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Category Archives: Relativity

Relativity posts

What Galileo really demonstrated

Galileo Galilei’s inclined plane experiment is described in his work Dialogues Concerning Two New Sciences, which I quote from the Dover edition. He speaks (through his character Salviati) of “those sciences where mathematical demonstrations are applied to natural phenomena, as is seen in the case of perspective, astronomy, mechanics, music, and others where the principles, once established by well-chosen experiments, become the foundations of the entire superstructure.” (p.178) This is the ancient method of science that Galileo applied to experiments, establishing the foundation of modern science.

Galileo states his Theorem II, Proposition II as:

The spaces described [i.e., traced] by a body falling from rest with a uniformly accelerated motion are to each other as the squares of the time-intervals employed in traversing these distances. (p.174 or p.142 on the OLL edition)

But it has just been proved that so far as distances traversed are concerned it is precisely the same whether a body falls from rest with a uniform acceleration or whether it falls during an equal time-interval with a constant speed which is one-half the maximum speed attained during the accelerated motion.

Then he describes his experiment:

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Science and Hypothesis excerpts

What follows are excerpts from the book Science and Hypothesis by Henri Poincaré, translated (1905) from La Science et l’hypothèse (1902).

p.xxiii The latter [definitions or conventions] are to be met with especially in mathematics and in the sciences to which it is applied. From them, indeed, the sciences derive their rigour; such conventions are the result of the unrestricted activity of the mind, which in this domain recognises no obstacle. For here the mind may affirm because it lays down its own laws; but let us clearly understand that while these laws are imposed on our science, which otherwise could not exist, they are not imposed on Nature. Are they then arbitrary? No; for if they were, they would not be fertile. Experience leaves us our freedom of choice, but it guides us by helping us to discern the most convenient path to follow.

p.xxv Space is another framework which we impose on the world. Whence are the first principles of geometry derived? Are they imposed on us by logic? Lobatschewsky, by inventing non-Euclidean geometries, has shown that this is not the case. Is space revealed to us by our senses? No; for the space revealed to us by our senses is absolutely different from the space of geometry. Is geometry derived from experience? Careful discussion will give the answer—no! We therefore conclude that the principles of geometry are only conventions; but these conventions are not arbitrary, and if transported into another world (which I shall call the non-Euclidean world, and which I shall endeavour to describe), we shall find ourselves compelled to adopt more of them.

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From racing to relativity

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

(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 length and duration. 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 length and duration. This is the situation of general relativity and transportation in general. Conversion of length and duration are local, not global, and the optimal route depends on whether length or duration are optimized.

Ignatowsky relativity

Vladimir Ignatowski (1875-1942) was a Russian physicist. “In 1910 he was to first who tried to derive the Lorentz transformation by group theory only using the relativity principle (postulate), and without the postulate of the constancy of the speed of light.” K M Browne gave a simplified derivation in the European Journal of Physics, 39 (2018) 025601, from which the key steps are presented below, followed by the corresponding steps for a dual transformation, switching space and time.

This is a derivation of the Ignatowsky transformation in which the axes x, y, and z are taken to represent space axes rx, ry, and rz with time t. The relativity postulate is taken to be: a valid relativistic transformation must be identical in all inertial frames.

Step 1. To find a valid transformation, we take the usual inertial reference frames S and S′ (the latter moving at velocity v in the +x direction relative to the former) for which intra-frame space is Euclidean but inter-frame space (measured from one frame to the other) may be non-Euclidean. Linear equations are necessary so that an event in one frame appears as a single event, without echoes, in the other. Initial conditions are x′ = x = 0 when time t′ = t = 0. We expect the generalised x equation to be the Euclidean equation x′ = xvt with an added multiplier, and if time is the fourth dimension, then the time equation will be similar but with two additional multipliers. The second of these, n, having the dimension of inverse velocity squared, is required to make the equation dimensionally correct. The y and z coordinates are not expected to be affected by x and t. The generalised transformation and its inverse are then

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Length and duration in space and time

The following derivations are based on the exposition by G. G. Lombardi here.

Time Dilation

Time dilation with a light clock

A clock is made by sending a pulse of light toward a mirror at a distance L and back to a receiver. Each “tick” is a round-trip to the mirror and back. The clock is shown at rest in the “Lab” frame in Fig. 1a, or any time it is in its own rest frame. Consequently, it also represents the clock at rest in Rocket #1. Figure 1b is the way the clock looks in the Lab when the clock is at rest in Rocket #1, which is moving to the right with velocity v and legerity u, hence speed v and pace u.

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Time and simultaneity

There are several ways of understanding the time of remote events. What follows is a summary of the basic ways of determining simultaneity.

As a way of comparing the different ways consider transmitting a light signal to a remote location where it is reflected back. What is the time when the signal is reflected back?

Observation time is an extension of ordinary perception. When we observe an event, we say that it is happening at the time of observation. So when a light signal is reflected and received back, the reflection observed is considered to have happened when it was observed. In effect the light observed is instantaneous. By implication the one-way speed of light transmitted is c/2 in order for the two-way speed of light to equal c.

Observation time is thus the projection of the time of observation to the entire observable universe. This way of understanding time is characterized by the Galilean transformation.

Transmission time is an extension of the ordinary transmission of light. When we shine a light on an event, we say that it is happening at the time of transmission. So when a light signal is aimed toward a reflector, the event of reflection is considered to have happened when the light was transmitted. In effect the light transmitted is instantaneous. By implication the observed one-way speed of light is c/2 in order for the two-way speed of light to equal c.

Transmission time is thus the projection of the time of transmission to the entire transmittable universe. This way of understanding time is characterized by the dual Galilean transformation.

Probe time is an extension of measurement by a probe (a “small, unmanned exploratory craft”) to the entire probeable universe. See previous post here. An event is said to occur when intersected by a probe that measures the duration of probe movement since a reference event. So when a probe comes upon the reflection of light, the probe measures the time of reflection as the time of the probe. If the probe is not moving at the speed of light, there may need to be multiple probes.

Consider a series of probes moving at a speed v over a distance d to the reflection event. The probe that leaves at time (d/c) – (d/v) is the probe that intersects the reflection event. If v = c, then the time is zero.

Because probes can measure the length or duration of motion, probe time is characterized by the Lorentz or dual Lorentz transformation.

Reference frame time measures time by a rigid reference frame that has clocks which were previously synchronized spread throughout. See the Relativity of Simultaneity and Einstein Synchronisation. These synchronizations are characterized by the Lorentz transformation.

Einstein exchanged

Albert Einstein’s book Relativity: The Special and General Theory was originally published in German and translated into English in 1920. In the second chapter he introduces “The System of Co-ordinates”. The following post gives Einstein’s text followed by a revision that exchanges length with duration and space with time. First, Einstein’s text, with alternative wordings in square brackets:

End of Chapter I – If, in pursuance of our habit of thought, we now supplement the propositions of Euclidean geometry by the single proposition that two points on a practically rigid body always correspond to the same distance (line-interval), independently of any changes in position to which we may subject the body, the propositions of Euclidean geometry then resolve themselves into propositions on the possible relative position of practically rigid bodies.

Chapter II – On the basis of the physical interpretation of distance [line-interval] which has been indicated, we are also in a position to establish the distance between two points on a rigid body by means of measurements. For this purpose we require a “distance” (rod S) which is to be used once and for all, and which we employ as a standard measure. If, now, A and B are two points on a rigid body, we can construct the line joining them according to the rules of geometry; then, starting from A, we can mark off the distance S time after time [again and again] until we reach B. The number of these operations required is the numerical measure of the distance AB. This is the basis of all measurement of length.

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Time, space, and order

There are three axes (dimensions) of motion with six degrees of freedom. There are two metrics of motion: a length metric and a duration metric. The length metric is the magnitude of the vector between two points, and is called distance. The duration metric is the magnitude of the vector between two instants, and is called distime.

If one conceives of this as two 3D metric geometries of motion, then there is a 3D length space geometry with a distance metric and a 3D duration space geometry with a distime metric. If the speed of light is an absolute conversion between distance and distime (which is essentially Einstein’s second postulate of special relativity), then there is one 6D length-duration metric geometry.

A 3D length space coordinate system is built from an origin point and three orthogonal axes with a distance metric. A 3D duration space coordinate system is built from an origin instant and three orthogonal axes with a distime metric. A 6D length-duration coordinate system is built from an origin event, three length space coordinates, and three duration space coordinates. Either the three duration coordinates may be converted to lengths, or the three length coordinates may be converted to durations.

A stanceline represents two opposite linear motions with a constant rate (i.e., inertial motions). The positive direction represents distances to events diverging away from the origin point. The negative direction represents distances to events converging toward the origin point (i.e., destination). Apart from motion a point has a distance but its sign is ambiguous. A stanceline represents the stance of a metreloge.

A time line represents two opposite straight motions with constant rate (i.e., inertial motions). The positive direction represents distimes to events diverging from the origin instant. The negative direction represents distimes to events converging toward the origin instant (i.e., destination). Apart from motion an instant has a distime but its sign is ambiguous. A time line represents the time or scalar time of a clock.

Events may be ordered by the stance or the time. Events ordered by stance are stancelined. Events ordered by time are timelined. All events that are equal distances (equidistant) from the origin point are puncstancious with it. All events that are equal distimes (equidistimed) from the origin instant are simultaneous with it.

6D Galilean spacetime

Here we expand 4D Galilean spacetime into 6D Galilean spacetime, based on section 1.3 Galilean spacetime of The Geometry of Relativistic Spacetime: from Euclid’s Geometry to Minkowski’s Spacetime by Jacques Bros (Séminaire Poincaré 1 (2005) 1 – 45).

[p.3] We start with a representation space whose points are interpreted as the “physical events”. Any motion of a particle which is physically possible between two given events A and B is represented by a certain world-line with end-points A and B. There is an absolute orientation of such worldline, which can be called its “time-arrow”: its physical meaning is that one of the end-point events, e.g. B, is in the future of the other one A.

[p.6] From the viewpoint of mathematical physics, the use of geometry in more than three dimensions turns out to be necessary, if one wishes to represent phenomena whose description necessitates more than three independent quantities. A typical example is the six dimensional space Rab6Ra3 × Rb3 of the positions (a; b) of pairs of material points (or pointlike particles) in mutual interaction. Trajectories of such pairs are represented by curves in R6, described in terms of a parameter t by equations of the form a = a(t); b = b(t).

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Motion measurements

As described in the previous post here, the three dimensions of motion are axes for traveling along (length) or revolving around (time).

A measure of motion may be either (1) dependent on the the target motion, or (2) independent of the target motion. A measure that is independent is either available prior to or separately from the target motion. For example, an independent measure may be determined by agreement, such as the length of a race, or it may measure another motion, such as the motion of a clock, which is then correlated with the target motion.

A standard clock measures time because it measures rotations around an axis as an angle. A length clock measures rotations about an axis as a length. With constant rates of rotation constant, there is a fixed ratio between the two kinds of clock.

A device that measures its own internal motion may be called an autometer. A clock is an example of an autometer. The internal motion of an autometer can be correlated with a target motion. For clocks this is called synchronization. For a length clock this is called synstancialization.

An odometer is a measurement device that depends on its target motion. The standard odometer measures length of travel. A time odometer, or trip-timer, measures time of travel. A trip-timer is a stopwatch that is on only while the target motion takes place. If there is a stop in the target motion, then the trip-timer also stops. So the trip-timer measures time of motion rather than elapsed time.

A device that measures a quantity of motion need not be attached to the moving body. The theory of relativity deals with the remote measurement of quantities of motion. A device that is attached to the moving body produces proper measures such as proper length or proper time.