Apsidal Precession in Kerr Space
The first test of the validity of the theory of General Relativity was the calculation of an extra term in astrophysicists’ calculation of the apsidal precession of the orbit of Mercury, a term due to the sun’s gravity warping spacetime. Because the sunMercury system displays relatively little angular momentum, the Schwarzschild solution of Einstein’s equation yields the correct result, the famous 43 seconds of arc per century. But a binary star possesses a large amount of angular momentum, so it deforms spacetime from a Schwarzschild metric into a Kerr metric. Do binary stars actually display such a metric?
With a few nearby exceptions, astronomers don’t see binary stars as two separate stars. They see a point of light in their telescopes and infer the existence of two stars by discerning a double Doppler shift in the stars’ combined spectrum. If the light dims periodically, astronomers infer the existence of an eclipsing binary, one of which we see the orbits edgeon so that from our point of view the stars take turns eclipsing each other. From timing the eclipses, astronomers can infer more information about the stars and their orbits about their common center of mass. That information includes the rate at which the orbits precess in space. In theory, then, eclipsing binary stars provide a means of testing the Kerr solution: in practice, such a test is not easy.
In order to make the test, we must know the actual rate at which the stars’ mutual line of apsides precesses. Astronomers derive that datum from their timing of the stars’ eclipses. We also need to know the classical, Newtonian contribution to the precession and that’s something astrophysicists can calculate from measurements of the stars’ motions and other properties. Subtracting the result of that calculation from the stars’ actual precession yields the relativistic contribution to the precession. If our version of General Relativity is correct, a calculation of the precession due to the Kerr solution will match that latter number.
To describe the orbits of a binary star in a way that makes the relativistic calculation straightforward, imagine two bodies connected by a spring and revolving about their common center of mass in such a way that the period of radial oscillation due to the spring matches the period of their revolution. That image lets us see fairly clearly the features of the system that will change when we introduce Relativity into the calculation of the orbital properties and thereby determine the rate at which the line of apsides precesses.
Special Relativity gives us two phenomena that we must take into account, though they have no actual effect on the precession. The massandspring system acts as a clock, so time dilation comes into play due to the bodies’ motions on their orbits. To an observer outside the system the clock appears to slow down, though the orbital motion is not affected, so the bodies go a small distance beyond their previous peritelions before they reach the next one: in consequence, the line of apsides precesses in the prograde direction. The LorentzFitzgerald contraction also applies to the lengths of the bodies’ orbits. That contraction makes the bodies return to their respective peritelions before they reach the locations of the previous ones, thereby making the line of apsides precess in the retrograde direction. For orbital speeds much slower than the speed of light, those effects produce the same amount of precession, so they cancel each other out. Thus Special Relativity is not a factor in apsidal precession, as we found out in discussing the Schwarzschild solution.
The Kerr solution comes to us as four transformation equations that convert measurements made by one observer into the equivalent measurements made between the same two events by another observer:
(Eq’ns 1)
In those equations the lowercase variables represent measurements made by an observer at or near the rotating, gravitating body and the uppercase variables represent measurements between the same two events made by an observer far enough from the body that the gravitational field is negligible. For convenience we use the Schwarzschild radius of the gravitating body,
(Eq’n 2)
in which M represents the mass of the body (for reference, the Schwarzschild radius of the sun is 2.956 kilometers). We also use the Kerr radius of the gravitating body,
(Eq’n 3)
in which R_{0} represents the body’s radius of gyration and Ω represents the body’s angular speed of rotation.
In the case of the apsidal precession of a binary star we want to restrict our calculations to events occurring in the plane of the stars’ orbits. For convenience we also make the reasonable assumption that the stars’ equatorial planes coincide or nearly coincide with the stars’ orbital plane. Thus we have the colatitudes in our calculations as Θ=90 and our working equations become:
(Eq’ns 4)
To calculate the relativistic contribution to the precession of a binary star’s apsides we must carry out four calculations for each star due to the influence of the other star. We then add together all eight results to obtain the total relativistic precession. We derive those contributions as:
I. Kerr time dilation. Gravitational distortion of time in the third of Equations 4 makes a clock moving with and near the forced star, due to the forcing star, count time more slowly relative to the far observer’s clock. The speed of the star on its orbit has the same value for both observers, so while the near observer sees the star return to its previous periastron, the far observer sees the star pass its previous periastron and go a little further before it reaches the next periastron. The near observer understands that because they see the Universe revolving at a slightlytoohigh angular speed, so it goes a little farther than 360 degrees between the star’s arrival at successive periastrons. The amount of the excess is the same for both observers, as it must be. We can state the proportionality by saying that the angular length of the orbit (as seen by the far observer) is to 360 degrees as the orbital period measured by the far observer is to the orbital period measured by the near observer, which gives us
(Eq’n 5)
Note that R represents the distance between the stars. Subtracting 360 degrees from that result gives us the amount that the orbit precesses on each revolution of the stars due to gravitational time dilation.
II. Kerr contraction of the orbit’s radius. As seen by the far observer, the radius of the forced star’s orbit about the forcing star is subject to the gravitational analogue of the LorentzFitzgerald contraction, subject to a minor modification due to the rotary motion of the forcing star. Here we use the first of Equations 4, so we see that the radius as seen by the far observer must be smaller than the same radius measured by the near observer. Again we don’t count any change between the observers in the orbit’s length, so the orbit appears to the far observer to overlap itself by a small amount. The angular length of the orbit thus appears to increase in the same proportion in which the orbit’s radius appears to decrease and
(Eq’n 6)
In this case R_{K} represents the Kerr radius of the forcing star. Again, subtracting 360 degrees from that calculation gives us the amount that the orbit precesses on each revolution of the stars due to gravitational contraction of the orbit’s radius.
III. Kerr contraction of orbit length, starbased. The rotation of the forcing star gives it a Kerr radius that alters the length of the forced star’s orbit indirectly through the Pythagorean theorem. The second of Equations 4 reflects that fact, so that’s what we use. We have already accounted for the precession due to gravitational contraction of the orbit’s radius, so we remove the factors of r and R from the equation (treating them as if r=R) and use the equation in the form
(Eq’n 7)
Subtracting 360 degrees from that result gives us a negative precession, a precession in the retrograde direction.
IV. Kerr contraction of the orbit length, systembased. The revolution of the two stars about their common center of mass creates a Kerr radius at that center. We calculate the Kerr radius in this case by imagining one of the stars held stationary and the other defining the radius of gyration of the system with its motion about it. For this contribution to the precession we use Equation 7 with R representing the mean distance between the star under consideration and the system’s center of mass.
First Example: DI Herculis
Floating in space 2000±200 lightyears from Earth in the constellation of Hercules, DI Herculis is an eclipsing binary consisting of a B4V star (DI HerA) and a cooler B5V star (DI HerB) revolving about their common center of mass. Astronomers have inferred the following properties for the system:
Period 
10.55 days 
Frequency 
3462 revolutions/century 
Angular speed 
6.89x10^{6} radian/second 
Mean Width 
30,067,000 kilometers 
Eccentricity 
0.489 
Kerr Radius 
20,776.8 kilometers 
(Table 1)
For the stars themselves we have:
DI HerA 
DI HerB 

Mass 
5.15 Sol 
4.52 Sol 
Schwarzschild Radius 
15.22 kilometers 
13.36 kilometers 
Radius 
2.68 Sol 
2.48 Sol 
Radius 
1,865,000 kilometers 
1,725,800 kilometers 
Rotation 
3.0 days 
2.8 days 
Angular Speed 
2.424x10^{5} radian/second 
2.6x10^{5} radian/second 
Kerr Radius 
281.24 kilometers 
258.32 kilometers 
Mean Radius of Orbit 
14,054,000 kilometers 
16,013,000 kilometers 
(Table 2)
With those data we calculate the relativistic contribution to the apsidal precession of the system.
For the precession of Star B due to Star A we have:
I. ΔΦ=0.3154 degree per century.
II. ΔΦ=0.3154 degree per century. The Kerr radius term has only a negligible effect on the calculation.
III. ΔΦ=5.45x10^{5} degree per century.
IV. ΔΦ=1.0489 degree per century.
For the precession of Star A due to Star B we have:
I. ΔΦ=0.277 degree per century.
II. ΔΦ=0.277 degree per century.
III. ΔΦ=4.6x10^{5} degree per century.
IV. ΔΦ=1.362 degree per century.
When we add those contributions together we get a net relativistic precession of 1.2261 degree per century. Based on what they know of the two stars, astrophysicists have calculated a Newtonian contribution to the precession of +1.93 degrees per century. Those figures, taken together, give us a net expected precession of +0.7039 degree per century.
By timing the eclipses of the stars over a long elapse of time, astronomers can determine the orientation of the stars’ line of apsides in space. Because the system is an eclipsing binary, we know that we are seeing the stars’ orbits more or less edgeon and that fact makes the determination straightforward. Based on 84 years of observations, astronomers announced in 1985 that the line of apsides was precessing at +0.65 degree per century. Eleven years later, after further observation and study, they revised that figure to +1.05 degree per century. Our calculated precession falls between those two values, which seems to imply that we have a correct result. That implication is an illusion, as we shall see.
Second Example: AS Camelopardalis
This eclipsing binary consists of B8V star (AS CamA) and a B9.5V star (AS CamB). For the system as a whole we have:
Period 
3.43 days 
Frequency 
10,648.7 revolutions/century 
Angular Speed 
2.12x10^{5} radian/second 
Mean Width 
11,932,800 kilometers 
Eccentricity 
0.164 
Kerr Radius 
10,070 kilometers 
(Table 3)
For the stars themselves we have:
AS CamA 
AS CamB 

Mass 
3.213 Sol 
2.323 Sol 
Schwarzschild Radius 
9.498 kilometers 
6.867 kilometers 
Radius 
? 
? 
Rotation 
? 
? 
Kerr Radius 
? 
? 
Mean Radius of Orbit 
5,007,200 kilometers 
6,925,600 kilometers 
(Table 4)
Both stars are smaller than those of DI Herculis, which means that their Kerr radii will be of the same order of magnitude at most and their effect on the system’s precession will, therefore, be negligible.
For the precession of Star B due to Star A we have:
I. ΔΦ=1.525 degrees per century.
II. ΔΦ=1.525 degrees per century.
IV. ΔΦ=7.752 degrees per century.
For the precession of Star A due to Star B we have:
I. ΔΦ=1.103 degree per century.
II. ΔΦ=1.103 degree per century.
IV. ΔΦ=3.832 degrees per century.
When we add those contributions together we get a net relativistic precession of 6.328 degrees per century. The measured precession comes to +13.3 degrees per century, which implies a Newtonian precession of +19.628 degrees per century. However, astrophysicists have calculated the Newtonian contribution to the precession at between +40 degrees per century and +87 degrees per century, depending on their assumptions about the structures of the two stars.
Something is clearly wrong with our understanding of this system and a hint at what that something may be comes from the fact that astronomers have inferred from their measurements the existence of a third massive body revolving about the two stars with a period of 2.2 years. That object makes its own contribution to the Newtonian component of the apsidal precession of the two stars. Other objects, planets, will also contribute. If a couple of hot Jupiters revolve around each of the stars rather than revolve about both of them, they could make a negative contribution to the Newtonian precession. Until astronomers can detect these hypothetical objects we won’t know.
Third Example: V541 Cygni
This eclipsing binary consists of two B9.5V stars. For the system as a whole we have:
Period 
15.34 days 
Frequency 
2381 revolutions per century 
Angular Speed 
4.74x10^{6} radian per second 
Mean Width 
30,038,000 kilometers 
Eccentricity 
0.479 
Kerr Radius 
15,385.75 kilometers 
(Table 5)
For the stars themselves we have:
V541 CygA 
V541 CygB 

Mass 
2.335 Sol 
2.26 Sol 
Schwarzschild Radius 
6.9 kilometers 
6.68 kilometers 
Radius 
1.859 Sol 
1.808 Sol 
Rotation Rate 
? 
? 
Kerr Radius 
? 
? 
Mean Radius of Orbit 
14,774,000 kilometers 
15,264,000 kilometers 
(Table 6)
For the precession of Star B due to Star A we have:
I. ΔΦ=0.09836 degree per century.
II. ΔΦ=0.09836 degree per century.
IV. ΔΦ=0.4354 degree per century.
For the precession of Star A due to Star B we have:
I. ΔΦ=0.0953 degree per century.
II. ΔΦ=0.0953 degree per century.
IV. ΔΦ=0.4648 degree per century.
When we add those contributions together we get a net relativistic precession of 0.5129 degree per century. The measured precession comes to +0.60 degree per century, but the calculated Newtonian precession equals +0.103 degree per century. That latter calculation falls short of what the Newtonian precession needs to be by 1.01 degree per century. Again, this system likely has planets that affect the precession, but until astronomers can detect them and measure them, nobody will be able to make a definitive calculation of the Newtonian component of the precession.
Fourth Example: PSR 1913+16
Also known as the HulseTaylor pulsar, this binary system floats 21,000 lightyears from Earth in the constellation of Aquila. It consists of a neutron star flashing every 59.03 milliseconds (implying a rotation rate of 106.44 radians per second) and a companion of similar mass, which astronomers assume is also a neutron star. By timing the flashing of the pulsar astronomers can determine the gross properties of the system with good accuracy. For the system as a whole we have:
Period 
7.752 hours 
Frequency 
113,080 revolutions per century 
Angular Speed 
2.25x10^{4} radian/second 
Mean Width 
1,950,100 kilometers 
Eccentricity 
0.617131 
Kerr Radius 
2854 kilometers 
(Table 7)
For the stars themselves we have:
the pulsar (A) 
the companion (B) 

Mass 
1.441 Sol 
1.387 Sol 
Schwarzschild Radius 
4.26 kilometers 
4.1 kilometers 
Radius 
10 kilometers (approx.) 
10 kilometers (approx.) 
Rotation Rate 
106.44 radians/second 
? 
Kerr Radius 
0.0355 kilometer 
? 
Mean Radius of Orbit 
956,432 kilometers 
993,668 kilometers 
(Table 8)
For the precession of Star B due to Star A we have:
I. ΔΦ=44.47 degrees per century.
II. ΔΦ=44.47 degrees per century.
IV. ΔΦ=167.9 degrees per century.
For the precession of Star A due to Star B we have:
I. ΔΦ=42.79 degrees per century.
II. ΔΦ=42.79 degrees per century.
IV. ΔΦ=181.24 degrees per century.
When we add those contributions together we get a net relativistic precession of 174.62 degrees per century. The measured precession comes to +422.6595 degrees per century, which implies that the Newtonian precession comes to +597.28 degrees per century. We expect any astronomical object, even a neutron star, to display some oblateness if it spins a full circle almost 17 times every second, so we can infer at least part of that Newtonian precession. There are certainly no planets to add to the precession: two supernovae would have blasted them into dust and blown them out into space. But the burntout remnant of another star might still revolve about the twins and add to the Newtonian precession. Someday we may know for certain and we can confirm the Kerr precession.
Note: The LenseThirring Effect
Discovered by Austrian mathematician Josef Lense (1890 Oct 28  1985 Dec 28) and Austrian physicist Hans Thirring (1888 Mar 23  1976 Mar 22) in 1918, the eponymous effect provides an alternative to the geometric approach to describing the effects due to a Kerr metric. Just as the Schwarzschild solution relates the gravitational force to a distortion of spacetime by a massive body, so too the Kerr solution relates a force to a distortion of spacetime by a massive spinning body: as the gravitational force is analogous to the electric force, so the force in the Kerr solution is analogous to the magnetic force. The LenseThirring approach exploits the existence of what Robert Forward called a protational field (and what others call a gravitomagnetic field) to calculate descriptions of effects, such as apsidal precession in a spinning system.
In making those calculations, physicists use the BoyerLindquist version of the Kerr metric equation. But that metric equation does not conform to the metric equation derived from the equations of the Kerr Transformation, which I have deduced, so there is nothing to be gained by pursuing the LenseThirring approach at this point.
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