The Failure of Galilean Transformations The Lorentz Transformation Time and Space in Special Relativity Relativistic Momentum and Energy. Special Relativity. The Galilean Transformations.
Consider the primed coordinate system moving along the x-axis at speed u. Consider events where clocks record the time at the location of the event.
Galilean Transformation between coordinate systems
Galilean Transformations not correct for light !!!!
A reference frame in which a mass point thrown from the same point in three different (non co-planar) directions follows rectilinear paths each time it is thrown, is called an inertial frame.
– L. Lange (1885) as quoted by Max von Laue in his book (1921) Die Relativitätstheorie, p. 34, and translated by Iro).
Lengths perpendicular toare unchanged
Coefficients determined by invoking symmetry arguments and Einstein’s postulates ….
Invoking the constancy of speed of light. Consider flash of light set off at t’=t=0 at common origins. At a later time t an observer in frame S will measure a spherical wavefront of light with radius ct, moving away from the origin and satisfying:
Similarly, at a time t’, an observer in frame S’ will measure a spherical wavefront of light with radius ct’, moving away from the origin O’ with speed c and satisfying :
Inserting equations 4.10-4.13 into 4.15 and comparing with 4.14
t’ should be the same if y-->-y or z-->-z.
Consider motion of the origin O’ of frame S’. Synchronized at t=t’=0. x-coordinate of O’ is given by x=ut in frame S, and x’=0 in frame S’.
At this point:
Thus the Lorentz transformations linking the space and time coordinates (x,y,z,t) and (x’,y’,z’,t’) of the same event measured from S and S’ are
When ,relativistic formulas must agree with Newtonian equations….
Space and time “mix” !!!!
for light wave x=ct,x’=ct’,x”=ct”,….
(interval)2=(separation in time)2-(separation in space)2
Space-time interval is invariant
Observer in S measures two flash-bulbs going off at same time t but at different locations x1and x2. Then an observer in S’ would measure the time interval between the two flashes as:
Events that occur in simultaneously in one inertial frame do NOT occur simultaneously in all other inertial reference frames