MBS TECHNICAL REPORT 17-02

Transcription

1 MBS TECHNICAL REPORT 7-02 On a meaningful axiomati derivation of some relativisti equations Jean-Claude Falmagne University of California, Irvine Abstrat The mathematial expression of a sientifi or geometri law typially does not depend on the units of measurement. This makes sense beause measurement units have no representation in nature. Any mathematial model or law whose form would be fundamentally altered by a hange of units would be a poor representation of the empirial world. This paper formalizes this invariane of the form of the laws as a meaningfulness axiom. In the ontext of this axiom, relatively weak, intuitive onstraints may suffie to generate standard sientifi or geometri formulas, possibly up to some numerial parameters. We give several example of suh onstrutions, with a fous of some relativisti formulas. When properly formalized, the invariane of the mathematial form of a sientifi or geometri law under hanges of units beomes a powerful meaningfulness axiom. Combining thismeaningfulness axiom with abstrat, intuitive, gedanken properties suh as assoiativity, permutability, bisymmetry, or other onditions in the same vein, enables the derivation of sientifi or geometrial laws possibly up to some parameter values). In the last part of this paper, I will show how, in the ontext of meaningfulness, the axiom LLl, v), w) = Ll, v w) ) yields speifi numerial expressions for the funtion L and the operation. Equation ) is an abstrat axiom representing the mehanisms possibly involved in the Lorentz-FitzGerald Contration Feynman, Leighton, and Sands, 963, Vol., 5-3) or the Doppler effet. The operation represents the relativisti addition of veloities. The left hand side of Equation ) formalizes an iteration of the funtion L. The equation states that suh an iteration amounts to adding a veloity via the relativisti addition of veloities operation. A. Motivating the meaningfulness ondition The trouble with an equation suh as v ) 2, Ll, v) = l 2) I thank Chris Doble, Jean-Paul Doignon, and Louis Narens for their ollaboration on various part of my work in this area.

2 representing the Lorentz-FitzGerald Contration is its ambiguity: the units of l, whih denotes the length of an objet, and of v and, for the speed of the observer and the speed of light, are not speified. Writing L70, 3) has no empirial meaning if one does not speify, for example, that the pair 70, 3) refers to 70 meters and 3 kilometers per seond, respetively. While suh a parenthetial referene is standard in a sientifi ontext, it is not instrumental for our purpose, whih is to express, formally, an invariane with respet to any hange in the units. To retify the ambiguity, I propose to interpret Ll, v) as a shorthand notation for L, l, v), in whih l and L on the one hand, and v on the other hand, are measured in terms of two partiular initial or anhor units fixed arbitrarily. Suh units ould be m meter) and km/se, if one wishes. The, index of L, signifies these initial units. Desribing the phenomenon in terms of other units means that we multiply l and v in any pair l, v) by some positive onstants α and, respetively. At the same time, L also gets to be multiplied by α, and the speed of light by. Doing so defines a new funtion L α,, whih is different from L = L, if either α or or both). But, from an empirial standpoint, L α, arries exatly the same information as L,. For example, if our new units are km and m/se, then the two expressions L 0 3,0 3.07, 3000) and L70, 3) = L,70, 3), while numerially not equal, desribe the same empirial situation. This points to the appropriate definition of L α, in the ase of the Lorentz-FitzGerald Contration. It turns out.f. Definition 3) that we should write: L α, l, v) = l v 2. The onnetion between L and L α, is atually: v 2 v ) 2 α L α,αl, v) = αl = l = Ll, v). α This implies, for any α,, ν and µ in R ++, α L α,αl, v) = ν L ν,µνl, µv), 3) whih is a speial ase of the invariane equation we were looking for, in the partiular ase of the Lorentz-FitzGerald Contration Equation or in the ases of the Doppler Effet or Beer s Law). Remark. Looking at Equation 3), one might objet that going in that diretion would render the sientifi or geometri notation very ompliate. But the ompliation is only temporary. When we have extrated all the useful onsequenes from the meaningfulness axiom, we an go bak to the usual notation. In fat, we already have the equation permitting to go bak to our usual notation. Indeed, Equation 3) implies α L α,αl, v) = L, l, v) = Ll, v). Note that the onept of meaningfulness is of ourse related to standard physial onepts suh as dimensional analysis. I will not deal with this issue here, but see Narens 98, 988, 2002, 2007). 2

3 B. Defining meaningfulness Our example of the Lorentz-FitzGerald equation makes lear that the onept of meaningfulness must apply to a olletion of sientifi or geometri funtions we all them odes), and not to a partiular funtion. 2 Definition. We write R ++ = ]0, [ for the set of positive real numbers, and R + for the non negative reals. Let J, J 2, and J 3 be three non-negative real intervals, and suppose that F = {F α, α, R ++ } is a olletion of odes, with for the initial ode F : F = F, : J J 2 onto J 3. Eah of α and indexing a ode F α, in F represents a hange of the unit of one of the two measurement sales. Let δ and δ 2 be two of rational numbers. The olletion of odes F defined above is δ, δ 2 )-meaningful if for any whih yields x, x 2 ) J J 2 and α, ), µ, ν) R 2 ++, we have α δ δ 2 F α, αx, x 2 ) = µ δ ν δ 2 F µ,ν µx, νx 2 ) = F, x, x 2 ) F α, αx, x 2 ) = α δ δ 2 F, x, x 2 ) = α δ δ 2 F x, x 2 ). The role of δ and δ 2 is to speify the measurement sale of the funtion F α, relative to those of its two variables. In the ase of the Lorentz-FitzGerald and similar equations, the measurement sale of the ode is the same as that of the first variable. The relevant definition is given below. 3 Definition. A meaningful olletion of odes, with F = {F α, α, R ++ } as in the previous definition, is alled, 0)-meaningful if it is δ, δ 2 )-meaningful with δ = and δ 2 = 0. We have then, for any x, x 2 ) J J 2 and α, ), µ, ν) R 2 ++, α 0 F α,αx, x 2 ) = µ ν 0 F µ,νµx, νx 2 ), α F α,αx, x 2 ) = µ F µ,νµx, νx 2 ), α 0 = α, µ ν 0 = µ) = F, x, x 2 ) whih yields F α, αx, x 2 ) = αf, x, x 2 ). Suh olletions are also alled ST-meaningful, with ST standing for self transforming. Many sientifi or geometri laws are self transforming. We give several examples in this paper. In this paper, we only deal with sientifi or geometri funtions in two variables, and with ratio measurement sales. 3

4 C. As an introdution: the Pythagorean Theorem One example of an abstrat axiom is the assoiativity equation: F F x, y), z) = F x, F y, z)) whih an be shown to hold for right triangles, with eah of F x, y), F x, z), F F x, y), z) and F x, F y, z)) denoting the measures of the hypothenuses of a right triangle. In the figure below, F x, y) denotes the length of the hypothenuse of the right triangle ABC, with sides lengths x and y, while F y, z) denotes the length of the hypothenuse of the right triangle BCD. The two remaining triangles: ABD, with sides lengths x and F y, z), and ACD, with sides lengths z and F x, y), have the ommon hypothenuse AD. Its length is This shows that the hypothenuse of a right triangle is an assoiative funtion of the lengths of) its two sides. Using funtional equations arguments Azél, 966, Chapter X), we an prove that the assoiativity equation 4) has a representation F x, y) = f fx) + fy)) whih is an equation generalizing the Pythagorean Theorem. F F x, y)z) = F x, F y, z). 4) A x FFx,y),z) = Fx,Fy,z)) B Fx,y) Under meaningfulness, and in the ontext of reasonable bakground onditions, we an prove that the funtion F has the form F x, y) = x η + y η ) η, see Theorem 5 below. The exat statement requires realling some onditions. 4 Definition. A ode F : R ++ R ++ R ++ is symmetri if F x, y) = F y, x) for x, y R ++. Suh a ode is homogeneous if F θx, θy) = θf y, x) for x, y, θ R ++. y Fy,z) C D z 5 Theorem. Suppose that F = {F α α R ++ } is a 2, 2 )-ST-meaningful olletion of onto odes, with F α : R ++ R ++ R ++ for all α in R ++. If one of these odes is symmetri, homogeneous and assoiative, then any ode F α F must have the form for some onstant θ R ++. F α x, y) = x θ + y θ) θ = F x, y), 4

5 For a proof, see Falmagne and Doble 205a, Theorem 7.., page 85). The fat that we must have θ = 2 an be derived from the Area of the Square Postulate and a ouple of other intuitively obvious postulates of geometry. The proofs of Theorem 5 and a ouple of other results given in this paper follow the shema illustrated by the next graph. Abstrat Axiom Example: Assoiativity F F x,y),z)=f x,f y,z)) Abstrat Representation Example: F x,y)=ff x)+f y)) Quantitative Formula Example: F x,y)=x θ +y θ ) θ Meaningfulness Condition Proofs shema: An abstrat axiom yields an abstrat representation. The latter, paired with a meaningfulness ondition leads, via funtional equation arguments, to one or a ouple of potential sientifi laws speified up to the values) of numerial parameters). D. Another example: The Translation Equation for Beer s law The Beer-Lambert law, also known as Beer s law, the Lambert-Beer law, or the Beer-Lambert-Bouguer law is an equation desribing the attenuation of light resulting from the properties of the material through whih the light is traveling. See the figure below.) Inoming light Outgoing light Following the guidelines of the Proof Shema, we first formulate the abstrat axiom. 6 Definition. Let J and J be two non-negative real intervals. A ode F : J J J is translatable, or equivalently, satisfies the translation equation 2 if F F x, y), z) = F x, y + z) x J, y, z, y + z J ). 5) 2 See Azél 966, page 245) for this onept and for the proof of Lemma 7. 5

6 An example of a translatable ode is Beer s Law: Indeed, we have Ix, y) = x e y. 6) IIx, y), z) = Ix, y) e z = x e y e z = x e y+z = Ix, y + z). Next, we need the abstrat representation in this ase. It is formulated in the next lemma. 7 Lemma. Let F : J J H be a ode suh that J =]d, [ for some d R +, and suppose that, for some a R +, and for some b R + or b =, we have J =]a, b] or J =]a, b[ with F x, y) stritly dereasing in y. Then, the ode F : J J H is translatable if and only if there exists a funtion f satisfying the equation F x, y) = ff x) + y). Injeting now the meaningfulness ondition, we obtain our quantitative formula. 8 Theorem. Let F = {F µ,ν µ, ν R ++ } be a, 0)-meaningful ST-olletion of odes, onto with F µ,ν : R ++ R ++ R ++. Suppose that one of these odes, say the ode F µ,ν, is stritly dereasing in the seond variable, translatable, and left homogeneous of degree one, that is: for any a in R ++, we have F µ,ν ax, y) = af µ,ν x, y). Then there is a positive onstant suh that the initial ode F has the form so for any ode F α, F F x, y) = x e y ; F α, x, y) = x e y. We summarize below the proof ontained in Falmagne and Doble see the proof of Theorem 7.4., page 98, 205a). Sketh of proof. We first show that, if one of the odes in the olletion F is translatable, then by the meaningfulness ondition, the translatability ondition propagates to all the odes in the olletion. Without loss of generality, we suppose that the initial ode F = F, is translatable. Suessively, we have for any ode F α, in F: Fα, x, y) F α, F α, x, y), z) = αf, z ) by ST-meaningfulness) α x = αf F α, y ), z ) by ST-meaningfulness) x = αf α, y + z ) by the translatability of F ) = F α, x, y + z) by ST-meaningfulness). So, F α, is translatable. By meaningfulness, we an also show that left homogeneity of degree one propagates to all the odes in the olletion F. We omit this part of the proof.) 6

7 Beause F α, is translatable, Lemma 7 implies that there exists a stritly dereasing funtion f α, : R ++ R ++ suh that F α, ax, y) = f α, f α, ax) + y) = af α, f α, x) + y) = af α,x, y) by left homogeneity of F α,. Set f α, x) = w, and so f α,w) = x. Applying f α, on both sides of the seond equation above, we get or with g a,α, = f α, af α,), f α, af α,)w) + y = f α, af α,)w + y), g a,α, w) + y = g a,α, w + y), a Pexider equation in the variables w and y. So,the funtion g a,α, is of the form g a,α, w) = w + Ba, α, ). for some funtion Ba, α, ) whih must be dereasing in a. Rewriting the last equation in terms of the funtion f α, yields or equivalently, with x = f α, w), we get f α, af α,)w) = w + Ba, α, ) f α, ax) = fα, x) + Ba, α, ), another Pexider equation.f. Azél, 966, page 4) that is, an equation of the form: hax) = hx) + ga). By funtional equations arguments, the equation f α, ax) = fα, x) + Ba, α, ), implies for some onstants kα, ) > 0 and bα, ), f α, x) = kα, ) ln x + bα, ) whih gives us, with t = f α, x), So, we get f α,t) = e t bα,) kα,). F α, x, y) = f α, f α, x) + y) = x y e kα,) after some manipulation. By the left homogeneity of F α, and the ST-meaningfulness of the family F, we must have The last equation shows that Defining = kα,) α F α,αx, y) = F α, x, y) = x e kα,), we finally obtain y kα,) = F x, y). does not depend upon α or. F x, y) = x e y. Aordingly, we obtain for any ode F µ,ν F, using left homogeneity of degree in the seond equation below F µ,ν x, y) = µf x µ, y ν = F x, y ) ν = x e y ν. 7

8 Various other results in the same vein are reported in Falmagne and Doble 205a) see also Falmagne, 205b). The last two lines of the table below summarizes some of these results. The funtional equations results mentioned in the seond abstrat representation) olumn of the table may be found, together with a onsiderable list of other results and extended referenes, in Janos Azél s lassi volume Azél, 966). Name and formula of abstrat axiom Abstrat representation: funtions f, m, g, et. Resulting possible sientifi laws 2 Assoiativity F F x,y),z)=f x,f y,z)) F x,y)=ff x)+f y)) F x,y)=y η +x η ) η Translatability F F x,y),z)=f x,y+z) F x,y)=ff x)+y) F x, y) = xe y Quasi-permutability F Gx,y),z)=F Gx,z),y) F x,y)=mfx)+gy)) F x,y)=x η +λy η +θ) η or F x,y)=φxy γ or x η +y η ) η Bisymmetry F F x,y),f z,w))=f F x,z),f y,w)) F x,y)=f q)f x)+qf y)) F x,y)= q)x η +qy η ) η or F x,y)=x q y q E. Meaningful derivation of relativisti laws The figure below illustrates, in the ase of the Lorentz-FitzGerald Contration, a possible empirial ontext of the equation: LLl, v), w) = Ll, v w). 7) Imagine that an observer is moving very fast along on a straight line, for example in a bullet train. Suppose that this observer measures the length of a rod plaed along a line parallel to the train trak. If the train moves very fast, at a sizable fration of the speed of light, the length of the rod will appear smaller than its atual length. train moving at speed v }{{} l l v Ll, v) the atual length of the rod at rest the speed of the train { } the length of the rod measured from the moving train 8

9 It may not be obvious why Equation 7) is onsistent with situation evoked by the figure. However we will see in Theorem 0 that Equation 7) is equivalent to the formula Ll, v) Ll, v ) Ll, v w) Ll, v w), 8) whih may seem more immediately ongruous with that situation. However, at this stage, we annot assume that taking Equations 7) or 8) as our abstrat axiom and ombining it with meaningfulness will deliver the exat funtional form of the Lorentz-FitzGerald Contration, that is, Equation 2). This is the reason why, in our next definition, we give the pair L, ) the unommitled name of abstrat LFD-pair for Lorentz-FitzGerald-Dopper). 9 Definition. Let L : R ++ [0, [ R ++ be a ode, with > 0 a onstant representing the speed of light. The ode L is a Lorentz-FitzGerald-Doppler Funtion, or a LFD funtion for short, if there is a binary operator : [0, [ [0, [ [0, [ suh that the pair L, ) satisfies the following five onditions:. The funtion L is stritly inreasing in the first variable, stritly dereasing in the seond variable, ontinuous in both variables, and for all l, l R + and v, v [0, ], and for any a > 0, we have 2. Ll, 0) = l for all l R lim v Ll, v) = 0. Ll, v) Ll, v ) Lal, v) Lal, v ). 4. The operation is ontinuous, stritly inreasing in both variables, and has 0 as an identity element. 5. Either Axiom [R] or Axiom [M] below is satisfied for l, l > 0, and v, v, w [0, [: [R] LLl, v), w) = Ll, v w); [M] Ll, v) Ll, v ) Ll, v w) Ll, v w). When these five onditions are satisfied, the pair L, ) is alled an abstrat LFD-pair. In words, Axioms [R] and [M] state the following ideas. Axiom [R]: One iteration of the funtion L involving two veloities v and w has the same effet on the pereived length as adding v and w via the operation. Axiom [M]: Adding a veloity via the operation preserves the order of the funtion L. 0 Theorem. Suppose that L, ) is an abstrat LFD-pair. Then the following equivalenes hold: [R] [DE ] & [AV ] ) [M], with for some stritly inreasing funtion u and some positive onstant ξ: [DE ] Ll, v) = l uv) ξ; +uv)) [AV ] v w = u uv)+uw). + uv)uw) 2 9

10 For a proof, see Falmagne and Doignon, 200). We now have the representation formulas for the abstrat axioms [R] and [M]. The next definition introdues the meaningful olletion with initial pair L, ). Definition. Let L = {L µ,ν µ, ν R ++ } be a ST-meaningful olletion of odes, with L µ,ν : R ++ [0, ν[ onto R ++ and R ++. Let O = { ν ν R ++ } be a 2, )- 2 meaningful olletion of operators, with v ν : [0, ν[ [0, ν[ onto [0, ν[ and v ν w = ν ν w ) v, w [0, ν[ ). ν Suppose that eah ode L µ,ν L is paired with a binary operation ν O, forming an ordered pair L µ,ν, ν ), with the initial ordered pair L,, ) = L, ). Then the pair of olletions L, O) is alled a meaningful LFD-system. Note that the measurement sale of the operation ν is the same as that of the seond variable of the funtion L µ,ν. 2 Remark. There is an equivoation in the definition of the funtion L as defined on the Cartesian produt R + [0, [. In the proof of the next lemma, we have as the first equation l L α, l, v) = αl α, v ) 9) whih is equivalent to L α, αl, v) = αll, v). 0) By definition, the domain of the funtion L in Equation 0) is R + [0, [ with v [0, [. But the domain of L in 9) annot be R + [0, [: we annot have v [0, [. Indeed, this would imply v [0, [ with possibly, if < with < v <, ontraditing v [0, [. This means that, while the two funtions L in 9) and 0) are idential, they don t have the same domain. To deal with this ambiguity, we should define in general the domain of the funtion L by the statement L : R + [0, θ[ : l, θv) Ll, θv), θ R ++ ), whih makes lear that the funtion L is the same in 9) and 0) but with different domains: with θ = in 0) and θ = in 9). 3 Propagation lemma for abstrat LFD-pairs. Suppose that one ordered pair L µ,ν, ν ) from a meaningful LFD-system L, O) is an abstrat LFD-pair, that is, L µ,ν, ν ) satisfies Conditions -5 of the definition of an abstrat LFD-pair. Then any ordered pair L α,, ), with L α, L and O, is also suh an abstrat LFD-pair. So, meaningfulness enables the propagation of all five onditions to all the ordered pairs L α,, ) in a meaningful LFD-system L, O). Sketh of proof. Without loss of generality, we an assume that the ordered pair L, ) of initial ode L is an abstrat LFD-pair. That is, L, ) satisfies the five onditions of Definition 9. Sine by meaningfulness, we have: 0

11 L α, l, v) = αl l α, v ) and v w = v w ), Conditions to 4 follow almost immediately. For example, Condition holds beause, suessively: L α, l, v) L α, l, v ) l αl α, v ) l αl α, v αl a l α, v ) ) αl a l α, v L α, al, v) L α, al, v ) by ST-meaningfulness) by Condition for L, )) by ST-meaningfulness). For Condition 3, we have lim v L α, l, v) = α lim v L l α, v = 0. We turn to Condition 5. Sine Axioms [R] and [M] are equivalent by Theorem 0, it suffies to prove that the ordered pair L α,, ) satisfies Axiom [R]. By the ST-meaningfulness of L, Lα, l, v) L α, L α, l, v), w) = αl, w ) = αl αl l α, v, w. α α Caneling the α s in the fration inside the parentheses in the r.h.s. gives l L α, L α, l, v), w) = αl L α, v ), w ) l = αl α, v w ) by Axiom [R] applied to L l = αl α, ) v w) by the meaningfulness of O) = L α, l, v w) by the ST-meaningfulness of L). 4 Representation Theorem. Suppose that one ordered pair L µ,ν, ν ) from a meaningful LFD-system L, O) is an abstrat LFD-pair, that is, L µ,ν, ν ) satisfies Conditions -5 of Definition 9. Then, Axioms [DE ] and [AV ] of Theorem 0 beome for the initial ode L: v ξ Ll, v) = l for some positive onstant ξ and v [0, [ ) ) + v and for the operation : v w = v + w + vw with v, w [0, [ ). 2) 2 Both of these equations are of ourse well-known in speial relativity. For example, with ξ = 2, Equation ) is related to the red shift phenomenon, and Equation 2) is the standard formula for the relativisti addition of veloities. Proof. Without loss of generality, we an assume that the initial ode of the meaningful LFD-system L, O) is an abstrat LFD-pair. That is, the funtion L satisfies the

12 five onditions of an abstrat LFD-pair. From meaningulness, we have for any ode L α, : l L α, l, v) = αl α, v ) l = α u ξ v α + u v with v [ 0, [) by the representation theorem for abstrat LFD pairs. So, we have l L α, l, v) = αl α, v ) = l u ξ v ) + u v with v [ 0, [). 3) The fat that the last equation requires that 0 v < rather than 0 v < has been disussed in Remark 2. Beause L α, l, v) in the l.h.s. of 3) does not depend upon, whih is just an indiation of the unit) the r.h.s. annot depend upon either. As the ratio ) u v + u v is a funtion of v only independent of, we must have ) u v gv) = + u v for some funtion g : [0, [ [0, [. Setting = z and rearranging, we get gv) uvz) = z + gv) and with hv) = gv) +gv), we get the Pexider equation uvz) = zhv), whose solution for the funtion u is, for some onstant θ: uv) = θv for all v ]0, [. Using the representation [DE ] from Theorem 0, we get Ll, v) = l u v) ξ θv ξ = l. + u v) + θv But the ode L must satisfy Condition 3 of an abstrat LFD-pair Definition 9), whih requires that lim v Ll, v) = 0. This implies lim l v θv ξ = l + θv θ ξ = l + θ θ ξ = 0 whih holds only if θ =. + θ We onlude that the funtion u must be the identity funtion: uv) = v. With the funtion u being the identity funtion, the two equations obtained in Theorem 0 from the representation of abstrat LFD-pairs [DE ] Ll, v) = l uv) ξ; +uv)) [AV ] v w = u uv)+uw) + uv)uw) 2 ). 2

13 beome: v ξ Ll, v) = l + v v w = v + w + vw. 2 I was a bit puzzled by this result, whose importane is debatable. However, it ould ertainly be taken as one more example of the power of meaningfulness in the derivation of sientifi or geometrial formulas. Note that I did obtain a representation theorem for the Lorentz-FitzGerald Equation, whih was using a different kind of meaningfulness onstraints based on the onept of meaningful transformations see Falmagne, 2004). Referenes Azél, J. Letures on Funtional Equations and their Appliations. Aademi Press, New York and San Diego, 966. Paperbak edition, Dover, Falmagne, J.-Cl. Meaningfulness and order invariane: two fundamental priniples for sientifi laws. Foundations of Physis, 9:34 384, Falmagne, J.-Cl. Deriving meaningful sientifi laws from abstrat, gedanken type, axioms: three examples. Aequationes Mathematiae, 89: , 205. Falmagne, J.-Cl. and Doble, C.W. On meaningful sientifi laws. Springer: Berlin, Heidelberg, 205. Falmagne, J.-Cl. and Doignon, J.-P. Axiomati derivation of the Doppler fator and related relativisti laws. Aequationes Mathematiae, 80 ):85 99, 200. Feynman, R.P., Leighton, R.B., and Sands, M. The Feynman letures on physis. Addison- Weisley, Reading, Mass, 963. Narens, L. A general theory of ratio salability with remarks about the measurementtheoreti onept of meaningfulness. Theory and Deision, 3: 70, 98a. Narens, L. Meaningfulness and the Erlanger program of Felix Klein. Mathématiques Informatique et Sienes Humaines, 0:6 72, 988. Narens, L. Theories of Meaningfulness. Lawrene Erlbaum Assoiates, New Jersey and London, Narens, L. Introdution to the Theories of Measurement and Meaningfulness and the Use of Symmetry in Siene. Lawrene Erlbaum Assoiates, Mahwah, New Jersey and London,