Apsidal Precession in Kerr Space

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    The first test of the validity of the theory of General Relativity was the calculation of an extra term in astrophysicistsí calculation of the apsidal precession of the orbit of Mercury, a term due to the sunís gravity warping spacetime. Because the sun-Mercury system displays relatively little angular momentum, the Schwarzschild solution of Einsteinís 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 donít 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 starsí combined spectrum. If the light dims periodically, astronomers infer the existence of an eclipsing binary, one of which we see the orbits edge-on 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 starsí mutual line of apsides precesses. Astronomers derive that datum from their timing of the starsí eclipses. We also need to know the classical, Newtonian contribution to the precession and thatís something astrophysicists can calculate from measurements of the starsí motions and other properties. Subtracting the result of that calculation from the starsí 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 mass-and-spring system acts as a clock, so time dilation comes into play due to the bodiesí 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 Lorentz-Fitzgerald contraction also applies to the lengths of the bodiesí 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:

(Eqíns 1)

In those equations the lower-case variables represent measurements made by an observer at or near the rotating, gravitating body and the upper-case 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,

(Eqín 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,

(Eqín 3)

in which R0 represents the bodyís radius of gyration and Ω represents the bodyís 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 starsí orbits. For convenience we also make the reasonable assumption that the starsí equatorial planes coincide or nearly coincide with the starsí orbital plane. Thus we have the colatitudes in our calculations as Θ=90 and our working equations become:

(Eqíns 4)

    To calculate the relativistic contribution to the precession of a binary starís 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 observerís 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 slightly-too-high angular speed, so it goes a little farther than 360 degrees between the starís 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

(Eqín 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 orbitís radius. As seen by the far observer, the radius of the forced starís orbit about the forcing star is subject to the gravitational analogue of the Lorentz-Fitzgerald 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 donít count any change between the observers in the orbitís 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 orbitís radius appears to decrease and

(Eqín 6)

In this case RK 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 orbitís radius.

    III. Kerr contraction of orbit length, star-based. The rotation of the forcing star gives it a Kerr radius that alters the length of the forced starís orbit indirectly through the Pythagorean theorem. The second of Equations 4 reflects that fact, so thatís what we use. We have already accounted for the precession due to gravitational contraction of the orbitís 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

(Eqín 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, system-based. 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 systemís center of mass.

First Example: DI Herculis

Floating in space 2000Ī200 lightyears from Earth in the constellation of Hercules, DI Herculis is an eclipsing binary consisting of a B4-V star (DI Her-A) and a cooler B5-V star (DI Her-B) 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 Her-A

DI Her-B

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 starsí line of apsides in space. Because the system is an eclipsing binary, we know that we are seeing the starsí orbits more or less edge-on 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 B8-V star (AS Cam-A) and a B9.5-V star (AS Cam-B). 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 Cam-A

AS Cam-B

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 systemís 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 wonít know.

Third Example: V541 Cygni

    This eclipsing binary consists of two B9.5-V 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 Cyg-A

V541 Cyg-B

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 Hulse-Taylor 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 burnt-out 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 Lense-Thirring 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 Lense-Thirring 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 Boyer-Lindquist 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 Lense-Thirring approach at this point.

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