Dimensions in Special Relativity Theory 
a Euclidean Interpretation*
R.F.J. van Linden
Smeetsstraat 56, 6171 VD Stein, NETHERLANDS
email rob@vlinden.com, web http://www.euclideanrelativity.com
September 2005
Abstract
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
03.30.+p.
1
1
Introduction
time dimension. Rewriting the usual Minkowski
invariant
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
(1)
a number of authors. Amongst others Montanus into Euclidean form:
[1,2], Gersten [3] and Almeida [4], 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 [5] ﬁrst proposed Eu one arrives at the temporal velocity component
clidean general relativity.
The version in the present paper emphasizes ex
χ = cdτ /dt
(3)
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 4vectors are estab as a parameter in Euclidean approaches similar to
lished.
special relativity, e.g. in [1,2], but here it will be
A
simpliﬁed
and
popularized
version
is treated as a genuine extra Euclidean dimension. A
available
that
will
get
you
in
the
’right general expression for speed in the time dimension
mood’.
It can be found on the web at (henceforth refereed to as timespeed) is now:
http://www.euclideanrelativity.com.
χ = cdx4/dx5
(4)
2
The Time Dimension
while the scalar value of timespeed χ is
χ =
c2 − v2
(5)
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 spacetime is
nent is given the opposite sign. Some alternative
interpretations (e.g. [14]) seek positive deﬁnite
vi = dxi/dx5
(6)
Euclidean metrics for spacetime. Also in this arti
cle, the time dimension will be treated as a regular
fourth dimension in Euclidean spacetime.
3
Using TimeSpeed in Special
If time is considered a fourth spatial dimension,
Relativity
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.
bcd
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 [6], 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 [1],
2
where this is referred to as a Relative Euclidean
SpaceTime. In the approach that follows now,
x4
x’
4
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
X
the motion. The diagrams in Fig. 1 and Fig. 2
C
should illustrate this.
l
0
x
V
i
x4
l’
A
x
l
4
’
4
(i =1, 2, 3)
x’
C
i
l
l’
x
Figure 2: Object A in motion relative to observer.
0
A
i
x
The dimensional axes of object A have rotated rel
i
’
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
x
i
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)
(7)
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
(8)
Fig. 2.
Deﬁnitions are as follows:
but this factor can also be written as
γ = c/ c2 − v2 = c/χ
(9)
• Vector C indicates the 4D velocity, having
magnitude c, of object A.
leading to
x = c(x − vt)/χ
(10)
• 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
(11)
• 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
(12)
3
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
dimension x
the relative motion.
i. This length is the manifestation of
the diﬀerence in proper time (the nonsimultaneity)
between the endpoints of object A in motion ac
x4
x’
4
cording to the Lorentz transformation equation for
x
time:
4’
’
t = γ(t − vx/c2)
(13)
and can be interpreted as a rotation ’out of space’
of the proper length l towards the negative axis of
x
0
x
4.
At t = 0 the propertime diﬀerence between
V W
i
tail and head of arrow l will be
U
t = −γvl/c2 = −lv/cχ
(14)
x’
i
From l = l χ/c and l4 = l v/c it follows that
l4 = −ct
(15)
which conﬁrms that l
Figure 3: Relativistic addition of velocities in three
4 represents the propertime
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
(16)
• 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.
(17)
• 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:
4
Relativistic Addition of Ve
u + v
w =
(18)
locities
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
1
tion for relativistic addition of velocities as used in
w = c cos(−α) = c sin( π + α)
special relativity. Although this particular point
2
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
(19)
4
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:
c
w
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
c
speed c for an observer in frame x will still have
that same speed c for an observer in frame x .
u = v
0
This is one of Einstein’s original postulates and
c
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
c
rotated along with that frame. The third ob
ject, having speed w, as measured from frame
x, is not a photon but a masscarrying 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
carrying particles.
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 spacelike
c
quadrants in the classical light cone). This fact
strongly suggests that the graph from Fig. 4
w
c
is an approximation of the real function.
Finally, both Eqs. (18) and (19) are plotted
together in Fig. 6.
u = v
0
Equation (19) is almost identical for speeds be
low about c/2 but begins to deviate at higher
c
speeds. The top of Eq. (19) corresponds to
√
u = v = c/ 2. From the circle diagram in
Fig. 3 it shows that the timespeed of the ob
c
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 timespeed for the object becomes
also near c, which is in accordance with the
negative. (This situation might be related to
5
c
c
New
w

w
c
Classic
c
Classic
New
c/ 2
u = v
0
0
u = v/3
0.949c
c
c
c
c
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.
antiparticles, 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
devices.
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 timespeeds 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
ticed.
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.
[7], 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
by Michelson.
involved. We call the proper time for the spaceship
The vast majority of experimental setups that τs and for the missile τm. For simplicity we set
are aimed at veriﬁcation of relativity theory the spacetime 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 setup with three reference frames. At
The
observer
on
Earth
calculates
time
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;
6
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 spacetime however the observer Doppler eﬀect in space and a frequency shift that
measures the timespeed χs of the spaceship as: results from the lower timespeed.
χs =
c2 − v2s =
c2 − (0.9c)2 = 0.4359c.
According to this observer the absolute value
of the timespeed χ
6
Mass, Energy and Momen
m of the missile is χm
=
c2 − (0.7846c)2 = 0.62c, but from the circle di
tum
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
x
of impact according to the observer on Earth for
4
x’
4
the spaceship: τ
M
s = tsχs/c = 0.4359s; and for the
’
missile: τ
K
m = 1.147(−0.62) = −0.7111s.
K
In proper time the missile hits the asteroid before
X
C
the spaceship does despite its lower spatial speed.
Causality is therefor not violated. The missile runs
0
backwards in proper time.
x
L
i
V
5
Relativistic Doppler Eﬀect
√
Using the identity χ =
c2 − v2 for the timespeed
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
(20)
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 timevelocities re
λ = λ0(c + v)/χ
(21) ﬂects the fourvelocities 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 timespeed).
m20c2 = m20χ2 + m20v2
(24)
Setting χ = c results in:
which, using the identities E = γm0c2 and p =
γm
λ
0v, is equivalent to the classical equation
v = λ0(1 + v/c)
(22)
E2/c2 = m2
which is the regular equation for the acoustic
0c2 + p2
(25)
Doppler eﬀect with moving source and stationary E being the total energy and p being the spatial
receiver. Setting v = 0 results in:
momentum.
λ
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 timespeed of the emitting particle.
spatial dimension of the rest frame of the observer.
7
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 spacetime:
If we multiply the factor γ2 into all three elements
of Eq. (24) we get:
Λ = m0c2
(30)
γ2m2
The latter is to be interpreted as the ’kinetic en
0c2 = γ2m2
0χ2 + γ2m2
0v2
(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
(25): γm
any solution is a constant in 4D spacetime.
0c = E/c, γm0χ = m0c, γm0v = p. The
factor γm
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
m
ment to a 3D subspace of 4D spacetime. 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 timemomentum. 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
0χ.]
The Lagrangian formal
ism for this situation has been worked out by Mon sures the object’s speed and energy as constants.
tanus in [2]. 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
and Momentum
to keep track of the progress of the object along its
worldline) appear in Goldstein [8]. 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
principle:
U of magnitude u relative to frame x .
(please refer also to Fig. 3 and the deﬁnitions
x5(2)
given there)
δI =
Λ(xµ, uµ)dx5
(27)
x5(1)
• 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 EulerLagrange equations of motion are:
• E = γ(u)m
∂Λ
d
0c2 is the energy of that same ob
−
(∂Λ/∂u
ject moving with velocity U of magnitude u
∂x
µ) = 0
(28)
µ
dx5
relative to frame x and measured from frame
x .
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
0uµuµ
(29)
8
x
where ds = cdτ . Fourvectors with the Euclidean
4
metric (+1, +1, +1, +1) as used in the previous Sec
x’
4
E
c
/c = (w)m c
(w)m
= m
0
0
0
tions use the 4D velocity of the moving object and
4D Euclidean distances as invariants, which is in
x4’
’
fact the essence of Eq. (2):
X
E /c
’ = (u)m c
0
c2 = v21 + v22 + v23 + χ2
(35)
0
x
V W
i
U
Multiplication with dt2 = dx2
Note:
5 yields (recall that
m =1
χ = cdτ /dt):
0
x’
i
c2dt2 = dx21 + dx22 + dx23 + c2dτ2
(36)
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 fourvectors
for
position,
velocity
and
en
mensional axes.
ergy/momentum:
Euclidean
Minkowskian
• 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.
equation
(34) written in a diﬀerent form, with as a main
E/E = γ(w)/γ(u)
(31) input the deﬁnition of χ, being the timespeed 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)
(32)
• It creates a new invariant c, being the universal
magnitude of the 4D velocity of an object.
For momentum a generic transformation equation
is
• It provides a Euclidean basis for the deﬁnition
p/p = wE/uE
(33)
of vectors in the direction of the time dimen
sion.
where:
• 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 fourvectors can be
0w is the momentum of the object
as measured from frame x.
derived from the Minkowski fourvectors by using
the time component in the Minkowski fourvector
as the invariant (the vector sum) for the new four
8
Euclidean FourVectors
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 [9] for a detailed treat
ds2 = c2dt2 − dx2 − dy2 − dz2
(34) ment of Minkowski and Euclidean fourvectors).
9
References
[1] H. Montanus, ”ProperTime Formulation of
Relativistic Dynamics”, Foundations of Physics
31 (9) 13571400 (2001).
[2] H. Montanus, ”Special Relativity in an Abso
lute Euclidean SpaceTime”, Physics Essays 4
(3) 350356 (1991).
[3] A. Gersten, ”Euclidean special relativity”, Foun
dations of Physics 33 (8) 12371251 (2003).
[4] Jose B. Almeida, ”An Alternative to Minkowski
SpaceTime” (arXiv:grqc/0104029 v2, 10 Jun
2001).
[5] Robert d’E Atkinson, ”General Relativity in
Euclidean Terms”, Royal Society of London
Proceedings Series A 272 (1348) 6078 (1963).
[6] Brian R. Greene, The elegant universe, page
391 note 5 (W.W. Norton & Company New
YorkLondon, 1999).
[7] C. Renshaw, ”The Experiment of Fizeau as a
Test of Relativistic Simultaneity”, available at:
renshaw.teleinc.com.
[8] H. Goldstein, Classical Mechanics, Second
edition, Chapter 79 (Addison Wesley, 1980).
[9] R.F.J.
van
Linden
”Minkowski
ver
sus
Euclidean
4vectors”,
available
at:
www.euclideanrelativity.com.
10
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