Derivation of the Lorentz transformation without the use of Einstein's second postulate Andrei Galiautdinov Department of Physics and Astronomy,University of Georgia,Athens,GA 30602,USA (Dated:January 3,2017) Derivation of the Lorentz transformation without the use of Einstein's Second Postulate is pro- vided along the lines of Ignatowsky,Terletskii,and others.This is a write-up of the lecture first delivered in PHYS 4202 E&M class during the Spring semester of 2014 at the University of Georgia The main motivation for pursuing this approach was to develop a better understanding of why the faster-than-light neutrino controversy (OPERA experiment,2011)was much ado about nothing. Special relativity as a theory of space and time Capsule All physical phenomena take place in space and time. The theory of space and time(in the absence of gravity) is called the Special Theory of Relativity. We do not get bogged down with the philosophical Floating Ball problems related to the concepts of space and time.We simply acknowledge the fact that in physics the notions of space and time are regarded as basic and cannot be reduced to something more elementary or fundamental. We therefore stick to pragmatic operational definitions: Touch-sensitive Time is what clocks measure.Space is what measuring surface rods measure. n FIG.1:(Color online.)A floating-ball inertial detector.After In order to study and make conclusions about the prop- [ erties of space and time we need an observer.A natural choice is an observer who moves freely (the one who is free from any external influences).An observer is not a Definition of clock synchronization single person sitting at the origin of a rectangular coor- dinate grid.Rather,it is a bunch of friends(call it Team K)equipped with identical clocks distributed through- It is pretty clear how to measure distances:the team out the grid who record the events happening at their simply uses its rectangular grid of rods. respective locations. It is also clear how to measure time intervals at a par- ticular location:the team member situated at that loca- How do we know that this bunch of friends is free tion simply looks at his respective clock.What's not so from any external influences?We look around and make clear,however,is how the team measures time intervals sure that nothing is pulling or pushing on any member between events that are spacially separated. of the bunch;no strings,no springs,no ropes are at- A confusion about measuring this kind of time inter- tached to them.An even better way is to use a collection vals was going on for two hundred years or so,until one of"floating-ball detectors"(Fig.1)distributed through- day Einstein said:"We need the notion of synchronized out the grid [1].When detector balls are released,they clocks!Clock synchronization must be operationally de- should remain at rest inside their respective capsules.If fined." any ball touches the touch-sensitive surface of the cap- The idea that clock synchronization and,consequently, sule,the frame is not inertial. the notion of simultaneity of spacially separated events, has to be defined (and not assumed apriori)is the single In the reference frame associated with a freely moving most important idea of Einstein's,the heart of special observer (our rectangular coordinate grid),Galileo's Law relativity.Einstein proposed to use light pulses.The of Inertia is satisfied:a point mass,itself free from any procedure then went like this: external influences,moves with constant velocity.To be In frame K,consider two identical clocks equipped able to say what "constant velocity"really means,and with light detectors,sitting some distance apart,at A thus to verify the law of inertia,we need to be able to and B.Consider another clock equipped with a light measure distances and time intervals between events hap- emitter at location C which is half way between A and pening at different grid locations. B(we can verify that C is indeed half way between A andDerivation of the Lorentz transformation without the use of Einstein’s second postulate Andrei Galiautdinov Department of Physics and Astronomy, University of Georgia, Athens, GA 30602, USA (Dated: January 3, 2017) Derivation of the Lorentz transformation without the use of Einstein’s Second Postulate is pro￾vided along the lines of Ignatowsky, Terletskii, and others. This is a write-up of the lecture first delivered in PHYS 4202 E&M class during the Spring semester of 2014 at the University of Georgia. The main motivation for pursuing this approach was to develop a better understanding of why the faster-than-light neutrino controversy (OPERA experiment, 2011) was much ado about nothing. Special relativity as a theory of space and time All physical phenomena take place in space and time. The theory of space and time (in the absence of gravity) is called the Special Theory of Relativity. We do not get bogged down with the philosophical problems related to the concepts of space and time. We simply acknowledge the fact that in physics the notions of space and time are regarded as basic and cannot be reduced to something more elementary or fundamental. We therefore stick to pragmatic operational definitions: Time is what clocks measure. Space is what measuring rods measure. In order to study and make conclusions about the prop￾erties of space and time we need an observer. A natural choice is an observer who moves freely (the one who is free from any external influences). An observer is not a single person sitting at the origin of a rectangular coor￾dinate grid. Rather, it is a bunch of friends (call it Team K) equipped with identical clocks distributed through￾out the grid who record the events happening at their respective locations. How do we know that this bunch of friends is free from any external influences? We look around and make sure that nothing is pulling or pushing on any member of the bunch; no strings, no springs, no ropes are at￾tached to them. An even better way is to use a collection of “floating-ball detectors” (Fig. 1) distributed through￾out the grid [1]. When detector balls are released, they should remain at rest inside their respective capsules. If any ball touches the touch-sensitive surface of the cap￾sule, the frame is not inertial. In the reference frame associated with a freely moving observer (our rectangular coordinate grid), Galileo’s Law of Inertia is satisfied: a point mass, itself free from any external influences, moves with constant velocity. To be able to say what “constant velocity” really means, and thus to verify the law of inertia, we need to be able to measure distances and time intervals between events hap￾pening at different grid locations. Floating Ball Capsule Touch-sensitive surface FIG. 1: (Color online.) A floating-ball inertial detector. After [1]. Definition of clock synchronization It is pretty clear how to measure distances: the team simply uses its rectangular grid of rods. It is also clear how to measure time intervals at a par￾ticular location: the team member situated at that loca￾tion simply looks at his respective clock. What’s not so clear, however, is how the team measures time intervals between events that are spacially separated. A confusion about measuring this kind of time inter￾vals was going on for two hundred years or so, until one day Einstein said: “We need the notion of synchronized clocks! Clock synchronization must be operationally de- fined.” The idea that clock synchronization and, consequently, the notion of simultaneity of spacially separated events, has to be defined (and not assumed apriori) is the single most important idea of Einstein’s, the heart of special relativity. Einstein proposed to use light pulses. The procedure then went like this: In frame K, consider two identical clocks equipped with light detectors, sitting some distance apart, at A and B. Consider another clock equipped with a light emitter at location C which is half way between A and B (we can verify that C is indeed half way between A and arXiv:1701.00270v1 [physics.class-ph] 1 Jan 2017