The Galilean transformation nevertheless violates Einstein’s postulates, because the velocity equations state that a pulse of light moving with speed
c along the
x -axis would travel at speed
$c-v$ in the other inertial frame. Specifically, the spherical pulse has radius
$r=ct$ at time
t in the unprimed frame, and also has radius
$r\prime =ct\prime $ at time
$t\prime $ in the primed frame. Expressing these relations in Cartesian coordinates gives
This cannot be satisfied for nonzero relative velocity
v of the two frames if we assume the Galilean transformation results in
$t=t\prime $ with
$x=x\prime +vt\prime .$
To find the correct set of transformation equations, assume the two coordinate systems
S and
$S\prime $ in
[link] . First suppose that an event occurs at
$(x\prime ,0,0,t\prime )$ in
$S\prime $ and at
$(x,0,0,t)$ in
S , as depicted in the figure.
Suppose that at the instant that the origins of the coordinate systems in
S and
$S\prime $ coincide, a flash bulb emits a spherically spreading pulse of light starting from the origin. At time
t , an observer in
S finds the origin of
$S\prime $ to be at
$x=vt.$ With the help of a friend in
S , the
$S\prime $ observer also measures the distance from the event to the origin of
$S\prime $ and finds it to be
$x\prime \sqrt{1-{v}^{2}\text{/}{c}^{2}}.$ This follows because we have already shown the postulates of relativity to imply length contraction. Thus the position of the event in
S is
This set of equations, relating the position and time in the two inertial frames, is known as the
Lorentz transformation . They are named in honor of H.A. Lorentz (1853–1928), who first proposed them. Interestingly, he justified the transformation on what was eventually discovered to be a fallacious hypothesis. The correct theoretical basis is Einstein’s special theory of relativity.
The reverse transformation expresses the variables in
S in terms of those in
$S\prime .$ Simply interchanging the primed and unprimed variables and substituting gives:
Spacecraft
$S\prime $ is on its way to Alpha Centauri when Spacecraft
S passes it at relative speed
c /2. The captain of
$S\prime $ sends a radio signal that lasts 1.2 s according to that ship’s clock. Use the Lorentz transformation to find the time interval of the signal measured by the communications officer of spaceship
S .
Solution
Identify the known:
$\text{\Delta}t\prime ={t}_{2}\prime -{t}_{1}\prime =1.2\phantom{\rule{0.2em}{0ex}}\text{s};\phantom{\rule{0.2em}{0ex}}\text{\Delta}x\prime =x{\prime}_{2}-x{\prime}_{1}=0.$
Identify the unknown:
$\text{\Delta}t={t}_{2}-{t}_{1}.$
Express the answer as an equation. The time signal starts as
$\left(x\prime ,{t}_{1}\prime \right)$ and stops at
$\left(x\prime ,{t}_{2}\prime \right).$ Note that the
$x\prime $ coordinate of both events is the same because the clock is at rest in
$S\prime .$ Write the first Lorentz transformation equation in terms of
$\text{\Delta}t={t}_{2}-{t}_{1},$$\text{\Delta}x={x}_{2}-{x}_{1},$ and similarly for the primed coordinates, as:
every matter made up of particles and particles are also subdivided which are themselves subdivided and so on ,and the basic and smallest smallest smallest division is energy which vibrates to become particles and thats why particles have wave nature
according to de Broglie any matter particles by attaining the higher velocity as compared to light'ill show the wave nature and equation of wave will applicable on it but in practical life people see it is impossible however it is practicaly true and possible while looking at the earth matter at far
Manikant
a centeral part of theory of quantum mechanics example:just like a beam of light or a water wave
Swagatika
Mathematical expression of principle of relativity
given that the velocity v of wave depends on the tension f in the spring, it's length 'I' and it's mass 'm'. derive using dimension the equation of the wave