ABSTRACT
Ψ∂ (4.1) The harmonic trial solution of this equation with respect to a fixed inertial frame ( is given by
)tx, ( ) ( ) ( )ctxikectxktx −Ψ=Ψ−Ψ=Ψ 00 orcos, In another inertial frame( ),, tx ′′ considered to be in uniform translational motion with velocity as described by the Galilean transformation (2.13), with respect to the fixed frame, the wave is represented by a wave function
,v ( ) ( ), related to ,x t x′ ′Ψ Ψ t as
2 2i.e., x t
x x x t x x x x ′ ′∂Ψ ∂Ψ ∂ ∂Ψ ∂ ∂Ψ ∂ Ψ ∂ Ψ= + = =′ ′ ′ ′∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
and
2 2i.e., 2 x t v v v
t x t t t x t x tt x ′ ′∂Ψ ∂Ψ ∂ ∂Ψ ∂ ∂Ψ ∂Ψ ∂ Ψ ∂ Ψ ∂ Ψ ∂ Ψ= + = − + = − +′ ′ ′ ′ ′ ′ ′∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
2t∂ ∂ ∂ where use has been made of known properties of partial derivatives. It follows that
obeys the following wave equation, with respect to the moving inertial frame ( tx ′′Ψ , )
( ) tx
v tx
vc ′∂′∂ Ψ∂−′∂
Ψ∂=′∂ Ψ∂−
2 22 2 (4.2)
which is obviously different from the wave equation (4.1) that holds in the fixed inertial frame. In other words the electromagnetic wave equation is not invariant with respect to a uniform translation of the inertial frame, using a Galilean transformation. Inserting into Eq.(4.2) harmonic solution written as
we obtain
2 2 2 2 2 2( ) ( ) 2 ( ) i.e., orc v k k w v k w w v c w c− − = − − = − ± = ± v (4.3) Hence Galileo's transformation predicts that the velocity of light should be different from c, if measured with respect to a moving inertial frame. We therefore have a choice to make before we can fully understand the electromagnetic phenomena. We can either accept Galileo's principle of relativity which states that all inertial frames are equivalent or the Galilean transformation (2.13). We can sacrifice Galileo's principle of relativity by saying that there is a privileged inertial frame, called the ether, in which it is supposed that the electromagnetic wave velocity is c as given by Maxwell's equations,
although they made no essential reference to the ether. A long series of investigations has been carried out, in order to detect some effect of the motion of the earth through the ether, always with negative results. All of the experiments (the best known being that of Michelson-Morley) have indicated that the speed of light is always the same in all directions and is independent of the relative motion of observer, source and transmitting medium. Einstein has reaffirmed Galileo's principle of relativity on the equivalence of all the inertial frames and, in addition, has assumed as a basic postulate the experimental fact that the speed of light is always constant. As a consequence, the Galilean transformation had to be replaced by another, in order to preserve the same speed of light in all systems. Einstein's basic assumptions are known as the postulates of special relativity:
1. The laws of physics take the same form in all inertial systems 2. The speed of light in empty space is a universal constant c and is the
same for all observers. It is independent of the state of motion of the emitting body
Special relativity is only concerned with transformations between inertial frames, as indicated by the restriction contained in the first postulate. General relativity covers reference frames in arbitrary relative motion, and is based on the principle of equivalence, confirmed experimentally, which unifies gravitational and inertial forces. The first of Einstein's postulates implies that it is only possible, by means of physical measurements, to demonstrate that two gravity-free reference frames are moving relative to each other. There is no physical basis to infer that any such frame is intrinsically at rest or in uniform motion. The notion of absolute rest is not consistent with the first postulate. The second postulate is in contradiction with the Galilean transformation which would preserve the form of Newton's laws of motion, as illustrated by Eq.(4.3). Therefore, we must find a transformation between two frames moving with uniform relative motion, which is consistent with the second postulate. We must then generalize the laws of Newtonian mechanics in a form which is invariant under that transformation, as required by the first postulate. A fundamental requirement of special relativity is that the laws of Newtonian mechanics are valid in the low speed limit and must be obtained as limiting cases of relativistic formulae for .cv << This assumption plays, for special relativity, a role analogous to that of Bohr's correspondence principle in quantum mechanics (see Chapter 28). 4.2. THE LORENTZ TRANSFORMATION Einstein's procedure is to locate an event in a reference frame by its spatial Cartesian coordinates x, y and z and a temporal coordinate t. Consider two frames S and S' moving with uniform relative motion, such that the Cartesian axis system in
coincides with that in S, Oxyz, at time ,S ′
zyxO ′′′′ .0==′ tt As illustrated in Figure 4.1, the
x and axes are parallel to the direction of relative motion: in moves in the x direction with velocity v, and in
x′ , the frameS ′S S, the frameS ′ moves with velocity in the
direction. A pulse of light emitted from O at time v−
x′ 0=t spreads as a spherical wave travelling with speed c, so that the equation of the wave front in S, at a time t will be (4.4) 22222 tczyx =++
0 0'
z'
y'
x,x'
z
y
Figure 4.1. Cartesian axes of two inertial frames in uniform relative motion The second postulate requires that the wave front in S ′ can be represented by (4.5) 22222 tczyx ′=′+′+′ In other words an observer in also sees a spherical wave travelling from its origin if not, any change of the wave front could be used to infer that the system is moving uniformly instead of being at rest. The invariance of the expression
S ′ ,O′ S ′
(4.6) 2222222222 tczyxtczyx ′−′+′+′=−++ is a basic requirement that the transformation between the two systems must satisfy. The Galilean transformation (2.13), which in our case reads
ttzzyyvtxx =′=′=′−=′ ,,, is not consistent with the invariance property (4.6), as it gives
xvttvtczyxtczyx 2222222222222 −+−++=′−′+′+′
The first postulate brings some simplification to the possible form of the desired transformation. Since a uniform rectilinear motion in S must go over into a uniform rectilinear motion in S', the transformation must be linear. A nonlinear transformation would predict acceleration in S' even if the velocity were constant in S. The choice of the origins in space and time has avoided constant terms, and the transformation will be homogenous. The directions perpendicular to the relative motion, which are effectively at rest, must be left unchanged by the transformation. We then take the general form
BtAxx +=′ yy =′
(4.7) zz =′ FtDxt +=′ Further simplification comes from the argument that the origin O', which has the coordinate moves along the x axis with velocity ,0=′x ,/ vdtdx = that is
ABvBtAx /or0 −=+= while the origin O, with coordinate ,0=x moves along x-axis with velocity Recalling that when , we can write
.'/' vdtdx −= 0=x 0=t
00 ort D B Bx A B t v
F F F ′ − ⋅′ ′= ⋅ + = − =
so that Inserting now Eqs.(4.7) into Eq.(4.5), we obtain the equation .FA =
( ) ( ) 02 22222222222222222 =−+⎟⎟⎠ ⎞
⎜⎜⎝ ⎛ −−++−=′−′+′+′ DcBAxt
c BAtczyxDcAtczyx
The identity (4.6) requires that
2222 =−=−=− DcB c BADcA
Since ,B vA= we obtain
2222 /1
/, /1
, /1
cv
cvD cv
vB cv
A −
−= −
−= −
= (4.8) so that Eqs.(4.7) now read
22 /1 cv
vtxx −
−=′ yy =′
(4.9) zz =′
/1
/
cv
cvxtt − −=′
The four equations (4.9) are known as the Lorentz transformation, which in this case applies to the relative motion of inertial frames along the x-axis. It is a straightforward matter to show that Eqs.(4.9) are consistent with the invariance relation (4.6), which is thus a sufficient as well as a necessary condition for the Lorentz transformation. With the usual abbreviations
( )1 /1
1, 222
≥ −
= −
== γ β
γβ cvc
v (4.10)
the Lorentz transformation simplifies to
( ) ( )x x ct x x ctγ β γ β′ ′= − = + ′ a)
yyyy ′==′ or b) (4.11)
zzzz ′==′ c)
⎟⎠ ⎞⎜⎝
⎛ ′+′=⎟⎠ ⎞⎜⎝
⎛ −=′ x c
ttx c
tt βγβγ d) The right hand side shows the inverse transformations where the primed and unprimed quantities are interchanged and the sign of the relative velocity is reversed, because its direction is the only difference between the systems S and S'. In the low velocity limit
the Lorentz transformation reduces to the Galilean transformation since the only difference consists in the presence of the factor γ and the term
,1<<cv ( )xc/β− in d). It is clear
that, in Eq.(4.11a), γ indicates a change of measuring stick length and in Eq.(4.11d) a change in the clock rate in the Lorentz transformation. The term ( )xc/β− in Eq.(4.11d) shows that the time is dependent on the position of a given event. Therefore, the immediate consequences of the Lorentz transformation require the revision of the usual concepts of simultaneity, time and length. Simultaneity Consider two events occuring at the same time ttt == 21 at two different positions and in the unprimed system S. According to Eqs.(4.11) the events are recorded in the primed system S' at different times
1x 2x
/ /, 1 1
t x c t xt t cβ ββ β − −′ ′= =
− −
so that the apparent time interval in S' is given by
( )
0 1
/ 2
21 12 ≠−
−=′−′=′∆ β
β cxx ttt (4.12)
and hence, it is a function of the relative velocity v. The events will not appear as simultaneous to observers in S'. We are then forced to abandon the intuitive idea of absolute simultaneity, which is based on our experience with objects of low velocity, and accept the relativity of simultaneity, which depends on the coordinate system and applies in the domain of high velocities. Length contraction Consider a rod at rest in the primed system S' lying along the x-axis and having the length 12 xx ′−′=λ given by the distance between its ends 21 and xx ′′ measured at the same instant of time t'. The length in an inertial frame in which the rod is at rest is called its proper length. To define length in a frame S in which the rod is moving, an observer must locate the position of both end points, 21 and xx ′′ , at the same time From the inverse equations (4.11) we obtain
1 2 .t t t= =
1 1 2 1
x ct x ct x xx x x xβ ββ β β − −′ ′ ′ ′= = − = − −
− −
so that the apparent length 12 xxl −= is
21 βλ −=l (4.13) This equation is known as the Lorentz contraction and states that a rod parallel to the direction of its motion is shortened by the factor .1 2β− It results from the fact that simultaneous events in the rest frame of one rod are not simultaneous in the rest frame of the other. A rod transverse to its direction of motion is unchanged in length. The proper length λ is independent of the orientation of the rod, as a manifestation of the isotropy of inertial rest frame S'. If the orientation of a rod is neither longitudinal nor transverse to the direction of relative motion, both the length and the inclination are changed in the laboratory frame S, as an effect of a contraction in the direction of the relative motion with no change of transverse dimensions.
