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On a New Paradox in Special Relativity
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On a New Paradox in Special Relativity
Stephan J. G. Gift Department of Electrical and Computer Engineering
Faculty of Engineering The University of the West Indies St. Augustine, Trinidad and Tobago, West Indies Email: Stephan.Gift@sta.uwi.edu
Abstract. In this paper a recently published paradox in special relativity referred to as the light speed paradox is discussed. This paradox takes the form of an inconsistency in the Lorentz transformations where light speed determined using the transformations in two different approaches yield two different answers. A more straightforward proof of the paradox is presented and Selleri’s resolution of the issue is again highlighted.
Keywords: Lorentz transformations, Selleri transformations, special relativity, light speed invariance postulate, Selleri’s paraox, one-way light speed, inconsistency
1. Introduction In the application of special relativity several paradoxes have been reported
including the length contraction paradox [1], the twin paradox [2] and Bell’s spaceship paradox [3]. The accepted view today is that these paradoxes have been resolved and the problem is now regarded as one of misinterpretation of concepts [4].
Selleri’s paradox [5, 6] however stands unresolved in the relativistic framework [7]. Here the ratio of the speeds of two light signals travelling in opposite directions around a rotating platform is a non-unity value that remains non-unity when the disc radius and angular speed are adjusted such that the system becomes inertial. This contradicts the light speed invariance postulate which requires that this light speed ratio be unity.
In support of Selleri’s arguments, a new paradox referred to as the Light Speed Paradox involving the Lorentz transformations was identified [8]. Specifically it is shown that using these transformations to determine light speed in an inertial frame employing two different approaches yields two different answers. In this paper the argument demonstrating the inconsistency is refined such that its validity is more easily verified.

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2. One-Way Light Speed Determination

Consider an inertial system So with space and time coordinates xo , yo , zo ,to in which the speed of light is c, and another inertial system S having space and time

coordinates x, y, z,t which is moving at speed v relative to So along the xo axis. The two

systems So and S are coincident at to = t = 0 . The Lorentz transformations which relate

coordinates in the two frames from So to S are given by [9-11]

x = γ (xo − vto )

(1a)

y = yo

(1b)

z = zo

(1c)

t

=

γ

(to

−

vxo c2

)

(1d)

where γ = 1/ 1 − β 2 and β = v / c .

Based on the Lorentz transformations (1), a rigid rod of length ∆x at rest along the

x-axis in S has a length ∆xo in So relative to which it is moving at velocity v given by

∆x = γ∆xo

(2)

This is the length contraction formula of special relativity [10 (p97), 11 (p62)]. Also

based on the Lorentz transformations (1), the time interval ∆t between two events at a

standard clock fixed in S corresponds to a time interval ∆to in So given by

∆t

=

1 γ

∆to

(3)

This is the time dilation formula of special relativity [10 (p100), 11 (p64)] which, according to French [10 (p100)] “is basically the consequence of comparing successive readings on a given clock with readings on two different clocks.”
Using these transformations the one-way speed of light in frame S is determined for light travelling in a direction opposite to that of v using two different approaches: a differential approach and a kinematic approach. 2.1 Differential Approach
Using the Lorentz transformations (1) the one-way speed of light cS (LT ) = dx / dt in frame S for light travelling in a direction opposite to that of v is

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found by differentiating the transformations. Hence differentiating equations (1a) and

(1d) gives

dx = γ ( dxo dto − v dto )

(4)

dt dto dt dt

dt dto

=

γ

(1

−

v c2

dxo ) dto

(5)

Setting dxo = −c since the light is travelling in a direction opposite to that of v gives dto

dx = γ dto (−c − v)

(6)

dt dt

dt = γ (1 + v )

(7)

dto

c

Substituting (7) in (6) gives

cS

(LT

)

=

dx dt

=

−cγ

(1 +

v

/

c)

/γ

(1 +

v

/

c)

(8)

which reduces to

cS (LT ) = −c

(9)

This is the well-known light speed invariance principle of special relativity which is used

to derive the Lorentz transformations [9-11].

2.2 Kinematic Approach

For light travelling in a direction opposite to that of v , the one-way speed of

light cR (LT ) = dx / dt relative to a receiver fixed in frame S is now found using the kinematic relation speed equals distance over time. Consider a light transmitter I fixed at

a point on the x axis of the inertial frame S and a receiver R fixed at the origin of the

inertial frame S as shown in fig.1. R therefore moves along the xo axis in the direction of

I with constant speed v relative to So . At time to = To on a clock Co fixed in So and located

at the position of the transmitter as shown in fig.2, I emits a light signal in the direction

of the receiver. Let the position of the receiver R as measured in S using a rigid

measuring rod fixed in S be such that the distance between receiver R and transmitter

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I is the proper length ∆x . Let the corresponding distance in So between R and I at the

instant the light is emitted be ∆xo . Equation (2) relates ∆x and ∆xo such that

∆x = γ∆xo

(2)

So

S

v

∆x R R′

I xo,x

∆xo

Fig.1 Inertial Frames in Relative Motion at time to = To

So

S

v

C ∆x
R R′
CA CB ∆xo

I xo,x Co

Fig.2 Synchronized clocks CA , CB and Co in So and clock C in S at time to = To

As the emitted light travels from transmitter I toward receiver R , the receiver moves to position R′ where it receives the light. From fig. 2 let the elapsed time measured by a clock C fixed at the receiver in S for the light to travel from the transmitter I to the receiver at R′ be ∆t . Measurement of ∆t on a single clock is necessary to satisfy the requirements of the theory. In order to accomplish this the instant of light transmission to = To indicated on clock Co at the position of the transmitter must be

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available at the position of the receiver so that the time on clock C at this instant can be

recorded. This is easily achieved using a clock CA fixed in So and also located (at this

instant) at the position of the receiver R . This clock CA , because light speed is c in So ,

can be (Einstein) synchronized with the clock Co in So thereby enabling the time to = To when light emission occurs to be known at the receiver. Therefore at the time of light

emission to = To as indicated on synchronized clock CA in So at the position of the receiver R , the time on clock C at the receiver is recorded. Following this as the receiver travels to position R′ , time on clock C at the receiver is recorded at the instant of arrival

of the light. In this way the proper time interval ∆t between the emission and reception of

the light is determined by successive readings on the same clock C fixed at the receiver

in S as required by the theory [10, 11].

Let the corresponding elapsed time measured in So between the emission and

reception of the light be ∆to . Time interval ∆to can be measured using the synchronized

clock CA in So at the receiver R to record the instant to = To and another similarly

synchronized clock CB fixed in So at the receiver position R′ as shown in fig.2 to record the time the light signal is received. Thus by “comparing successive readings on a given

clock [ C in S ] with readings on two different clocks [ CA and CB in So ]” we satisfy the

conditions of the theory [10 (p100)] and specifically equation (3) which relates ∆t and

∆to such that

∆t

=

1 γ

∆to

(3)

Using (2) for the distance ∆x in S and (3) for the time ∆t in S for the light to

traverse that distance and arrive at the receiver, the speed cR (LT ) of the received light relative to the receiver as determined in the frame S of the receiver can be calculated and

is given by

cR (LT )

=

−

∆x ∆t

=

−γ

2

∆xo ∆to

(10)

Because the receiver moves from R to position R′ during the light transmission, the

actual distance in So travelled by the light is ∆xo − v∆to where v∆to is the distance moved

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in So between R and R′ in time ∆to . Since the light speed in So is c it follows that

∆xo − v∆to = c

(11)

∆to

from which

∆xo = c + v

(12)

∆to

Substituting (12) in (10) yields

cR (LT )

= − ∆x ∆t

=

−

c+v 1− β 2

= −c 1− β

(13)

The result in (13) deduced here by indirect application of the Lorentz transformations is

the same result obtained by direct application of these transformations [8]. This confirms

the correctness of (13) as following both directly and indirectly from these

transformations in the kinematic calculation.

2.3 Inconsistency

This light speed value cR (LT ) = −c /(1− β ) calculated in (13) using distance travelled ∆x divided by elapsed time ∆t must, because of the requirement of consistency,

be equal to the light speed cS (LT ) = −c in (9) derived from the Lorentz transformations

by differentiation. This requires that

− c = −c

(14)

1− β

which for v ≠ 0 cannot be satisfied and therefore represents an inconsistency [8].

3. Discussion Thus it has been shown that the Lorentz transformations contain an inconsistency
where light speed determined using the transformations in two different approaches yield two different answers. The Lorentz transformations are the accepted transformations in space-time physics and therefore the inconsistency demonstrated in (14) must be removed. We argue that the resolution of this problem involves the approach proposed by Selleri [5, 6] who, on the basis of the examination of all possible linear transformations using his set of “equivalent” transformations, effectively modified the time component (1d) of the Lorentz transformations given by

7

tL

=

γ

(to

−

vxo c2

)

=

to γ

−

vx c2

(15)

by removing the term vx / c2 in (15) resulting in

tS

=

to γ

(16)

Here subscripts are used to differentiate between the Lorentz time transformation in (15) and the Selleri time transformation in (16).
Guerra and de Abreu [12] refer to the revised transformations involving (16) as

synchronized transformations since the clocks measuring tS in S can be externally

synchronized using synchronized clocks in So where the light speed is c . They view the effect in (16) of subtracting the quantity vx / c2 (thereby resulting in (15)) as delaying

these synchronized moving clocks “by a factor that is proportional to their distance x to

the reference position x = 0 ”and describe this process as “de-synchronizing” the

synchronized measuring clocks which now measure time tL in S . These revised transformations (1a-c) and (16) which we shall refer to as the

Selleri transformations, also contain the length contraction and time dilation formulas (2)

and (3) [5 , 13] and therefore produce the light speed value cR (ST ) of the received light relative to the receiver as determined in the frame of the receiver in (13) given by

cR

(ST

)

=

−c 1− β

(17)

While this speed is the same as cR (LT ) = −c /(1 − β ) given by the Lorentz transformations in (13), the two transformations are not equivalent as Guerra and de Abreu believe since they make different light speed predictions based on the differential approach. In particular while the Lorentz transformations using the differential approach predict light

speed cS (LT ) = −c as given in (9), the Selleri transformations based on the differential

procedure in section (2.1) predict a different light speed cS (ST ) ≠ −c [5]. Using the Selleri transformations involving (16) instead of (15) as equation (1d), the one-way speed

of light cS (ST ) = dx / dtS in frame S in a direction opposite to that of v is found by differentiating the transformations giving

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dx = γ ( dxo dto − v dto )

(18)

dtS

dto dtS dtS

dtS = 1

(19)

dto γ

Again with dxo = −c since the light is travelling in a direction opposite to that of v and dto

using (19) we get

dx dtS

=

γ

2

(−c

−

v)

=

−

c+v 1− β2

= −c 1− β

(20)

Hence

cS (ST ) =

dx dtS

= −c 1− β

(21)

This light speed value (21) predicted by the Selleri transformations using the differential

approach is the same as the light speed value (17) predicted by these transformations

using the kinematic approach. These results are summarized in Table 1:

Table 1. Light Speed Calculations

Space-Time Transformations

Speed of Light: Differential Approach

Speed of Light: Kinematic Approach

Lorentz Transformations

cS (LT ) = −c

cR

(LT

)

=

−c 1− β

Selleri Transformations

cS

(

ST

)

=

− 1−

c β

cR

(ST

)

=

−c 1− β

There is therefore no inconsistency in the Selleri transformations as occurs in the Lorentz transformations and Selleri has shown [13] that these revised transformations predict the confirmed relativistic effects associated with the Lorentz transformations.

4. Conclusion We have discovered a paradox in special relativity that is the subject of this paper.
We have shown using elementary analysis that the Lorentz transformations of special

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relativity contain an inconsistency since using these transformations, light speed determined utilizing a differential approach is different from light speed determined using a kinematic approach. A critical factor in the kinematic calculation is the existence of (Einstein) synchronized clocks in So where light speed is c such that the instant of light emission to = To is available both at the transmitter and the receiver. This enables the legitimate application of the time dilation formula (3) which, along with the length contraction formula (2) yields the inconsistency represented in (14).
This use of the length contraction formula (2) and time dilation formula (3) in the kinematic calculation is new and represents an indirect application of the Lorentz transformations from which they originate. It produces the light speed result (13) given by cR (LT ) = −c /(1 − β ) which is also obtained by direct application of the Lorentz transformations in the kinematic calculation [8]. The validity of the kinematic result (13) is therefore beyond question. The problem for special relativity is that this light speed result (13) determined using the kinematic approach is different from the light speed result (9) given by cS (LT ) = −c determined using the differential approach.
Removal of this inconsistency requires an adjustment in the temporal component of the transformations such that the term vx / c2 is excluded. This was done by Selleri who showed that the resulting transformations make predictions that closely accord with observation. These Selleri transformations do not suffer from the inconsistency (14). Moreover they dispose of Selleri’s paradox since they predict that the ratio of the speeds of two light signals travelling in opposite directions around a rotating disc is Selleri’s non-unity value as required [5, 6, 13]. This is consistent with the light speed values on a rotating platform obtained using the GPS [14, 15] as well as with the fact that one-way light speed constancy is unconfirmed [16]. We therefore advance the Selleri transformations as the correct space-time transformations of modern physics [17].
References
1. Rindler, W., Length Contraction Paradox, American Journal of Physics, 29, 365, 1961.

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2. Debs, T.A. and Redhead, M.L.G., The twin “paradox” and the conventionality of simultaneity, American Journal of Physics, 64, 384, 1996.
3. Bell, J.S., Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press, Cambridge, 1987.Einstein, A.
4. Gu, Y., Some Paradoxes in Special Relativity and Resolutions, Advances in Applied Clifford Algebras, 21, 103, 2011.
5. Selleri, F., Noninvariant One-Way Speed of Light and Locally Equivalent Reference Frames, Foundations of Physics Letters, 10, 73, 1997.
6. Selleri, F., Sagnac Effect: End of the Mystery, in Relativity in Rotating Frames, edited by G. Rizzi and M. L. Ruggiero, Kluwer Academic Publishers, London,
7. Gift, S.J.G., On the Selleri Transformations: Analysis of Recent Attempts by Kassner to Resolve Selleri’s Paradox, Applied Physics Research, 7, 112, 2015.
8. Gift, S.J.G., The Light Speed Paradox in Special Relativity, Physics Essays, 30, 450, 2017.
9. On the Electrodynamics of Moving Bodies, in The Principle of Relativity by H.A. Lorentz, A. Einstein, H. Minkowski and H. Weyl, Dover Publications, New York, 1952.
10. French, A.P., Special Relativity, Nelson, London, 1968. 11. Rindler, W., Relativity, 2nd edition, Oxford University Press, New York, 2006. 12. Guerra, V. and de Abrue, R., On the Consistency between the Assumption of a
special system of reference and Special Relativity, Foundations of Physics, 36, 1826, 2006. 13. Selleri, F., La Relativita Debole, Edizioni Melquiades, Milano, 2011. 14. Gift, S.J.G., One-Way Light Speed Measurement Using the Synchronized Clocks of the Global Positioning System (GPS), Physics Essays, 23, 271, 2010. 15. Gift, S.J.G., Light Transmission and the Sagnac Effect on the Rotating Earth, Applied Physics Research, 5, 93, 2013. 16. Zhang, Y.Z., Special Relativity and its Experimental Foundations, World Scientific, Singapore, 1997. 17. Gift, S.J.G., The Selleri Transformations and the One-way Speed of Light, Physics Essays, 28, 474, 2015.
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