Superluminal jets. Fig. (a) shows the path by a knot in a jet of ionized gas tha

Question

Superluminal jets. Fig. (a) shows the path by a knot in a jet of ionized gas that has been expelled from a galaxy. The knot travels at constant velocity at angle ? from the direction of Earth. The knot occasionally emits a burst of light, which is eventually detected on Earth. Two bursts are indicated in Fig . (a), separated by time t as measured in a stationary frame near the bursts. The bursts are shown in Fig. (a) as if they were photographed on the same piece of film, first when light from burst 1 arrived on Earth and then later when light from burst 2 arrived. The apparent distance Dapp traveled by the knot between the two bursts is the distance across an Earth-observer's view of the knot's path. The apparent time Tapp between the bursts is the difference in the arrival times of the light from them. The apparent speed of the knot is then Vapp = Dapp/Tapp.

(a) What are Dapp and Tapp? (Use the following as necessary: t, v, ?, and c for the speed of light.)

Dapp =
Tapp =

(b) Evaluate Vapp for v = 0.963c and ? = 28°. When superluminal (faster than light) jets were first observed, they seem to defy special relativity-at least the correct geometry (Fig (b)) was understood.

Explanation / Answer

(a)   The spacial separation between the two bursts is   vt We project this length onto the direction perpendicular to the light rays headed to Earth and obtain Dapp    = vtsin Burst 1 is emitted a time t ahead of burst 2. Also, burst 1 has to travel an extra distance L more than burst 2 before reaching the Earth    is L = vtcos   the extra time taken to travell this distance is   t '   = L /C hance apparent time Tapp      =   t   - t '      = t - (vtcos  /c)   = t ( 1- (vcos  /c) ) (b) vapparent     = Dapp / Tapp      = vsin   / ( 1- (vcos  /c) ) Here given v =   0.963c   and   = 280 plug all values we get vapparent     = 3.02 c      L = vtcos   the extra time taken to travell this distance is   t '   = L /C hance apparent time Tapp      =   t   - t '      = t - (vtcos  /c)   = t ( 1- (vcos  /c) ) (b) vapparent     = Dapp / Tapp      = vsin   / ( 1- (vcos  /c) ) Here given v =   0.963c   and   = 280 plug all values we get vapparent     = 3.02 c     

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