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3a) Use the Lorentz transformation to derive the expression for the speed measured in the primed frame u, dx/dt and the relative frame velocity B = v/c dx' / dt' in terms of Ux 3b) If an object moves at speed ux = c in the unprimed frame, what is its speed u, in the primed frame? 3c) Calculate the transformation for the component of velocity along the vedirection u' du' I dt' in terms of 1 - du / d+ and u

3a) Use the Lorentz transformation to derive the expression for the speed measured in the primed frame u, dx/dt and the relative frame velocity B = v/c dx' / dt' in terms of Ux 3b) If an object moves at speed ux = c in the unprimed frame, what is its speed u, in the primed frame? 3c) Calculate the transformation for the component of velocity along the vedirection u' du' I dt' in terms of 1 - du / d+ and u

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IN ASTROPHYSICS WE SOMETIMES ENCOUNTER MATERIAL moving at very near the speed of light. Accreting black holes, for example, can expel jets of hot gas moving outward with Lorentz factors of y ~ 100 or more. Any light emitted by the hot gas will be relativistically beamed in the forward direction, such that you will only see the jet if it is coming right at you. Let's see how. Imagine an object that, in one inertial frame, moves at speed ux = dx/dt in the x-direction. The coordinates in another inertial reference frame are related by the Lorentz transformation x' = r(x – ß ct) ct' = r(ct – Bx) y' = y z' = z (19) (20) dx' /dt' in this other From which we can calculate the speed uz reference frame 3a) Use the Lorentz transformation to derive the expression for the speed measured in the primed frame u, dx' /dt' in terms of dx/dt and the relative frame velocity B = v/c Ux 3b) If an object moves at speed ux = c in the unprimed frame, what is its speed u, in the primed frame? 3c) Calculate the transformation for the component of velocity along the y-direction, u', = dy' / dt' in terms of uy = dy/dt and ux
Transcribed Image Text:IN ASTROPHYSICS WE SOMETIMES ENCOUNTER MATERIAL moving at very near the speed of light. Accreting black holes, for example, can expel jets of hot gas moving outward with Lorentz factors of y ~ 100 or more. Any light emitted by the hot gas will be relativistically beamed in the forward direction, such that you will only see the jet if it is coming right at you. Let's see how. Imagine an object that, in one inertial frame, moves at speed ux = dx/dt in the x-direction. The coordinates in another inertial reference frame are related by the Lorentz transformation x' = r(x – ß ct) ct' = r(ct – Bx) y' = y z' = z (19) (20) dx' /dt' in this other From which we can calculate the speed uz reference frame 3a) Use the Lorentz transformation to derive the expression for the speed measured in the primed frame u, dx' /dt' in terms of dx/dt and the relative frame velocity B = v/c Ux 3b) If an object moves at speed ux = c in the unprimed frame, what is its speed u, in the primed frame? 3c) Calculate the transformation for the component of velocity along the y-direction, u', = dy' / dt' in terms of uy = dy/dt and ux
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