Relativity of a Spinning Wheel
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Imagine a large wheel floating in deep space far from any gravitating bodies. An observer sitting on the hub (the inner observer, using lower-case letters to represent measured or derived quantities) can tell that the wheel rotates with a certain angular speed. They do so by looking over a fixed point on the wheelís rim and watching it pass the images of distant quasars. If they measure the time that elapses between the fixed point passing a given quasar and passing that same quasar again and divide that time into 360 degrees (or 2ū radians), they will calculate the angular speed ý at which the wheel turns.
We donít state or imply what the wheel rotates at ý relative to: we donít have to. With rotary motion we have the closest that we can come to actual absolute motion in this Universe. If we have a telescope and the far distant quasars in its field of view donít appear to move, then the telescope has a straight line motion, which is a purely relative thing; if the quasars appear to move across the field of view, then the telescope has a rotary motion that will be essentially the same for all observers.
The fundamental phenomenon of Relativity is the distortion of time between reference frames. Once two sets of observers figure out how their clocks differ from one another, they can then work out the associated distortions of space. Once those distortions are known, the observers can work out equations that transform measurements made of paired events in one frame into the measurements that would be made of those same events in another frame.
To measure the time elapsed between pairs of events, the inner observer sets up a Feynman clock with its internal light path lying on the axis about which the wheel turns. The outer observer (sitting on the wheelís rim and using upper-case letters to represent measured and derived quantities) sets up a series of Feynman clocks on the rim of the wheel, each clock with its light path perpendicular to the spoke on which it is mounted. A proxy observer occupies a Minkowski frame in which the center of the wheel floats at rest: looking down on the wheel from a point not far above what we identify as the wheelís north pole, that observer sees the wheel rotating counterclockwise and sees the light pulses in the rim clocks tracing a sawtooth pattern on a circle against the sky.
Again we can use the length of light paths in the Feynman clocks and the constancy of lightspeed to work out the relative rates at which the clocks in the system count time. The inner observer and the proxy observer discover that their clocks count time at the same rate, doing so because both observers occupy the same inertial frame of reference. They also see that the outer observerís clocks count time more slowly than theirs do: between any two events the clocks on the wheelís rim count fewer microseconds than the clock on the wheelís hub does, which means that, relative to the inner observerís clocks the outer observerís clocks count dilated time. If the outer observer measures dT microseconds between two events, then the inner observer will count dt microseconds between the same two events in accordance with
The dT microseconds last longer than the dt microseconds, so fewer of them will fit between two events.
Because the motion between the inner and outer observers does not change the distance between the observers, the distortion between those observersí clocks must be asymmetrical. That fact means that both observers will use Equation 1 when they convert each otherís temporal measurements.
But how does the outer observer justify that usage? The outer observer knows that the inner observer is not moving, because the inner observer displays none of the centripetal acceleration that would be required to maintain a circular motion. The outer observer knows that they are moving, because they can feel the centrifugal force that is the inertial reaction to centripetal acceleration. The outer observer feels as if they lie at the bottom of a gravity well with the inner observer above them. That means that there must be a virtual radial velocity between the two observers.
Imagine that the inner observer allows a body of a certain mass m to slide freely down one of the spokes that connect the wheelís hub to its rim. They can calculate the bodyís radial speed at any point on the spoke by equating the momentum that the body gains to the centrifugal force acting on it and then calculating the change in energy,
After integrating that equation, dividing out the mass, and multiplying by two, they get
That equation gives the observers the virtual radial velocity between their frames of reference.
When the body reaches the outer observer it has gained mass in accordance with
In concept we can convert the particles that comprise the body into photons and send them back to the inner observer, where, by conservation of energy, they must have frequencies that correspond to m by way of Planckís theorem and Einsteinís mass-energy relation,
in which n represents the number of photons that we get from the conversion. If we letϕ represent the frequency of the photons at the outer observerís location, then we have
Comparing that result with Equation 4 gives us Equation 1, as we require.
Equation 3 relates the radial motion of the dropped body to the circular motion of the wheel, so letís see how our two observers see that circular motion.
A circle consists of an infinite set of points, certainly, but we use a finite measure to denumerate angular displacements, angles given in degrees or radians. We also use finite measure to express distance on straight lines, but that measure can be distorted by the Lorentz Transformation because a straight line extends to "infinity", where the properties of infinity absorb the distortion. We donít have that option available in reference to circular displacement, because all observers must see a full traverse of the circle coming back to the same point lying on a straight line extending from the center of the circle to some far distant quasar. Thus we infer that no distortion of circular measure exists: between the points at which any two events occur two observers will measure the same angular displacement, regardless of how they are rotating relative to each other. Because our observers refer all angular measure to distant objects on the sky, we must have the statement
standing true to Reality.
We know that when two observers have a relative motion between them, the relative speed must have the same value for both of them, so we know that the statement
must stand true to Reality. If we look at that equation in light of Equation 1 and then Equation 7, we get
Those equations tell us that longitudinal distances and radial distances measured by the outer observer contain fewer millimeters relative to the inner observerís measurements of the same intervals: the outer observerís measuring rods appear dilated relative to the inner observerís measuring rods. Because the wheel itself can be used to measure longitudinal distances and it would appear normal to the outer observer, it must appear enlarged to the inner observer; it would appear to spread across more of the inner observerís and proxy observerís common Minkowski frame.
Thatís what we get in the Lorentz Transformation for linear motion: moving objects dilate in the direction of relative motion. The Lorentz-Fitzgerald contraction comes about when that dilation is combined with the temporal offset that has the rear of an object partially overtaking the front. Likewise, what appears to the inner observer to be a dilation of radial distances measured by the outer observer matches what we see in the Schwarzschild Transformation of a lower observerís radial measurements as seen by an upper observer, such as we see in the gravitational redshift.
Another way to see that change involves the proxy observer extending a cantilever from the wheelís hub to a point over the rim and placing a clock at that point. The cantilever does not move with the wheel, so the clock at its end keeps time with the inner observerís clock on the hub. Now the outer observer and the proxy observer want to measure the time elapsed between two events, the events being two clocks on the rim passing the proxy clock.
The proxy observer measures a time interval dt directly as both of the rim clocks pass by the proxy clock. They then multiply that interval by the speed at which the rim clocks pass them and get Vdt =r(dθ/dt)dt = rdθ for the distance between the rim clocks.
The outer observer watches the proxy clock move past the forward rim clock and then past the rearward rim clock and sees the clocks display the same times that the proxy observer saw. Because the rim clocks are counting dilated time relative to the inner observerís clock, the proxy clock, which counts time at the same rate as does the inner observerís clock, appears to the outer observer to be counting contracted time: it appears to be running faster than the rim clocks in accordance with Equation 1. The observers all calculate the same speed between their respective frames, so the outer observer calculates VdT = R(dΘ/dT)dT = RdΘ for the distance that the proxy clock has moved between the rim clocks. That result comes out consistent with Equations 9.
But both observers use the same equations. Instead of the symmetry that we see in the Lorentz Transformation, we have the asymmetry of the Schwartzschild Transformation. There doesnít seem to be a Lorentz-Fitzgerald contraction appearing on the rim of the wheel.
How is that possible? It implies that there is no temporal offset between the clocks, in which two observers canít agree on whether two clocks are synchronized. How would either of our present observers know whether two clocks are synchronized or not? The observers answer that question by illuminating the clocks. They shine a light on the clocks and form the reflected light into an image of the numbers that the clocks are displaying. To ensure that they are recording instants of time, the observers use the briefest pulses of intense light for the illumination.
For consistency in our analysis we want the outer observerís pulses to follow the rim of the wheel, their motion being purely longitudinal. To that end we imagine an array of flat mirrors on the rim with the mirrors all facing the wheelís hub. By making the mirrors narrow enough and putting them edge to edge, we can make light follow a path arbitrarily close to a circular arc. The outer observer can then send two pulses of light simultaneously in opposite directions along that mirror path to illuminate clocks equidistant from their location to see whether those clocks are synchronized. Are they? The answer to that question depends on how we start the clocks.
All of our clocks start when we project a pulse of light of a certain frequency onto them (we use light of another frequency to read the clocks). We preset the clocks to show a certain time (a certain number that will increase when the clocks are running) and then illuminate them all simultaneously. So we set all of the rim clocks to the same time and illuminate them with a circular pulse of light emitted from the wheelís hub. The clocks will then all tell the same time.
The outer observer disagrees. They have sent out pulses simultaneously to a forward and a rearward clock, both equidistant from the outer observerís location, and when those pulses return, again simultaneously, they show the clocks displaying different numbers: the forward clock shows a time later than the time displayed on the rearward clock. If the light travels in both directions at the same speed, according to the outer observerís measurements, then the forward clock appears to be displaced into the future relative to the rearward clock.
The inner observer sees the outer observerís experiment differently. The light pulses travel at the same speed of light, certainly, but the clocks move with the wheelís rotation. The forward clock moves away from the light pulsesí emission point and the rearward clock moves toward the emission point. It is as if, in the inner observerís calculation, the light pulse travels to the forward clock at the speed c-rω and the pulse travels to the rearward clock at the speed c+rω. For the reflected light the speeds are interchanged, so the pulses end up crossing the same round trip distance and come back to the outer observer simultaneously. If the clocks both lie a distance rdθ from the outer observer, as measured by the inner observer, then the pulses will appear to illuminate the clocks at times displaced from each other by an interval equal to
But the clocks are counting dilated time in accordance with Equation 1, so we know that the outer observer sees them offset from each other by an interval
That looks like the standard temporal offset, the product of a velocity, a distance, and the Lorentz factor divided by the square of the speed of light.
But itís not so much a temporal offset as it is a time lag. The light takes longer to reach the forward clock than it does to reach the rearward clock. This fact is the basis for the laser gyroscope. Also note that if two events illuminate the clocks next to them when they occur, the time will be correct for both observers - a simple subtraction of one clockís reading from the otherís, regardless of when the observers receive those readings. We donít need to account for the time lag in that case.
If the outer observer synchronizes the clocks so that they display the same time for that observer, then the inner observer will see the rearward clock leading the forward clock by that offset. Thus, if two events occur at those clocks and the inner observer makes the measurements on them, they must add that offset to the dilated time that they calculate directly from their own measurements.
But if the inner observer sees the outer observerís clocks offset from each other and those clocks are moving, then shouldnít they see the following clock partly overtake the leading clock? Should the inner observer not detect the Lorentz-Fitzgerald contraction shrinking the rim of the wheel?
This brings us to Ehrenfestís paradox. In 1909 Paul Ehrenfest presented a paradox based on his understanding of Special Relativity (the general theory wasnít available yet). He asserted that someone riding the rim of a spinning wheel would see no distortion of the wheel because they would see no relative motion. On the other hand, an off-wheel observer would see the rim of the wheel display the Lorentz-Fitzgerald contraction while the spokes, oriented perpendicular to the direction of relative motion, display no distortion. Thus, the off-wheel observer would see either the spokes buckle or the rim tear apart, though the on-wheel observer sees no such damage.
So how do we get a Lorentz-Fitzgerald contraction on the wheel? For that matter, how do we know that we must have one?
Our inner observer, equivalent, through the proxy observer, to Ehrenfestís off-wheel observer, sees both the spokes and the rim dilate. The outer/on-wheel observer sees no change in the wheel from its stationary (proper) state. Thatís the basic distortion of distance that we see in Special Relativity. The Lorentz-Fitzgerald contraction comes from combining that distance dilation with the temporal offset between clocks separated from each other in the direction of relative motion.
The resolution of the paradox comes from noticing that the observers do not disagree on the temporal distortion visible in the rim clocks. And they both also agree on the cause of the visible offset. Recall to mind the fact that in the situations requiring the Lorentz Transformation the two observers cannot agree on which one is moving and which one is stationary. In the present case there is perfect agreement: both observers agree that the outer observer occupies a moving frame. The outer observer knows that they are in a rotating frame, because they need only move an object "up" or "down" and note the play of the Coriolis force (which is based on conservation of angular momentum). Thus, when the outer observer sees two of their clocks out of synchrony, they know that itís because their system has moved since the illuminating pulses were emitted. If both observers agree on that temporal offset, as observers in the linear case do not, then there is no actual distortion of time to create the Lorentz-Fitzgerald contraction.
Although we see no distortion of the angular measure between the inner and outer observers, there will be a difference in the longitudinal displacement between events. As seen by the inner observer, the outer observer will move some longitudinal distance between the occurrence of two events. If we measure positive temporal differences in the counterclockwise direction (subtracting the time on a point from the time on a point ahead of it), then the inner observer must add to the distance that the outer observer measures (corrected for distance dilation) a correction equal to the velocity of the outer observer multiplied by the temporal difference between the events. That is, we have
Thus we have the transformation equations for a rotating system. Effectively all we need are Equations 1 and 12. There is no change in the axial direction between the reference frames.
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