- #1
Achmed
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Consider two events that take place at the origin of the frame of an inertial observer [itex] O' [/itex]. At times [itex] t_1 ' = 0 [/itex] and [itex] t_2 ' = T [/itex]. [itex] O' [/itex] moves with a constant speed [itex] v [/itex] w.r.t. another inertial observer [itex] O [/itex].
1. Use the Lorentz-transformations to show that these events occur at [itex] x=vt [/itex] in the frame of [itex] O [/itex], at the times [itex] t_1 = 0 [/itex] and [itex] t_2 = \gamma T[/itex]. Show furthermore that the events do not occur at the same place in [itex] O [/itex], but at [itex] x_1=0 [/itex] and [itex] x_2 = \gamma vT [/itex].
My attempt below. I am very confused, so I have no idea if what I'm doing is even remotely correct:
Since both events take place at the origin of [itex] S' [/itex], we get that [itex] x' = 0 [/itex]. From here it follows using the Lorentz transformation that [itex] x=vt [/itex].
After this the confusion starts. If we use the Lorentz transformation for time, and we use [itex] t'_1 = 0 [/itex], we actually get [itex] t_1 = \dfrac{vx}{c^2} [/itex]. But this doesn't correspond with what they ask, right? If we do the same for [itex]t_2' = T [/itex], we get the very same problem. What am I doing incorrectly here?
Using the same tactic above, if I fill in [itex] x' = 0 [/itex] and [itex] t_1 = 0 [/itex] and [itex] t_2 = \gamma T [/itex], I do get the correct answers for [itex] x_1 [/itex] and [itex] x_2 [/itex]! So, that confuses me..
1. Use the Lorentz-transformations to show that these events occur at [itex] x=vt [/itex] in the frame of [itex] O [/itex], at the times [itex] t_1 = 0 [/itex] and [itex] t_2 = \gamma T[/itex]. Show furthermore that the events do not occur at the same place in [itex] O [/itex], but at [itex] x_1=0 [/itex] and [itex] x_2 = \gamma vT [/itex].
My attempt below. I am very confused, so I have no idea if what I'm doing is even remotely correct:
Since both events take place at the origin of [itex] S' [/itex], we get that [itex] x' = 0 [/itex]. From here it follows using the Lorentz transformation that [itex] x=vt [/itex].
After this the confusion starts. If we use the Lorentz transformation for time, and we use [itex] t'_1 = 0 [/itex], we actually get [itex] t_1 = \dfrac{vx}{c^2} [/itex]. But this doesn't correspond with what they ask, right? If we do the same for [itex]t_2' = T [/itex], we get the very same problem. What am I doing incorrectly here?
Using the same tactic above, if I fill in [itex] x' = 0 [/itex] and [itex] t_1 = 0 [/itex] and [itex] t_2 = \gamma T [/itex], I do get the correct answers for [itex] x_1 [/itex] and [itex] x_2 [/itex]! So, that confuses me..