Suppose a particle orbit as a function of time (t) is given in an inertial frame K and the integral dt r(t) is evaluated. Using the invariance of the space-time interval, show that if the same calculation is done in a frame K'moving uniformly with respect to K, then t' = T. In other words, the proper time r along a particle orbit is a Lorentz invariant quantity.
Suppose a particle orbit as a function of time (t) is given in an inertial frame K and the integral dt r(t) is evaluated. Using the invariance of the space-time interval, show that if the same calculation is done in a frame K'moving uniformly with respect to K, then t' = T. In other words, the proper time r along a particle orbit is a Lorentz invariant quantity.
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Transcribed Image Text:Suppose a particle orbit as a function of time (t) is given in an inertial frame K
and the integral
dt
T =
r(t)
1 - B:(1)- B; (t)– B¿ (1)dt
is evaluated. Using the invariance of the space-time interval, show that if the same
calculation is done in a frame K'moving uniformly with respect to K, then r'=r.
In other words, the proper time r along a particle orbit is a Lorentz invariant
quantity.
Expert Solution
Definition:
The space time interval between two events which are time dt and distance dr=(dx,dy,dz) apart is defined as,
In proper frame i.e. the rest frame of particle the particle never changes position so dr=0, this gives,
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