Thought-Experimental Principles of Relativity

0\. Introduction and Goal

In his considerations concerning simultaneity, Albert Einstein indicated and employed certain very basic abilities of each individual observer, and relations between observers:

* that one may perceive and distinguish various signals,
* that, given one's observations of any two distinct signals, one may judge whether one perceived these two signals together ("coincidently"), or separately ("first" vs. "afterwards"),
* that one may distinguish and consistently recognize others by their signals, which one observed,
* that one's signals may in turn be observable and recognizable to others, and
* that one may recognize a signal as someone else's response ("echo") to an own signal.

These abilities and relations may not be readily exhibited in every practical instance and circumstance. However, being asserted, they may be taken as equally and unambiguously comprehensable. For example (cmp. section (3.)), one cannot even consider the problem of how two observers might come to an agreement on which, if any, of their observations had been "simultaneous" to each other, without thereby already admitting an understanding of the fundamental abilities and relations listed above. The mutual assumption of these observational abilities and experimental relations, even if it presents only an idealization, is consequently also uniquely suited as a starting point towards solving this and related problems.

The descriptions to characterize relations between observers, and first of all, how mutual agreement (measurement) may be reached between those involved, give rise to a great variety of relativistic thought experiments; concerning notably (in historic order)

* how to measure "simultaneity" (as already indicated by Einstein; that is, requiring or assuming the measurement that "curvature" had been negligible, and a procedure to agree on which clock parametrizations are "good"),

* how to measure "distance" and "curvature" (by John L. Synge, as "Chronogeometry"; that is, requiring or assuming a procedure to agree on which clock parametrizations are "good"),

* how to measure whether two observers had "moved parallel" to each other (by Alfred Schild, known as "Schild's Ladder"; that is, assuming or requiring a procedure to decide whether or not a certain auxiliary observers had been "free", and which clock parametrizations are "good" at least for such "free" auxiliaries),

* how to decide which clock parametrizations of a "free" observer are "good" (by Robert F. Marzke and John Archibald Wheeler; assuming or requiring all procedures employed for "Schild's Ladder"),

* how to decide whether a given observer had been "free" (by Udo Schelb; assuming or requiring procedures to distinguish "opposite directions").

The goal of this course is to review and motivate the observer abilities which are to be assumed, and strictly on their basis to present the relativistic thought experiments in a systematic development.

As its title indicates, this is a course in experimental physics (evidently concerning its foundational aspects, rather than accounts of specific obtained values). Starting from what any one observer can assert individually, procedures are sought and derived on how several observers may reach mutual agreement on their relations with each other.

Particular attention is naturally given to providing a thought-experimental definition to the notion of "opposite directions"; or more directly concerning observers, and in association with Einstein's usage: to the identification of which one, if any, had constituted a "middle between" certain given observers. The sequence of thought-experimental definitions which are obtained includes further

* how to characterize a suitable system of observers by their "curvature"; and in particular, to decide whether, or to which accuracy, two observers had been "local" to each other,

* how to decide whether, or to which accuracy, a given observer was "free" (or synonymously: "inertial", or in the context of measured curvature: "geodetic") among a suitable system of auxiliary observers,

* how to characterize suitable systems of observers through geometric quantities such as "distance", "direction", "speed", and so on; especially

* how to measure whether several observers belonged together to an "inertial frame"; and eventually

* how to determine the geometric distribution of "mass/energy", "momentum" or "stress" among a suitable system of auxiliary observers.

The student (who can of course be presumed to possess and therefore to understand the indicated fundamental observer abilities as well, at least in principle) will thereby be prepared to encounter the indicated notions where they appear without consideration for a definition in terms that are accessible to observers.

The systematic development of thought-experimental definitions also lends itself to recognizing relations between measured quantities that arise by definition (for instance: the equality of light speed in regions of equal refractive index, the effects of "Time dilation", and "Pericenter advance"), and to distinguish them from relations that are subject to experimental test (such as correspondingly, whether two regions had equal refractive index, whether two twins were aging at equal proper rates, or whether, or to which accuracy, someone moved freely in some particular region and whether, or to which accuracy, observers in this region would find their curvature compatible with the Schwarzschild metric).

In emphasizing these distinctions, and in demonstrating overall that relativity provides definitions for how to measure geometric relations in principle, based on abilities which are unambiguous for everyone involved and which are justified from the outset, this course aims to teach the larger lesson that any measured values which may thereby be obtained are incapable of providing in turn tests (or possibly even falsifications) of these underlying assumptions and definitions.

Outline:

## Contents

* 1 The Fundamental Abilities of Observers Reviewed
* 1.1 Distinguishing and Recognizing Signal Sources
* 1.2 Ordering One's Observations
* 1.3 Recognizing Signals as Echos to Own Signals
* 2 From Observations to Measurements
* 2.1 Shared Abilities Allow Mutual Agreements
* 2.2 Procedures and Values can be Copied (Anything Else is Presumed Unique)
* 2.3 The Fundamental Abilities of Observers as Minimal Mutual Assumptions
* 3 The Making of a Middle
* 3.1 Where (Experimental) Physicist Fear to Tread: A clock, by any other ticks, did still advance
* 3.2 Retreat into the Railway Car (Bouncing Echos Off Two Ends)
* 3.3 Ambiguity Identified - And Resolved (1): Dangling Among Four Corners
* 3.4 Ambiguity Identified - And Resolved (2): Stuck Between Six Vertices
* 3.5 Why Stop There? (If One Pair Fits in 1 Plus 3 Dimensions, So Should All)
* 4 Observer Systems in which Middles were Found
* 4.1 The Measurement of Simultaneity and of Duration
* 4.2 The Measurement of Distance, Speed, and Acceleration
* 4.3 The Ambiguity of Concentric Expansion or Contraction
* 4.4 Relations between Measured Values: Length-Contraction and Time-Dilation
* 4.5 The Measurement of Spatial Curvature and Refractive Index
* 5 Observer Systems in which Middles were Not Found
* 5.1 The Failure to Identify a Middle in a System of Accelerated Observers
* 5.2 What a System of Accelerated Observers Succeeds Measuring Nevertheless
* 5.3 Determination of Curvature Invariants in the Limit of Iterated Measurements
* 6 Application: The Definitions of Dynamic Quantities and Potentials
* 6.1 Invariants of Closed Systems, and their Dependence on Velocity
* 6.2 From Measurements to Expectations: The Principle of Stationary Action and the Field Equations
* 7 Practicalities and Outlook
* 7.1 Representing Measured Relations through Coordinates
* 7.2 Estimates and Systematic Uncertainties Given Limited Observational Data
* 7.3 Composite Observers and Composite Signal Sources

## The Fundamental Abilities of Observers Reviewed

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### Distinguishing and Recognizing Signal Sources

[edit | edit source]

### Ordering One's Observations

[edit | edit source]

### Recognizing Signals as Echos to Own Signals

[edit | edit source]

## From Observations to Measurements

[edit | edit source]

### Shared Abilities Allow Mutual Agreements

[edit | edit source]

### Procedures and Values can be Copied (Anything Else is Presumed Unique)

[edit | edit source]

### The Fundamental Abilities of Observers as Minimal Mutual Assumptions

[edit | edit source]

## The Making of a Middle

[edit | edit source]

### Where (Experimental) Physicist Fear to Tread: A clock, by any other ticks, did still advance

[edit | edit source]

### Retreat into the Railway Car (Bouncing Echos Off Two Ends)

[edit | edit source]

### Ambiguity Identified - And Resolved (1): Dangling Among Four Corners

[edit | edit source]

### Ambiguity Identified - And Resolved (2): Stuck Between Six Vertices

[edit | edit source]

### Why Stop There? (If One Pair Fits in 1 Plus 3 Dimensions, So Should All)

[edit | edit source]

## Observer Systems in which Middles were Found

[edit | edit source]

### The Measurement of Simultaneity and of Duration

[edit | edit source]

### The Measurement of Distance, Speed, and Acceleration

[edit | edit source]

### The Ambiguity of Concentric Expansion or Contraction

[edit | edit source]

### Relations between Measured Values: Length-Contraction and Time-Dilation

[edit | edit source]

### The Measurement of Spatial Curvature and Refractive Index

[edit | edit source]

## Observer Systems in which Middles were Not Found

[edit | edit source]

### The Failure to Identify a Middle in a System of Accelerated Observers

[edit | edit source]

### What a System of Accelerated Observers Succeeds Measuring Nevertheless

[edit | edit source]

### Determination of Curvature Invariants in the Limit of Iterated Measurements

[edit | edit source]

## Application: The Definitions of Dynamic Quantities and Potentials

[edit | edit source]

### Invariants of Closed Systems, and their Dependence on Velocity

[edit | edit source]

### From Measurements to Expectations: The Principle of Stationary Action and the Field Equations

[edit | edit source]

## Practicalities and Outlook

[edit | edit source]

### Representing Measured Relations through Coordinates

[edit | edit source]

### Estimates and Systematic Uncertainties Given Limited Observational Data

[edit | edit source]

### Composite Observers and Composite Signal Sources

[edit | edit source]