The Rotating-Frame Metric
Superimpose upon a Cartesian coordinate grid marking a Minkowskian inertial frame a non-inertial frame marked by a polar coordinate grid rotating counterclockwise with angular velocity
ω. Ensure that the origins of both grids coincide and place a master clock at that common center of measurement. Suppress the third spatial dimension for simplicity so that each point that we consider occupies only the spatial coordinates (x,y) and (r,θ) such that
(Eq’ns 1)
Differentiating those equations yields
(Eq’ns 2)
which equations give us the transformation between the differential elements of distance in the two frames. Applying Minkowski’s theorem to generate the spatio-temporal analogue of the Pythagorean equation yields
(Eq’n 3)
That equation implies the existence of a radial time dilation effect, but we can rewrite that equation as
(Eq’n 4)
In that equation the presumed time dilation vanishes and we see in its place a statement that the longitude of a point changes with the elapse of time. Which of those interpretations of the differential distance element stands true to Reality?
In passing from unaccelerated straight-line motion to rotary motion, we pass from Special Relativity into the realm of General Relativity. Thus the Equivalence Principle comes into play in our analysis of the coordinate frames described above. If we imagine putting an observer inside a windowless vehicle, then the principle tells us that making the vehicle accelerate yields one of two outcomes: if the observer detects the acceleration, then a Lorentzian distortion of space and time comes between the beginning and ending of the acceleration; and if the observer cannot detect the acceleration, then the spatio-temporal points at the beginning and ending of the acceleration occupy the same inertial frame, albeit a deformed one.
As a relevant example consider a thin ring of mass M carrying angular momentum L=MVR, in which R represents the radius of the ring and V represents the longitudinal velocity of the ring’s rotation. Centered on the origin of the grid, the ring starts with a pseudo-infinite radius, so V approaches zero, thereby putting the ring arbitrarily close to the inertial frame marked by our Cartesian coordinate grid. Someone then makes the ring shrink down until its radius has such a value that V/R=
Ω, the angular velocity of our polar coordinate grid as measured by an observer on the Cartesian grid. Conservation of angular momentum makes the longitudinal velocity increase in accordance with
(Eq’n 5)
which leads to a description of the Coriolis acceleration,
(Eq’n 6)
Because we could divide the inertial mass out of the equation, we know that an observer riding on the ring will not detect the Coriolis acceleration directly: the system lacks the inertial difference that would make objects move differently. Thus we infer, in accordance with the Equivalence Principle, that in the longitudinal direction the rotating frame remains arbitrarily close to the non-rotating frame.
On the other hand, something must apply a force to make the ring shrink and then to keep it shrunk. Unless the force comes from gravity, in which case we can divide out the inertial mass as we did above, an observer on the ring will detect the acceleration directly. Newton’s first law of motion necessitates the application of a force to make any body move on any path other than a straight line at constant velocity and Newton’s second law of motion equates the force that acts on a body to the rate at which the product of the body’s mass and velocity (quantity of motion or linear momentum) changes with the elapse of time. To travel in a circle a body must accelerate continuously toward the center of the circle at a rate of A=
Ω2R: the forced body reacts against that acceleration with what physicists call centrifugal force.In order to make the ring described above shrink its radius, the applied force must do work on the ring. In the absence of a longitudinal component of the applied force (which absence we assume) the ring’s angular momentum remains constant (L=M
ΩR2=const.), so the work done against the centrifugal force comes out as
(Eq’n 7)
The expression on the last line of that equation represents the work that the applied force must do on the ring to shrink it down until it spins with our rotating coordinate grid. The negative sign, necessitated by the rules of integration, indicates that we have actually calculated the amount of energy lost by the mechanism that applies the force to the ring.
If the applied force had acted on a simple body floating in free space, that body would have shifted between two inertial frames separated each from the other by the velocity V. In that case, as in the case of shrinking the spinning ring, a blind-wrapped observer will detect the acceleration directly, so the Equivalence Principle requires that the observer end up in a frame different from the frame they occupied at the beginning of the process.
So now imagine that we have two observers and their measuring equipment. One observer occupies a point on the ring (we call him the Ringrider) and between any two events he measures a radial displacement r, a longitudinal displacement
θ, and a time interval t. At the Farpoint, a point lying a pseudo-infinite distance from the center of the double coordinate grid and motionless relative to the Cartesian grid, the other observer measures of those same events R, Θ, and T. We want now to inter-relate those two trios of measurements. We note in passing that for the Farpoint observer Equations 1 take the form
(Eq’ns 8)
Between the Farpoint observer and the Ringrider there exists a virtual velocity in the radial direction, but no virtual velocity in the longitudinal direction, while there exists no net real radial velocity but a non-zero real longitudinal velocity between them. In light of those facts, we begin our analysis, much as we did with Special Relativity, with our observers measuring the properties of light.
To start, the Ringrider transmits a pulse of monochromatic light in the prograde direction such that it will pass through the Farpoint. He also tells the Farpoint observer the frequency of the radiation. When the pulse reaches the Farpoint the observer measures a higher frequency in the radiation, due to the fact that the transmitter moved toward the Farpoint with a speed of V=
ΩR when it emitted the pulse. On first impression the observers attribute the change to the Doppler shift, but then they notice that, according to the Equivalence Principle, the transmitter doesn’t move relative to the receiver. Unable to invoke the Doppler shift, our observers must necessarily infer that the Ringrider’s clocks count time faster than the Farpoint observer’s clocks do, so they use an inverted version of the standard time dilation formula,
(Eq’n 9)
Thus, if our observers measure a time interval between two events, the Ringrider’s clocks count more time than the Farpoint clocks do, so the transformation from one observer’s measurements to the other’s must diminish the number derived from the Ringrider’s clocks.
To prove and verify that proposition the Farpoint observer projects a pulse of radiation on a trajectory that will just graze the ring. She projects the radiation in the prograde direction, the direction in which the ring spins, because only in that direction do the spinning frame and the Cartesian frame come arbitrarily close to each other. When the radiation grazes the ring the Ringrider’s instruments capture it and measure its frequency. In this instance the radiation undergoes a redshift, indicating that the Farpoint observer’s clocks count time more slowly than the Ringrider’s clocks do, thereby confirming an inverted version of Equation 9.
The inference of a temporal distortion between the observers raises the possibility of another warping of time coming into play. In Special Relativity we have that other warping occurring between clocks separated from each other in the direction of relative motion. The constancy of the speed of light necessitates that between clocks synchronized in one inertial frame there appears a temporal displacement in other frames. But that displacement won’t occur between clocks on a spinning ring and we have two reasons for making that statement.
Recall to mind the fact that in Special Relativity a timekeeper synchronizes two clocks by setting them to show the same time and then using a pulse of light originating midway between the clocks to start them running. Another observer, moving parallel to the line separating the clocks, sees the following clock start before the leading clock does, because light travels at the same speed in both directions in that observer’s frame; thus, that observer infers the existence of a temporal displacement or offset between the two frames. However, in a rotating frame marked by a spinning ring, a timekeeper riding on that ring can synchronize all of the clocks on the ring by setting them to show the same time and then starting them with a pulse of light emanating from the center of the ring’s rotation, the origin of the coordinate grids. The clocks will all start simultaneously for any observer for whom the origin of the frame does not move, so both the Ringrider and the Farpoint observer will see the clocks all in perfect synchrony.
Another way of confirming the lack of a temporal offset between clocks on the ring consists of taking a picture. Imagine suspending a camera above the common origin of our two frames and using it to take a picture of the ring and its clocks. Now assume that on the developed picture we see a temporal offset between two of the clocks. As we look at clocks progressively further around the ring we must see that offset grow. But that fact necessarily entails the consequence that we must see, depending on the direction we followed around the ring, the first clock that we saw running fast or slow relative to itself. Reality does not permit the manifestation of absurdities, so we must dismiss one of the premises that led us to the one above. Thus we infer that no temporal offset comes between any two clocks on a spinning ring.
That photograph also reveals something else. Imagine that the Ringrider had evenly spaced 360 clocks around the ring. Each pair of neighboring clocks has one degree of angle between them. In the photograph the Farpoint observer must see the same thing. The lack of a temporal offset between neighboring clocks means no spatial offset in the distance between the clocks as in the Lorentz-Fitzgerald contraction. Thus, if our observers measure the angular distance between two points, they have
(Eq’n 10)
If we divide that equation by Equation 9, we get a mathematical description of the angular velocity of the ring as measured by both observers;
(Eq’n 11)
With that equation and one other we can use the law of conservation of angular momentum to work out the rest of the rotating-frame transformation.
Although we want to concentrate on the rotating frame, we must still acknowledge that the ring exists as a body spinning freely in Minkowski space. In particular we must note that its rotary motion gives it a kinetic energy that, in accordance with Special Relativity, comes manifest in the increased mass of the ring. Of course, the Ringrider does not experience that mass increase; only the Farpoint observer, occupying the Minkowski frame, does. Thus, if the Ringrider measures the mass of an object on the ring as m, then the Farpoint observer will infer of that object a mass M in accordance with
(Eq’n 12)
The work done in shrinking the ring makes it and the things on it more ponderous.
At the beginning of our imaginary experiment, when the ring has a pseudo-infinite radius, it occupies a rotating coordinate frame negligibly different from the Cartesian inertial frame measuring the Minkowski space occupied by the Farpoint observer. There both observers calculate the ring’s angular momentum from their measurements of its properties and find that it has essentially the same value for both of them;
(Eq’n 13)
When the radius of the ring shrinks the conservation law necessitates that the equality remain unaffected. Applying Equations 11 and 12 to that equation thus gives us
(Eq’n 14)
which necessitates that
(Eq’n 15)
stand true to Reality. Thus, if the Ringrider measures a distance dr between two points in the radial direction, then the Farpoint observer must measure a shorter distance between those same two points.
The relation in Equation 15 certainly applies to measurements made in the radial direction, but we must also ask whether it applies as well to the measurements made in the longitudinal direction. Although angles measured in the longitudinal direction come out the same for both observers, according to Equation 10, do distances measured between two points in that direction actually conform to the expected relation
(Eq’n 16)
or should we infer a different relation? We can discern several reasons for accepting the above equation as a correct description of longitudinal distances in our rotating frame.
If we divide the above equation by Equation 9 and interpret the result through Equation 11, we get
(Eq’n 17)
Likewise, if we divide Equation 15 by Equation 9, we get VR=vr. Thus, we see that both observers measure the same velocity between their grids, as we expect of observers occupying two different aspects of what we must comprehend as the same reference frame.
Further, both Equations 15 and 16 must stand true to Reality in order to resolve a paradox attributed to Paul Ehrenfest (1880 Jan 18 – 1933 Sep 25). Ehrenfest imagined a spinning disc and asserted that its longitudinal dimension would suffer a Lorentz-Fitzgerald contraction while its radial dimension would suffer no alteration. In essence he imagined taking a Lorentz Transformed inertial frame and wrapping it around a pole. The paradox came from the dilemma in which the spacetime occupied and marked by the disc, as well as the disc itself, must either crumple in the radial direction or tear apart in the longitudinal direction. That dilemma vanishes if both the radial and the longitudinal dimensions deform in the same way, as Equations 15 and 16 indicate. That fact stands as strong evidence for the validity of the above analysis.
The validity of Equation 16 also comes to us by way of an application of Equation 17. Beginning with Equation 9, I have used R
Ω as the velocity in the Lorentz factor. Without the equality in Equation 17 that use would have created a serious problem for, if not completely illegitimated, the above analysis. Fortunately, our observers can write the Lorentz factor between their frames as
(Eq’n 18)
That equality conforms to what Relativity leads us to expect, that two observers should always calculate the same value for the Lorentz factor between their reference frames. And, as we expect, that equality persists after the Ringrider falls off the ring and flies away from it along a straight line.
That latter proposition shows us a kind of temporal boundary, with General Relativity applicable on one side and Special Relativity on the other. Further, we know that we cannot have a discontinuity at that boundary. The phenomena that we infer from our General Relativistic analysis of the system must merge smoothly with the phenomena of Special Relativity at that boundary.
But we seem to have a subtle discontinuity anyway. Imagine that we have two identical chains wrapped around our ring and that at some instant they began to unroll from the ring, coming off opposite sides (to keep the forces balanced), and fly off into space in straight lines. Certainly the longitudinal contraction of the chains as they cling to the ring segues easily into the Lorentz-Fitzgerald contraction of the free-flying chains. But the Lorentz-Fitzgerald contraction comes from a dilation of length modified by a temporal offset between clocks and we have no temporal offset between clocks on either of the chains.
However, on the ring the clocks count contracted time while on the free-flying chain they count dilated time. That fact will produce exactly the temporal offset that we need. Imagine two clocks a distance x apart on the chain. First one will come off the ring and fly straight and then an interval of time x/V later the other will come off the ring and fly straight. During that interval the leading clock will count dilated time while the following clock counts contracted time, so the Farpoint observer will see the following clock gain over the leading clock an offset of
(Eq’n 19)
in which V=
ΩR. That equation does, in fact, give us the temporal offset that we get from the Lorentz Transformation, so as the clocks come off the ring they automatically reconfigure themselves to conform to the requirements of Special Relativity. And looking at that result, showing how the Special and General theories merge smoothly one into the other, gives us an exquisite example of the ætherial beauty that we can find in the theory of Relativity.If the observers measure the longitudinal distance between two events, they cannot use Equation 16 as we have presented it. They must account for the fact that a relative rotation comes between their measurement grids. We have Equation 10 in the form
(Eq’n 20)
which obliges us to modify Equation 16 to read
(Eq’n 21)
Finally we can gather up and present together the rotating-frame transformation and metric equations. For the transformation we have Equations 9, 15, and 21
(Eq’ns 22)
and for the metric equation we have, via Minkowski’s theorem,
(Eq’n 23)
Now we have the means to answer a question that I posed at the beginning of this essay. Equation 23 looks more like Equation 4 than it looks like Equation 3, differing only by multiplication by the square of the reciprocal of the Lorentz factor. Indeed, for rotational velocities very much less than the speed of light Equation 23 becomes Equation 4, which means that Equation 4 represents the non-relativistic relation between the reference frames.
Looking forward, we can see that we now have the means to augment the Schwarzschild metric to include rotation and thereby devise a description of the Kerr metric.
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