The Kerr Spacetime
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In 1963 mathematician Roy P. Kerr devised a metric equation describing the region of space and time surrounding a simple, spherically-symmetric body pondering a mass M and spinning with an angular momentum of J. This added the complication of rotary motion to the already-known Schwarzschild solution of Einstein’s equation.
In 1966 mathematician Robert H. Boyer and physicist Richard W. Lindquist demonstrated that, when the angular momentum of a gravitating body goes to zero, Kerr’s metric equation does, indeed, become Schwarzschild’s equation (see Appendix below). Then Boyer and Lindquist produced an all polar coordinates version of Kerr’s equation that clearly resembles Schwarzschild’s equation. I now want to deduce that version of Kerr’s equation in the same way I deduced Schwarzschild’s equation.
Let’s begin by recalling to mind the proposition that any collection of bodies or particles not attached to each other (either directly or indirectly) and neither moving nor accelerating relative to each other occupies and marks an inertial frame of reference. Using that statement as a definition of an inertial frame gives us the perspective to appreciate how the equivalence principle describes a warped inertial frame. We have already used that perspective to look at a collection of bodies falling freely in a gravitational field and to deduce the transformation equations and the metric equation of Schwarzschild space. Now we have another phenomenon that comes within the purview of the equivalence principle.
I deduced the Schwarzschild solution of Einstein’s field equation from the equivalence principle, which Einstein discovered when he read a newspaper article about a carpenter falling from the roof of a house in or near Bern, Switzerland. I got the inspiration for my simpler version of General Relativity from a similar image of a body falling freely in a gravitational field. Guided by the equivalence principle and the concept of virtual relative motion, I derived four equations describing what I call the Schwarzschild Transformation, the General Relativistic analogue of the Lorentz Transformation. Then I combined those four equations, as Hermann Minkowski had combined the equations of the Lorentz Transformation, into the four-dimensional analogue of the Pythagorean theorem and got Schwarzschild’s metric equation. That metric equation describes a region of spacetime deformed by the gravitational field emanating from a uniform, spherical body of mass M located at the origin of a polar coordinate frame, the description coming from contemplation of a minuscule body falling freely in the gravitational field, the falling body both occupying and marking the deformed equivalent of an inertial frame of reference.
We understand the equivalence principle to mean that acceleration only truly exists for any observer who can detect it directly when enclosed in an opaque container. Of course, we don’t actually need the container. Even though he was in the open air, Einstein’s carpenter, moving with increasing velocity towards Earth’s surface, was not truly accelerating, but, rather, occupied a warped inertial frame: the carpenter’s statement that he saw his tools floating weightless before him, as if unaffected by any applied force, gave Einstein the clue he needed. That interpretation reflects the fact that the force exerted upon a body by Earth (or any other gravitating body) stands in direct proportion to the forced body’s inertial mass, so all bodies accelerate at the same rate. Any other force that has that property will also produce a warped inertial frame. Basically, we can describe a warped inertial frame (or non-inertial frame) by saying that it looks like we’re accelerating, but feels like we’re not.
We have an example of such a force in the law that decrees that conservation of angular momentum necessitates, as a direct consequence, the existence of a longitudinal force that accelerates and decelerates revolving bodies as they move closer to and farther from their center of revolution. Physicists call that force, named after Gaspard-Gustave de Coriolis (1792 May 21 - 1843 Sep 19), a fictitious force, primarily to distinguish it from forces, such as gravity, electricity, or nuclear attraction, which inhere in properties of matter and, therefore, seem more real. Where the existence of matter necessitates the existence of those real forces, Coriolis force and centrifugal force appear in matter moving in a particular way. But instead of calling it a fictitious force, we would better describe Coriolis force as an obligatory force, one whose existence Reality requires in order to enforce the conservation law of angular momentum.
In form, the mathematical description of the Coriolis force has one feature in common with our description of gravitational force: in magnitude it is directly proportional to the inertial mass of the body whose curved motion makes it manifest. That fact means that Coriolis acceleration, like gravitational acceleration, has no dependence whatsoever on the inertial mass of the accelerated bodies. All bodies following the same trajectory will undergo the same acceleration at the same point on the trajectory. Thus, a flock of unattached bodies following a given trajectory will stay together (neglecting tidal effects) as they would if they occupied an undistorted inertial frame in deep space. We must infer, then, that the equivalence principle applies to Coriolis acceleration as it applies to gravitational acceleration; that is, just as gravitation distorts inertial frames in the radial direction, so Coriolis force distorts them in the longitudinal direction. Now we need only devise a mathematical description of that distortion.
If we establish two observers, stationary relative to each other, at different distances from a gravitating body whose mass we represent as M, then the equivalence principle obliges them to translate their measurements of the same two events before communicating them to each other. They must base the translation on the premise that they occupy two different inertial frames that have a virtual velocity between them, even though the actual velocity between them equals zero. To make the translation, they would use a version of the Lorentz Transformation modified by Newton’s law of gravity. If the lower observer measures their position a distance r from the center of the gravitating body and the upper observer at pseudo-infinity (at a distance far enough from the gravitating body that the body’s gravity approaches arbitrarily close to zero) measures the same distance as R from the center of the gravitating body, then those observers would use in their translations the equations of what I call the Schwarzschild Transformation:
In applying those equations to our observers’ measurements we must remember that we use the upper-case variables to represent measurements that the observer at pseudo-infinity (the upper observer) makes and that we use the lower-case variables to represent measurements that the lower observer (stationed at the distance r from center of the gravitating body) makes. On the assumption that the system under consideration has spherical symmetry, I have written the transformation equations in spherical polar coordinates, in which the letter phi represents the latitude of a point in the system and the letter theta represents that point’s longitude. If we add together the squares of the first three of those equations and subtract the square of the fourth, we get the Schwarzschild metric equation: the left side looks like the Minkowski metric equation in polar coordinates and the right side looks like a deformed version of that metric.
If we modify that situation by making the gravitating body rotate with angular momentum J (without distorting the body itself), then our observers will have before them the task of modifying Equations 1 in a way that incorporates a description of the Coriolis warping of the already gravitationally warped inertial frame. First they must discern a number that truly represents the source of the Coriolis warping. To that end, to discern the correct measure of the longitudinal warping of the inertial frame by the Coriolis force, our observers review the creation of a space dominated by the Schwarzschild metric.
Imagine that someone has assembled a collection of particles into a thin spherical shell of pseudo-infinite radius and mass dM and that they allow that shell to collapse under the force of its self-gravitation. Then that someone creates another shell, identical to the first, and allows it to collapse onto the body formed by the collapse of the first shell and then repeats the process again and again. In time a body of mass M forms and the last shell to collapse onto it occupies and marks the warped inertial frame that defines the Schwarzschild metric of the space around the body. At any distance R from the center of the body the shell moves inward at a speed calculated by equating the kinetic energy gained by a unit mass in the shell to the gravitational potential energy that unit mass loses; that is,
Absorbing dM into M (since dM is negligibly small), our upper observer uses that equation to make the substitution onto the Lorentz factor that converts that factor into the analogous Schwarzschild factor in the first and fourth of Equations 1; to wit
Now to create a rotating body someone gives each spherical shell an angular momentum dJ and allows it to collapse under the influence of its self-gravitation and the cumulative gravitation of the previous shells. Due to its angular momentum, each shell turns with a uniform angular speed (that is, each element of the shell has the same angular speed, though they all have different longitudinal velocities). As the shell collapses the radii of gyration of all the elements change in the same proportion, so the angular speeds of the elements also increase in the same proportion (due to conservation of angular momentum expressed as ùr2=constant). We give each shell an angular momentum such that when the shell comes down on the growing sphere it and the sphere rotate at the same angular speed. For the completed sphere, of mass M, angular momentum J, and radius r0, we calculate the angular speed of rotation as
in whichI represents the sphere’s moment of inertia. Note that I have tacitly assumed that the sphere rotates as a solid body.
The equivalence principle tells us that an observer riding the last shell to collapse onto the sphere with feel no acceleration; neither radial acceleration, due to the gravitational force, nor longitudinal acceleration, due to the Coriolis force. That observation means that the observer occupies a reference frame insignificantly different from the Minkowski inertial frame that originally occupied the space around the sphere and which still effectively exists at pseudo-infinity. That observer has both a radial velocity and a longitudinal velocity relative to an observer hovering just above the surface of the sphere, an observer who remains actually motionless relative to an observer at pseudo-infinity. Thus the observer at pseudo-infinity must treat any measurements made by the hovering observer as if there were a velocity between the two of them. That virtual velocity is the negative of the velocity gained by the free-falling observer.
We obtain the virtual velocity from equating the kinetic energy gained by a free-falling body to the potential energy that it has lost. That calculation must acknowledge that we have two components of velocity – the radial component v and the longitudinal component w=Rω. We thus have for our calculation
Using that, we calculate
In making that calculation I have assumed that the velocities do not go into the relativistic realm of speeds achieving a substantial fraction of the speed of light, so this derivation will only give us a semi-classical treatment of Kerr space.
Whatever transformation equations our observers end up using, those equations must have a form that reverts to Equations 1 as a limit as the angular speed of the gravitating body’s rotation tends toward zero. That reversion must proceed smoothly and continuously, so that the theory will satisfy the correspondence principle, the statement that any theory modified to accommodate new conditions must revert to the original theory when those new conditions go away.
Likewise, if we have a rotating system and the mass goes to zero, the transformation equations must revert to
We can imagine that this is what we have left when the massive body has diminished to a thin frame of negligible mass. If we compare these equations to Equations 1, we see that the temporal term is going to give us a special problem to solve.
Recall to mind the fact that we obtained the first and fourth of Equations 1 from the Lorentz-Fitzgerald contraction and time dilation, respectively, by replacing the square of the relative velocity in the Lorentz factor with twice the gravitational potential between two observers. If we make the analogous substitution from Equation 6 as a first guess in the present case, we get
In those equations the omega term corresponds to the centrifugal force that opposes the gravitational force. Clearly those equations revert to the first and fourth of Equations 1 when the angular speed tends to zero.
In the equatorial plane of our system the third of Equations 1 becomes
but we suspect that we may have to modify it to express relativistic effects due to the system’s angular motion. Though we have two observers with no actual longitudinal motion between them, the spinning body’s rotation puts a virtual longitudinal motion between them. How does that fact oblige us to modify Equation 10?
Between any two points both observers measure the same angular displacement. We see how that statement must be true to Reality if we ask our observers to measure the shortest angular distance between the two points and the longest. For both observers both distances must add up to two pi radians (360 degrees) and the ratio of the two distances must be the same, so it must be true to Reality that dè=dÈ. In light of Equation 8, then, we can rewrite Equation 10 as
Note that when M goes to zero, that equation does not coincide with the first of Equations 7. We should not expect it to do so, because the two equations represent different phenomena. That fact tells us that Equation 11 is not complete. We will complete it further along.
Through appropriate surveying techniques, the upper observer measures a differential radial distance dR between two events and the lower observer measures dr. In order to communicate their measurement to the lower observer, the upper observer converts the length into wavelengths of a pulse of monochromatic light that they send to the lower observer. But the gravitational redshift subjects descending light to a blueshift, making the wavelengths shorter, so the upper observer must multiply their measurement by a compensating factor before transmitting the data. That factor comes from the relativistic Doppler shift,
In this case v/c in the numerator equals zero because there’s no actual motion between the observers. Using Equation 6 for the square of the radial velocity gives us Equation 8, so if the upper observer measures one meter between events in the radial direction, the lower observer will measure more than a meter, in accordance with Equation 8.
Temporal distortion is a little different. The gravitational redshift diminishes the frequency of light rising through a gravitational field, so the upper observer sees the lower observer’s clocks running slower than theirs do. The time that the upper observer measures between the two events must thus be diminished in order to match what the lower observer measures. In this case, though, the Doppler coefficient must use the whole virtual velocity and not only the radial component of that velocity, so we have
If the upper observer measures one second between the events, then the lower observer’s clocks will measure less than one second. Note how that equation differs from Equation 9, which was predicated on the idea that the upper observer has a virtual motion only in the radial direction.
Let’s look at the effects due to a virtual motion in the longitudinal direction and see how we need to modify Equation 11. We make two events occur near the lower observer, who measures an interval (dθ,dt) between them. The upper observer measures an interval (dΘ,dT) and proposes using Equations 11 and 12 to translate that interval for the lower observer. But the virtual relative velocity between the observers will add a spacial offset to the longitudinal component of the interval, reflecting the distance that the loci of the events appear to move in the time interval between them.
We didn’t need an offset due to virtual relative velocity in the radial direction, so why do we need it in the longitudinal direction? In the radial case there exists no actual motion between either of the observers and the gravitating body, only a virtual relative motion between the upper observer and both the lower observer and the gravitating body, so no correction for actual motion is necessary. In the longitudinal case there exists an actual motion between both observers and the gravitating body due to that body’s rotation.
If we calculate the differential time elapsed between two nearby events by subtracting the time of the aft event from the time of the fore event (going from west to east, with the rotation of the body), then the upper observer must subtract the corresponding longitudinal displacement from the distance measured by the lower observer (adding it to their own measurement), which means that Equation 11 becomes
Is there also a temporal offset due to virtual longitudinal motion in the system? There was none in the rotating-frame metric, but the addition of gravity might change that fact. We need to test that proposition and we do so by the usual method of synchronizing clocks.
Imagine laying clocks along the equator of the spinning body M and on a ring hovering just a fraction of a millimeter above them and motionless relative to the lower observer. Imagine further that the body M is transparent and that a very brief flash of light emanates from its exact center. Clearly that flash starts all of the clocks simultaneously for all observers. Thus we see no basis for a temporal offset to appear among the clocks in either array as seen by either of our observers.
One minor point deserves comment. If we assume that the body M has the optical properties of vacuum, then we know that the light emanating from its center will propagate straight outward, purely in the radial direction. But an observer moving with the equator clocks would see the light aberrated, apparently coming from a point displaced from the center of the body M by a distance that we can call the Kerr radius,
That result comes from comparing two similar right triangles: in the first triangle the hypotenuse (tilted away from the radial direction by an angleα) represents the speed of light and the side opposite the angle á represents the speed of the clocks due to the rotation of the body M and in the second triangle the hypotenuse represents the radius of the body M and the side opposite the angle α represents the apparent displacement of the source of the light. Equation 15 comes from the equality between the ratios of the sides.
Suppose we want to measure the longitudinal distance between two points at some colatitudeϕ. Classically we expect to have
In this case we refer the radial distance to the distance measured by the shortest route from the axis of rotation of the system; hence, the reference to the sine of the colatitude. Now we want to develop the relativistic version of that equation. We can measure the longitude and the colatitude easily enough: we anticipate some problem in our measurement of the radial distance that we must use in the calculation.
Imagine that the body M is perfectly transparent and that a reflective thread occupies its axis of rotation. The stationary lower observer sends a pulse of light at the thread and measures the time it takes the pulse to return. From that datum they calculate rcosϕ.
The moving lower observer, the upper observer’s proxy (riding one of the equator clocks), makes the same measurement, but does so using aberrated light. Both going and coming, the light is displaced longitudinally by the Kerr radius, so the radial distance used in calculating the longitudinal positions of the events conforms to
But that’s only true in the equatorial plane; at different colatitudes that observer must use
We know that both moving and stationary observers must measure rsinϕ with the moving observer accounting for aberration. That’s true because the pulse of light used in surveying the distance must strike the reflective thread and be reflected off it at a right angle, so the pulse must have no latitudinal component in its motion. The amount of aberration stands in direct proportion to the radial distance that the pulse travels, so the compensating term is RKsinΦ, from the inclusion of which in the Pythagorean theorem we get Equation 18. That means that we get the relativistic version of Equation 16 as
For distances measured in the latitudinal direction we don’t expect to see any displacement effects, but there may be proportionality effects; that is, we likely won’t get the simple
of classical spherical geometry. We will likely have a different coefficient on the right side of that equation, but no extra terms.
If our big gravitating body does not rotate, then r represents the radial distance that the lower observer measures between the center of the body and the point at which the event under consideration occurs. We refer that distance to the recticurvaceous reference grid (with spherical polar coordinates) that we have established with the center of the body at the origin. If the body then begins to rotate, the reference frame will begin to swirl as a cylindrical vortex. The longitudinal speed of the vortex increases as the radial distance decreases and, if the body is sufficiently compressed, reaches the speed of light at the Kerr radius.
As the body’s spin rate goes from zero to its full value, the Kerr radius goes from zero to its full value, taking the origin of the frame with it. As a consequence the upper observer must modify their measurement of radial distance by adding the Kerr radius through the Pythagorean theorem. Keeping the measurement in the plane defined by the statementΦ=constant, the observer sees the projected circle of the Kerr radius becoming narrower as the colatitude goes from zero (at one of the poles) to 90 degrees (on the equatorial plane), which means that they see the Kerr radius diminished by the cosine of the colatitude. Thus they calculate
and Equation 20 becomes
Equation 8 must also be modified to become
So now we have the four equations of the Kerr Transformation, Equations 23, 19, 22, and 13. If we combine those equations through Minkowski’s theorem, we get the Kerr metric equation,
That equation differs from the Boyer-Lindquist equation (Equation A-15, with a=RK), but not by much. So we will accept this provisionally for now and examine its consequences in other essays to see if we can discover how to fix it.
Appendix: The Kerr Metric Equation (Boyer-Lindquist)
In "Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics" [Physical Review Letters, Vol. 11, No. 4, 1963 Sep 01, Pp 237 and 238] Roy P. Kerr presented a new solution of Einstein’s field equation of General Relativity,
in which Tik represents the stress-energy tensor and Rik represents the Ricci tensor,
in which the Christoffel symbols (or affinities) take the form
in which gik represents the metric tensor describing the region of space and time under consideration and the contravariant version of the metric tensor conforms to
Kerr’s solution describes a region of space and time deformed by the stress-energy associated with a body of mass M spinning with angular momentum J with the spin axis coinciding with the z-axis of a Cartesian coordinate system. At a great distance from the body or when M and J go to zero that solution must revert to Minkowski’s metric equation,
To meet that criterion Kerr devised a metric equation that, when translated into a system of asymptotically flat coordinates, became
and r defined by
yields the standard geometric radial distance. In essence Kerr simply modified the Minkowski equation by adding a fifth term to it, one that represents the superposition on the flat, Cartesian-coordinate Minkowski spacetime a cylindrically symmetric distortion due to the spinning, gravitating body.
When the value of J goes to zero, Equation A-6 must become the equation described by the Schwarzschild metric, the description of the region of space and time deformed by the presence of a simple gravitating mass. If we let a=0, then Equation A-6 becomes
If we convert that equation into polar coordinates, which Schwarzschild used, with the center of mass of the gravitating body occupying and defining the origin of the spatial part of the 4-grid, then we get
which doesn’t look like Schwarzschild’s equation at all.
(At this point I need to make a slight digression. As if the study of General Relativity did not provide enough mathematical difficulty, physicists who specialize in this field have added another complication. In the polar coordinates used in the rest of physics phi represents the latitude of a point on the 3-grid, measured from minus ninety degrees at the south pole, through zero in the equatorial plane, to plus ninety degrees at the north pole of the grid, and theta represents the longitude of the point. In General Relativity, on the other hand, phi represents the longitude of a point on the grid and theta represents the colatitude, measured from zero at both poles to ninety degrees in the equatorial plane. To maintain proper consistency I will use the latter system, but also express a hope that someday the community of mathematicians and physicists will reconcile the two systems so that we can do all of our physics in one system of polar coordinates.)
In "Maximal Analytic Extension of the Kerr Metric" [Journal of Mathematical Physics, Vol 8, No 2, Feb 1967, Pages 265 - 281] Robert H. Boyer and Richard W. Lindquist explained that the difference between Equation A-12 and the usual expression of Schwarzschild’s metric equation consists of a simple coordinate transformation. Kerr’s coordinates (r,θ, ϕ, ct) differ from Schwarzschild’s coordinates (r’, θ’, ϕ’, ct’) by a temporal offset expressed as a function of radial position; thus we have
If we make the appropriate substitution of that transformation into Equation A-12 and work through the algebra (and I covered a full page with my calculations to prove and verify the following), then we get the standard Schwarzschild metric equation,
Boyer and Lindquist then laid out their version of the Kerr metric equation,
By simply rearranging Equation A-15 we can put it into a form more similar to that of Schwarzschild’s metric equation,
We want to use this form, the Boyer-Lindquist version of Kerr’s metric equation, to verify our derivation of the Kerr solution.
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