6.3 Quantity equations
6.3.3 Four-momentum formulation
By a formal rescaling of the physical energy values 1
c2 ·E(i) and substitution v(AB)=−v(BA)
the induced relation between Alice and Bob’s rescaled energy-momentum values coincides with the familiar Lorentz transformation.
The Lorentz transformation from a proper time interval of object O t
Corollary 9 Alice rescaled total energy-momentum values for free body m v 1
Though after substitution the new values do not refer to the directly countable energy sources S
v=0 and impulse standards 1 vS and the dimensions are not physically connected by the vivid reference process S
0(A),1 0(A),1 0(A) wS⇒ (A) 1−vS(A),1 vS(A). Despite the appeal of a simpler calculus, without the operationalization thephysical interpretation is concealed.
19One can argue whether to regard a ”photon” (propagation of electromagnetic interactions) as particle since one can only determine its potential energy-momentum (Feynman attributes the latter strictly to the emitting material source [50]) upon absorption, i.e. after its destruction in the absorbing body.
Chapter 7 Discussion
One usually explains physics axiomatically. One derives a formalism from a manageable system of initial propositions which are logically independent from one another. That is good for purposes like formal mathematical elegance, if one wants to teach a concise formalism [34]. Though this approach already begins in the abstract. It obscures the physical meaning and vivid understanding (Anschaulichkeit). Students find it hard to relate the mathematical structure to the physical phenomena.
The axiomatic explanation of mechanics assumes mathematical terms (mass m, force F, impulsep) as known elements of its description, postulates properties (m = const,F/modv) and basic equations (F = m· a, p = m · v). The mathematical theory defines derived quantities in terms of basic quantities and proves the equations from a system of postulated basic propositions. This logical mathematical reasoning though starts from undefined basic elements and from unproven postulates. Helmholtz [7] demands a further justification and derivation. He does not accept the axioms (of geometry, arithmetics) as unprovable and not proof requiring propositions. After all physics is a measuring natural science and as such distinguished from mathematics.1
One can also look into what actually happens in measurement practice. The main objec-tive for Mach’s account of the historic evolution of mechanical principles [10] was ”to dispel metaphysics out of physics... because we are used to call those notions metaphysical, from which we have forgotten how we arrived at them.”2 It was also motivated pedagogically [9]:
1Ruben stresses ”A pure mathematical relation (alone) says nothing about a particular sensual concrete domain, thus has nomechanical meaning at all. The latter comes only about if symbols (for measures and operations), which are connected in a mathematical relation, symbolize properties which are tangible in the experimental conduct” [22]. Aside from developing the quantitative notions one must also specify the physical and methodical prerequisites.
2According to H¨orz, Wollgast [7] explaining known laws in scientific theories also requires the description of methods by means of which they were found. New concepts are not given naturally. The development is always connected with particular operations, which determine the actual meaning. The measurement theoretical foundation of physics plays a big role for the ongoing relativization of absolute physical notions even today (e.g. Einstein’s revision of the concept of simultaneity). To prevent the continuation of prejudices we always connect the conceptual critique with the inspection of operations which lead to the concepts.
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”An insight is grasped best by the learner through the same way along which it was found.”
The primary requirement in education, Plank [16] explains, ”is not so much the amount of material but rather the way of its treatment... A single mathematical proposition, which is truly understood by students, has more value for them than ten memorized formulas...
School is not supposed to convey technical routines but consequent methodical thinking.”
7.1 Basic measurement operations
Our novel approach to the foundation of physics starts from the underlying measurement practice. We are long familiar with basic measurements, as in a length measurement by placing copies of a ruler side by side. Wallot [19] defines ”physical measures are never tangible things, but always attributes of things, properties, which we can notice on the things of our experience”. Helmholtz regards attributes of objects, which in a comparison allow the difference of larger, equal or smaller. The tangible procedure for finding ”how many times” more is the measurement.
Helmholtz [7] starts from counting same objects. He begins with basic questions:
1. ”What is the physical meaning if we declare two objects asequal in a certain relation?”
2. ”Which character must the physical concatenation of two objects have, that we may consider comparable attributes thereof as connectedadditively?”
In this way Helmholtz elucidates familiar examples like the weight mO of a materialObject, the length sAB of a straight line AB, the duration tP of a physical Process etc. Thereby we notice, that the way of concatenation generally depends on the kind of measure. ”We add e.g. weights, by simply placing them on the same weighing-pan. We add time periods, by letting the second begin at exactly the moment, where the first stops; we add lengths, by placing them next to each other in a certain way, namely in a straight line etc.” For a basic measurement we specify a method of comparison for the energy (of directly observable phenomena) and a method for their physical concatenation.
In everyday work experience one becomes aware of what is meant by ”length”, ”duration”,
”capability to work” (Wirkungsverm¨ogen) and ”impact (in a collision)” (Wucht). We use the familiar notions for the relative motion of nearby rigid bodies and the behavior in work actions to prevent a premature anticipation of the mathematical formalism. Our foundation is circularity free. We neither presuppose equations of motion nor formal proportionalities, conservation laws, symmetries etc. Every new basic measure has to be explained in words or by examples because definition-equations for basic quantities do not exist [19].3 We define the basic observables from practical comparison ”longer than”, ”heavier than”, ”more impact than” etc. Without a word of mathematics one can assess
• >l if two extended objects lie on top of each other: one will cover the other
3One defines derived quantities by equations. Helmholtz [7] calls them more accurately coefficients.
”Basic quantities cannot be deduced by equations onto other already explained quantities.”