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Acta Polytechnica Vol. 51 No. 1/2011

The Velocity Tensor and the Momentum Tensor

T. Lanczewski

Abstract

This paper introduces a new object called the momentum tensor. Together with the velocity tensor it forms a basis for
establishing the tensorial picture of classical and relativistic mechanics. Some properties of the momentum tensor are
derived as well as its relation with the velocity tensor. For the sake of clarity only two-dimensional case is investigated.
However, general conclusions are also valid for higher dimensional spacetimes.

Keywords: relativistic classical mechanics, velocity tensor, momentum tensor.

1 Introduction
In [1], an object called the velocity tensor V μν (v) was described. It comes from a generalization of the equation

dx(t)− v (t) dt =0 (1)
into a generally covariant form

V μν (v) dx
ν =0. (2)

The two-dimensional matrix of the classical velocity tensor takes the form

V (v)= V 01

(
−v 1
−v2 v

)
, (3)

while in the relativistic case

V (β)= γ2V 01

(
−β 1
−β2 β

)
, (4)

where V 01 is some arbitrary constant, β = v/c and γ =
(
1− β2

)−1/2
.

As was shown in [1], the tensorial description has an obvious advantage over a standard description since it
does not use the notion of the proper time

τ = t

√
1−

v2 (t)
c2

(5)

and therefore it allows a description of non-uniformmotions and systems with an arbitrary number of material
points. It also provides a cornerstone for formulating a generally covariant mechanics. However, the velocity
tensor deals solely with kinematical issues. To make the tensor description complete we need to introduce
another tensorial object called the momentum tensor Πμν(v). By means of this tensor it is possible to solve
dynamical problems.

2 Definition of the momentum tensor

In classical and relativistic mechanics the following formula holds true [2]:

dp(x, t)
dt

= F(x, t). (6)

The tensorial equivalent of Eq. (6) is presumed to be

∂μΠ
μ
ν(x, t)=Φν(x, t), (7)

where Πμν(x, t) is the momentum tensor and Φν(x, t) is an influence of the exterior on a body. It should be
stressed here that we do not assume a priori the relationship between F(x, t) and Φν(x, t). The choice of
the form of the mixed tensor Πμν(x, t) comes from the assumption that the momentum tensor should be some
function of the velocity tensor. Since the velocity tensor is a function of a classical velocity v, the momentum
tensor is Πμν(x, t) :=Π

μ
ν(v).

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Acta Polytechnica Vol. 51 No. 1/2011

3 General construction of the momentum tensor

In general, the momentum tensor Πμν(v) is represented by a square matrix

Π(v)=

⎛
⎜⎜⎜⎜⎜⎝
Π00(v) Π

0
1(v) · · · Π

0
n(v)

Π10(v) Π
1
1(v) · · · Π

1
n(v)

...
...

...
...

Πn0(v) Π
n
1(v) · · · Π

n
n(v)

⎞
⎟⎟⎟⎟⎟⎠ ,

where the elements Πμν(v) are some functions of velocity v variable with time. In order to determine them,
we make use of the transformation relation for a mixed tensor. Passing from an inertial reference frame S
to an inertial system S′ that moves with velocity u relative to S, the momentum tensor Πμν(v) transforms in
accordance with the following formula

Πμν(v) → Π
μ′

ν′(v
′)= Lμ

′

μ (u)Π
μ
ν(v)L

ν
ν′(u), (8)

or in matrix notation
Π(v) → Π′(v′)= L(u)Π(v)L(−u). (9)

Assuming that Π(v) is form-invariant, i.e. Π′(v′)= Π(v′), we arrive at a functional equation for Π(v) in the
form

Π(v′)= L(u)Π(v)L(−u), (10)
where v′ is the velocity of a material point in the system S′ and v is its velocity in S. It is easy to prove [1]
that after some simple substitutions and rearrangements in Eq. (10) we get the solution

Π(v)= L(−v)Π(0)L(v), (11)

where Π(0) is an arbitrary square matrix formed by constant elements.

4 Two-dimensional momentum tensor

4.1 Non-relativistic case

In this case we substitute in Eq. (11) the Galilean transformation in the form

G(v)=

(
1 0

−v 1

)

and hence we get

Π(v)=

(
1 0

v 1

)(
Π00 Π

0
1

Π10 Π
1
1

)(
1 0

−v 1

)
=

(
Π00 − vΠ

0
1 Π

0
1

Π10 + v
(
Π00 −Π

1
1

)
− v2Π01 Π

1
1 + vΠ

0
1

)
, (12)

where all elements Πμν in Eq. (12) are constant. Since the above equation is only time-dependent, Eq. (7) leads
to the expression

∂0Π
0
ν(v)=Φν , (13)

where ∂0 = d/dt, and therefore we get

Φ0 = ∂0Π
0
0(v)= ∂0

(
Π00 − vΠ

0
1

)
= −v̇Π01 (14)

and
Φ1 = ∂0Π

0
1(v)= ∂0Π

0
1 =0. (15)

Hence, in order to reconstruct the classical Newtonian equation of motion we have to assume that

Π01 = m and Φ0 = −F, (16)

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Acta Polytechnica Vol. 51 No. 1/2011

where m is mass of a material point and F is a classical Newtonian force in a two-dimensional spacetime. The
choice of the sign in Eq. (16) results from considerations in higher dimensional spacetimes.
It results fromEqs. (12), (14) and (15) that only the element Π01 takes part in dynamical processes since no

other coefficient appears in Eq. (14). Therefore, the other elements may take arbitrary values and each specific
choice among them will lead to the same dynamics. In particular, we may choose them in such way that the
relation

Π(v)= mV (v) (17)

is satisfied. Keeping in mind that V (v) is given by Eq. (3), we get that

Π(v)=Π01

(
−v 1
−v2 v

)
. (18)

The fact that in the considered caseΦ1 =0 leads to the general assumption that the componentΦ0 plays a key
role in the dynamics, and the components Φk are auxiliary quantities that provide the formalism covariance.

4.2 Relativistic case

In the case of substituting into Eq. (11) the Lorentz transformation given by

L(β)= γ

(
1 −β
−β 1

)
we get that

Π(β)= γ2
(
Π00 + β(Π

1
0 −Π

0
1)− β

2Π11 Π
0
1 + β(Π

1
1 −Π

0
0)− β

2Π10
Π10 + β(Π

0
0 −Π

1
1)− β

2Π01 Π
1
1 + β(Π

0
1 −Π

1
0)− β

2Π00

)
. (19)

According to Eq. (13) we obtain that

Φ0 = ∂0Π
0
0(β)= ∂0γ

2
[
Π00 + β(Π

1
0 − Π

0
1)− β

2Π11
]

= γ4β̇
[(
1+ β2

)(
Π10 −Π

0
1

)
+2β

(
Π00 −Π

1
1

)]
, (20)

Φ1 = ∂0Π
0
1(β)= ∂0γ

2
[
Π01 + β(Π

1
1 − Π

0
0)− β

2Π10
]

= γ4β̇
[(
1+ β2

)(
Π11 −Π

0
0

)
+2β

(
Π01 −Π

1
0

)]
. (21)

As we can observe, generally all coefficients Πμν take part in the dynamics in this case since all of them are
present in Eq. (20).
In order to illustrate the role of parametersΠμν let us consider a general case of dynamics where Φ0 =const.

After the integration of Eq. (20) we find that

γ2
[
Π00 + β(Π

1
0 −Π

0
1)− β

2Π11
]
=Φ0t + C, (22)

where C is an integration constant. Taking into consideration the initial condition for t =0 we obtain that

C = γ20
[
Π00 + β0(Π

1
0 −Π

0
1)− β

2
0Π
1
1

]
,

where β0 and γ0 are the values for t = 0. If we additionally assume that β0 = 0 (i.e. γ0 = 1) then C = Π
0
0.

Substituting this into Eq. (22) and making simple rearrangements we arrive at the following:

β2
(
Φ0t +Π

0
0 −Π

1
1

)
+ β

(
Π10 −Π

0
1

)
−Φ0t =0. (23)

The solutions of the above equation are of the form

β± =

(
Π01 −Π10

)
±
√
(Π01 −Π10)

2
+4Φ0t(Φ0t +Π00 −Π11)

2(Φ0t +Π00 −Π11)
. (24)

In the standard formalism of the Special Theory of Relativity [3], when a constant force F is applied to a body
one gets the following solutions of the equations of motion for a velocity:

βST R± = ±
√

F2t2

m2c2 + F2t2
. (25)

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Acta Polytechnica Vol. 51 No. 1/2011

If we expect that Eq. (23) also has two symmetric solutions, we have to assume that Π01 = Π
1
0. Hence in this

case we find that

β± = ±
√

Φ0t
Φ0t +Π00 −Π11

. (26)

 0

 0.1

 0.2

 0.3

 0.4

 0.5

 0.6

 0.7

 0.8

 0.9

 1

 0  1  2  3  4  5  6  7

v/
c

t

Fig. 1: Comparison of β(t) (green dashed) and βST R(t) (red). F =Φ0 =1, m
2
c
2 =Π00 −Π

1
1 =1 are assumed here

It should be stressed here that the asymptotes of Eqs. (25) and (26) are identical, i.e.:

lim
t→∞

βST R± = lim
t→∞

β± = ±1 and lim
t→0

βST R± = lim
t→0

β± =0.

As we can see from Eq. (26), the constant Π11 plays the role of a “renormalization” constant for Π
0
0, hence it

can be discarded without losing the generality of considerations. Then matrix (19) takes the form

Π(β)= γ2
(

Π00 Π
0
1 − βΠ

0
0 − β

2Π01
Π01 + βΠ

0
0 − β

2Π01 −β
2Π00

)
. (27)

Matrix (27) can also be rewritten as

Π(β)= γ2Π00

(
1 −β
β −β2

)
+Π01

(
0 1

1 0

)
, (28)

where the second matrix on the right hand side of Eq. (28) is constant in time.
Assuming that Π01 =Π

1
0 and Π

1
1 =0, Eqs. (20) and (21) turn into

Φ0 = ∂0Π
0
0(β)= ∂0γ

2Π00 =2γ
4β̇βΠ00,

(29)

Φ1 = ∂0Π
0
1(β)= ∂0γ

2
(
−βΠ00

)
= −γ4β̇

(
1+ β2

)
Π00.

In order to compare it with the standard formalism of the Special Theory of Relativity, let us recall that in the
standard description the equation of motion is given by [3]

F =
dp
dt
=

mcβ̇

(1− β2)3/2
= γ3mcβ̇,

and therefore

β̇ = γ−3
F

mc
.

Substituting this expression into Eq. (29) we get

Φ0 = 2γ
F

mc
β̇βΠ00,

(30)

Φ1 = −γ
F

mc
β̇
(
1+ β2

)
Π00.

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Acta Polytechnica Vol. 51 No. 1/2011

This indicates that the assumption that β̇ in this formalism and the standard description is the same leads to
the conclusion that for a force F constant in time the component Φ0 is not constant in time, and vice versa.
However, the uniform motion (β̇ =0) in both formalisms occurs simultaneously.
The non-trivial part of the matrix (28) can also be expressed by means of well-known relativistic quantities

such as energy and momentum:
E = γmc2, p = γmcβ.

Therefore we get

Π(β)=
Π00

m2c4

(
E2 −Epc
Epc −p2c2

)
+Π01

(
0 1

1 0

)
. (31)

It should be highlighted here that — as was mentioned before — it is possible to choose a different special
form of the relativistic velocity tensor matrix and — consequently — a different description of dynamics. For
instance, by analogy with the non-relativistic solution, we can assume that the relation between the velocity
tensor describedbyEq. (4) and themomentumtensor is givenbyEq. (17). Hence in order to reproduceEq. (17)
the general form of the momentum tensor matrix (19) has to be reduced to the matrix

Π(β)= γ2Π01

(
−β 1
−β2 β

)
, (32)

where — as we have shown for the non-relativistic case — the constant Π01 can be identified with mass m of a
material point. It is easy to observe that form (32) is obtained from Eq. (19), where all coefficients with the
exception of Π01 vanish. Therefore, Eqs. (20) and (21) can be written down as:

Φ0 = −γ4
(
1+ β2

)
β̇Π01,

Φ1 = 2γ
4ββ̇Π01.

5 Conclusions

The aim of this paper was to introduce a new dynamical object called the momentum tensor as an analogue to
the kinematical velocity tensor, and therefore to complete the tensorial description of classical and relativistic
mechanics. Calculations show that the choice of constants in the momentum tensor matrix results in different
models of dynamics in the relativistic case. Another important fact is that the naturally assumed relation
between the tensors: Π(v)= mV (v) is just one amongmany. Further investigationswill focus on verifying the
other models.

Acknowledgement

I would like to thank Prof. Edward Kapuścik for his scientific advice, and also for useful comments and ideas
on this subject.

References

[1] Kapuścik, E., Lanczewski, T.: On the Velocity Tensors, Physics of Atomic Nuclei, 72 (2009) 809.

[2] Goldstein, H.: Classical mechanics, Addison-Wesley, Reading, 1980.

[3] Landau, L. D., Lifshitz, E. M.: Classical Theory of Fields, PWN Warsaw, 1980.

Tomasz Lanczewski
E-mail: tomasz.lanczewski@ifj.edu.pl
H. Niewodniczański Institute of Nuclear Physics
Polish Academy of Sciences
Radzikowskiego 152, PL 31342 Kraków, Poland

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