Programa de mantenimiento a las plantas de tratamiento de la Ciudad

Edward Kapu´scik

Department of Physics, University of L´od´z L´od´z, Poland

It is rather unlikely that the physics community will accept the approach presented by L. Ferreira in his paper. The paper is how- ever a beautiful illustration of the fact that an exciting idea (to view antiparticles as traces of tachyons) may fail if one tries to incorpo- rate it into traditional standard frame. I am deeply convinced that completely new ideas may be developed only after drastic change of existing theories.

All extensions of Special Theory of Relativity always stumble on the notion of proper timeτ related to physical time by the relation

dτ =dt s 1− − →_{v (t)}2 c2 . (1)

where −→v (t) is the velocity of the moving body the motion of which is investigated. Clearly, it is the square root which restricts the ve- locities of motion to the domain

The appearance of the proper time comes from the generalization of the Galilean invariant Newton equations

d−→x(t) dt = − →_{v (t), (3)} d−→p (t) dt = − → F(t) (4) to Lorentz invariant equations

dxµ_(τ) dτ =u µ_{(τ), (5)} dpµ_(τ) dτ =F µ_{(τ). (6)}

The above and commonly used passage from Galilean-Newton mechanics to Einstein one has at least two serious disadvantages:

1) the parameter τ is not a monotonically increasing function of physical timet,especially for periodic motions,

2) the scheme cannot be generalized to many body systems where each particle has its own proper time.

Fortunately enough, there exists another, quite different, way of passing from the Galilean to Einsteinian mechanics in which the proper time never appears. In fact, let us first rewrite eq. (3) in the form

d−→x(t) =−→v (t)dt (7) and then use its relativistic analog in the form

dxµ=Vνµ(x)dxν, (8) where Vµ

ν (x) is some tensor field which we shall call the velocity tensor. In the Galilean case this tensor depends only on the time variable and in the matrix representation is given as

V (t) =     1, 0, 0, 0 vx(t), 0, 0, 0 vy(t), 0, 0, 0 vz(t), 0, 0, 0     . (9)

It is easy to check that (9) indeed represents a Galilean tensor and using (9) in (8) we immediately get (7).

In general, (8) is an eigenequation for the matrix V(x) with eigenvalue 1. We want to have this eigenvalue as the only non-zero and non-degenerated eigenvalue. Otherwise, the relation (8) will not uniquely determine the eigenvector dx0_{, dx}1_{, dx}2_{, dx}3

and the mo- tion will not be uniquely described. In order to fulfill this require- ment, the characteristic equation for the eigenvaluesλof the matrix V : (−λ)n+ (−λ)n−1T r1V + (−λ) n−2 T r2V +.... + (−λ)2T rn−2V + (−λ)T rn−1V +T rnV = 0 (10) must reduce to the simple form

λn−1(λ−1) = 0. (11)

Here, nis the dimension of spacetime andT rjV are the sums of diagonal minors of the dimensionj. Clearly, 1≤j ≤nand

T r1V =T rV, T rnV = detV. (12) The eigenvaluesλ= 0 all correspond to the rest of the bodies.

From (11) and (10) the matrixV must obey the conditions T r1V = 1,

T rjV = 0,

(13) for 2≤j ≤n. Unfortunately, it is not easy to solve these conditions for arbitraryn. But forn= 2 it can be done explicitly because in this case the two-dimensional matrixV must satisfy only two conditions T rV = 1 (14) and

detV = 0. (15)

It is easy to check the matrix V (x) = 1−k(x)β(x), k(x) [1−k(x)β(x)]β(x), k(x)β(x) (16) indeed satisfies conditions (14). Herek(x) is some arbitrary function of spacetime variables whileβ(x) was introduced in order to get from (8) the relation

which identifiesβ(x) with the ordinary velocity in (7).

Since we want to have the velocity tensor depending only on the ordinary velocityβ(x) we shall further assume that the functionk(x) in (16) is a function of the velocityβ(x), i.e.:

k(x) =K(β(x)). (18)

It turns out that the shape of this function can be fixed from the transformation rule of the velocity tensor

Vνµ(x)→V µ0 ν0 (x0) = ∂xµ0 ∂xαV α β (x) ∂xβ ∂xν0 (19)

under the change of coordinatesxµ _→xµ0_{. The explicit form of these} transformation rule depends on the symmetries of spacetime. In the Galilean case from (16), (17) and (18) we get the functional equation K(β−βr) =K(β), (20)

where βr is the relative velocity of two inertial systems of reference. Clearly, eq. (19) means that K(β) is a constant. In the Newtonian mechanics this constant vanish.

In the Minkowski spacetime from (16), (17) and (18) we get the functional equation K _β −βr 1−ββr = βr(1−ββr) 1−β2 r +K(β)(1−ββr) 2 1−β2 r (21) with the solution

K(β) = κ−β

1−β2, (22)

whereκis an arbitrary constant. The relativistic velocity tensor has therefore the form

V (t) = 1 1−β(t)2 1−κβ(t), κ−β(t) [1−κβ(t)]β(t), [κ−β(t)]β(t) . (23)

It is seen that this expression has a singularity at β(t)2 = 1. This singularity, however, does not forbid to consider the case when β(t)2>1.

To formulate the dynamical laws we may use the fact that the only generally covariant equation, similar to the Newtonian equation (4), has the form

∂µπµν(x) =Fν(x), (24) whereπµν_{(x) is an antisymmetric tensorial density andF}ν_{(x) is the} density of force.

The next step is the use of the constitutive relation

πµν(x) =Mµρ(x)Vρν(x)−Mνρ(x)Vρµ(x) (25)

which is the analogy of the non-relativistic relation

−

→_{p(t) =m}−→_{v (t). (26)}

Assuming that the mass tensorMµρ_{(x) is a function of the velocityβ} and proceeding exactly as above from the relativistic transformation rule ofMµρ_{(x) we get the following general shape of the mass tensor}

M00_{(β) =} 1 1−β2 M00−βM01−βM10+β2M11 M01_{(β) =} 1 1−β2 M01−βM00−βM11+β2M10 M10(β) = ₁₋1_β2 M 10_−βM11_−βM00_+β2_M01 M11(β) =₁₋1_β2 M 11_−βM01_−βM10_+β2_M00 . (27)

As it is seen there is no square root p1−β2 _{either but there is} a singularity atβ2_{= 1.This singularity can be however removed by} the choice

M11_=−M00_,

M10_=−M01_. (28) In this case the mass tensor has the simple form

M = M00 M01 −M01 _−M00 (29) and the inertial properties of bodies are described by two mass pa- rameters.