1. Introduction
In special relativity, the word “special” alludes to invariance under transformations relating constant relative velocity frames of reference, which are known as Lorentz transformations, and a Lorentz-invariant quantity is one that assumes an identical form under a Lorentz transformation. A very attractive notion is that the fundamental structures and mechanisms of the universe are somehow connected with the invariances of the underlying model. A curious fact associated with general relativity is that while spiral galaxies are common in the universe, there appear to be no simple exact solutions of general relativity that reflect these structures. Yet, in both fluid and solid mechanics, logarithmic spirals arise from the invariance of the underlying equations under simple one-parameter stretching and rotations. In this paper, we examine those special relativistic three-dimensional motions for which the three velocity components in the
,
and
directions are invariant under arbitrary three-dimensional Lorentz transformations. These results might find physical application in cosmological theories with background vector fields, such as proposed in [
1,
2].
While Lorentz invariance and its consequences are well established in special relativity, it seems to have been overlooked that the imposition of a Lorentz-invariant velocity field restricts the functional form of the velocity to the solution of a certain partial differential equation. Here, for three-dimensional motion with velocity components , and in the , and directions, respectively, the requirement that the three velocity equations , and remain invariant under an arbitrary three-dimensional Lorentz transformation implies that the three velocity components , and satisfy certain coupled first-order partial differential Equations (3.1) or (3.3), leading to their functional forms. We note that, by the Lorentz invariance of the three differential equations , and , we mean that, under an arbitrary Lorentz space-time transformation, the same three differential equations are obtained in the transformed space-time variables. Further, since the Lorentz transformation forms a one-parameter Lie group, we may deduce the governing partial differential equations from an examination of the infinitesimal version of the one-parameter Lie group. Only in the derivation of the partial differential equations do we assume an infinitesimal frame velocity.
We determine general functional forms of the velocity components in terms of four arbitrary functions , , and . Assuming that the function is determined from the relation , we note in particular the singular case for which the magnitude of the particle velocity is the speed of light, which means that there are infinitely many families of paths for which the particles are moving at the speed of light. The existence of these infinite families of paths with particles travelling at the speed of light indicates the endless possibilities existing at the speed of light.
In [
3,
4], the author has given corresponding results for the cases of one- and two-dimensional special relativistic motions, respectively. For a single spatial dimension
x, the one-dimensional velocity
satisfies the first-order partial differential equation
while, for planar motions, using plane rectangular Cartesian coordinates
, the velocity components
and
satisfy the following coupled partial differential equations
where
is the planar angle corresponding to
in the three-dimensional formulation. In terms of solutions, the one-dimensional Equation (1.1) derived in [
3] is far more restrictive. Here, we follow closely the development [
4], and many of the basic formulae and calculations presented here does not differ significantly from those presented in [
4], except, of course, that the results here are fully three-dimensional. Accordingly, here, we present the full formulae, but as concisely as is feasible.
We remind the reader that, for those problems involving partial differential equations and boundary or initial conditions, in order for the present analysis to be useful, it is necessary to ensure the invariance of both the equation and any associated conditions under a one-parameter Lie group of transformations. If this is the case, then, generally speaking, invariance under a one-parameter Lie group of transformations implies the major simplification of the problem (see, for example, [
5]). In the present context, any solutions of the coupled partial differential Equations (3.3) will generate solutions of those special relativistic problems provided that any boundary or initial conditions also remain invariant under Lorentz transformation. This means that any associated boundary or initial conditions must be assumed to be expressible in terms of invariants of the full three-dimensional Lorentz group (2.1).
In the following section, we summarise the essential results of special relativity theory that are needed in order to deduce the partial differential Equations (3.1) or (3.3) for the velocity components
,
and
. For fully three-dimensional motion, these partial differential equations are derived in the subsequent section, and the corresponding partial differential equations for energy and momentum are derived in the section thereafter. The calculation details for the solutions (3.6) for
,
,
and
in terms of four arbitrary functions
and
where
are presented in
Appendix A, and the derived solutions are summarised and illustrated in the final sections of the paper.
2. Results from Special Relativity
Special relativity has become a standard subject such that almost every text on physics or mechanics has a dedicated chapter on special relativity. The older texts are closer to the original motivating issues and the developments that gave birth to the subject. Dingle [
6] and McCrea [
7] provide student texts, while more comprehensive accounts can be found in Bohm [
8], French [
9] and Resnick [
10]. Both Moller [
11] and Tolman [
12] provide standard works of reference, and the reader may wish to consult [
13], containing the original papers of Einstein, Lorentz, Minkowski and Weyl with notes by Arnold Sommerfeld.
The notion of invariance with respect to frames moving with a constant relative velocity underlies special relativistic mechanics, and particularly those transformations of space and time leaving the wave equation unchanged, referred to as Lorentz transformations. We consider a rectangular Cartesian frame and another rectangular Cartesian frame moving with constant frame velocity relative to the first frame, where , and are the direction cosines of the frame velocity such that and denotes the magnitude of the frame velocity .
We view the magnitude of the relative velocity as a measure of the departure of the current frame from the rest frame , and, throughout, we adopt consistent notation, employing lowercase for variables associated with the moving frame and uppercase or capitals for those variables associated with the rest frame. Accordingly, time is measured from the frame with the variable T and from the frame with the variable t so that and are the variables of principal interest and we assume that the two frames coincide initially.
For
, from [
11] (p. 42), the standard fully three-dimensional Lorentz transformations with fixed cosine angles
,
and
are given by
where
, with the inverse and the identity transformations characterised by
and
, respectively. From the above relations for
, we may deduce
and the geometric identity arising as a consequence of zero relative motion perpendicular to the direction of motion
A second relation arises as a consequence of relative motion in the direction of motion:
The invariance (2.4) describes the measurement of the distance perpendicular to the direction of relative motion and confirms the isotropy of space independently of time, and it is most easily proven by writing each of the spatial components of (2.1) in the form
and squaring and adding. The invariance (2.4) describes the time-dependent coupling in the direction of travel and is easily established using the working variables
and
and evaluating
from the two relations
and
. Together, the relations (2.3) and (2.4) yield the well-known special relativistic identity equivalent to the Minkowski line element, namely
The three invariances (2.3), (2.4) and (2.5) subsequently arise in the solution of the coupled partial differential Equations (3.6) given in
Appendix A.
With velocity components
,
and
in the
,
and
directions, respectively, defined by
we may deduce, by the division of the differential versions of (2.1), the Einstein addition of velocity laws in the
and
directions, respectively; thus,
noting that we have the relations
By squaring and adding the above relations (2.7), we may show that
and, from this equation, it is clear that
if and only if
. Further, we have the important relation
With energy and momentum in the three spatial directions in the two frames defined, respectively, by
where
is the rest mass and
, we may deduce from (2.8) and the velocity relations (2.6) the Lorentz-invariant energy momentum relations
and the identities
3. Lorentz-Invariant Velocity Components , and
In this section, we determine the most general three-dimensional velocity field
with velocity components
,
and
in the
,
and
directions, respectively, that remain invariant under the Lorentz transformation (2.1) . Equivalently, we determine the velocity components
,
and
such that the three differential problems
,
and
transform into
,
and
, respectively, under the general three-dimensional Lorentz transformation (2.1). Since the Lorentz transformation (2.1) forms a one-parameter group of transformations in the frame velocity
, we need only to expand (2.1) to the first order in
and equate the corresponding infinitesimals to obtain the first-order partial differential equations for
,
and
. For infinitesimally small velocities
, on retaining only the linear terms involving
, the Lorentz transformation (2.1) becomes simply
so that, for example, on expanding
, we obtain
which, on expanding and equating the first-order terms in
and then reverting to the
variables, and with similar calculations for
and
, we may deduce the following coupled partial differential equations for
,
and
:
On making use of the inverse relations to (2.1), namely
and (2.2), we may show, using the chain rule and by direct substitution, that the linear partial differential operator
L is Lorentz-invariant, namely
and the coupled partial differential Equation (3.1) becomes simply
In terms of a vector and matrix notation
,
and
defined, respectively, by
Equation (3.3) becomes
and it is a simple matter to show that
, so that
if and only if the particle is travelling at the speed of light. This particular characteristic is shared by both the corresponding one- and two-dimensional Equations (1.1) and (1.2) derived, respectively, in [
3] and [
4].
On introducing
AB,
C and
D through the relations
we may show that Equation (3.3) takes on the remarkably simple form
With the substitution
for some function
, these equations become
and the final three equations may all be re-written since, for example,
so that Equation (3.3) is finally simplified to become
where
. In the following section, we present the corresponding partial differential relations for energy and momentum. The calculation details for the solutions of the coupled partial differential Equation (3.6) for
,
,
and
are presented in
Appendix A, and the solutions are summarised in the subsequent section.
4. Partial Differential Relations for Energy and Momentum
From the relations (2.9) for energy and momentum, we may deduce the following expressions for the partial derivatives
with similar expressions for the partial derivatives with respect to
y,
z and
t. On making use of these relations, and with some rearrangement and division of (3.1) by
, we may deduce the partial differential relations connecting the partial derivatives of the momentum and energy
p,
q and
e; thus,
where
.
In terms of the invariant operator
L defined by (3.2) , these partial differential relations become more transparent and are simply
which arise using the definitions (2.9) and the relations (3.3) as follows:
Formally, we may use the results (4.2) to obtain
and
Further, for (2.11), we may apply the operator L to to confirm the validity of the equation , as might be anticipated. The partial differential relations (3.1) are also fully compatible with the Lorentz-invariant energy–momentum relations (2.10). Since L is a Lorentz-invariant operator, the application of L to the first equation of (2.10) yields a linear combination of the second and third equations of (2.10), while its application to both the second and third yields the first relation.
5. Summary of Solutions of Coupled Partial Differential Equations (3.3)
Equations (3.3) constitute first-order partial differential equations, which are formally solved in
Appendix A using Lagrange’s characteristic method, leading to (3.6). The final details are as follows:
where
, and
,
,
and
all denote arbitrary functions of the indicated arguments
as given below,
noting that
. As described in
Appendix A, the final solutions of (3.5) for
A,
B,
C and
D as defined by (3.4) are given by
where
and
denote arbitrary functions of the indicated arguments, for
and
.
From the definitions (3.4) of
A,
B,
C and
D, the general Lorentz-invariant velocity components
,
and
inherit the particular functional forms determined from the relations
where
and
denote four arbitrary functions of the indicated arguments, for
and
. From the above equations, we may readily deduce
By squaring and adding the four relations in either (5.4) or (5.5), we obtain
from which it is clear that if
is determined from the relation
, then
, and there exist infinitely many families of singular paths with particles moving at the speed of light
that arise from the special case
.
From Equations (2.9) and (5.6) and with the abbreviation
, we obtain the following expression for the particle energy
e,
while, from the expressions for the momenta
p,
q and
r
we obtain
In the final section of the paper, we provide some illustrations of these formulae assuming a specific dependence on the arbitrary functions and for .
Finally, in this section, we note an interesting connection with the covariant curvature tensor
. For general
, on using the expressions (5.5) with
and
, we have the differential relations
noting that we have used
, and, in the derivation of the second equation, we have used the first. Thus, from (5.6), we have
and, therefore, with the abbreviation
, we may deduce the metric
. For this metric in the two independent variables
, from [
14] (p. 56), there is only one non-zero component of the covariant curvature tensor
, namely
, which is given by
For a flat space, we expect , so that necessarily has the structure , where and denote arbitrary functions of .
6. Some Illustrations of the Solutions (5.1)
and the Momenta Expressions (5.9)
In this section, for the purposes of illustration and to verify the analysis at least for a special case, we assume a particular dependence on the four arbitrary functions
and
, where
. Specifically, we examine the case when these functions depend only on
and we use the notation
, where
. We first illustrate the solutions (5.1) for
and
and then give an application of the momenta expressions (5.9) . We consider the development of special relativity formulated in [
15], which predicts that the momenta
,
and
and the wave energy
each satisfy the planar classical wave equation, namely
where
denotes an applied external potential that is generating conventional spatial forces
,
and
and a non-conventional force
in the direction of time, such that
and
is more commonly recognised as the mass or energy production term. We refer the reader to [
15] for further details of this particular extension of special relativity.
We first assume that the arbitrary functions
and
are functions of
only, so that, with
A and
B defined by
where
, we might deduce the following expressions for the partial derivatives
with similar expressions for the partial derivatives with respect to
y and
z and for the functions
C and
D. On making use of these expressions for the partial derivatives, together with the definition (3.2) of the operator
L, it is easy to verify that the equations
and
are correctly satisfied.
The special relativity theory formulated in [
15] predicts that the components of momentum as given by (5.8) satisfy the classical three-dimensional wave equation (6.1). Assuming that the arbitrary functions
and
are functions of
only and that
, we see from (5.8) that the momentum
p in the
direction has the structure
, where the functions
and
here are defined by
and where
denotes
. The question therefore arises as to whether it is possible to choose the functions
and
(or equivalently the functions
and
) such that
satisfies the wave equation. On making use of partial differential expressions such as
with similar expressions for
and the partial derivatives in the
y and
z directions, the condition that
p satisfies the wave equation becomes
Thus, the functions
and
must be such that
and therefore any expressions of the form
and
where
for
denote four arbitrary constants will ensure that the wave equation is correctly satisfied.
As a simple illustration of this result, we consider the case
,
and
for certain constants
and
. In this case, the momentum is given by
which clearly satisfies the classical wave equation. The functions
and
are obtained by solving the two equations
from which we may readily deduce
so that
is a constant, and, evidently, the constants
and
must be of the same sign in order to ensure that
is well defined.
7. Conclusions
For fully three-dimensional motion with rectangular Cartesian coordinates
, we have shown that the requirement that the three velocity equations
,
and
remain invariant under the general three-dimensional Lorentz transformation (2.1) gives rise to the three coupled partial differential Equations (3.1) for the three velocity components
,
and
in the
,
and
directions, respectively. These first-order partial differential equations are solved using Lagrange’s characteristic method to deduce the solutions
in terms of four arbitrary functions
and
for
,
and
for
denote four independent integrals of the solution procedure that are defined by (5.2), namely
With the velocity components given by (7.1) , the corresponding particle energy e and momenta p, q and r are given, respectively, by (5.7) and (5.8).
We observe the very curious fact that with A, B, C and D given by (A.3) involving the four arbitrary functions , , and , the singular case is such that for all arbitrary functions , , and satisfying the constraint. The existence of these infinitely many singular families of paths with particles moving at the speed of light is indicative of the abundant possibilities that might exist in the “fast lane”. To the author’s knowledge, neither the coupled partial differential Equations (3.1) and (3.3), nor the particular functional forms determined from either (5.3) or (5.5), nor the fact that the energy e and momenta p, q and r satisfy the partial differential relations (4.1) or in operator form (4.2) have been previously given in the literature.