Dimensions in Special Relativity Theory -
a Euclidean Interpretation*
R.F.J. van Linden
Smeetsstraat 56, 6171 VD Stein, NETHERLANDS
e-mail email@example.com, web http://www.euclideanrelativity.com
A Euclidean interpretation of special relativity is given wherein proper time τ acts as the fourth
Euclidean coordinate, and time t becomes a ﬁfth Euclidean dimension. Velocity components in both
space and time are formalized while their vector sum in four dimensions has invariant magnitude c.
Classical equations are derived from this Euclidean concept. The velocity addition formula shows a
deviation from the standard one; an analysis and justiﬁcation is given for that.
* c Copyright Galilean Electrodynamics Vol 18 nr 1, Jan/Feb 2007. Printed with permission. PACS
time dimension. Rewriting the usual Minkowski
Euclidean relativity, both special and general, is
steadily gaining attention as a viable alternative
c2 = (dct/dτ )2 − (dx/dτ )2 − (dy/dτ )2 − (dz/dτ )2
to the Minkowski framework, after the works of
a number of authors. Amongst others Montanus into Euclidean form:
[1,2], Gersten  and Almeida , have paved the
way. Its history goes further back, as early as 1963
c2 = (cdτ /dt)2+(dx/dt)2+(dy/dt)2+(dz/dt)2 (2)
when Robert d’E Atkinson  ﬁrst proposed Eu- one arrives at the temporal velocity component
clidean general relativity.
The version in the present paper emphasizes ex-
χ = cdτ /dt
tending the notion of velocity to the time dimen-
sion. Next, the consistency of this concept in 4D This clearly deﬁnes τ as the coordinate for the
Euclidean space is shown with the classical Lorentz fourth Euclidean dimension, and it says that the
transformations, after which the major inconsis- velocity components in all four dimensions involve
tency with classical special relativity, the velocity derivatives with respect to t, which then can no
addition formula, is addressed.
Following para- longer represent the fourth dimension. It can only
graphs treat energy and momentum in 4D Eu- be an extra, ﬁfth dimension, x5 (provided we index
clidean space, partly using methods of relativistic the other four x1, x2, x3, and x4 respectively, with
Lagrangian formalism already explored by others τ = x4). This ﬁfth dimension is sometimes treated
after which some Euclidean 4-vectors are estab- as a parameter in Euclidean approaches similar to
special relativity, e.g. in [1,2], but here it will be
is treated as a genuine extra Euclidean dimension. A
’right general expression for speed in the time dimension
It can be found on the web at (henceforth refereed to as time-speed) is now:
χ = cdx4/dx5
The Time Dimension
while the scalar value of time-speed χ is
c2 − v2
Minkowski interpretations of special relativity treat
time diﬀerently from spatial dimensions, showing The general expression for spatial velocity compo-
from the Minkowski metric where the time compo- nents in 4D Euclidean space-time is
nent is given the opposite sign. Some alternative
interpretations (e.g. [1-4]) seek positive deﬁnite
vi = dxi/dx5
Euclidean metrics for space-time. Also in this arti-
cle, the time dimension will be treated as a regular
fourth dimension in Euclidean space-time.
Using Time-Speed in Special
If time is considered a fourth spatial dimension,
then it must show properties similar to those found
in the other three. In there we encounter properties It will be shown that the Lorentz transformation
like length, speed, acceleration, curvature etc., ex- equations for length and time can be reproduced
pressed respectively as s, ds/dt, d2s/dt2, Ra
etc. from the Euclidean context.
Of those properties, a single one can be measured
Maintaining orthogonality for all Euclidean di-
relatively easily in the time dimension: the ’length’ mensions, Eqs. (2) and (5) imply that the axes
or timeduration ∆t. That raises the question of for the proper time dimension and the spatial di-
how a hypothetical speed in time, let us call it χ, mension in the direction of the initial motion must
should be expressed mathematically. In , Greene have rotated for the moving object, as seen from the
has given a derivation of an expression that can be rest frame of the observer, in fact deﬁning Lorentz
used as the velocity component in the Euclidean transformations as rotations in SO(4). See also ,
where this is referred to as a Relative Euclidean
Space-Time. In the approach that follows now,
these axes will therefor (unlike in the Minkowski di-
agram) both rotate in the same direction, clockwise
or counter clockwise, depending on the direction of
the motion. The diagrams in Fig. 1 and Fig. 2
should illustrate this.
(i =1, 2, 3)
Figure 2: Object A in motion relative to observer.
The dimensional axes of object A have rotated rel-
ative to the observer.
(i =1, 2, 3)
• l and l4 are, respectively, the projections of this
proper length on the spatial dimensions and
the proper time dimension of the observer.
Figure 1: 4D representation of an observer at O
In Fig. 2, object A moves with speed v relative
and an object A, both at rest.
to the observer. This leads to a relative rotation of
dimensions x4 and xi such that V is the projection
Figure 1 depicts an object A at rest together with of the original 4D velocity C of object A on the
an observer at O, also at rest. The horizontal axis xi axis of the observer at rest. The situation is
shows both the spatial dimensions x
examined at the instant where x
i, i = 1, 2, 3,
i = xi = x4 =
for the object A as well as the spatial dimensions x
4 = 0.
for the observer. The vertical axis shows both time
The Lorentz transformation equation for x is
dimensions with notation conform Eq. (2), so x4 =
cτ . Due to object A being at rest, relative to the
x = γ(x − vt)
observer, the axes overlap. The circle is just a tool where
to better show the rotation that will be depicted in
γ = 1/ 1 − v2/c2
Deﬁnitions are as follows:
but this factor can also be written as
γ = c/ c2 − v2 = c/χ
• Vector C indicates the 4D velocity, having
magnitude c, of object A.
x = c(x − vt)/χ
• Vector V, of magnitude v, and X, of magni-
tude χ, are the projections of this velocity C At t = 0, the length of object A will be contracted,
on, respectively, the spatial dimensions and the as measured by the observer, according to
proper time dimension of the observer.
x = x χ/c
• l indicates the proper length of object A in the so the contraction of length l can be written as
spatial direction xi in the rest frame of object
A (in this example l is also set to c).
l = l χ/c
which shows that l, as measured by the observer
Figure 3 depicts a situation with three reference
at rest, is indeed the goniometric projection of the frames: a stationary unprimed frame x, a moving
proper length l on the xi axis.
primed frame x and a third, double primed frame
Arrow l4 is the projected ’length’ component of x of an object that moves relative to both other
the moving object A on the proper time axis x4 frames, x and x . Each frame has dimensional axes
of the observer as a result of the rotation of the rotated relative to the other frames as a result of
the relative motion.
i. This length is the manifestation of
the diﬀerence in proper time (the non-simultaneity)
between the endpoints of object A in motion ac-
cording to the Lorentz transformation equation for
t = γ(t − vx/c2)
and can be interpreted as a rotation ’out of space’
of the proper length l towards the negative axis of
At t = 0 the proper-time diﬀerence between
tail and head of arrow l will be
t = −γvl/c2 = −lv/cχ
From l = l χ/c and l4 = l v/c it follows that
l4 = −ct
which conﬁrms that l
Figure 3: Relativistic addition of velocities in three
4 represents the proper-time
diﬀerence in object A. The factor c results from the reference frames, each with rotated dimensional
choice of units for space and time.
axes relative to each other.
Summarizing, from the perspective of the ob-
server, the proper length l of object A is decom-
• Vector V of magnitude v is the spatial velocity
posed in the components l and l4 according to:
of an observer with rest frame x as measured
by an observer with rest frame x.
l 2 = l2 + l24
• Vector W of magnitude w is the spatial veloc-
and so is also the 4D speed c of the object decom-
ity of a third object as measured by the ob-
posed in the components χ and v:
server with rest frame x.
c2 = χ2 + v2.
• Vector U of magnitude u is the spatial velocity
of that same object but now as measured by
Equation (16) thus combines Eqs. (7) and (13) into
the observer with rest frame x .
a single Pythagorean equation in four dimensions.
When u, v, and w are parallel, the classical rela-
tion between them is:
Relativistic Addition of Ve-
u + v
1 + uv/c2
If we apply the approach as used consistently until
It appears that the Euclidean approach as used in now it yields the expression:
the previous Section does not yield the same equa-
tion for relativistic addition of velocities as used in
w = c cos(−α) = c sin( π + α)
special relativity. Although this particular point
may be a serious obstacle to the acceptation of this
= c sin(β + ϕ) = c(cos ϕ sin β + cos β sin ϕ)
proposal, it obviously is necessary to point it out.
= u 1 − v2/c2 + v 1 − u2/c2
This expression is not nearly similar to the classical
classical view. But if (as a matter of math-
expression in Eq. (18).
ematical experiment) the range of u and v is
Like Eq. (18), Eq. (19) still limits the speeds
extended beyond the maximum value of c then
as measured by both observers to the maximum of
the plot looks like depicted in Fig. 5.
c, which is also clear by inspection of the Figure.
Some remarks will be made now on the probability
of either of the equations to be the right one:
1. Equation (18) is in fact based on the univer-
sality of light speed and the basis for reason-
ing is that an object, e.g. a photon, having
speed c for an observer in frame x will still have
that same speed c for an observer in frame x .
u = v
This is one of Einstein’s original postulates and
also in this Euclidean approach it will still be
maintained as a valid postulate, which essen-
tially means that the photons velocity vector,
as measured from the moving frame, must have
rotated along with that frame. The third ob-
ject, having speed w, as measured from frame
x, is not a photon but a mass-carrying parti- Figure 5: Classical graph for relativistic addition of
cle for which such a rotation apparently does velocities with hypothetical (superluminal) exten-
not apply. It must therefor be emphasized that sions.
Eq. (19) for now may only be applied to mass-
The part from Fig. 4 can still be recognized
but it is clear now that this actually forms part
2. Equation (18) shows a discontinuity that is un-
of a continuous function that extends beyond
usual in physics. In Fig. 4, Eq. (18) is plotted
c. The part beyond u = v = c may not be used,
for the situation where u always equals v.
solely because the classical function is not de-
ﬁned, nor ever shown to be valid, for such su-
perluminal extensions (actually the space-like
quadrants in the classical light cone). This fact
strongly suggests that the graph from Fig. 4
is an approximation of the real function.
Finally, both Eqs. (18) and (19) are plotted
together in Fig. 6.
u = v
Equation (19) is almost identical for speeds be-
low about c/2 but begins to deviate at higher
speeds. The top of Eq. (19) corresponds to
u = v = c/ 2. From the circle diagram in
Fig. 3 it shows that the time-speed of the ob-
ject, as measured from frame x, then becomes
zero. Equation (19) further shows decreasing
values for w in situations where the values of u
Figure 4: Graph of classical equation for relativistic
and v go beyond c/ 2 (the frame of the mov-
addition of velocities.
ing object then rotates beyond π/2 relative to
frame x). It turns out that in that case the cor-
With u and v nearing c, the resulting w will
responding time-speed for the object becomes
also near c, which is in accordance with the
negative. (This situation might be related to
u = v
u = v/3
Figure 6: Classical [Eq. (18)] and newly derived Figure 7: Classical [Eq. (18)] and newly derived
graph [Eq. (19)] for relativistic addition of veloci- graph [Eq. (19)] for relativistic addition of veloci-
ties plotted together for u = v.
ties plotted together for u = v/3.
anti-particles, running ’backwards in time’.).
on the order of 10−5 m/s, which might be no-
The situation where u equals v gives the max-
ticeable using adequately accurate measuring
imum possible deviation relative to the clas-
sical graph. Other ratios between u and v
give (much) smaller deviations and the tops
A hypothetical case will now be used to show
of Eq. (19) will shift outwards towards c as that Eq. (19) does not necessarily lead to causality
can be seen in Fig. 7 where the ratio between conﬂicts as a result of the negative time-speeds that
u and v equals 3:1. At a ratio 10:1 both plots can occur.
are practically identical. Virtually all practi-
A spaceship travels relative to Earth at speed
cal situations that require the velocity addition vs = 0.9c and heads toward an asteroid that is at
formula to be used exist under such circum- rest relative to Earth. The ship launches a missile
stances, which indicates that a deviation from at the asteroid at vm = 0.9c relative to the ship.
the classical graph is likely to remain unno- An observer on the ship watches the missile destroy
the asteroid. According to Eq. (19), an observer
on Earth would see the missile traveling at only
3. Some interpretations of Fizeau’s experiment 0.7846c so the missile’s spatial speed is lower than
give rise to doubt concerning the correctness that of the spaceship. It seems therefor that this
of Eq. (18). If Eq. (19) is used in the anal- observer would see the ship hit the asteroid before
ysis of Fizeau’s experiment done by Renshaw the missile.
, it yields better results than Eq. (18), al-
The explanation of this paradox can be found in
though still not within the margins as claimed the comparison of the proper times of all objects
involved. We call the proper time for the spaceship
The vast majority of experimental set-ups that τs and for the missile τm. For simplicity we set
are aimed at veriﬁcation of relativity theory the space-time event of the launch at t = τm =
are using two reference frames. These exper- τs = 0 and the distance between the spaceship and
iments are not suitable for the veriﬁcation of the asteroid at that moment at 0.9 light second (as
the velocity addition formula. One would have measured by the observer on Earth).
to use a set-up with three reference frames. At
speeds on the order of 104 m/s the diﬀerence in coordinates of the impact (against the asteroid)
resulting values between Eqs. (18) and (19) is using his own time t for the spaceship: ts = 1s;
and for the missile: tm = 0.9/0.7846 = 1.147s, so it
The relativistic Doppler eﬀect can thus be inter-
seems as if the spaceship reaches the asteroid ﬁrst. preted as a combination of the normal ’acoustic’
In 4D Euclidean space-time however the observer Doppler eﬀect in space and a frequency shift that
measures the time-speed χs of the spaceship as: results from the lower time-speed.
c2 − v2s =
c2 − (0.9c)2 = 0.4359c.
According to this observer the absolute value
of the timespeed χ
Mass, Energy and Momen-
m of the missile is χm
c2 − (0.7846c)2 = 0.62c, but from the circle di-
agram (Fig. 3) it shows that we must now take
the negative root so its value is χm = −0.62c. Figure 8 depicts a moving object with spatial veloc-
Note that the cyclic nature of γ now also implies ity V of magnitude v, as measured by an observer
that in this situation γ has a negative value in at point L, at rest.
τm = tm/γ = tmχm/c for the missile.
We calculate the proper times at the moment
of impact according to the observer on Earth for
the spaceship: τ
s = tsχs/c = 0.4359s; and for the
m = 1.147(−0.62) = −0.7111s.
In proper time the missile hits the asteroid before
the spaceship does despite its lower spatial speed.
Causality is therefor not violated. The missile runs
backwards in proper time.
Relativistic Doppler Eﬀect
Using the identity χ =
c2 − v2 for the time-speed
variable in the wavelength equation for the rela-
tivistic Doppler eﬀect
1 + v/c
Figure 8: 4D velocity of magnitude c in x4 of an
λ = λ0
1 − v/c
object at L. An observer at rest at L has velocity
of magnitude c in x4.
simpliﬁes this expression to
The vector sum of spatial and time-velocities re-
λ = λ0(c + v)/χ
(21) ﬂects the four-velocities of the observer (along x4)
It is possible to identify the individual contribu- and the moving object (along x4). It follows natu-
tions of the factors v and χ to the total Doppler rally that the Lorentz invariant m0c (m0 is the rest
eﬀect by considering χ = c (which isolates the ef- mass) in the moving object A can be decomposed
fect of the spatial speed) and v = 0 (which isolates in
the eﬀect of the time-speed).
m20c2 = m20χ2 + m20v2
Setting χ = c results in:
which, using the identities E = γm0c2 and p =
0v, is equivalent to the classical equation
v = λ0(1 + v/c)
E2/c2 = m2
which is the regular equation for the acoustic
0c2 + p2
Doppler eﬀect with moving source and stationary E being the total energy and p being the spatial
receiver. Setting v = 0 results in:
The components in the right part of Eq. (24)
χ = λ0c/χ
(23) cannot simply be interpreted as, respectively, the
which simply makes the photon’s frequency propor- object’s momenta in the time dimension and the
tional to the time-speed of the emitting particle.
spatial dimension of the rest frame of the observer.
There is an obvious problem in the fact that the which equals, as a result of the universal velocity
factor γ is involved in the expressions for E and p. magnitude c for the free particle in 4D space-time:
If we multiply the factor γ2 into all three elements
of Eq. (24) we get:
Λ = m0c2
The latter is to be interpreted as the ’kinetic en-
0c2 = γ2m2
0χ2 + γ2m2
(26) ergy’ of the particle in four dimensions, which is
which describes triangle LK’M (if m
a fundamentally diﬀerent concept than kinetic en-
0 is set to 1).
This alternative form for Eq. (24) immediately ergy in three dimensions. It corresponds to the
shows the meaning of its components. They now total energy of a particle at rest. Other solutions
correspond one to one with the components in Eq. for Λ are possible but the essential element is that
any solution is a constant in 4D space-time.
0c = E/c, γm0χ = m0c, γm0v = p. The
The relativistic Lagrangian Λ shows that the fac-
0c is however not invariant under rota-
tions in SO(4), while m
tor γ in Eq. (26) must be a result of our conﬁne-
0c is. [Note that although
ment to a 3D subspace of 4D space-time. In order
0c is indeed Lorentz invariant from the perspec-
tive of the observer, its physical meaning in its own to maintain conservation laws for energy and mo-
rest frame is the moving object’s time-momentum. mentum, while only being able to measure their
The same invariant value can be found in the rest ’projections’ to our 3D space, the factor γ is an
frame of the observer (see also Fig. 9) but should artiﬁcial necessity. It vanishes for a hypothetical
then be read as γm
observer with full 4D observational skills, who mea-
The Lagrangian formal-
ism for this situation has been worked out by Mon- sures the object’s speed and energy as constants.
tanus in . The reader is therefore referred to
this source for the detailed derivation. The generic 7 Transformation of Energy
principles used for such 5D situations (or more gen-
erally 4D with the addition of an extra parameter
to keep track of the progress of the object along its
world-line) appear in Goldstein . The latter how- The generic transformation equations for energy
ever uses the classical indeﬁnite Minkowski metric and momentum depend indirectly on the equation
as a basis for the development of the relativistic La- for relativistic addition of velocities. Because a new
grangian Λ where Montanus uses a positive deﬁnite one replaces this equation, it is necessary to rework
metric like in this article. A short overview of the the transformation equations for energy and mo-
main equations is given here.
mentum as well.
In agreement with classical mechanics it is as-
Figure 9 depicts an object moving with velocity
sumed that the variation according to Hamilton’s W of magnitude w relative to frame x and velocity
U of magnitude u relative to frame x .
(please refer also to Fig. 3 and the deﬁnitions
• E = γ(w)m0c2 is the energy of an object that
moves with velocity W of magnitude w relative
is an extremum, where uµ = dxµ/dx5. The corre-
to frame x and measured in frame x.
sponding Euler-Lagrange equations of motion are:
• E = γ(u)m
0c2 is the energy of that same ob-
ject moving with velocity U of magnitude u
µ) = 0
relative to frame x and measured from frame
leading to a possible relativistic Lagrangian for a
free object in the absence of a forceﬁeld (so the
• Frame x moves with velocity V of magnitude
potential energy equals zero):
v relative to frame x.
Λ = m
• γ(u) = 1
1 − u2/c2
where ds = cdτ . Four-vectors with the Euclidean
metric (+1, +1, +1, +1) as used in the previous Sec-
/c = (w)m c
tions use the 4D velocity of the moving object and
4D Euclidean distances as invariants, which is in
fact the essence of Eq. (2):
’ = (u)m c
c2 = v21 + v22 + v23 + χ2
Multiplication with dt2 = dx2
5 yields (recall that
χ = cdτ /dt):
c2dt2 = dx21 + dx22 + dx23 + c2dτ2
where the factors c2dτ 2 and c2dt2 from Eq. (34)
have switched roles.
Figure 9: Generic transformation of energy and mo-
The Euclidean metric thus gives rise to
mentum in three reference frames with rotated di- four-vectors
• γ(v) = 1
1 − v2/c2
(x1, x2, x3, cτ)
(x1, x2, x3, ct)
(v1, v2, v3, χ)
γ(v1, v2, v3, c)
• γ(w) = 1
1 − w2/c2
(m0v1, m0v2, m0v3, m0χ)
(p1, p2, p3, E/c)
For energy this leads to a generic transformation
Equation (36) is not really new. It is merely Eq.
(34) written in a diﬀerent form, with as a main
E/E = γ(w)/γ(u)
(31) input the deﬁnition of χ, being the time-speed of
which can be written in diﬀerent forms using Eq. an object as measured by an observer at rest, which
(19). With u = 0 this reduces to the classical form: has three eﬀects:
E/E = γ(v)
• It creates a new invariant c, being the universal
magnitude of the 4D velocity of an object.
For momentum a generic transformation equation
• It provides a Euclidean basis for the deﬁnition
p/p = wE/uE
of vectors in the direction of the time dimen-
• It enables these new vectors to be summed
• p = γ(u)m0u is the momentum of the object
with existing vectors in the spatial dimensions.
as measured from frame x .
• p = γ(w)m
In general, the new Euclidean four-vectors can be
0w is the momentum of the object
as measured from frame x.
derived from the Minkowski four-vectors by using
the time component in the Minkowski four-vector
as the invariant (the vector sum) for the new four-
vector. It is essentially doing Pythagoras “the other
way around”, i.e., calculating the hypotenuse from
The traditional Minkowski line element with metric the rectangular sides, instead of calculating a rect-
(+1, −1, −1, −1) is:
angular side from the hypotenuse and the other
rectangular side (refer to  for a detailed treat-
ds2 = c2dt2 − dx2 − dy2 − dz2
(34) ment of Minkowski and Euclidean four-vectors).
 H. Montanus, ”Proper-Time Formulation of
Relativistic Dynamics”, Foundations of Physics
31 (9) 1357-1400 (2001).
 H. Montanus, ”Special Relativity in an Abso-
lute Euclidean Space-Time”, Physics Essays 4
(3) 350-356 (1991).
 A. Gersten, ”Euclidean special relativity”, Foun-
dations of Physics 33 (8) 1237-1251 (2003).
 Jose B. Almeida, ”An Alternative to Minkowski
Space-Time” (arXiv:gr-qc/0104029 v2, 10 Jun
 Robert d’E Atkinson, ”General Relativity in
Euclidean Terms”, Royal Society of London
Proceedings Series A 272 (1348) 60-78 (1963).
 Brian R. Greene, The elegant universe, page
391 note 5 (W.W. Norton & Company New
 C. Renshaw, ”The Experiment of Fizeau as a
Test of Relativistic Simultaneity”, available at:
 H. Goldstein, Classical Mechanics, Second
edition, Chapter 7-9 (Addison Wesley, 1980).