Relativistic Radar

Math

An electromagnetic (EM) pulse is sent from a moving (speed=v) ship initially at distance r toward a stationary distant object, is reflected back, and then is received by the ship.  What is the time between emission and reception? 

δ t Out = r c δ t refl = r−v(δ t out +δ t refl ) c cδ t refl +vδ t refl =r−v t out =r(1−v/c) δ t refl = r(1−v/c) c+v δ t sum = r c + r(1−v/c) c+v = r(c+v)+rc(1−v/c) c(c+v) =r 2 (c+v) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeqiTdq MaamiDamaaBaaaleaacaWGpbGaamyDaiaadshaaeqaaOGaeyypa0Za aSaaaeaacaWGYbaabaGaam4yaaaaaeaacqaH0oazcaGG0bWaaSbaaS qaaiaadkhacaWGLbGaamOzaiaadYgaaeqaaOGaeyypa0ZaaSaaaeaa caWGYbGaeyOeI0IaamODaiaacIcacqaH0oazcaWG0bWaaSbaaSqaai aad+gacaWG1bGaamiDaaqabaGccqGHRaWkcqaH0oazcaWG0bWaaSba aSqaaiaadkhacaWGLbGaamOzaiaadYgaaeqaaOGaaiykaaqaaiaado gaaaaabaGaam4yaiabes7aKjaadshadaWgaaWcbaGaamOCaiaadwga caWGMbGaamiBaaqabaGccqGHRaWkcaWG2bGaeqiTdqMaamiDamaaBa aaleaacaWGYbGaamyzaiaadAgacaWGSbaabeaakiabg2da9iaadkha cqGHsislcaWG2bGaamiDamaaBaaaleaacaWGVbGaamyDaiaadshaae qaaOGaeyypa0JaamOCaiaacIcacaaIXaGaeyOeI0IaamODaiaac+ca caWGJbGaaiykaaqaaiabes7aKjaadshadaWgaaWcbaGaamOCaiaadw gacaWGMbGaamiBaaqabaGccqGH9aqpdaWcaaqaaiaadkhacaGGOaGa aGymaiabgkHiTiaadAhacaGGVaGaam4yaiaacMcaaeaacaWGJbGaey 4kaSIaamODaaaaaeaacqaH0oazcaWG0bWaaSbaaSqaaiaadohacaWG 1bGaamyBaaqabaGccqGH9aqpdaWcaaqaaiaadkhaaeaacaWGJbaaai abgUcaRmaalaaabaGaamOCaiaacIcacaaIXaGaeyOeI0IaamODaiaa c+cacaWGJbGaaiykaaqaaiaadogacqGHRaWkcaWG2baaaiabg2da9m aalaaabaGaamOCaiaacIcacaWGJbGaey4kaSIaamODaiaacMcacqGH RaWkcaWGYbGaam4yaiaacIcacaaIXaGaeyOeI0IaamODaiaac+caca WGJbGaaiykaaqaaiaadogacaGGOaGaam4yaiabgUcaRiaadAhacaGG Paaaaiabg2da9iaadkhadaWcaaqaaiaaikdaaeaacaGGOaGaam4yai abgUcaRiaadAhacaGGPaaaaaaaaa@B64B@  

(1.1)

This result is entirely in keeping with the requirement that the pulse travels at speed c while the ship travels at speed v.  The same result would apply to a sonar pulse from a moving ship to a stationary object when the water has no current.

 

How would this time change if the range was given in a frame that is stationary respect to the distant object?  Then the range would be longer:

r 0 = r 1− (v/c) 2 =γr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkhadaWgaa WcbaGaaGimaaqabaGccqGH9aqpdaWcaaqaaiaadkhaaeaadaGcaaqa aiaaigdacqGHsislcaGGOaGaamODaiaac+cacaWGJbGaaiykamaaCa aaleqabaGaaGOmaaaaaeqaaaaakiabg2da9iabeo7aNjaadkhaaaa@441D@  

(1.2)

so the time would be

δt=γr 2 (c+v) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKjaads hacqGH9aqpcqaHZoWzcaWGYbWaaSaaaeaacaaIYaaabaGaaiikaiaa dogacqGHRaWkcaWG2bGaaiykaaaaaaa@4117@  

(1.3)

This derivation relates to the twin paradox.  The space twin can first measure her speed, v, using Doppler shift  of waves reflected from nearby objects that are stationary with respect to the distant object.  Then, having already learned that the distance from earth to the distant object is r0, she can compute the distance in her frame of reference using r from equation (1.2).  And from that she can compute the time (equations (1.1)) it will take to get a radar return from the distant object.