Musings
This page discusses my thoughts on various topics.
On the meaning of Law (of physics) and the principle of relativity.
The word Law appears throughout physics. In particular, Special relativity' first postulate is often cited as:The laws of physics are the same for all inertial reference frames. However, if one does a quick search for the definition of Law (of physics), one obtains but few hits and often with unclear definitions. Since the word Law is an important concept, it should be clearly defined. Reading the various definitions we can get a good idea on its intended meaning. The general idea of a Law may be defined as,
:= A rule, expression or equation(s) which remain valid (keep the same form) in any admissible frame.
The definition itself may need further explanation. Take a set of frames. Find expressions (rules or equations) which are valid in all these frames. Then by definition, these expressions are Laws. A frame (or reference frame) is a system by which we can compare (measure) or attribute the coordinates of a concept. Usually, the concepts are position, time, etc… Since we wish to analyze or measure our "physical" world, our concepts ( as position, time…) must have operational definitions; our frames must be coordinated by a physical procedure (otherwise we would be unable to do measurements!). Therefore, to verify if an expression is a Law, the frames in question must be coordinated.
In the context of SR, the admissible frames are the inertial frames. The definition of inertial frame is treated in another section below.
The principle of relativity, as the first postulate of SR, is often stated that: "The laws of physics are the same for all inertial reference frames". In essence, SR's first postulate is nothing more than a word per word restatement of the definition of Law; it is a tautology and thus necessarily true.
From Wiki: "In physics, the principle of relativity is the requirement that the equations, describing the laws of physics, have the same form in all admissible frames of reference". This too is just a restatement of Law. The conclusion is that SR's first postulate (as formulated above) is redundant because it is just a restatement of the definition of Law.
Conversely, the principle of relativity may be taken as to define Law. To discover the Laws, we need to find which principles (or equations) describing physics keep the same form in all inertial frames. This means that with our chosen coordinated systems, we need to discover which equations (quantities, principles…) will retain the same form (in all inertial frames). We are free to choose any coordinate system we wish. We will choose the coordinate systems ( the operational definitions) that yield simple equations, or, simple Laws. [H. Poincare, La mesure du temps, 1898].
But, "simple" as in the "beauty" of an equation, is in the eye of the beholder. Modern physics has embraced the "beauty" of the definition of Law (the relativity principle).
See wiki Physical Law for further discussions.
La Mesure du temps 1898. Poincare.
Mais quelle est la nature de ces règles ? Pas de règle générale, pas de règle rigoureuse ; une multitude de petites règles applicables à chaque cas particulier. Ces règles ne s’imposent pas à nous et on pourrait s’amuser à en inventer d’autres ; cependant on ne saurait s’en écarter sans compliquer beaucoup l’énoncé des lois de la physique, de la mécanique, de l’astronomie.
Nous choisissons donc ces règles, non parce qu’elles sont vraies, mais parce qu’elles sont les plus commodes, et nous pourrions les résumer en disant :
« La simultanéité de deux événements, ou l’ordre de leur succession, l’égalité de deux durées, doivent être définies de telle sorte que l’énoncé des lois naturelles soit aussi simple que possible. En d’autres termes, toutes ces règles, toutes ces définitions ne sont que le fruit d’un opportunisme inconscient. »
On the various formulations of the principle of relativity of Special relativity.
There are many formulations of the principle of relativity as one can see from a quick search on the net. Following are three formulations. The first is taken from Wiki. The second a simplified formulation of the prior and the third is Einstein's original formulation.
The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems in uniform translatory motion relative to each other
The laws of physics are the same for all inertial reference frames
All inertial frames are totally equivalent for the performance of all physical experiments
One needs to read several times the first formulation to begin to grasp the idea. Moreover, several expressions need to be understood as "uniform translatory motion". Although "uniform translatory motion" is not necessarily equivalent to "inertial frame", we will for simplicity admit that they have the same meaning. We will therefore admit that the first two formulations of the principle of relativity are equivalent and begin our discussion with the simpler one, the second formulation.
As stated in the previous discussion we have argued that this formulation is superfluous since it is the restatement of the definition of Law (of physics). Definitions are "non-discussible" and thus this ends the discussion on the first two formulations.
The last formulation is a little more interesting. Pondering on its intended meaning, we conclude the following. No matter what experiment is performed in an inertial frame (I-F), performing the same experiment in another I-F will yield the same result (outcome). But, is this too, not just a tautology?
The expression "same experiment" means that all parameters, from the beginning of the experiment to its end, must be identical to the parameters of the other identical experiment performed in another I-F. We can not perform an experiment and then perform it again changing an intermediary parameter without expecting a different outcome!
Can two identical experiments have different outcomes? This depends on the definition of "identical experiment"; is the outcome of the experiment part of the meaning of "same experiment". If so, then we do indeed have a tautology (and the requirement for an I-F becomes useless). If not, then this raises a series of other questions:
How can two experiments be totally identical except for its outcome. Somewhere between where the experiments are identical an their different outcomes, there must be some effect, some parameters, which change the outcome. Effects or parameters that have this quality cannot be modeled by deterministic equations; they must be inherently undeterminictic (probabilistic). Quantum theories have this quality.
If all parameters remain identical except for the "last instant", then how do we treat causation? What caused the different outcomes? Can the causal process be identical for two experiments (in all its parameters at all times) but at the last instant differ? Such thoughts lead to hefty discussions in causality, which we will not go into in this discussion.
We therefore have two ways to go. (1) Either we define "same experiment" as all parameters be identical from beginning to end inclusively for the two experiments. In this case, Einstein's original formulation of the principle of relativity is simply a tautology. (2) The outcome of the experiment is not part of the meaning of "same experiment" in which case brings up several questions in causality and (un)deterministic theories.
Accepting (1) is the simpler way to go since it does not raise the complications of (2). But accepting (1) also makes the (third formulation) principle of relativity superfluous.
By defining Law as discussed on this page and defining "same experiment" as in this discussion, makes the principle of relativity superfluous. Definitions are more fundamental than postulates.
On the Second Postulate of special relativity
Around the 1900 it was noted that when one tried to measure the speed of light, all inertial observers obtained the same value (to within accuracy of course). This "constancy" of the speed of light was then used by Einstein as a postulate in his theory of special relativity in 1905. However, unbeknownst at the time (a few knew…), this constancy of the speed of light was a result not of a "constant speed of light" but a direct result of the not fully understood measurement procedures. It was H. Poincare in his 1898 work of "La mesure du temps" who pointed out that the usual procedures to measure distances and times necessarily lead to a constant value for the speed of light. He went on discussing that other procedures can yield non-constant speed of light. However, the simple (and only accessible) procedure to measure time was the usual procedure, the telegrapher's procedure (often called Poincare's synch procedure and since 1905 called Einstein's synch procedure). Therefore, by an unconscious definition of time (and distance) the speed of light was made constant.
Since 1983, these unconscious definitions were made clear. The BIMP has explicitly defined the meter in such a way that it makes the speed of light constant. The meter is (operationally) defined in essence as that same old telegrapher's procedure.
The seventeenth CGPM defines the metre as equal to the distance travelled by light in vacuum during a time interval of 1/299,792,458 of a second.
Its theoretical operational definition is the following, performed in an vacuum and in an inertial frame: Send an electromagnetic signal (light) from point A to point B where it is reflected back to A. Denote by T the total time traveled by the signal as indicated by the unique clock at A. Divide this value by 2 and multiply by the integer 299792458. The value obtained is the distance between points A and B, in meters. D = (T/2)*299792458 . Its practical operational definition is much more involved but is made to be as close as possible to the theoretical operational definition to within accuracy of the instruments. For more information, see the various standards organizations ( BIMP, NIST…), wiki metre and wiki redefinition of the metre.
When measuring the speed of light, one must insure that the distance (or other parameters) used makes the speed equal to 299792458 m/s, otherwise the speed measurement procedure is declared flawed or unconventional. With the defined conventional measurement procedure, the speed of light is constant by construction. Its a simple application of a (operational) definition!
With such a definition which makes the speed of light constant, there is no longer need to use the postulate of constancy of the speed of light in special relativity.
The postulate of constancy of the speed of light of special relativity is redundant and thus superfluous since 1983 and was unknowingly so since the years ~ 1900 !
Coordinate system and (Inertial) reference frame
The folowing definitions generalize and reflect the general ideas of a coordinate system etc.
:= Coordinate System (CS)
A coordinate system is a system by which we attribute a quality (quantitative or not) to a certain notion.
Examples with a mathematical connotation.
If the elements of a set are assigned elements of another set, the assignation is the CS.
From wiki. In mathematics …, a coordinate … system is a system for assigning an n-tuple of numbers or scalars to each point in an n-dimensional space.
Examples without a mathematical connotation.
If the elements of a set are assigned a color, the assignation is the CS.
If the temperature of the day(s) is assigned by {very cold, Brrr, ok, Hot} as expressed by a child then we have a procedure to coordinate the temperature.
Above is an example where the coordinate attributed to the daily temperature depends on the child chosen; a child may give "Brrr" whereas another may give "ok". We therefore can have many different "values" to coordinate a particular element (a day) of a set (the days) depending on which CS is used (which child is used).
:= Reference Frame (RF)
A reference frame is a CS which uses a physical process to determine the coordinates of a notion.
In physics, the usual notions are position, time, speed, etc. The procedure by which we attribute values to these notions is the CS. Since the procedures are physical, the CS is called a RF. Simply put, a RF is a CS with a physical connotation. We say that the notions (position …) are operationally defined. The act of performing the procedure to obtain a value is the measurement. The notions of CS, RF and measurement are very similar and generally there is no harm in taking them as synonyms. For more on this topic, see wiki http://en.wikipedia.org/wiki/Reference_frame_(physics)
Since physically we are limited to observations (measurements), quantities such as position and time are operationally defined; Their values are found via a RF.
Examples: The first figure: two different Reference frames set up by two different observers. In the red Cartesian grid, the coordinates of the point is (2,6) whereas in the "warped" grid it is (1,4). The second figure: A too small to see ant is well-holding on a fluffing flag. In the hypothetical grid of the flag, the ant, as well as the stars, remain at the same location; are fixed in space. Relative to the exterior Cartesian grid fixed to the ground, the ant and stars are continuously changing position.
In special relativity (SR), position and time are coordinated by a particular procedure and is the following.
Setup a (more or less rigid1) latticework of clocks.
Choose an origin for position and time: i.e choose a master clock and define its position as the spatial origin. Find/define/calibrate the distance of the other clocks from the origin: Send an EM from the clock to the mater clock where it is reflected back to t he clock. Take the interval of time as indicated by this clock, divide by two and multiply by the integer 299792458. The value obtained is the position (or distance from) of the clock in question. Do this for every clock. (There are simpler variants of this procedure).
Synchronize the clocks to the master clock: As the master clock sets itself to zero it sends an EM throughout the space. On reception of this signal a clock sets itself to its distance divided by the integer 299792458.
This (hypothetical) latticework of clocks will give the coordinates (position and time) of an event relative to this RF. Other observers may setup their own latticework but must use the same procedures (and can synch their master clock to another observer's master clock for simplicity). To use SR, this RF must also be an inertial frame, subject of the next section. Note that these RF's of SR imply that the speed of light will always be 299792458 as measured by any inertial observer.
1The latticework need not be rigid and the clocks may be flying around in all directions. However, with the specified coordination procedure, the positions and time sync wrt the master clock will continuously change. Hardly a way to locate events and would make the equations of physics quite convoluted!
:= Inertial Frame (IF)
A RF in which for any free test particle, there is no detectable change in its velocity.
Here we have opted to use Newton's first law to define an IF instead of the second law "F=ma" since the first law does not explicitly make reference to "force". In fact we can simplify the definition of an IF as one in which any free particle at rest remains at rest. We emphasize a few points:
A RF is a CS by which we define position, etc. The RF can be a material one, as a physical grid. It can be a non material one as a lattice produced by beams of light. It can be an operationally defined one as the definition of meter. It can also be an abstract mathematical one (in which case it can not be observed nor detected). There are no provisions for the mathematical structure of the coordinate system; Cartesian, polar or any other type may be chosen.
Since Newton's first "rule" is true in all IF by construction, this rule is a Law as per defined in a previous discussion.
A test particle is a physical particle, an atom, molecule or macroscopic object (further philosophical/metaphysical discussion needed).
Since the definition of IF uses test particles, the definition is operationally defined.
The adjective free refers to a situation where there is no detectable entity or force influencing the motion of the test(s) particle(s). Discussed below.
The term detectable refers to measurements made locally i.e within the RF. The use of measurements imply within their limits of accuracy. This too makes the definition of IF an operational one and depends on the current technology. What was an IF may no longer be an IF.
Since the CS/RF is set up, "velocity" is well defined. e.g: delta_X / delta_T .
A change in velocity implies that at least two successive measurements of the velocity have been performed. Hence a frame is inertial during a certain time interval.
Now back to a free particle. A particle is said to be free if we do not attribute to it an exterior or physical influence. For instance, in Newtonian physics, a particle near Earth will be "influenced" by the Earth. We attribute the change in velocity of the particle by the "gravitational force", hence the particle is not free. However, in general relativity, there is no such "gravitational force"; the change in velocity of the particle is due to the "curvature" of spacetime, not due to a "physical" influence, hence the particle is free. Is "curvature of spacetime" an exterior influence? a physical influence ? We must define these words first... thats for a future topic..
Therefore, for a RF to be an IF we must (1) decide if all test particles are free and (2) verify that there is no detectable change in their velocity.
Examples: A) Dropping a particle in the vicinity of the ground. Having set up our RF (rulers/clocks) to the ground, we notice that the velocity of the particle changes (it accelerates towards the ground). If we declare the particle to be free then this RF is by definition not an IF. If we declare the particle to be not free, then we cannot conclude if our RF is an IS for we need free particles to decide so. B) Dropping an elevator box (containing test particles) in the vicinity of earth. Inside the box we have set up the usual RF. Declaring that the test particles are free and noticing that they all maintain their respective velocities (relative to the box/RF) then this box/RF is an IF. If we declare that the test particles are not free then we can not conclude if the box is an IF.
We come to understand that an inertial frame depends on the concept of a free particle, the concepts of an influence or force.
TBC.
What's a PARADOX?
There are two common definitions of 'paradox' as one can verify from various dictionaries:
1- An apparent contradiction; Which defies intuition.
2- A self inconsistency; Statement(s) that lead to a contradiction.
Both meanings of 'paradox' are used in physics and unfortunately the incorrect meaning is often taken by the simple reader which is why in special relativity there are so many 'paradoxes'. Part of the blame is also the authors who are often not explicit enough for their target audience. Special relativity is plagued with paradoxes if 'paradox' is taken to mean "An apparent contradiction...", apparent being the important word here. However, if one adopts the second definition of paradox, then special relativity is of date devoid of such paradoxes.
Bug-Rivet Paradox
Since a picture is worth a thousand words, perhaps these animated pictures will be worth 24 thousand words!
The bug-rivet paradox is extensively discussed on the internet and therefore I will only summarize its resolution. There are two events in consideration: The tail of the rivet hits the bottom of the well (bug gets squished) and the head of the rivet hits the entry of the well. In the frame of the rivet (first picture), the bug gets squished then the head of the rivet smashes into the entry point. In the frame of the bug it is the head of the rivet that initially hits the entry point then the bug gets squished. This can be shown by applying the Lorentz equations to these two events. Relativity of simultaneity at play. In this description we used SR to only give us the location and time of the events; we did not use SR to describe the physical processes involved leading to our paradoxical conclusion, subject of the following paragraph.
What seems paradoxical in the bug-rivet paradox is that from the bug's frame, the rivet's head initially hitting the entry point should stop the rivet from continuing on to squish the bug. But this is just a false intuition we have. The tail of the rivet has no 'idea' that its head has stopped and thus continues on to squish the bug. Its a little more complicated than that but does reflect the idea of what's going on. For a more in depth analysis one must consider applying SR to the electromagnetic forces (Laws) affecting the atoms of the rivet: the Rigidity (stresses, strains, pressure wave...) as viewed from the respective reference frames. This analysis will describe the dynamical and causal processes involved.
Bell's spaceship paradox
Click on image for animated gif
There are many 'resolutions' to this paradox. Here I offer a simple visual explanation. The two ships leave their mutual initial frame (the ground) simultaneously and have identical motions (hence identical accelerations) relative to the ground. Therefore, their separation remains constant wrt the ground. A ruler (also representing the trailing spaceship's frame) is attached to the trailing spaceship and hence has a certain motion wrt the ground. A little mark (15 units) on the ruler is drawn showing where the leading ship was initially along the ruler. As the ships accelerate and gain speed (wrt ground), the ships and the ruler Lorentz contract wrt the ground. Since the ruler contracts, the leading spaceship is further away as marked on the moving ruler; the spaceships have moved apart wrt the ruler, wrt the trailing ship. If instead of a ruler we had string as in the usual scenario then the 15 unit string would stretch and break. A full discourse on the subject involves simultaneity, Born rigid motion and so forth...which is not the aim of this simplistic text and archaic animation.
Math: Discovered or invented?
The Question: Is Mathematics discovered or invented?
In my view, it is discovered. From past philosophers to modern mathematicians this question still arouses hefty debates. (Platonic Realism, Philosophy of math)
One side claims that mathematics has been and always will be. It is "there", in its abstract realm, just waiting to be discovered. All those mathematical theorems have been searched and discovered by us humans. The other side claims that mathematics, all those theorems, were the fruit of human intellect and that without humans such theorems would not exist.
The debate on whether math is discovered or created can be somewhat (or fully?) settled if one concisely defines the terms used.
The words at play in this debate are: mathematics, discovered, invented and exist. There are other terms which would need concise definition but here we shall interest ourselves on the first word. Once we have defined math(thematics), then we will be in a better position to answer The Question.
Definition. Math: The implications of axioms (rules).
One might define it as the study of the implications of axioms. However, this latter definition has the idea that something (eg. human) is studying and working out the implications of the axioms. If we impose that only a human can "study" then the latter definition of math implicitly contains the "human" aspect in mathematics and therefore there is no math without humans. This consideration raises an interesting question. Do we need "humans" as the entities that study? Can not another creature have a primitive capacity of reason or can not there be other far away aliens with superior intellect than ours? If so, then the latter definition of math needs to be further expanded. More down to earth, computers can "study"; computers are often used to solve end even prove formal systems. Giving the rules to the computer, the computer churns away and "discovers" the answer or the sought theorem. Therefore, considering the latter definition of math and that computers can outlive the human race, will math still "exist" in a world with only computers?
The bolded definition voids the above considerations and simplifies the debate. In the bolded definition, the implications of axioms need not be found nor expressed by humans nor by any other calculating "creature". This definition is not contingent to the existence of humans and thus math is not "invented". The results of the axioms may be found by intelligent creatures. Sentient beings have the luxury to be aware of axioms and their deduced implications. In this definition of math, math is "out there", discovered and appreciated by sentient and intelligent beings.
A note on the axioms and the deductive reasoning leading to the implications. The axioms are rules to follow. The deductive reasoning follows some logic. This logic too is implicitly part of the axioms. The logic need not be Boolean Logic but any type of Logic which is nothing more that a set of specified rules to follow. Once these rules are in place then the implications may be found by intelligent creatures.
A Shot at the Arrow of Time
The wineglass shattering to the ground never reassembles itself back onto the table, or does it?
The Sandow Theorem
The Sandow Theorem pertains to a property of a squared circle (not to be confused with squaring the circle).
It states that the necessary and sufficient conditions to attain the maximum within a squared circle is by use of both the physical processes and computing processes. Symbolically, I >u. An alternate representation making explicit use of one's capacity to change and mass relative to the rest of the universe is I'm > U. In biology, Darwinism is a manifestation of the Sandow Theorem. In popular culture, the wrestler Damien Sandow embodies the theorem that bears his name.
History: The symbolism "I >u" and the general concept of the theorem has been known for some time1. However, only recently has it been formally proven by BauneS. The proof consist in acknowledging the physical attributes and scholarly successes of yours truly, hence I > u. Explained for the lay (these are the R.H.S of the ">"), it may be loosely stated as "brain will best brawn".
WWE's Damien Sandow
The 2-Piecewise Function
This short text (pdf here) gives a general formula to express a 2-piecewise function in a "one liner".