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Category Archives: Relativity

Relativity posts

Light clocks with multidimensional time

A previous post on this subject is here. One reference for this post is V. A. Ugarov’s Special Theory of Relativity (Mir, 1979).

A light clock is a device with an emission-reflection-reception cycle of light that registers the current time and stance in units of cycle length and duration. Consider two identical light clocks, at first in their reference frames at rest, K, K´ (left). Then, as the light clock in K´ moves relative to K with uniform motion at velocity v (right), from K observes the following:

light clocks, at rest & in motion

The left illustration shows one cycle length of the light path (i.e., wavelength), L, and one cycle duration (i.e., period), T, at rest in reference frames K, K´ (left). For the reference frame K´, in motion relative to reference frame K, call the arc length of one cycle of the light path x<. Call the distance between the beginning and ending points of one cycle x. For the reference frame K´ relative to reference frame K, call the arc time of one cycle of the light path t<. Call the distime between the beginning and ending instants of one cycle t.

Following Ugarov: Observing clock time rates in the two frames K and K´ moving relative to each other, one can only compare the reading of one clock time from one frame with readings of several clock times from another frame, because two clock times from different reference frames occur at the same point in space only once. In one of the frames there must be at least two clock times which are supposed to be synchronized. For the sake of definiteness we shall be comparing one clock time, t<, from the frame K´ with two clock times from the frame K, at the point in the beginning and end of a cycle.

Let a clock and a light source be located at the origin O´ of the frame K´. A mirror is set perpendicular to the L axis at the distance L/2 from the light source (and the clock). A light signal is transmitted from the source to the mirror from which it is reflected back and returns to the origin point O´ with the period TL/c. Both the light source and the mirror are at rest in the frame K´ and the signal travels there and back along the same straight line.

Now let us consider the propagation of the same signal in the frame K relative to which the source and the mirror move to the right together with the frame K´ at the velocity v. Although the signal was sent from the two coincident origins, O and O´, the reflection from the mirror will occur at another point x/2 of the frame K and the reception of the reflected signal at the point x of the axis. In this way the path of the signal in the frame K traces out two sides of an equilateral triangle.

As the path travelled by light in the frame K is greater than that in the frame K´, one can expect that the period T between the sending and reception of the signal, when measured in the frame K, will be greater than t. Indeed, the observer from the frame K will certify that the two events, i.e. the emitting of light from the origin O´ and its return to the origin O´, occur at the two different points of space. The period T between these two events in the frame K will be measured in this case by the two clocks removed from each other by the distance vt along the motion direction. The velocity of light is equal to c in all reference frames. Therefore, we obtain:

(x</2)² = (ct</2)² = (vt/2)² + (L/2)².

Given t< = t and collecting t< from this equation, we get

t<²(1 − v²/c²) = (L/c)²,

  t<= (L/c)/√(1 − v²/c²) = γ (L/c)

where γ = 1/√(1 − v²/c²).

Considering that L/c = T, then

t< = γ T.

Since both events occurred at the same point in the frame K´, they were registered by means of the same clock. A time interval between events registered by means of the same clock (which implies that the events occurred at the same point of space) is referred to as a proper-time interval between these events. Of course, a time interval of which the initial and the final moments are registered at different points of the reference frame and, consequently, by means of different clocks will not be a proper-time interval between events.


Following Ugarov but with Euclidean time: Observing clock stance rates in the two frames K and K´ moving relative to each other, one can only compare the reading of one clock stance from one frame with readings of several clock stances from another frame, because two clock stances from different reference frames occur at the same instant in time only once. In one of the frames there must be at least two clock stances which are supposed to be synstancized. For the sake of definiteness we shall be comparing one clock stance x from the frame K´ with two clock stances from the frame K, at the instant of the first and last points of a cycle.

Let a clock and a light source be chronated at the origin instant O´ of the frame K´. A mirroring event occurs parallel to the t⊥ axis at the distime T/2 from the light source (and the clock) perpendicular to the relative motion. A light signal is transmitted from the source to the mirror from which it is reflected back and returns to the origin instant O´ with the wavelength L = cT. Both the light source and the mirror are at rest in the frame K´ and the signal travels there and back along the same straight line.

Now let us consider the propagation of the same signal in the frame K relative to which the source and mirror move to the right together with the frame K´ at the velocity v. Although the signal was sent from the two coincident origin instants, O and O´, the reflection from the mirror will occur at another instant t/2 of the frame K and the reception of the reflected signal at the instant t. In this way the path of the signal in the frame K traces out two sides of an equilateral triangle.

As the time path travelled by light in the frame K is greater than that in the frame K´, one can expect that the wavelength L between the sending and reception of the signal, when measured in the frame K, will be greater than x<. Indeed, the observer from the frame K will certify that the two events, i.e. the emitting of light from the origin O´ and its return to the origin O´, occur at the two different instants of time. The wavelength L between these two events in the frame K will be measured in this case by the two clock stances removed from each other by the distime x/v along the motion direction. The velocity of light is equal to c in all reference frames. Therefore, we obtain:

(t</2)² = (x</2v)² = (x/2c)² + (T/2)².

Given x< = x and collecting x< from this equation, we get

x<²(1 − v²/c²) = (cT)²,

x<= (cT)²/√(1 − v²/c²) = γ (cT),

where γ = 1/√(1 − v²/c²).

Considering that cT = L, then

x< = L/γ.

Since both events occurred at the same instant in the frame K´, they were registered by means of the same clock stance. A length interval between events registered by means of the same clock stance (which implies that the events occurred at the same instant of time) is referred to as a proper-length interval between these events. Of course, a length interval of which the initial and the final moments are registered at different instants of the reference frame and, consequently, by means of different clock stances will not be a proper-length interval between events.

~

The moving light clock has x< = vt. and x< = ct<. Note: if c = ∞, then t|| = 0; x< = x; and t< = t. If v = 0, then t = 0; x< = x||; and t< = t||.

From the Euclidean metric for space we have: (x</2)² = (x/2)² + (x||/2)² . Combine this with the above to get:

(ct</2)² = (vt/2)² + (ct||/2)².

Divide by c² to get:

(t<)² = (βt)² + (t||)².

If x< = x above, then

(x<)² = (x</β)² + x||², or

(x<)² (1 − 1/β²) = x||².

From the Euclidean metric for time we have: (t2)² = (t)² + (t||)². Combine this with the above to get:

(x</c)² = (x/v)² + (x||/c)².

Multiply by c² to get:

(x<)² = (x/β)² + x||²,

which is a weighted metric.

Can we infer x|| = ct||?

Space as time and time as space

Galileo parabola

Galileo used the length of uniform motion as a measure of duration, i.e., time (Dialogues Concerning Two New Sciences Tr. by Henry Crew and Alfonso de Salvio, 1914):

Accordingly we see that while the body moves from b to c with uniform speed, it also falls perpendicularly through the distance ci, and at the end of the time-interval bc finds itself at the point i. p.199

Without getting into the details of the Figure 108, notice the shift of language: “the body moves from b to c” [i.e., a length-interval], then “the time-interval bc“. Galileo uses a length interval to measure a time-interval, which is justified since the motion is “with uniform speed”.

Let there be a ball dropped out the window by a passenger on a train in uniform motion. Consider the following four scenarios, in which the length or duration of a uniform motion is measured: (1) looking down above the moving ball, measuring the length of fall; (2) looking down above the moving ball, measuring the (uniform) duration of fall; (3) looking from the side, measuring the length of motion in two dimensions; and (4) looking from the side, measuring the (uniform) duration of motion in two dimensions.

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Simultaneity and simulstanceity

Max Jammer’s book Concepts of Simultaneity (Johns Hopkins UP, 2006) describes the significance, meaning, and history of simultaneity in physics. Here are a few excerpts from his Introduction:

… Einstein himself once admitted: “By means of a revision of the concept of simultaneity in a shapable form I arrived at the special relativity theory.” p.3

That not only temporal but also spatial measurements depend on the notion of simultaneity follows from the simple fact that “the length of a moving line-segment is the distance between simultaneous positions of its endpoints,” as Hans Reichenbach … convincingly demonstrated. Having shown that “space measurements are reducible to time measurements” he concluded that “time is therefore logically prior to space.” p. 4-5

P. F. Browne rightly pointed out that all relativistic effects are ultimately “direct consequences of the relativity of simultaneity.” p.5

One might give the dual to the second statement as: That not only spatial but also temporal measurements depend on the notion of simulstanceity follows from the simple fact that “the duration of a moving line-segment is the time interval between simulstanceous chronations of its endpoints. Space is therefore logically prior to time.

In the next chapter, Terminological Preliminaries, Jammer clarifies the relevant concepts. It is ironic that he gives an early example of the metonym “of spatial terms to denote temporal relations that is frequently encountered both in ancient and in modern languages.” (p.9) Space has priority in language.

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From spacetime to space and time

This relates to the post here.

There are three dimensions of motion with two measures of the extent of motion, which makes a total of six metric dimensions of motion. But these six metric dimensions collapse into two structures of one and three dimensions as the conversion factor approaches infinity.

The invariant proper length, , is:

dσ² = dr²dt²/ç² = dr1² + dr2² + dr3² – dt²/ç² = dr² – dt1²/ç² – dt2²/ç² – dt3²/ç² = dr1² + dr2² + dr3² – dt1²/ç² – dt2²/ç² – dt3²/ç².

As the conversion factor, ç, the pace of light, approaches infinity, this becomes

dσ² = dr² = dr1² + dr2² + dr3².

That is, the time coordinates separate from the invariant length, which becomes the Euclidean distance of three dimensional space. Time is left as an invariant scalar called the time.

The invariant proper time, , is:

dτ² = dσ²/c² = dr²/c² – dt² = (dr1² + dr2² + dr3²)/c² – dt² = dr²/c² – dt1² – dt2² – dt3² = (dr1² + dr2² + dr3²)/c² – dt1² – dt2² – dt3².

As the conversion factor, c, the speed of light, approaches infinity, this becomes

dτ² = – dt² = – dt² = – dt1² – dt2² – dt3².

That is, the length coordinates separate from the invariant time, which becomes the Euclidean distime of three dimensional time. Space is left as an invariant scalar called the stance.

The result is that six dimensional spacetime collapses into 3D space with scalar time or 3D time with scalar space.

Duality as a convention

Is color an absorption phenomena or an emission phenomena? The answer is that it’s both. But absorption works subtractively whereas emission works additively. The question then is whether color is subtractive or additive. Again the answer is that it’s both. Color is a duality.

Does an artist work with subtractive colors or additive colors? Here the answer is one or the other. A painter works with pigments that are subtractive, whereas a glass artist works with stained glass that is additive. Even though absorption and emission are operating in both cases, working with color requires picking one or the other (except for mixed media).

A simultaneity convention can also be a duality. What has been called apparent simultaneity is the convention that the backward light cone is simultaneous. But it is possible to adopt a complementary convention in which the forward light cone is simultaneous (see here). Either of these is something of a combination of Newton’s and Einstein’s physics.

One could recover Newtonian physics by adopting a combination of the backward and forward light cone simultaneity conventions. For an absorption event the backward light cone is simultaneous. For an emission event the forward light cone is simultaneous. This is like half-duplex communication (push to talk, release to listen). Such a duality convention recovers Newtonian physics because it is as if the speed of light is instantaneous in all directions.

Mean speed of light postulate

Einstein stated his second postulate as (see here):

light is propagated in vacant space, with a velocity c which is independent of the nature of motion of the emitting body.

Since the one-way speed of light cannot be measured, but only the round-trip (or two-way) speed, let us modify this postulate to state:

The measured mean speed of light in vacant space is a constant, c, which is independent of the nature of motion of the emitting body.

This is the most that can be empirically verified. Then for convenience sake, let us adopt the following convention:

The final observed leg of the path of light in empty space takes no time.

Since the (harmonic) mean speed of light is c, the speeds of the other legs of light travel are at least c/2 such that the mean speed equals c. In this way, the Galilean transformation is preserved for the final leg. And interchanging length and duration leads to an alternate version of the Galilean transformation.

This accords with common ways of speaking. Even astronomers speak of where a star is now, rather than pedantically keep saying where it was so many years ago. Physical theory should be in accord with observation of the physical world as much as possible. This is an example of how amateur scientists can help re-integrate science and common life.

Half-duplex relativity

Galilean relativity requires the speed of light to be instantaneous (i.e., zero pace). Because the one-way speed of light is not known, it may be instantaneous as long as the mean speed of light is finite. Such a situation is possible if light is conceived as in half-duplex telecommunications: one direction at a time is observed or transmitted, but never both simultaneously.

Consider a light clock in this context:

light at restSaw-tooth light path

Let Δt be the time for one cycle of light at rest (top diagram). Let Δt’ be the time for one cycle of light traveling at relative velocity v (bottom diagram). The mean speed of light is c. Then

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Galileo’s reciprocity

From Galileo’s Dialogue Concerning the Two Chief World Systems, translated by Stillman Drake (UC Press, 1967):

Salv. Now imagine yourself in a boat with your eyes fixed on point of the sail yard. Do you think that because the boat is moving along briskly, you will have to move your eyes in order to keep your vision always on that point of the sail yard and to follow its motion?

Simp. I am a sure that I should not need to make any change at all; not just as to my vision, but if I had aimed a musket I should never have to move it a hairsbreadth to keep it aimed, no matter how the boat moved.

Salv. And this comes about because the motion the ship confers upon the sail yard, it confers upon you and also upon your eyes, so that you need not move them a bit in order to gaze at the top of the sail yard, which consequently appears motionless to you. [And the rays of vision go from the eye to the sail yard just as if a cord were tied between the two ends of the boat. Now a hundred cords are tied at different fixed points, each of which keeps its place whether the ship moves or remains still.] p.249-250

Galileo is portraying motion as viewed by a human observer. The implication is that the observer in another ship would be observing the same kinds of things. Then two observers in motion with respect to one another who observe one another must face one another. That is, they are positioned opposite one another, effectively each turned 180º from the other.

This is the Galileo Reciprocity Principle, the convention that an observed frame has the opposite orientation as the frame from which it is observed, which ensures that corresponding velocities are equal. It is the opposite of the Einstein Reciprocity Principle, the convention that an observed frame has the same orientation as the frame from which it is observed.

Relativity of orientation

The Principle of Relativity states that the laws of physics are the same in all inertial frames of reference (IRF). Since a frame of reference includes an orientation, that is, a convention as to which rectilinear semi-axes are positive (and so which are negative). Since there is no preferred frame of reference, each frame has its own orientation, not the orientation of a particular frame. That means IRF orientations are what is called “body-fixed” orientations.

A frame of reference is called “body-fixed” if it is conceptually attached to a rigid body, such as a vehicle, watercraft, aircraft, or spacecraft. Body-fixed frames are inertial frames if the body to which the frame is affixed is in inertial motion. The body is usually referenced in anthropomorphic terms, such as its left, right, face, or back, although some craft have their own terms, notably, ships with port, starboard, fore, and aft.

Consider the following scenario of cars in five lanes, oriented so that their forward direction is positive, with their unsigned speeds shown relative to the two parked cars in the middle lane:Six cars in five lanesCompare the direction of cars B, C1, C2, and D according to the frames attached to the five cars:

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Velocity reciprocity clarified

This is a follow-on to posts here and here.

It is common to derive the Lorentz transformation assuming velocity reciprocity, which seems to say that if a body at rest in frame of reference is observed from a frame of reference S that travels with relative velocity +v, then a body at rest in frame of reference S will be observed from the frame of reference to be traveling with velocity –v. But that’s not the case.

Consider the typical scenario in which a person standing on the earth (embankment, station) with frame of reference S observes a person sitting in a railway car with frame of reference . Say they are both waving their right hands and their frame of reference follows a right-hand orientation: the positive direction is toward their right.

Person waves to train

The first illustration shows the scenario from behind the observer standing on the earth in frame S, who observes the passenger sitting in the train moving to their right with velocity +v. The scenario is typically presented from only this perspective, that of an observer at rest in frame A, even if the perspective of an observer at rest in frame is described.

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