Finally....Finally....Finally..The Big Suspense is Over.. THE LORENTZ TRANSFORMATION
Hello friends , The day has come...Today we will learn the major breakthrough in the Theory in relativity and that is the Lorentz transformation.
All the incompatibilities of Relativity with Propagation of light arose due to the two unjustifiable hypothesis which are :
1)The time interval between two events is independent of the condition of motion of the body of the reference.
2) The Space- interval between two points i independent of the state of motion of he body of reference.
If we drop the above two considerations then all our problems disappear.
In this section we get to know place and time of an event with respect to train when we know it with respect to the embankment.It appears to be an easy task first but let me remind you that theorem of addition of velocities is not valid for us now.
Upto the present we have just considered events taking place along the embankment, which had mathematically to assume the function of a straight line . now we will imagine the reference body supplemented laterally and in vertical direction by means of a framework of rods, So that an event which takes place anywhere in the space can be localized with reference o this framework.Similarly, we can imagine the train traveling with the velocity V to be continued across the whole space,so that every event, how far it may be can be localized with respect to this second framework.Without committing any fundamental error, we can disregard the fact that in reality these frameworks would continually interfere with each other, owing to the impenetrability of the solid bodies,In every such framework, we imagine three surfaces perpendicular to each other marked out and designed as coordinate planes. A coordinate system K then corresponds to the embankment and another coordinate system K' corresponds to The Train. An event where it may have taken place , would be fixed in space with respect to K by three perpendiculars x,y,z and time value t. Then the same vent with respect to K' would be fixed be x',y',z' and time value t' and of course are not identical to those of K.
But it is a obvious fact that there must be some correlation between the values x,y,z,t and x',y',z',t'.This relation was to be formulated in such a way that law of propagation of light is fully satisfied.
Let us take two coordinate systems K and K' where K is at rest and K' is moving with velocity v in positive x direction with respect to K.
Then the Lorentz transformation give us the following equations
x=ct,
i.e. with the velocity c. According to the equations of the Lorentz transformation , this simple relation between x and t involves relation between x' and t'. In point of fact if we substitute for x the value ct in the first and the fourth equations of Lorentz transformation we get
x'=(c-vt) /(1-v^2/c^2)^1/2
t'=(1-v/c)t/(1-v^2/c^2)^1/2
for which the expression x'=ct' immediately follows.If referred to the system K', the propagation of light takes place according to this equation . We thus see that the velocity of transmission relative to the reference body K' is also equal to c. the same result is obtained for rays f light advancing in any other directions whatsoever. It is actually what we all had expected.
All the incompatibilities of Relativity with Propagation of light arose due to the two unjustifiable hypothesis which are :
1)The time interval between two events is independent of the condition of motion of the body of the reference.
2) The Space- interval between two points i independent of the state of motion of he body of reference.
If we drop the above two considerations then all our problems disappear.
In this section we get to know place and time of an event with respect to train when we know it with respect to the embankment.It appears to be an easy task first but let me remind you that theorem of addition of velocities is not valid for us now.
Upto the present we have just considered events taking place along the embankment, which had mathematically to assume the function of a straight line . now we will imagine the reference body supplemented laterally and in vertical direction by means of a framework of rods, So that an event which takes place anywhere in the space can be localized with reference o this framework.Similarly, we can imagine the train traveling with the velocity V to be continued across the whole space,so that every event, how far it may be can be localized with respect to this second framework.Without committing any fundamental error, we can disregard the fact that in reality these frameworks would continually interfere with each other, owing to the impenetrability of the solid bodies,In every such framework, we imagine three surfaces perpendicular to each other marked out and designed as coordinate planes. A coordinate system K then corresponds to the embankment and another coordinate system K' corresponds to The Train. An event where it may have taken place , would be fixed in space with respect to K by three perpendiculars x,y,z and time value t. Then the same vent with respect to K' would be fixed be x',y',z' and time value t' and of course are not identical to those of K.
But it is a obvious fact that there must be some correlation between the values x,y,z,t and x',y',z',t'.This relation was to be formulated in such a way that law of propagation of light is fully satisfied.
Let us take two coordinate systems K and K' where K is at rest and K' is moving with velocity v in positive x direction with respect to K.
Then the Lorentz transformation give us the following equations
x’=(x-vt)/( 1 - v^2/c^2 )^1/2
y'=y ; z'= z
t' = ( t - vx/c^2)/(1-v^2/c^2)^1/2
If in place of the law of transmission of light we had taken as our basis the tactic assumption of the older mechanics as to absolute character of times an lengths, then instead of the above we will get t he following equations
x'=x-vt
y'= y
z' = z
t'=t
This system of equations is often termed the Galilee transformation . the Galilee transformation can be obtained from Lorentz transformation by submitting an infinitely large value for the velocity of light c in latte transformation. Aided by the following illustration w can readily see that in accordance with the Lorentz transformation the law of transmission of light is satisfied for both the reference bodies K and K'. A light signal sent along the positive x axis, and this light stimulus advances with accordance to the equationx=ct,
i.e. with the velocity c. According to the equations of the Lorentz transformation , this simple relation between x and t involves relation between x' and t'. In point of fact if we substitute for x the value ct in the first and the fourth equations of Lorentz transformation we get
x'=(c-vt) /(1-v^2/c^2)^1/2
t'=(1-v/c)t/(1-v^2/c^2)^1/2
for which the expression x'=ct' immediately follows.If referred to the system K', the propagation of light takes place according to this equation . We thus see that the velocity of transmission relative to the reference body K' is also equal to c. the same result is obtained for rays f light advancing in any other directions whatsoever. It is actually what we all had expected.
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