The Rotating-Frame Metric Tensor

Assume that we have marked the flat spacetime of the Minkowski metric with two coordinate grids, one Cartesian and one polar cylindrical, that have a common origin. For convenience we suppress the z-direction, so that we only have to describe two spatial dimensions. Imagine a platform set on a pivot at the origin of the coordinate grids and rotating at a uniform and constant angular speed ω (omega) in the counterclockwise direction. To measure the elapse of time we use a clock located at the center of the platform.

Given the polar coordinates (r,θ) of a point on the platform at some initial time t=0, we can calculate the x- and y-coordinates of that point through the transformation equations

(Eq’ns 1)

In those equations theta represents the angular distance between the point and an arbitrarily defined prime meridian (zero degrees east or west) on the polar grid. Usually we define that prime meridian to coincide with the x-axis.

With those equations we can devise a description of the differential transformation of the three dimensions available to us: we need only multiply each differential coordinate by the derivative taken with respect to that coordinate, so we have

(Eq’ns 2)

From those equations we can extract a transformation matrix

(Eq’n 3)

such that

(Eq’n 4)

If we multiply the transformation matrix by its own transpose, we obtain the corresponding metric tensor,

(Eq’n 5)

In that equation I have left out the arguments of the sines and cosines because they are all the same as the ones in Equation 3. With that tensor we can write out the metric equation, the spatio-temporal analogue of the Pythagorean theorem,

(Eq’n 6)

The third line in that equation reveals the fact that, although the elements of the metric tensor are functions of the coordinates, the metric tensor in this case does not indicate a deformation of space and time. The metric tensor encodes the fundamental difference between rectilinear coordinates and curvilinear coordinates, as well as between linear motion and rotary motion. Proof and verification of that proposition comes easily enough through the Christoffel symbols associated with the metric tensor.

First we must obtain the contravariant version of the metric tensor. We calculate the ik-th element of the contravariant metric tensor from the covariant metric tensor through the relation

(Eq’n 7)

and obtain

(Eq’n 8)

We confirm the correctness of that result by calculating the matrix product

(Eq’n 9)

the identity matrix based on the Kronecker delta. To carry out the calculation imagine obtaining the im-th element of the product matrix by superimposing the m-th column of the covariant metric tensor on the i-th row of the contravariant metric tensor, multiplying the superimposed elements together, and summing the partial products thus created. The result is a purely diagonal matrix whose non-zero elements all equal one, as required.

For the calculation of the Christoffel symbols,

(Eq’n 10)

the covariant metric tensor gives us four non-zero derivatives,

(Eq’ns 11)

With∂_{r}=∂_{1}, we have eight Christoffel symbols, six of
them in equal pairs,

(Eq’ns 12)

In those equations (11 & 12) I’m still using the numeral four to indicate the temporal coordinate because spatial dimension #3 is implicit in the matrices, the column and row for that dimension consisting of all zeros in this restricted case.

A basic description of acceleration comes from the geodesic equation in the form

(Eq’n 13)

In that equation we have the four-velocity

(Eq’n 14)

and the three-velocity

(Eq’n 15)

As usual, the Latin indices take the values 1, 2, 3 and the Greek indices take the values 1, 2, 3, 4. That three-vector gives us measurements of a point made by an observer who sits on the platform and turns with it: the three-velocity describes the motion of the point relative to the platform. Equation 13 then gives us the two components of the acceleration of the point as

(Eq’n 16)

and

(Eq’n 17)

Both of those results are functions of the sum of the motion of the point across the platform in the longitudinal direction and the motion of the platform in the same direction. The first (Equation 16) describes the centripetal acceleration that must be imposed on the point to keep it at a constant radial distance from the center of the platform: an observer on the platform at that point would experience a centrifugal force in consequence. The second (Equation 17), when multiplied by the radial distance between the point and the center of the platform, represents the Coriolis acceleration of the point (we have to multiply by the radial distance because the acceleration comes out as an angular change per unit of elapsed time). Physicists refer to both of those results as fictitious forces, but we would better denote them as obligatory forces, forces that rotary motion obliges to occur.

In that analysis there are no relativistic effects. We have tacitly assumed that no part of the platform reaches a relativistic speed. Now we want to assume that the outer parts of the platform move at speeds that are a substantial fraction of the speed of light. How will that assumption change our analysis?

We must modify our imagery to put an observer on the rim of the platform and then we must devise a way of translating measurements that such an outer observer would make between two events into the measurements that an observer at the center of the platform would make between the same two events.

The fundamental phenomenon of Relativity is the distortion of time between inertial 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.

Our observers begin by setting up their clocks and getting them running. The outer observer (riding the rim of the platform and using upper-case variables to represent their measurements) sets up a number of clocks along the rim of the platform and sets them all to show the same time. The inner observer (sitting on the axis about which the platform rotates and using lower-case variables to represent their measurements) has only one clock. All of the clocks are Feynman clocks; that is, they operate by counting the number of times that a pulse of light passes between two mirrors, the light path being oriented perpendicular to the platform (that is, in the spatial dimension that we are ignoring).

The inner observer starts all of the clocks by producing a brief, intense, omnidirectional flash of light from a point on the platform’s axis. The inner clock starts immediately and the outer clocks all start simultaneously with one another at a time r/c later. Clearly the outer clocks, because they are moving, will display the effect of time dilation: if they measure dT microseconds between two events, then the inner clock will measure dt microseconds between the same two events in accordance with

(Eq’n 18)

The dT microseconds last longer than the dt microseconds, so fewer of them will fit between the two events. Because the outer clocks all started simultaneously, they display no temporal offset from one another, as we might expect from our knowledge of the symmetry of the situation.

But the situation is not entirely symmetric. Because the motion between the inner and outer observers does not change the distance between the observers, the distortion between those observers’ clocks must be asymmetrical. That fact necessitates that both observers use Equation 18 when they convert each other’s temporal measurements. The physical reason for that asymmetry lies in the fact that the outer observer must accelerate toward the inner observer, but the inner observer undergoes no acceleration at all. The centrifugal force that the outer observer feels is equivalent to gravity, so the reference frame that the rim of the platform occupies and marks is deformed away from a Minkowskian inertial frame of reference, which the inner observer occupies and marks. On the axis of the platform the Minkowskian frame and the deformed frame coincide.

Distinct from a Minkowskian inertial frame, the rotating frame does not display a temporal offset. That means that we don’t have to modify Equation 18. We know this statement is true to Reality because of the finite nature of the longitudinal dimension.

A circle consists of an infinite set of points, certainly, but when we observe rotations we use a finite measure, angles given in degrees or radians, a finite number of which measure a complete circle. 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 toward "infinity", where the properties of infinity absorb the distortion. That option is not available for circular displacement, so we must 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 revolving about some axis relative to each other. We thus infer that

(Eq’n 19)

That statement must stand true to Reality because we can mark all of the clocks on the rim of the platform with their angular positions. With both observers looking at a clock that’s set on the platform opposite the position occupied by the outer observer, we know that both observers must agree on when that clock appears to pass a certain quasar: the clock must display the same time for both of them. If the inner observer uses a mirror to reflect the image of any clock to the outer observer, the same analysis will apply. It may be dilated time, but it will be the same for both observers (the inner observer’s clock will show something different, of course). Thus, we infer that no temporal offset appears on the rim clocks for either observer.

Further, if there were such an offset, it would lead to a clear absurdity. If the outer observer holds a master clock, then clocks located away from that master clock in one direction would be retarded relative to the master clock and clocks located away from the master clock in the opposite direction would appear advanced. Going around the platform, we would eventually reach a clock that must be both advanced and retarded while appearing to pass a certain quasar, a clear absurdity. Thus, we infer that there is no temporal offset in rotating frames.

If some velocity exists between two observers, it must have the same value for both of them. The outer observer, surveying the Universe, determines that they are moving around the circle occupied by the rim of the platform (and, thus, relative to the inner observer) at a speed of

(Eq’n 20)

The inner observer must calculate the same speed,

(Eq’n 21)

Setting those equations equal to each other and substituting from Equations 19 and 18 tells us that

(Eq’n 22)

In measuring radial distance and longitudinal distance between two given points, the outer observer uses dilated space relative to what the inner observer uses. Because the observers can use the platform itself (or, more precisely, markings on it) to measure distances and because the platform would appear normal to the outer observer, it must appear enlarged to the inner observer; it appears to spread across more of the inner observer’s Minkowski frame.

Although there is no distortion of the angular measure between the inner and outer observers, there will be a 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 the point ahead of it in the direction of the platform’s motion), then the inner observer must add to the longitudinal distance that the outer observer measures between two events a correction equal to the velocity of the outer observer multiplied by the temporal difference between the events. The inner observer would then calculate

(Eq’n 23)

as the longitudinal distance between the events.

Now gather together Equations 22, 23, and 18;

(Eq’ns 24)

This is the rotary analogue of the Lorentz Transformation (with the z-direction suppressed because both observers measure the same distances in that direction.) From these equations we can devise a transformation matrix.

I will note that Equations 24 stand as a resolution of Ehrenfest’s paradox. Stated by Paul Ehrenfest in 1909, the paradox comes from applying a misguided version of Special Relativity to a rotating disc. Ehrenfest assumed that the rim of the disc would be subject to the Lorentz-Fitzgerald contraction while the radius, oriented perpendicular to the relative motion would remain unaffected. Thus, he expected that the disc would either buckle or tear itself apart for outside observers, while a person riding the rim of the disc would see no such distortion. Ehrenfest’s analysis came before the advent of General Relativity, which, as we see, dissolves the paradox.

We can express the rotating-frame transformation matrix (bearing in mind the fact that we have suppressed the z-direction for convenience) through the transformation equation,

(Eq’n 25)

With that transformation matrix we can devise the associated metric tensor by multiplying it on the left by its transpose. If we look at the metric equation itself,

(Eq’n 26)

we can see how to draw the imaginary coefficients into the matrix. Using appropriate simplifiers, we get

(Eq’n 27)

That tensor describes the spacetime of a rotating frame relative to a non-rotating frame, all measured in polar coordinates. Thus we obtain a description of the relativistic kinematics of rotation.

Appendix: Rotary Pseudo-gravity

How can we understand the temporal distortion created by rotation? Imagine that the inner observer in the above-described situation allows a small body of rest mass m to slide freely down one of the spokes that support the platform, holding it rigidly to its axle. That observer can calculate the body’s 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 over a minuscule radial distance,

(Eq’n A-1)

After integrating that equation, dividing out the mass, and multiplying by two, they get

(Eq’n A-2)

Inside the small body there is a tiny chamber with perfectly reflective walls. Photons flitter to and fro within that chamber. As the body slides down the spoke, the frequency of the photons must increase, as does the mass of the body. We know that statement must stand true to Reality because we know that, at least in concept, we can convert the photons into particles whose masses are related to the photons’ frequencies through Planck’s theorem and Einstein’s mass-energy theorem,

(Eq’n A-3)

And, of course, we assume that energy obeys the conservation law.

The inner observer calculates the relativistic mass of the body as

(Eq’n A-4)

They also have Mc^{2}=h(dn/dT), in which dn represents a number of
cycles of a photon’s vibration. Both observers agree to use the same number for
dn, so they calculate

(Eq’n A-5)

That equation, with a little algebraic reworking, yields Equation 18.

In this example we have used the centrifugal force as if it were the gravitational force and we got a result like the one we got for the Schwarzschild solution of Einstein’s space-warping equation.

eabf