aa r X i v : . [ phy s i c s . g e n - ph ] F e b Lorentz Transformation Equations in Galilean Form
Sadanand D. AgasheDepartment of Electrical EngineeringIndian Institute of Technology Bombay, Powai,Mumbai-76India - 400076email: [email protected]
Abstract
Using the definition of “position” given in an earlier paper, we show that the Lorentz trans-formation equations for position can be put in a particularly simple form which could be said tobe “Galilean”. We emphasize that two different reference frames use their individual definitionof position and distance. This fact gets obscured in the usual “rectangular Cartesian co-ordinatesystem” approach.
Einstein, in his pioneering paper , insisted that when talking about motion, we must give a physical meaning to “time”. He showed how this can be done by introducing his idea of “synchronizedclocks”. However, he took for granted the concept of “position”. He wrote:Let us take a system of co-ordinates in which the equations of Newtonian mechanicshold good. . . . If a material point is at rest relatively to this system of co-ordinates,its position can be defined relatively thereto by the employment of rigid standards ofmeasurement and the methods of Euclidean geometry, and can be expressed in Cartesianco-ordinates.Thus, he did not describe how position could be defined in a general, physically meaningful way.Nobody subsequently has done so. A little further in , he said:Let us in “stationary” space take two systems of co-ordinates, i.e.,two systems, each ofthree rigid material lines, perpendicular to one another and issuing from a point.1re the co-ordinate axes, then, material bodies? Do we need to think of what might happen tothem when they move? Or are they merely conceptual? In a sense, then, Einstein’s Kinematicswas incomplete. In a recent paper , we suggested how Einstein’s Kinematics could be completed. We proposedthat an “observation system” could have the following ingredients. An observer, S , say, capable ofsending and receiving light signals, is equipped with a single clock and three passive “reflecting”stations, say, S , S , S . Using a radar-like approach, the observer could obtain data sets as follows.He sends a signal in all directions at a time t in his clock, and then records the times of arrivalsof the echoes of this signal by reflection in the following four different ways: time t ′ of arrivalafter reflection at the place P, say, of an event being observed (path SP S ); time t of arrival afterreflection at the event P first, followed by a reflection at the station S S (path SP S S );similarly, time instants t , t . Thus, he would obtain 5-tuples of data items of time, t , t ′ , t , t , t .From each 5-tuple, he is to decide by definition what could be meaningfully called the “time ofoccurrence” and “place” of the event. Following Einstein, we chose t = t + t ′ as the definition ofthe time of occurrence.There remained the problem of deciding what could meaningfully be called the place of theevent. Here, we proposed to go beyond the classical 3-dimensional rectangular Cartesian co-ordinatesystem idea which was only conceptual; we chose instead to think of the place of an event as anelement of a 3-dimensional vector space to be equipped with a suitable scalar or inner product. Theplace of an event could thus be thought of as a “position vector”. Any 3-dimensional vector space, V ,say, would do. (Today, we know the possible advantages of such “abstract” representation .) Sincethe reflecting stations deserved “places” of their own, it was natural to represent them by vectors s , s , s , say, forming a basis of V . Again, any basis would do. Of course, S itself would be assignedthe zero vector. Now, the reflecting stations could not be allowed to be totally arbitrary; they hadto remain at fixed “distances” from S and from one another. But what are distances? These hadto be physically determinable. S has only a clock- no measuring rods. As in radar, one coulddefine “distance” in terms of time interval through a parameter called the velocity of light, c . Bydoing some further signalling, involving various reflections, S could obtain a set of 6 time intervalsthat would correspond to 6 transition times, SS , SS , SS , S S , S S , S S . These multiplied by c would be taken as the lengths or “norms” of the 6 vectors s , s , s , s − s , s − s , s − s . These6 numbers would then uniquely determine the scalar or inner product on V , thus making it into2n inner product space. Note that although V and a basis for it were arbitrarily chosen, the scalarproduct was determined by the observation system itself. The problem of obtaining from the 5-tuple of an event a representing vector p , say, in V was then a problem of linear algebra (see fordetails, with a slightly different notation). There was a “technical” hitch, however. Not any set of6 numbers would do; this was explored in an “addendum” .The stage was set to admit another observation system, an observer S ′ , say, with a clock andreflecting stations S ′ , S ′ , S ′ . This observer could choose a vector space, V ′ , say, not necessarilythe same as V of S , basis vectors s ′ , s ′ , s ′ to represent its stations, and using the same “velocityof light” constant c , determine a scalar product on it, and finally obtain the representing vector p ′ , say, and time t ′ , say, of the same event for which S had obtained p as position vector and t as time. To relate p, t with p ′ , t ′ , one assumed, as usual, that the system S ′ was in uniformmotion relative to system S with velocity v . This motion would be observed by S , and thus, v would be a vector in V . S would observe the motions of S ′ , S ′ , S ′ , S ′ to be given by the vectors d + tv, d + tv + d , d + tv + d , d + tv + d , say, d , d , d , d being all vectors in V . To go further,one needed some relation between the clocks of S and S ′ in the following sense. Suppose that as S ′ moves, the clocks of S and S ′ at S ′ show values t and t ′ , respectively. We need some relationbetween these two “times”. We assume, with Einstein, linearity of this relation: t ′ = β t , where β is some constant. This is the only assumption of linearity that we make. We then prove(see ) thatthe following linear relations hold between the times and places of the events in the two systems: t ′ = β (cid:20) t − ( p − d − tv, v ) S c − v (cid:21) (1)where the symbol ( u, w ) S denotes the scalar product of the vectors u and w in V , and p ′ = T ( p − ( d + tv )) (2)where T is a linear transformation on V onto V ′ such that it maps each vector d i of V to the vector s ′ i of V ′ , i.e., the vectors in V representing the relative positions of the stations of S ′ are mappedto the vectors in V ′ representing the stations of S ′ . This completes a summary of our derivationof the Lorentz transformation in . We call it “Einstein’s Lorentz transformation” because we havefollowed an Einsteinian approach - except with respect to the meaning of “position”. Although we allowed the possibility that the representation vector spaces V and V ′ could bedifferent, they could be chosen to be the same. Further, the vectors d i representing the relative3ositions in V of the stations of S ′ could be chosen to be the basis vectors for S ′ . Thus, we couldchoose s ′ i = d i . Of course, the scalar products could be different, as they are dictated by theobservational data. The transformation T then becomes the identity transformation, and for thevectors representing the place of the event in S and S ′ , we obtain the following simple Galilean relation: p ′ = p − ( d + tv ) (3)which can be seen as the Galilean position vector of P relative to S ′ . It must be emphasized,however, that there are still two representations because there are possibly two different scalarproducts for S and S ′ , and these scalar products relate to two different calculations of distancesin the two systems. Both are Euclidean in the sense they are both based on a scalar product.The normal “Euclidean” co-ordinate systems use the distance p x + y + z , which is related to aspecial scalar product.Let us consider in this context the Einstein form of the Lorentz equations: x ′ = β ( x − vt ) , y ′ = y, z ′ = z, t ′ = β ( t − vxc ) . (4)One could argue that one could put them in the Galilean form by using new variables ¯ x, ¯ y, ¯ z :¯ x = x − vt, ¯ y = y, ¯ z = z (5)and explicitly defining a new distance for S ′ given in terms of norm by || ( p, q, r ) || S ′ = β p + q + r , (6)choosing the constant β equal to 1 /β , but this would have looked like a mathematical “trick”. Incontrast, here we have envisaged the possibility of a different scalar product and distance in thevery notion of representation by a vector. The Galilean form for the position vector has the advantage that the Einsteinian factor β does notappear in it, making manipulations easier. Also, length contraction and time dilatation disappear.However, β does remain in a slightly different appearance in the equation relating the times. Theadditional factor β could be chosen as unity. The main point of this paper is, however, that theconcept of “position” has to be defined and that there is a choice in representing position. Thepresent paper could be read as a postscript to the earlier papers and .It would be interesting to apply the coordinate-free vector representation of position to Maxwell’sequations and to see how the Galilean form of the Lorentz transformation works out.4 EFERENCES [1] A. Einstein, “On the Electrodynamics of Moving Bodies,” pp. 37-65, in H. A. Lorentz, A.Einstein, H. Minkowski, and H. Weyl,
The principle of relativity: a collection of originalmemoirs on the special and general theory of relativity, with notes by A. Sommerfeld, translatedby W. Perrett and G. B. Jeffery , (Methuen, London, 1923); reprinted (Dover, New York, 1952).[2] S. D. Agashe, “Einstein’s “Zur Electrodynamik...” (1905) revisited, with some consequences,”
Found. Phys. , 955-1011 (2006).[3] S. D. Agashe, “Addendum to “Einstein’s “Zur Elektrodynamik . . . ” (1905) revisited, withsome consequences” by S. D. Agashe,” Found. Phys.37