A Relativistic Trolley Paradox

A Relativistic Trolley Paradox is a paradox within the Special Theory of Relativity involving a trolley with relativistic velocity and its rolling wheels.

The situation involves a combination of translational and rotating motion—the translational motion of a trolley and the rotational motion of its wheels. This example makes clear the importance of taking the relativity of simultaneity into account when predicting the behaviour of objects moving relativistically.

It also sheds light on the physical reality of the Lorentz contraction and provides a method for measuring the increase of the rest length of the circumference of a wheel with constant radius and increasing rotational motion.

Imagine that an observer on a trolley is spinning the drive wheel to a certain angular velocity $\Omega$, and that this wheel is rolling along the rail without slipping, thus driving the trolley. Let's denote the rest frame of the trolley by $K$. Upon reaching the angular velocity $\Omega$, the linear velocity $v$ of the rotating rim of the drive wheel becomes

$v=R\Omega$ (1)

where $R$ is the radius of the wheel rim (Fig. 1).

The Relativistic Trolley with light pulse generator

Because it is impossible for the wheel rim speed to exceed the speed of light $c$, the angular velocity of the wheel cannot be higher than $c/R$. When the angular velocity of rotation of the wheel approaches the value $c/R$, the translational speed of the rim of the wheel approaches the speed of light $c$. In the absence of slippage between the rail and the drive wheel, the speed of movement of the rail in reference frame $K$ also tends to the speed of light $c$.

Suppose now that on the trolley next to the wheel rim there is a generator of laser light, and on the drive wheel there is a sensor. Each time the sensor passes the laser generator it detects a signal. The laser and sensor act like a clock ticking with a period equal to the time it takes for the sensor to pass the upper position of the wheel on two consecutive occasions.

If the wheel makes $f$ revolutions per second, the frequency of the signals is $f=\Omega /2\pi$. The speed of the wheel rim and of the rail relative to the trolley can in this case be expressed by the frequency $f$ of the signals as $v=R\Omega =2\pi Rf$ . The upper limit of the frequency of the signals is $c/2\pi R$.

Consider now the motion of the trolley in reference frame $K'$ rigidly linked to the rail. Inertial observers must agree on their relative velocity, so the speed of the trolley in this reference frame must also be equal to $v$. In the reference frame $K'$ the frequency of the signals is

$f'=\gamma ^{-1}f=\gamma ^{-1}(v/2\pi R)$ (2)

due to the relativistic time dilation, where $\gamma =(1-v^{2}/c^{2})^{-1/2}$.

When the speed of the trolley approaches light speed $c$, the frequency of the signals tends to zero. The signal frequency $f'$ tending to zero means termination of the drive wheel rotation in reference frame $K'$.

Consequently, it seems as if the trolley must move along the rails with wheel slipping in reference frame $K'$. However, the effect of the wheel slipping on the rail is not relative but absolute. Hence, in reference frame $K'$ slipping cannot occur because it disagrees with the absence of slipping in $K$. So there is a contradiction, and this is what is called “The Trolley Paradox.”

Solution of the Paradox with constant radius of the wheels

In this resolution paradox is resolved by nothing that rest length of the circumference of the wheel increases towards infinity as the speed of the trolley approaches that of light. Hence, as observed in the rail system, there is no slipping even if the angular velocity of the wheel approaches zero in the velocity of light limit. This solution presupposes that the radius of the wheel is constant. Consider a point $P$ that is at the origin of the coordinate system and at the bottom of the wheel at time $t=0$. The distance $l'$ between the points on the rail where point $P$ on the wheel touches the rail

$l'=\gamma 2\pi R$ (3)

The frequency of the wheel clock as measured in $K'$ is related to the angular velocity of the wheel as measured in $K$ by

$f'=\gamma ^{-1}\Omega /2\pi$ (4)

Because one revolution of the radius corresponds to an angular increase of $2\pi$ both in $K$ and $K'$, the angular velocity in $K'$ is related to the frequency in $K'$ by

$\Omega '=2\pi f'$

Equations (3) and (4) lead to

$\Omega '=\Omega \gamma ^{-1}$ (5)

If we let $\Omega _{c}\equiv c/R$, then

$\Omega '=\Omega {\sqrt {1-(\Omega /\Omega _{c})^{2}}}$ (6)

In $K'$ the velocity of trolley is

$v'=\gamma R\Omega '=R\Omega =v$ (7)

It is in agreement with the requirement that the velocity of the trolley as observed in the rest frame of the rail must be equal to the velocity of the rail as observed in the rest frame of the trolley.

Solution of the Paradox with contracting radius of the wheels

Assume now that a wheel is permitted to contract freely in the radial direction so that no tension develops in tangential direction. Then the rest length of the rim of the wheels must remain constant during the accelerated motion of the trolley. This means that the rim Lorentz contracts, and that the radial extension of the wheels contracts accordingly. The result is that the wheels become infinitely small in the limit that the trolley moves with the velocity of light.

If $v$ is velocity on the rim in the rest frame $K$ of the trolley, we have $\Omega =v/R$, where $R=R_{0}/\gamma$ is the contracted radus of the rotating wheels, and $R_{0}$ is their radius when they are at rest. The angular velocity of the rotating wheels is then

$\Omega =\gamma v/R_{0}$  (8)

Hence, in this case the angular velocity $\Omega$ must approach an infinitely great value in $K$ when the speed of the rail approaches that of light. As observed in the rail frame $K'$, the distance between the marks on the rail each time a point on a wheel touches it is

$l'=\gamma 2\pi R=2\pi R_{0}$ (9)

and this distance is independent of the speed of the trolley, even if the radius of the wheels decreases with increasing velocity, because the distance between the marks depends upon the rest length of the rim of the wheels and not their Lorentz contracted length. Also in this frame the angular velocity of the wheels remains finite even if the wheels have a vanishing radius when the velocity of the trolley approaches that of light,

$\Omega '=\gamma ^{-1}\Omega =v/R_{0}$ (10)

and hence $lim_{v\rightarrow c}\Omega '=c/R_{0}$, which is finite.

There is no slipping of the wheels neither in the rest frame of the trolley nor in the rest frame of the rails.

References

• Vadim N. Matvejev; Oleg V. Matvejev; Øyvind Grøn (2016). "A Relativistic Trolley Paradox". Amer. J. Phys. 84 (6): 419. doi:10.1119/1.4942168.

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