The Lorenz Transform

Transcription

1 The Lorenz Transform Flameno Chuk Keyser Part I The Einstein/Bergmann deriation of the Lorentz Transform I follow the deriation of the Lorentz Transform, following Peter S Bergmann in Introdution to the Theory of Relatiity After an introdutory explanation inoling obserers, trains, rods and loks (in Cartesian oordinates), and with the assumption that the speed of light is a onstant, and defines inertial motion relatie to, the atual analysis (pp. 38ff) begins with the hypothesis that the laws of physis are the same in eery inertial frame. Howeer, I interpret some of Bergmann s haraterizations a bit differently. The onstany of the speed of light implies there is an absolute oordinate frame in whih is defined as x = t and x = t x ' = t ' and x = = t x' t' (-x,t ) (-x,t ) (x,t ) (x,t ) Spae (-x,-t ) (-x,-t ) (x,-t ) (x,-t ) Time In this diagram, time moes up ertially, and the position oordinates of a photon modeled as a oordinate partile moe right or left as the time axis rosses their red dotted world lines. Sine the

2 photons do not interat at the point (0,0) shown, they ontinue their straight world lines after they oinide. For a pure oordinate system, it is immaterial when the oordinate photons are reated or destroyed as long as their world lines exist at their ommon point (at that point, they are simultaneous, at any other instant of time, they are separated. x The eloity of light is then defined as = sine is assumed to be isotropi (independent of t diretion) and homogeneous. If a medium is assumed, then the eloity of the medium is then assumed x to be defined as = along some axis passing through the origin at whih is defined t A Mihaelson-Morley apparatus onsists of two orthogonal and equal arms whih were rotated to test the effet of a motion in some diretion with respet to in terms of an interferene pattern, whih would hange along the arms as the system was rotated. Sine no effet was obsered, Lorentz hypothesized that the arms of the apparatus had atually hanged to ompensate for the interation of the medium with the apparatus. Lorentz assumed that the atual length of the arm in the diretion of eloity would be hanged by a fator linear α that would ary as the system was rotated, ausing a ariation in spae and time by interation with the medium. x ' = α( x t), y ' = y = 0, z ' = z = 0 (sine y is orthogonal to this axis, and z is irreleant) In partiular, for = 0 (an orientation orthogonal to the relatie eloity of the medium), x' = α x, = 0. For the orientation in line with the relatie eloity of the medium, x = t, so that x = 0, with α to be determined. In the diretion orthogonal to the position haraterized by α he then assumed that the measurement of time would be affeted by a linear fator of both spae and time, and replaing y with x : t' = γx+ βt so that x ' = t ' = ( γx + βt) Equating the two haraterizations of x, we hae the relation: α( x t) = ( γx + βt) Sine these results apply to arms that are orthogonal, there are no interating terms, and the apparatus is desribed by the lengths of the rotating arms in a medium where is homogeneous and isotropi, and

3 the arm length hange as the system is rotated: The area of the irle an then be alulated by the relation: πα ( x t) = π ( γ x βt) Eliminating π and expanding and rearranging results in the expression: ( β α ) = ( α γ ) + ( α + βγ ) t x xt

4 Equating the oeffiients of x and t ( is a onstant independent of area where s x = x = xx and s t = t = tt),and setting the oeffiient of xt equal to 0 (no interation between spae and time i.e., x and t are orthogonal; this is equialent to using dot and ross produts in the integral form of Maxwell s equations, and the elimination of the salar field to form the EM field tensor in Quantum Eletrodynamis) results in the expressions: 1: β α = : α γ = 1 3: α + βγ = 0 Equating α from 1,: (1 β ) 1: α = : α = 1+ γ (1 β ) = 1+ γ β 1 (1 ) 4: γ = 1 Equating α from (,3) : α = 1+ γ βγ 3: α = βγ,3: 1+ γ = From 1,3 5: γ = (1 β ) ( ) (1 β ) β γ = β Squaring 5 and equating with 4:

5 (1 β ) 1 (1 β ) = 1 β After a bit of algebra, this results in the expression: β = 1 Substituting in 5: 1 (1 ) (1 β ) 1 (1 β ) 1 5: γ = = β β = β 1 = = = 1 1 ( + )( ) ( + )( ) γ = = = = = β β 1 ( + )( ) ( + )( ) = = = = ( + )( ) ( + )( ) ( + )( ) ( + )( ) 1 β = = =

6 β β That is, γ = = β Substituting in Equation 3: 3: α βγ β β = = = β These expressions for α, β andγ are then substituted into the original equations x ' = α( x t) = β( x t) β x t' = γx+ βt = x+ βt = β t β = 1 = We hae the Lorentz transforms (from Bergmann): x ' = ( x t) t ' = x t

7 Important Note: Bergmann s parameters γ and β are reersed from that of onentional notation in my experiene, and in the following analyses the notation will be reersed as well. The result is the Lorentz transform: x ' = ( x t) γ z' = z ( = 0) x t' = ( t ) γ 1 with γ =, β = β (where t has replaed y in a spatial oneption of the model) The transform in (x,t) is: x ' = ( x t) γ x t' = ( t ) γ 1 with γ =, β = β The Time Dilation Equation Note that for x = t, x t' = ( t ) γ x x ' = t ' = ( t ) γ = ( t t) γ = ( x t) γ That is, from a purely algebrai perspetie, under the ondition x=t, the time and spae transforms are idential. This is beause has not been speified for both positie and negatie alues ( parity ), whih means we must also onsider the relation x ' = ( x + t) γ Adding these two together gies: x = xγ so that x = xγ For x = t and x =t, this gies the so-alled time dilation equation (notie that x has been expliitly eliminated by the relations t = x/ and t =x /:

8 t t' = = tγ Time an then be haraterized as a then a saling fator that desribes a eloity in relation to a eloity for a gien length or radius that relates to, with all parameters independent of whether a Galilean or a Radial oordinate system is seleted. I indiate this by using apital letters for these parameters, so that: T / T = 1 V C Time is then said to be oariant with eloity for either oordinate system, sine it transforms in diret proportion to eloity.

9 Part II - The Coariant formulation of STR The relatiisti Equations of Speial Theory of Relatiity an be deried independently of a oordinate system, where C and V are haraterized as mass reation rates and T and T are onsidered as reation times, with a hange in the total mass of the (single) system from M 0 =CT to M =CT is haraterized by a V perturbation VT where V is defined in terms of C by the relation β =. C We assume that C is deried from Coulomb s and Ampere s laws by Maxell s equations, so that 1 C = εµ 0 0 If perturbation V is independent of C, so that the system mass before and after the perturbation an be haraterized by ( CT ) ( VT ) ( CT ) = + or ( M ) = ( VT ) + ( M ) = ( M ) + ( M ) 0 0 V (M V is the mass added by during the perturbation T at the reation rate V) ( M ) = ( VT ) + ( M ) C C C ( ) C ( ) C ( ' ) 4 VT = VCT = M V C = P C, where P= MV ' is the relatiisti momentum. Then the relatiisti energy equation is ( MC ) = PC + ( M0C ) E' = PC + E 0 The ratio of mass reation times is gien by; T ' 1 =, that is T ' = 0 T 0 V V C T C

10 The following diagrams show these relationships: M 0 =CT M =CT E 0 =M 0 C E =M C M V = VT PC=(M V)C=h I hae inluded h as a preursor to relatiisti quantum mehanis. Here the interpretation is that if there is a hange of state (from E to E 0 ) in a distant soure emitting an photon of energy h that Is deteted loally, and there is a hange in the sensor of the same amount, then nothing has interated with it along its path. Howeer, if there is a hange due to interation, the obsered energy of the photon will derease, whih is obsered as a hange in h, and is responsible for the red shift. Note that Doppler (first or seond order) is not an issue here, sine the oherene length of the photon is tiny ompared to the length of the journey.

11 Part III Co-Variane and Contra-Variane We hae seen that the Lorentz transform an be interpreted as a time dilation equation in a spaetime oordinate system (x,t) with the assumption of the onstant eloity of light if both positie and negatie eloities are inluded in the analysis (whih satisfies the ondition of isotropy of both and for physial laws), where all other eloities are determined by the relation /. We hae also seen that the same equation an be deried independently of a oordinate system in the Energy-Momentum system (E,K) for a single partile (system) with an initial M 0 perturbed by an additional mass M V to a final mass M, with the transition desribed by saling fators (T,T ) The two domains (x,t) and (E,K) are independent of eah other; the Lorentz transformation was aomplished without referene to mass, and the Mass-Energy relationships were deried without referene to a oordinate system. If we set the ratios / = V/C with =C and = V we an then relate the (E,K) domain to the (x,t) domain by breaking and into their spae-time omponents. The ratio of eloities in Galilean oordinates is: lengths and time interals are defined. x t x t = = x x t t, where indiates that disrete In the relatiisti (E,K) domain, the mass reation rates V and C are related by the saling fators T and T by: V T0 = C T ' T' T, 0 x t V T0 CT0 M = = = = = x t C T ' CT ' M ' For a hange in /, the relation =C an be presered for ariations in energy proided that one of the parameters C and hae a oordinate in ommon. That means there are two possible solutions; that in whih a ommon spatial interal is assumed, and that in whih a ommon interal of time is assumed.

12 Case 1: x = x (o-ariant transformation) In this ase, we hae t M0 = = t M ', so that t = t M 0 M ' For M = M 0, we hae that t = t As M inreases along with V, t also inreases. Sine M in (E,K) is diretly proportional to t in the oordinate system, we say that the transformation is o-ariant. (We hae ignored parity, sine we assume equal alues set at the midpoint by squaring the parameters in the solution)., The interpretation of oariane is that an inertial objet (for < ) will take longer to traerse a gien distane than a photon, with an inreased total mass determined by the ratio M 0 /M. Here the fator 1 1 M ( ) 0 M ' is interpreted as an inreased density of the system as a whole for V/C.

13 Case : t = t (ontra-ariant transformation) x V M0 In this ase, we hae = = = x C M ', so that M 0 x = x M ' For M ' = M0, we hae x = 0, whih means that no additional mass has been added to the rest mass. As M ' inreases, x dereases, meaning that an inertial objet for < will not trael as far in M 0 a gien time as a photon. Again, the fator an be interpreted as a density, whih M ' inreases as inreasing mass (relatie to that of a photon); howeer, sine the result is inersely proportional to V/C, the transformation is ontra-ariant with respet to the oordinate system.

14 The Cartesian s. Radial oordinates At this point in the analysis, it is eident that the ontext is that of an area; either that of a square where eah element on the left and right side are interpreted as its sides, or as a irle, in whih eah side is interpreted as a radius, and: Area Square = Area Cirle = s = α ( x t) = ( γx βt) r = πα ( x t) = π[ ( γ x βt)] The two are, of ourse, related by π, sine t r = α( x t) = ( γx βt) -x x -t

15 The analysis of the Lorentz transform in Cartesian oordinates iolates the assumption of the homogeneity of spae and time, sine the distane from the origin at (0,0) to the perimeter of the square is only equal to that of the irle at the horizontal and ertial axes of the diagram aboe (t = t = 0 or x = x = 0). This means that is only independent of at these points. Howeer, if is a onstant, then must be independent of for all alues of, whih an only be true if the relation x ' = α( x t), y ' = y, z ' = z is taken to be that of a radius, in whih ase is orthogonal to as a tangent to the irumferene. r ' = α( r t), θ' = θ = π, ϕ' = ϕ = π t' = γr βt and r = t r ' = t ' The same analysis fan be performed, with the result that r = t = ( r t ) Also, note that is no longer defined by the relation = x/ t but rather by the relation: ( t ) ( t ') t r = 1 1 1, t' t t' = = r' If = 0 the denominator (or equialently, positie and negatie radii are inluded), then r r = and r is o-ariant with eloity, as are mass and energy. Howeer the density at a point on the irumferene is ontra-ariant with r.