It is generally recognized that there are two types of simultaneity conventions: One defines it as the same point in space, while the other between observers moving at a constant speed in an inertial frame. The former was described by Einstein (1905) as: "We could content ourselves with evaluating the time of events by stationing an observer with a clock at the origin of the coordinates, who assigns to an event to be evaluated by the corresponding position of the hands of the clock when a light signal from that even reaches him." This says nothing about the motion of the source and observer.

According to his second definition of simultaneity, Einstein imagines a light signal sent out at time t_1 at a, reflected back at b, and arriving back at a at time t_2. "An event at a that is simultaneous with the time of arrival of a light signal from b is the one with (t_1+t_2)/2 as its temporal coordinate." Moreover, the distance covered is proportional to their time difference, (t_2-t_1)/2. This convention assumes that light travels at a constant velocity in all directions, and forms Einstein's second postulate of his special theory of relativity.

The two conditions of simultaneity both involve two events, x and y. Any sequence of three numbers x, a, and y form an arithmetic progression if a is the arithmetic mean, A, a geometric progression if a is the geometric mean, G, or a harmonic progression if a is the harmonic mean, H. It is also well-known that the means satisfy the fundamental inequality:

A > G > H,

and, in the case of two events, such as outward and backward journeys, the fundamental equality holds:

A x H =G^2.

The equality is derived from the common difference:

1/H-1/x=1/y-1/H.

Which form of simultaneity we choose to satisfy depends on whether we identify the arithmetic or the geometric mean with the speed of light, which we set equal to 1.

In the former case, we are constrained to use the speed of light as the largest speed attainable, while, in the latter case, there is no limit on the speed of light. Both cases have unity as the average round-trip speed: the former says that the speed of light is constant and isotropic so it doesn't make a difference in which direction we are traveling, while the second case allows the one-way speed to be variable. The problem has been, and still is, how to measure the one-way speed of light.

With A=1 as the arithmetic mean, consider the case of Michelson and Morley where the speed in the direction of the ether is c_1=1+v, and the speed in the reverse direction is c_2=1-v, where v is the speed of the ether. The total transit time for a laser beam to be emitted from its source travel to the mirror where it is reflected and return to the photo detector is

1/(1-v) + 1/(1+v)= 2/(1-v^2).

Dividing through by 2, the left-hand side becomes the inverse of the harmonic mean, 1/H, while the right-hand side is the ratio of the arithmetic mean, A=1, to the square of the geometric mean, G=(1-v^2)^(1/2).

When the interferometer was rotated 90 degrees so that it was perpendicular to the ether flow, the ratio was found to be A/G. And since both gave the null result, they had to coincide which prompted Lorentz and FitzGerald (1894) to formulate their hypothesis of length contraction where the length of the interferometer arm would be shortened by a factor of G in the direction of flow, since the time interval was the ratio of the arm length to the speed.

The round trip had an average speed, (c_1+c_2)/2=1, the arithmetic mean, which is consistent with the assumption that the speed of light is constant and isotropic.

Now, consider the inverse case where the velocities in forward and backward direction are c_1=1/(1-v), and c_2=1/(1+v). It is now the harmonic mean, H=1, which is equal to the speed of light, so we can immediately anticipate that the arithmetic and geometric means will be greater than the speed of light. And, in fact, we find from (c_1+c_2)=2/(1-v^2) and (1/c_1+1/c_2)=2, that the arithmetic mean is 1/(1-v^2) and the geometric mean is 1/(1-v^2)^(1/2), since the harmonic mean is 1, the speed of light.

What is attractive to biblical scholars is the second case where v=1. Both the arithmetic and geometric means are infinite while the harmonic mean is still unity. From the book of Genesis (1:15), it can be inferred that the creation of the stars were nearly simultaneous with the light reaching the earth. So it would appear that God took advantage of using spatial coincidence in defining simultaneity so that the speed of light from the stars to the earth was infinity, while the reverse trip occurred at half the speed of light so as to satisfy the harmonic mean. In biblical circles, this is referred to as ASC, or anisotropic synchrony convention, for which observers at the same location agree on which events are simultaneous regardless of the speed of the observer [J P Lisle, "Anisotropic synchrony convention--A solution to the distance starlight problem," Answers Research Journal 3 (2010) 191.]

According to the Einstein convention, only those observers that have the same velocity--not position--can agree on which events are simultaneous. Yet, it is the velocity-independent convention that leaves open the possibility of a variable one-way speed of light. It is argued that it is the ASC convention that enables God to carry out his problem of creating the world in six days where on the fourth day, He made things such that the light of distant stars reaches earth almost instantaneously with their creation.

Whereas the Einstein convention involves length contraction in the form of the geometric mean G=(1-v^2)^(1/2), the ASC convention is related to time contraction where the geometric mean is G=1/(1-v^2)^(1/2). The slowing down of clocks enable greater velocities to be reached for which, in the ASC case, time comes to a complete half in the forward direction. That is, the one-way speed of light becomes infinite, and, in so doing, makes relativity compatible with the Bible by enabling God to complete his chores in the allotted amount of time!

I have remarked in Seeing Gravity that the eccentricity, e, plays the role of the relative velocity. The definition of an ellipse is that the sum of the distances between the foci and a point on the curve is constant allows me to carry over what has been said for the velocities to these distances. The sum of the distances,

r_1+r_2 = 2a,

where a is the semi-major axis of the ellipse. Moreover, if the geometric mean is the semi-minor axis, then r_2=a(1-e) is the perihelion distance, and the semi-latus rectum is p=r_2r_1/a. The semi-latus rectum is equal to the square of the angular momentum per unit mass divided by the gravitational parameter. It is also referred to as the semi-focal width, which, obviously, must be less than the semi-minor axis. The semi-latus rectum is none other than the harmonic mean since H=G^2/A!

Consequently, the arithmetic mean A=a, is the semi-major axis, the geometric mean G=b, is the semi-minor axis, and the harmonic mean H=p, the semi-latus rectum. This takes out the mystery of the significance of the sem-latus rectum. Indeed, the eccentricity plays the role of a relative velocity! And just like the eccentricity, the relative velocity is indicative of the presence of deformities caused by the motion.

Going from a slicing a cone to slicing a torus, we come to Cassini ovals. These curves, according to Cassini (1680) [see Seeing Gravity], were supposed to be better approximations to Kepler's ellipses for curves of low eccentricity. The earth would be placed at one of the foci while the sun would trace out a curve in the shape of an oval for given values of the eccentricity. Now, instead of the sum of the distances to the foci being constant, it its their product which is constant. This mean a constant value of the geometric mean.

If c is the distance to a focus, the distances are

{(x-c)^2+y^2}^(1/2) and {(x+c)^2+y^2}^(1/2).

Their product, b, is a constant. Transforming to polar coordinates x=r cosA, y=r sinA, the condition that the geometric mean be constant is

R^4-2R^2 cos(2A)=e^2-1,

where R=r/c and e=b/c. For e<1/2^(1/2), the curve is a single loop that intersects the x-axis and the greatest distances: plus or minus {b^2+c^2}^(1/2). Higher order curves should have the condition that the harmonic mean be constant.

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