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/1-5300737x108.png" xlink:type="simple"/></inline-formula>
We use the general form of the curl of a four-vector from equation (10) to obtain
after applying standard three-dimensional space vector identities
Subtracting Equation (17b) from Equation (17a) and applying standard three-dimensional vector identities, together with appropriate rules of differentiation of vector products, we obtain the final result
We now use the four-vector cross product from Equation (8b) to obtain
which we expand as appropriate and apply standard three-dimensional space vector identities
to obtain the final result
Substituting Equation (17f) into Equation (18c) gives the four-vector identity
This four-vector identity generalizes the corresponding vector identity in standard three-dimensional Euclidean space [5] [6] given earlier in Equation (13e).
3.6. The Curl of a Four-Vector: <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/1-5300737x119.png" xlink:type="simple"/></inline-formula>
Let us start by taking the four-vector curl of the curl of the general complex four-vector in Equation (10) to obtain
which we on expansion takes the form
Next, we take the four-vector gradient of the divergence of the general complex four-vector in Equation (9c) to obtain
which we expand in the form
We apply standard three-dimensional Euclidean space vector identities giving
which we substitute into Equation (19d) as appropriate to obtain the final form
Subtracting Equation (19b) from Equation (19h) and using standard three-dimensional Euclidean space vector identities giving
we obtain
which on reorganizing
takes the form
We easily obtain
which we substitute into Equation (20c) to obtain
Setting
in the general four-vector dot product in Equation (7b) gives
which we substitute into Equation (24e) and reorganize to obtain the four-vector identity
This four-vector identity generalizes the corresponding vector identity in standard three-dimensional Euclidean space [5] [6] given earlier in Equation (19g).
3.7. Gradient of Dot Product of Four-Vectors: <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/1-5300737x138.png" xlink:type="simple"/></inline-formula>
For general complex four-vectors, , we have
Expanding these term by term gives
We apply three-dimensional Euclidean space vector identities to obtain
Substituting Equations (24a)-(24e) into Equation (23e), adding the result to Equation (23f) and reorganizing gives
where we have applied a standard three-dimensional Euclidean space vector identity to introduce
Rewriting
we express Equation (24f) in the form
which on substituting the definition of the spacetime derivative four-vector from Equation (6a) and the four- vector dot product obtained in Equation (11c), gives the desired four-vector identity in the final form
This four-vector identity generalizes the corresponding vector identity in standard three-dimensional Euclidean space [5] [6] given earlier in Equation (24g).
3.8. Triple Cross Product of Four-Vectors: <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/1-5300737x158.png" xlink:type="simple"/></inline-formula>
We introduce a third four-vector defined by
and use the four-vector cross product to obtain
which we expand as
We apply standard three-dimensional Euclidean space vector identities to write
which we substitute into Equation (28b) and collect like terms to obtain
We rewrite the last two terms in Equation (28g) by subtracting and adding appropriate terms according to
which we substitute back and collect like terms to express Equation (28g) in the form
Finally, we apply the usual definitions of the four-vectors, and introduce the four-vector dot products
in Equation (29c) to obtain the desired four-vector identity in the form
This four-vector identity generalizes the corresponding vector identity in standard three-dimensional Euclidean space given earlier in Equation (28f).
We easily apply the four-vector identity obtained in Equation (30a) to establish the cyclic property of the triple four-vector cross product in the form
which generalizes the corresponding vector identity in standard three-dimensional Euclidean space [5] [6] .
The four-vector identities derived in Equations (12), (14), (15g), (16d), (18d), (22), (26), (30a) and (30b) confirm the consistency of the definitions of the complex four-component vectors and corresponding mathematical operations within the complex four-dimensional spacetime frame. This means that complex four-dimensional spacetime frame characterized by complex four-component vectors is a consistent mathematical extension of the standard three-dimensional space characterized by the usual three-component vectors. The other four-vector identities can be derived following similar procedure.
4. Contravariant and Covariant Four-Vectors
To complete the mathematical formalism within complex four-dimensional spacetime frame, we introduce contravariant and covariant forms, which are useful in carrying out general mathematical operations with four-vec- tors. A contravariant four-vector is specified by positive spatial components, while a covariant four-vector is specified by negative spatial components. We represent the four-vector in a contravariant form by and in a covariant form by, where with 0 labeling the temporal component, while 1, 2, 3 label the spatial components, respectively. We define and below.
Denoting the four unit vectors by as presented in this paper, we define the contravariant coordinates of the general four-dimensional complex spacetime frame in the form [7] [8]
The corresponding covariant coordinates are defined in the form
We then express the contravariant spacetime displacement four-vector and the corresponding covariant form as
which we introduce the position vector according to Equations (31a)-(31b) to express in the final forms
The spacetime event interval takes the contravariant and covariant forms
A general four-vector as defined earlier is expressed in contravariant and covariant forms according to
with
which we express in the final forms
The contravariant and covariant four-vectors are related through complex conjugation in the form
We use this contravariant-covariant four-vector conjugation relation to obtain
which provides the definition of the invariant length of the general four-vector or according to
We express this in the general form
Using, from Equation (32d), noting the relation in Equation (32e), we apply Equation (32g) or (32h) to obtain the invariant length in the explicit form
which is modified by a factor arising from the general orientation of the temporal unit vector relative to the spatial unit vectors according to the definition
The invariant length of the spacetime event interval is then modified according to
where is the velocity, with speed defined as usual. This result has important implications for current theoretical and experimental research in physics [9] [10] .
Tensors in the Complex Spacetime Frame
We now develop the procedure for defining tensors [7] [8] within the general four-dimensional complex spacetime frame. To put the presentation in familiar form, we adopt the standard contravariant and covariant four- vector notation to express and from Equation (20d) in the form
with complex conjugates taking the form
where the usual four-vector mathematics is applied, but now taking account of the general orientation of the temporal unit vector relative to the spatial unit vectors to obtain the general results presented above.
Using the complex conjugation relation from Equation (33b) gives
from which a definition of complex contravariant and covariant rank-2 tensors and follows according to
In addition,
provides a definition of complex rank-2 mixed tensors and in the form
The definition of more general tensors of higher rank follows easily. Some mathematical properties of the rank-2 tensors defined above can be obtained by interchanging the indices, or taking complex conjugation or carrying out both operations simultaneously.
The complete definition of contravariant and covariant complex four-vectors, which can be used to define tensors of general ranks in contravariant, covariant or mixed forms, provides the necessary foundation for more general vector and tensor analysis, leading to reformulation of differential geometry a complex four-dimensional spacetime frame. This is indeed the origin of a new framework for studying physics, mathematics and related disciplines in the 21st-century and beyond. Some important implications for physics are presented in [1] .
5. Conclusion
All the basic four-vector identities which we have derived in this work take exactly the same form as the standard vector identities established in the familiar three-dimensional space. This confirms the consistency of the definition of complex four-component vectors and corresponding mathematical operations within a complex four-dimensional spacetime frame with an imaginary temporal axis. The contravariant and covariant forms introduced here lead to consistent definitions of complex tensors, which are the basic quantities for reformulation of differential geometry within complex spacetime frame. This new mathematical framework has important implications for various models of relativistic mechanics, quantum field theory and general relativity as a theory of gravitation and cosmology.
Acknowledgements
I thank Maseno University, Kenya, for supporting this work by providing facilities and working environment.
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