This preview shows page 1. Sign up to view the full content.
Unformatted text preview: could be the same in t' = γ
every frame. o In going from one inerCal frame to another, both x and t transform. o The Cme is diﬀerent in diﬀerent inerCal frames of reference. o Derived the previously stated Lorentz transformaCon from requirement that the speed of light is the same in every inerCal frame. γ= v
#
%t − 2
$
c &
x(
' 1
2 v
1− 2
c 25 The Lorentz TransformaCon u༇ Deﬁne dimensionless quanCCes β and γ. u༇ “Boost” with velocity v along the x direcCon. o that is transform to a frame moving in the x direcCon with velocity v u༇ Note that ct and x have the same units. o Natural units would have c=1 Derive Lorentz TransformaCon 26 Inverse TransformaCon u༇ The K system moves with a velocity –v compared to the K’ system. u༇ Plug in –v for v in the LT to get the inverse transformaCon. u༇ Check whether we get the idenCty by applying one a^er the other. Derive Lorentz TransformaCon 27 Simple DerivaCon u༇ LT must be linear like rotaCon in order for Newtonian moCon (x=vt, x’=vt’) to correct. o also γ cannot depend on x or t. γ’=γ since laws of physics same in each frame u༇ consider wavefront at x=ct and x’=ct’ u༇ We have the basic LT for x. u༇ For more rigor, look at u༇ x’=γ(x βct) u༇ x=γ’(x’+βct’) u༇ replace x with ct for light wave u༇ ct’=γ(ct βct) u༇ ct=γ(ct’+βct’) u༇ divide by c u༇ t’=γ(1 β)t u༇ t=γ(1+β)t’ u༇ t’=γ2 (1 β2)t’ u༇ γ2 =1/(1 β2) u༇ Derive Lorentz TransformaCon 28 “Events” are 4D Space Cme Points u༇ Each Frame must have its own meter sCcks and clocks which are at rest in the frame. o clocks can be synchronized in a frame o clocks in diﬀerent frames can be synchronized when they are at the same place but will diﬀer later as they move apart. u༇ To measure the posiCon and Cme of an event, an observer must be right there at the event. u༇ Since we no longer have a universal Cme to parameterize the other variables, problems are harder to deﬁne and to solve. o We can (carefully) use the Lorentz transformaCon to get the answers we need. 29 Time DilaCon u༇ The measured Cme interval between two events will diﬀer in diﬀerent inerCal frames. u༇ The shortest measured Cme...
View
Full
Document
This note was uploaded on 02/24/2014 for the course PHYS 2D taught by Professor Hirsch during the Winter '08 term at UCSD.
 Winter '08
 Hirsch
 Physics

Click to edit the document details