Question

1a)  Show that time dilation is a consequence of the Lorentz transformations

b) Use the Lorentz transformations to find the relationship between the speed u of a moving object measured by one observer and the speed U measured by a second observer moving at a constant speed v relative to the first. Assume that the observers and the object move in the same straight line.

c) Show that the measured speed of an object can never exceed a maximum value, whatever the speed of the observer who makes the measurement.

Answer 1

1) lets suppose we associate a partice moving with velocity v in x direction

if the time interval measured in moving frame then

can be calculated using the lorentz transformation

so

if the time measurement made at the same location then x2=x1

so,

so because

so

so we get dilated time in rest frame

2) so using lorentz transformation positon for observer 1 w.rt 2

we write

so diffrentiating

and time coordinate

or

so dividing we get

or by rearrangement

3) Lets assume photon 1 is moving with -c velocity and photon to with c

than there relative speed using velocity addition formula

so we see that relativistically in any refrence frame we can not move faster than speed of light