Length and duration

June 22, 2019 12:38 pm ⋅ rag

Let us begin with (1) the motion of a body between two events and (2) two ways of measuring the extent of that motion: length and duration (or time). The measurement of length and duration is coordinated so that both measures are of the same motion. Length and duration are measured by a rigid rod and a stopwatch, respectively. A smooth manifold of length is called space (or 3D space), and a smooth manifold of duration is called time (or 3D time).

The length and duration of a motion are commonly measured along the trajectory (or arc) of the motion. The length along the trajectory of motion is the arc length (or proper length or simply length). The duration along this trajectory is the arc time (or proper time or simply time).

Once the length and duration are in hand, the next step is to form their ratio. The ratio with arc time as the independent variable and arc length as the dependent variable is the speed. Which is to say, speed is the time rate of arc length change.

Note that the ratio could just as well be formed in the opposite way, with the arc length as the independent variable and the arc time as the dependent variable. This ratio is called the pace from its use in racing, in which an arc length is first set and then the racer’s arc time is measured. As another way to state this, pace is the space rate of arc time change.

The reference trajectory for measuring the length of a motion is the minimum length trajectory between two event points. The length along this trajectory is the distance between the two event points, which forms the metric of space. Distance is represented as a straight line on a length-scale map.

The reference trajectory for measuring the duration of a motion is the minimum time trajectory between two event instants. The duration along this trajectory is the distime between the two event instants, which forms the metric of time. On a map, two isochrons are separated by a constant distime. Distime is represented as a straight line on an time-scale map.

Motion has direction as well as extent, and direction may also be measured in two ways. Consider the motion of rotation, which can be measured as a proportion of a circle and as a proportion of a cycle. For example, in an analogue clock a minute hand that moves the length of a right angle correspondingly moves a duration of 15 minutes and vice versa. The direction of motion may be measured by either length or duration.

A motion measured with length direction and distance comprises a vector displacement. A motion measured with time direction and distime comprises a vector dischronment. The ratio with time as the independent variable and displacement as the dependent variable is called the velocity. The magnitude of the velocity vector is the speed. The ratio with distance as the independent variable and dischronment as the dependent variable may be called the legerity. The magnitude of the legerity vector is the pace.

There are three dimensions of motion, and correspondingly three dimensions of length and duration. The three dimensions of length comprise space. The three dimensions of duration comprise time. The three dimensions of time come as a surprise, since the distime is often a parameter for ordering events. But the scalar distime should not be confused with the vector dischronment, which has three dimensions of motion measured by duration.

Posted in: Knowing, space & time ⋅ Tagged: physics, space & time

Historians and scientists

June 3, 2019 3:08 pm ⋅ rag

Historians establish the facts of history, of what happened in the past. They do this with a variety of sources, some documentary, some physical, and whatever else they find is relevant. Key particulars are more significant than universals in establishing the facts of history. Historians may consider scientific theory in doing this, but they may also conclude that some things happened that don’t fit well with current scientific theory. Whether or not there was an earthquake in 1755 that destroyed Lisbon is a matter of history, not science.

Scientists are dependent on historians for the facts of history. Scientists do not get to establish the facts of history, nor the limits of what could have happened in the past. The latter restriction is difficult for scientists to observe. If historians establish facts that don’t fit well with current scientific theory, then scientists are likely to react defensively rather than revise their theories.

Biblical (or creation) scientists consider the Bible as the key to history, and limit science to that which is consistent with biblical chronicles. As with all scientists, they depend on historians for facts about the past but not all historians have a high view of the biblical record. Disagreements among historians lead to variations in science, since they are working with different facts about the past.

The different rôles of historians and scientists are often confused. Astronomy is a case in point. Astronomical historians may work with documents produced by those who could be considered scientists from the distant past. But the interpretation of ancient or medieval scientific documents is not part of science. Astronomical historians deal with the particulars of history, in which universals play only an indirect rôle.

Astronomical scientists deal with universals, as all scientists do, and make use of the facts of history along with recent observations. Scientists may advise historians but science is dependent on history for facts about the past, not the other way around.

Posted in: history, Knowing ⋅ Tagged: history, history of science, science

Space and time as opposites

May 31, 2019 11:28 am ⋅ rag

A theme of this blog is that space and time are dual concepts, which means they are two ways of understanding the same thing. But in what ways are space and time opposite concepts?

Space is oriented toward its origin, the place that motion begins. Time is oriented toward its destination, the time that motion ends. Both length and duration are measured from an “origin,” a reference point, which is a zero point for each, but zero speed leaves a body in space at the beginning, whereas zero pace puts the body in time at its destination.

Length in the denominator of speed is a measure of the progress from the origin to the current location in space, whereas time in the denominator of pace is a measure of the lag from the destination to the current location in time. A body at zero speed will remain at its origin and never reach its destination, whereas a body at zero pace will arrive at its destination in literally no time. A body with a small speed will take a long time to reach its destination, whereas a body with a small pace will reach its destination quickly.

Large quantities in space correspond to small quantities in time. Large quantities in time correspond to small quantities in space. A high speed is fast, and a small speed is slow. A small pace is fast, and a large pace is slow. Mass and vass are inverses, as are energy and lethargy.

The origin in space corresponds to the destination in time. Time in space flows from the past toward the future. Space (stance) in time flows from the future toward the past.

Posted in: Knowing, space & time ⋅ Tagged: physics, space & time

Logical centrism

May 30, 2019 10:31 am ⋅ rag

Arcloge

May 22, 2019 10:33 am ⋅ rag

An arcloge (arc’-loje) is a continuous, independent measurement of length. That is, it measures an ever-increasing length, which is the stance, similar to how a clock shows the time. The term is a combination of arc (as in arc length) and loge (as in horologe, a clock).

What does an arcloge look like? Start with a sector, which is a geometric figure fixed to the center of a circle that sweeps out an angle and a curved edge:

Posted in: Knowing, space & time ⋅ Tagged: physics, space & time

Basic definitions

May 21, 2019 2:52 pm ⋅ rag

Independent variable is a quantity that is not dependent on another quantity, which is either (a) a quantity chosen before an experiment or race, or (b) an ever-increasing quantity. Dependent variable is a quantity that whose value is a function of another variable.

Space is (1) length; (2) a 3D differentiable manifold of length; (3) the order of events on a stance line; (4) the stance, the reading on an arcloge.

Time is (1) duration; (2) a 3D differentiable manifold of duration; (3) the order of events on a time line; (4) the time, the reading on a clock.

Length is measured by a rigid rod. 3D space is length measured in three directions of motion.

Duration is measured by a stopwatch, timer, or clock. 3D time is duration measured in three directions of motion.

Spacetime is the 6D manifold formed from 3D space and 3D time. Worldline is the path in spacetime traced out by an object in motion.

Displacement is the vector between two points (events) on a worldline. Distance is the magnitude of a displacement.

Dischronment is the vector between two points (events) on a worldline. Distime is the magnitude of a dischronment.

Reference frame (or frame) is an abstract coordinate system and set of reference points in 3D space that uniquely fix the coordinate system and standardize measurements. Rest frame of a body is the reference frame in which the body is moving at zero speed, which is the time conversion pace.

Reference timeframe (or timeframe) is an abstract coordinate system and set of reference points in 3D time that uniquely fix the coordinate system and standardize measurements. Freeflow frame of a body is the reference timeframe in which the body is moving at zero pace, which is the stance conversion speed.

Proper length is the length of a body measured by a rigid rod moving with it. Proper time is the time of a body measured by a clock moving with it.

Lorentz transformation is a set of equations that relate space and time coordinates of reference frames moving at a constant velocity relative to each other.

Dual Lorentz transformation is a set of equations that relate space and time coordinates of reference timeframes moving at a constant legerity relative to each other.

Posted in: Knowing, space & time ⋅ Tagged: physics, space & time

From racing to relativity

May 20, 2019 1:17 pm ⋅ rag

There are three different contexts for 3D time, depending on whether stance is continuously increasing and, if so, whether there is a conversion factor between space and time:

(A) Stance is not continuously increasing. This is the situation of a race or sport in which game time has a definite beginning and ending. For example, in many sports the game lasts a specific time. In a race, the length of the course is set and the time for each contestant ends when they cross the finish line. The average pace of a contestant is their race time divided by the course length.

(B) Stance is continuously increasing and there is a general conversion factor between space and time. This is the situation of the special theory of relativity and some transportation settings in which the conversion pace is the minimum pace (and maximum speed).

In this case, there is an increasing stance whether or not a positive time interval is measured. Without a time interval increase the pace is at a minimum (or the speed is a maximum). As the amount of time measured increases, the pace increases (or the speed decreases). Remember that a small amount of time per unit distance is a fast motion, whereas a large amount of time per unit distance is a slow motion.

In this way, the pace increases indefinitely. A pace of infinity would be at rest. A pace of zero is the minimum pace, which in relativity is the speed of light. That is, the speed counts down from the speed of light. This has been misinterpreted as a transformation with superluminal speeds, but because speed decreases as pace increases, object speeds are subluminal.

The dual Lorentz transformation (see here) is

$x'=\lambda (x - ur);\; y'=y;\; z'=z;\; r'=\lambda (r - c^2ur) \; \textup{with}\; \lambda = 1/{\sqrt{1-cu}}$

with the understanding that c represents the inverse of the pace of light. The cu in λ is the pace of the object divided by the pace of light, with the stance increasing at the conversion rate. As the time of motion increases, the pace increases (and the speed decreases) from that of light toward the pace or speed of rest. So, the square root never becomes negative here.

(C) Stance is continuously increasing but there is no general conversion factor between space and time. This is the situation of general relativity and transportation in general. Conversion of space and time are local, not global, and the optimal route depends on whether space or time are optimized.

Posted in: Knowing, space & time ⋅ Tagged: physics, space & time, transportation

Notable webpages

May 15, 2019 4:10 pm ⋅ rag

One of the original purposes of a blog was a web log, that is, a log of the interesting sites one has visited. This posting is a list of links to webpages that are notable in some way.

* Voting and Elections *

Approval voting – Voting each candidate yea-or-nea

Demeny voting – This is the idea that parents (or legal guardians) have a proxy vote for their children so that children have a political representation.

2016 Presidential Election Results – No candidate won the popular vote in 2016 since no one received a majority of the votes cast.

Washington Initiative 1000 – Washington State legalizes discrimination based on race, sex, color, ethnicity, age, etc. George Wallace would be pleased

* Reverse Chronological Order *

Medieval cure for modernity – As modernity ends, pre-modernity is looking better

The New Family Violence – Canada criminalizes sanity

A Revolution in Time – Time as a timeline

Truth-telling and Big Abortion – Pro-choice deception

Neo-Segregation at Yale – A new form of racial segregation

It’s time to recognise anti-Christianism – Persecution today

Has the Postmodern Revolution Come Full Circle? – Post-truth politics is failing

Posted in: politics, Relating ⋅ Tagged: Christianity, politics

History and theology

May 12, 2019 3:46 pm ⋅ rag

What follows are excerpts from Ramsay MacMullen’s book Christianizing the Roman Empire, A.D. 100-400 (Yale, 1984). He begins with historiography pointers relevant to religious history.

My subject here is the growth of the church as seen from the outside, and the period is the one that saw the church become dominant, and Europe Christian. p.vii

My object is history. It might be, but it isn’t, theology. Accordingly, my view focuses naturally upon significance, the quality of weight that distinguishes historical phenomena from the (sometimes much more engrossing or at least more diverting) items of merely human interest that we see in the headlines of certain newspapers: ‘‘Mom Axes Babe” or the like. Significance, in its turn, indicates the degree to which many people, not just a few, are made to live their lives differently in respects that much engage their thoughts, not in respects they do not think about very carefully. … Significance must be compounded of both “many” and “much,” in a sort of multiplicand of the two elements. p.1

This is all elementary. Still, it needs to be said in order to explain the inclusion in my account of scenes not usually given much attention in books about church growth, scenes in which large numbers of persons are brought to a change in their religious allegiance, but namelessly—they are just ordinary folk of no account—and without great dramatic, further consequences in their manner of life. I think these scenes need to be included, along with Saint Augustine and a handful like him, because otherwise we would see only a church all head and no body, a phenomenon that affected only a few lives, a change without mass and therefore without historical significance. And that is the exact opposite of the truth. p.1

The process we are tracing, of the slow but gigantic growth of a community of believers, seems thus to have had at its heart a psycho-logical moment that might have been, though it was not always, quite uncomplicated; and that fact belongs by right, and not by later development, to the whole long process of ecclesiastical maturing. From the very beginning, Jesus’ disciples followed him instantly, without instruction; new adherents, by supernatural actions, were won to instantaneous belief, or trust (πιστις, “commonly mistranslated, ‘Your faith’ …,” with implications of doctrine, as has been pointed out). p.3-4

There is an obvious connection between simplicity of belief and rapidity of conversion: the simpler the set of ideas with their attendant feelings, the shorter must be the period of transition to the new. Which is not to say that much longer, complicated transitions may not also have had their abrupt moments, like Saint Augustine’ in the garden near Milan. The point is worth stressing because the more richly intellectual and dramatically interesting conversions naturally hold our attention best, and are most written about. p.4

Posted in: history, Knowing ⋅ Tagged: Christianity, history

Invariant intervals

May 8, 2019 11:01 am ⋅ rag

The spacetime interval is invariant over the Lorentz transformation (LT). The following is a proof of this for the inverse LT with spatial axes x, y, and, z; temporal axis t (time line), velocity v, and maximum velocity c, along with β = v/c and γ = 1/√(1 − β²):

$\Delta x=\gamma (\Delta x'+\beta c\Delta t');\; \Delta y=\Delta y'\; \Delta z=\Delta z';\; c\Delta t=\gamma (c\Delta t'+\beta \Delta x').$

The invariant interval is

$(\Delta s)^2 = (c\Delta t)^2 -(\Delta x)^2 -(\Delta y)^2 -(\Delta z)^2$

$=\gamma ^2(c\Delta t'+\beta\Delta x')^2 -\gamma ^2(\Delta x'+ \beta c\Delta t')^2 -\Delta y'^2 -\Delta z'^2$

Expand the squares and cancel the middle terms to get:

$=\gamma ^2(c^2 \Delta t'^2(1-\beta ^2) -\Delta x'^2(1-\beta ^2)) -\Delta y'^2 -\Delta z'^2$

$=\gamma ^2(c^2 \Delta t'^2(1/\gamma ^2) -\Delta x'^2(1/\gamma ^2)) -\Delta y'^2 -\Delta z'^2$

$=(c\Delta t')^2 -(\Delta x')^2 -(\Delta y')^2 -(\Delta z')^2.$

The spacetime interval is invariant over the dual Lorentz transformation (DLT). The following is a proof of this for the inverse DLT follows with temporal axes x, y, and, z; spatial axis r (stance line), legerity u, and maximum legerity κ, along with ζ = u/κ and λ = 1/√(1 − ζ²):

$\Delta x=\lambda (\Delta x'+ \zeta \kappa \Delta r');\; \Delta y=\Delta y'\; \Delta z=\Delta z';\; \kappa \Delta r=\lambda (\kappa \Delta r'+\zeta \Delta x').$

Posted in: Knowing, space & time ⋅ Tagged: physics, space & time

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Length and duration

Historians and scientists

Space and time as opposites

Logical centrism

Arcloge

Basic definitions

From racing to relativity

Notable webpages

History and theology

Invariant intervals

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