Time Equivalence

Gravitational Time Versus Inertial Time

Physical time exists as two separate arrows or time flows that are measured by two completely different kinds of clocks. Inertial time can be measured by the daily rotation of the Earth on its axis or by the vibration of atoms in an atomic clock. Gravitational time can be measured by the yearly revolution of the earth around the sun or by the swing of a grandfather clock’s pendulum. The ability of a particular clock to measure the effects of one type of time is dependent on its ability to be isolated from the effects of the other type of time. A thought experiment is proposed to illustrate the mechanism by which an inertial clock is slowed by an increase in velocity and by which the rate of a gravitational clock increases as it gains velocity.

Metaphysical Time

Time does not have a real physical existence like matter or even the void of space that has a negative existence. Time is merely an idea used to quantify the motion of matter through the void. The idea of time is a dichotomy between metaphysical time and physical time. Metaphysical time is the perception of time as a continuous flow that is without interval and is thus immeasurable. It is the comprehension of motion and therefore of time’s passage. Once time becomes quantified into intervals by the cyclical motion of a clock it becomes Physical time.

Physical Time

Physical time does not have a single homogenous flow but rather is measured as two distinct and unrelated phenomena that are measured with two completely different kinds of clocks. The two separate types of time flow that we cut into equal intervals with our clocks are gravitational time, that is derived from the constancy of a gravitational acceleration and inertial time that is derived from the constancy of momentum (mv) or angular momentum (mvr) of a body in motion, either in a straight line or rotating about an axis. Physical time is the concept used to generalize these two opposite time flows into the idea of a single one-dimensional temporal motion. Inertial time and gravitational time are the quantitative intervals measured by clocks and metaphysical time is the qualitative and dimensionless ideal principle common to both.

Clocks do not measure time directly; they monitor the relationship between mass and space. Time is merely the relationship between mass and space, and its existence can only be established and quantified by the measurement of these two parameters. Since time cannot be measured independently of mass and space, any change in the values of mass and space will cause the values of the intervals recorded by clocks to be transformed as well.

Only two different kinds of clocks are used to measure time: inertial clocks and gravitational clocks. Each of these clocks measures a different relationship between mass (M) and its motion through space (S). Gravitational clocks measure Force (F=MS/T2) and inertial clocks measure momentum (MS/T) and angular momentum (Iw = MSR/T). Inertial clocks divide the constant inertial motion of a body into intervals and derive their standard of time from the conservation of momentum and angular momentum. Gravity clocks measure the inertial forces produced when a body’s motion is constantly changed by gravity and derive their standard of time from the constancy of this force. Inertial time units are the cycles produced by a rotating or vibrating body and gravitational time units are the cycles produced by the constant acceleration of gravity such as the swinging of a pendulum or the revolving of the moon. The accuracy of both clocks depends on an unchanging relationship between mass and space.

Gravity clocks measure the motion of gravity as force and inertial clocks measure inertial motion as momentum. Gravity clocks must be isolated from the effects of momentum and inertial clocks must be isolated from the effects of force. Pendulum clocks don’t work well on a boat because of the continual changes in momentum, and inertial clocks must be free of the effects of friction and energy transfer on their oscillators. The inertial clock of the earth’s rotation repeats a cycle of day and night, that over long periods of time, is slowed by tidal friction that causes the Earth’s rotation to slow and the length of the day to increase. In this process energy and angular momentum from Earth’s rotation are transferred to the moon. This slows its orbital velocity and increases the orbit’s size proportionally to the square of the velocity that is gradually slowed down. Also, the day/night cycle is not pure inertial time because it is contaminated by one part in 365 because the gravity clock of Earth’s revolution around the sun. One day/night cycle is lost every time the Earth goes around the sun. Likewise, the gravity clock of the moon’s monthly phases looses one Earth day of inertial time per month as the moon orbits Earth.

The history of time keeping is a steady parade of ingenious devices, all of which are either gravity clocks or inertial clocks. Ever since humans first developed the idea of time and began devising clocks to measure and record its passage they have used both gravity and inertial clocks. Hourglasses, water clocks, and pendulum clocks are examples of gravity clocks. An electronic digital clock, that records the rapid cyclical motion of groups of atoms, or a pocket watch, that measures the cyclical motion of a balanced weight, are examples of inertial clocks. For three hundred years the pendulum gravity clock was the most accurate of clocks, but recently the electronic inertial clock has far exceeded it in accuracy.

These two kinds of clocks can never be combined because each measures a different quantity, and these two quantities are mutually exclusive. For example, the inertial clock provided by the Earth’s rotation would not be affected by a sudden change in the Earth’s acceleration of gravity, but would be slowed down by an increase in mass. A pendulum clock would be speeded up by an increase in gravitational acceleration but would not be affected by the addition of mass to its pendulum. In fact it can be said that gravitational time and inertial time are opposites that flow in different directions. For example, a pendulum clock would run slower at the Equator than at the South Pole because, at the Equator, the downward acceleration of inertial time, caused by the Earth’s rotation, would counteract the upward acceleration of gravitational time. The accuracy of any clock depends on isolating the kind of time being measured from the kind not being measured.

The Equivalence Principle maintains that units of duration of inertial time and gravitational time are equal to one another. However, this principle does not hold up when we consider the transformations that occur to time-keeping devices when they are accelerated to extremely high velocities. When mass is accelerated, these two measures of time diverge from one another, and in doing so provide an example of the Equivalence Principle leading to an incorrect conclusion. Conversely, Absolute Motion Theory maintains that there is no equivalence of gravitational time and inertial time except at the position of absolute rest. As their absolute motion is increased, gravitational clocks run faster and inertial clocks run slower. These diverging time flows are not even symmetrical in that gravitational clocks increase at a rate that is the square root of the rate by which inertial clocks slow down.

To understand how these two measures of time diverge from one another with the transformation of mass, we will perform a thought experiment in which the spinning Earth is an example of an inertial clock and Earth’s acceleration of gravity is an example of a gravitational clock. Earth makes one revolution on its axis every day while a clock’s pendulum swings back and forth a certain number of times each day.

We will use two different accelerometers to monitor the two measures of time. One accelerometer is placed vertically at the North Pole to measure the gravitational acceleration. The second, much more sensitive accelerometer, is placed vertically at Earth’s Equator with its acceleration vector pointing at the center of the earth. This accelerometer measures the centripetal force caused by the rotation of Earth on its axis. We will assume that Max has been able to design this accelerometer in such a way that the opposite acceleration of gravity can be canceled from the reading. Both accelerometers are calibrated to read one unit of space per one unit of time squared (S/T2). Each of these three accelerometers is equipped with a pendulum-type clock that indicates the passage of time according to the acceleration experienced by it. An inertial Cesium-133 atomic clock is also shown that registers time based on the cyclical inertial motions of atoms contained within it. The following illustration entitled Lead World shows three “Earths”: Lead Earth, Earth at Rest, and Kinetic Earth.

On Earth at Rest all three clocks remain synchronized and register the passage of time identically so that both pendulums swing with a period of one second. For our thought experiment, let’s suppose that Max is able to fashion an exact replica of Earth out of solid lead so that it is the same size as Earth but has twice the mass. Once Max completes this task, he finds that all three clocks are running at different rates.

On Lead Earth the doubling of Earth’s mass also doubles its acceleration of gravity. This causes the gravity clock’s pendulum to speed up and decrease the period of its swing from one second to 1/2 = .70711 second.

In order to conserve the angular momentum (Iw) = (MVR) inherent in Earth’s rotation, the period of its rotation would be doubled and its equatorial velocity would be cut in half. This slowing of Earth’s rotation would cause the Equator inertial clock to register one quarter of its original rate and its pendulum to double the period of its swing to two seconds. The rate of the atomic clock would slow only very slightly by about one part in a billion due to the increased gravity, but it would register the passage of each day as 48 hours. If all three clocks were synchronized at six o’clock the atomic clock would read 6:10 when the inertial Equator clock read 6:05 and the gravity clock read 6:14.14. The observer on Lead Earth would weigh twice as much as he did on Earth at Rest but his internal bodily clock would remain in sync with the atomic clock and he would easily perceive that the days were twice as long as usual.

For the second part of our thought experiment we will imagine that Max is able to fabricate another Earth replica with a mass the same as Earth and then, accelerate it to 86.6% the speed of light. This is a very difficult task and requires 4 x 1041 joules of energy, which is equal to the sun’s total output for 10,000,000 years! At this velocity, a body’s kinetic mass is exactly equal to its rest mass. Even though this increase in velocity doubles the Earth’s mass to equal that of the Lead Earth, the transformation of the clocks is not quite the same.

On Kinetic Earth, Earth’s acceleration of gravity has doubled and the gravity clock is running at the same rate as the identical clock on Lead Earth, however, the Kinetic Earth accelerometer shows twice the acceleration. This is because the accelerometer’s movable weight has doubled in mass and now exerts twice as much force on the spring with the same amount of acceleration. The inertial clock at the equator is running at the same rate as the one on Lead Earth but the centripetal accelerometer is reading twice as much acceleration as the clock’s rate would indicate. This is again because of the doubling of the movable weight’s mass exerts twice as much force on the spring.

The inertial atomic clock has slowed to one half of its rate at rest and on Lead Earth. This slowing is caused by the same inertial process that slowed Lead Earth’s rotation. This atomic clock slows as its cesium atoms conserved angular momentum (Iw = mvr) as their masses increased with their velocity. As the linear velocity of a body is increased, the equatorial velocity of the circlon-shaped particles making up the body’s matter is reduced according to the formula: (V = Iw/mr). The atomic clock and the inertial equator clock both run at the same speed and both mistakenly show that the day is 24 hours long. However, the observer on Earth at Rest sees Kinetic Earth rotating every 48 hours, the same as Lead Earth.

The internal body clock of the observer on Kinetic Earth has slowed to one half of its rest rate along with the atomic clock and to him, the days seem to pass at their normal 24 hour rate, but he sees his gravity clock running at 2.28 times the rate of the inertial clocks. When he looks back at Earth at Rest he sees that it is rotating in just twelve hours. The observer’s weight has increased from 1 on Earth to 2 on Lead Earth and to 4 on Kinetic Earth.

In the case of the Earth accelerated to 86.6% the speed of light, the values for the equatorial velocities of both Earth and the particles of its matter would be reduced to one half of their values at rest. It is this effect that is responsible for the concept of Relativistic Time Dilation whereby an observer in motion experiences the passage of inertial time at a slower rate than an observer at rest. At rest the equatorial velocity of the particles making up matter is the speed of light. As they are accelerated, their equatorial velocity slows down until at 86.6% the speed of light, this velocity is reduced to one half the speed of light.

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