Spacetime Diagrams (1D in space)

Transcription

1 PH300 Modern Phsics P11 Last time: Time dilation and length contraion Toda: pacetime ddition of elocities Lorent transformations The onl reason for time is so that eerthing doesn t happen at once. /1 Da 6: - lbert Einstein Questions? pacetime Thursda: ddition of Velocities Relatiistic Momentum & Energ 1 Lorent Transformations Thursda: Relatiistic momentum and energ HW03 due, beginning of class; HW04 assigned Net week: Intro to quantum Eam I (in class) pacetime Diagrams (1D in space) pacetime Diagrams (1D in space) In PHY I: In PH300: c t Δt Δ = Δ / Δt t t pacetime Diagrams (1D in space) Recall: Luc plas with a fire cracker in the train. Rick watches the scene from the track. In PH300: c t obje moing with 0<<c. Worldline of the obje obje moing with 0>>-c L R c t obje at rest at =1 c t obje moing with = -c. =0 at time t= Luc Rick 1

2 Eample: Luc in the train Eample: Rick on the tracks Light reaches both walls at the same time. L R Light traels to both walls L R Rick concludes: Light reaches left side first. In Luc s frame: Walls are at rest Luc concludes: Light reaches both sides at the same time In Rick s frame: Walls are in motion Frame as iewed from =0.5c Frame is moing to the right at = 0.5c. The origins of and coincide at t=t =0. Which shows the world line of the origin of as iewed in? C D These angles are equal This is the time ais of the frame This is the space ais of the frame Frame as iewed from In : (=3,=3) In : ( =1.8, =) pacetime Interal oth frames are adequate for describing eents but will gie different spacetime coordinates for these eents, in general.

3 Distance in Galilean Relatiit Remember Luc? The distance between the blue and the red ball is: Luc h If the two balls are not moing relatie to each other, we find that the distance between them is inariant under Galileo transformations. Eent 1 firecracker eplodes Eent light reaches deteor Distance between eents is h Remember Rick? Eent 1 firecracker eplodes Eent light reaches deteor Distance between eents is cδt ut distance between -coordinates is Δ and: (cδt ) = (Δ ) + h We can write cδt Δ h nd Luc got since pacetime interal a we hae two eents: ( 1, 1, 1,t 1 ) and (,,,t ). Define the spacetime interal (sort of the "distance") between two eents as: With: Δ = 1 Δ = 1 Δ = 1 Δt = t t 1 pacetime interal The spacetime interal has the same alue in all reference frames! I.e. Δs is inariant under Lorent transformations. pacetime interal pacetime Here is an eent in spacetime. ( Δs' ) = ( cδt ') ( Δ' ) ( Δ' ) ( Δ' ) ( Δs' ) = ( Δs) The spacetime interal has the same alue in all reference frames! I.e. Δs is inariant under Lorent transformations. n light signal that passes through this eent has the dashed world lines. These identif the light cone of this eent. 3

4 pacetime pacetime Here is an eent in spacetime. Here is an eent in spacetime. The blue area is the future on this eent. The pink is its past. The ellow area is the elsewhere of the eent. No phsical signal can trael from the eent to its elsewhere! pacetime pacetime Now we hae two eents and as shown on the left. The space-time interal (Δs) of these two eents is: ) Positie ) Negatie C) Zero D C (Δs) >0: Time-like eents ( è D) (Δs) <0: pace-like eents ( è ) (Δs) =0: Light-like eents ( è C) If (Δs) is negatie in one frame of reference it is also negatie in an other inertial frame! ((Δs) is inariant under Lorent transformation). à Causalit is fulfilled in R. (Δs) is inariant under Lorent transformations. Eample: Waefront of a flash t=0 t>0 Waefront = urface of a sphere with radius : () = 0 pacetime interal for light-like eent: (Δs) = 0 Einstein: 'c' is the same in all inertial sstems. Therefore: (') - ' - ' - ' = 0 in all inertial sstems! (Here we assumed that the origins of and ' oerlapped at t=0) 4

5 Velocit transformation (1D) Velocit transformation (1D) Δ ,, where Δ= 1, Δ = 1 Use Lorent: = γ (-t) n obje moes from eent =( 1,t 1 ) to eent =(,t ). s seen from, its speed is with: Δ = 1 Δt = t t 1 Galilean result New in special relatiit s seen from, its speed is with: Δ = 1 Δt = t t 1 Velocit transformation in 3D Velocit transformation (3D) ' (,,,t) ' ' u (',',',t') ' The elocit u=(u, u, u ) measured in is gien b: u = Δ / Δt, u = Δ / Δt, u = Δ / Δt, where Δ= - 1 To find the corresponding elocit components u, u, u in the frame, which is moing along the -aes in with the elocit, we use again the Lorent transformation: In a more general case we want to transform a elocit u (measured in frame ) to u in frame. Note that u can point in an arbitrar direion, but still points along the -aes. 1 =γ( 1 -t 1 ), and so on t 1 =γ(t 1-1 /c ), and so on lgebra Velocit transformation (3D) (aka. Velocit-ddition formula ) ome applications 5

6 Relatiistic transformations Velocit transformation: Which coordinates are primed? u is what we were looking for! (i.e. elocit measured in ) uppose a spacecraft traels at speed =0.5c relatie to the Earth. It launches a missile at speed 0.5c relatie to the spacecraft in its direion of motion. How fast is the missile moing relatie to Earth? (Hint: Remember which coordinates are the primed ones. nd: Does our answer make sense?) Earth (,,,t) ' ' ' u (',',',t') pacecraft ' a) 0.8 c b) 0.5 c c) c d) 0.5 c e) 0 Transformations If is moing with speed in the positie direion relatie to, and the origin of and ' oerlap at t=0, then the coordinates of the same eent in the two frames are related b: The obje could be light, too! uppose a spacecraft traels at speed =0.5c relatie to the Earth. It shoots a beam of light out in its direion of motion. How fast is the light moing relatie to the Earth? (Get our answer using the formula). Lorent transformation (relatiistic) Velocit transformation (relatiistic) a) 1.5c b) 1.5c c) c d) 0.75c e) 0.5c note of caution: The wa the Lorent and Galileo transformations are presented here assumes the following: n obserer in would like to epress an eent (,,,t) (in his frame ) with the coordinates of the frame ', i.e. he wants to find the corresponding eent (',',',t') in '. The frame ' is moing along the -aes of the frame with the elocit (measured relatie to ) and we assume that the origins of both frames oerlap at the time t=0. ' ' ' (,,,t) (',',',t') ' pplication: Lorent transformation t 0 = 0 t 1 = 1s Two clocks (one at and one at ) are snchronied. third clock flies past at a elocit. The moment it passes all three clocks show the same time t 0 = 0 (iewed b obserers in and. ee left image.) What time does the third clock show (as seen b an obserer at ) at the moment it passes the clock in? The clock at is showing t 1 = 1s at that moment. à Use Lorent transformation! ) γ (t 1 -t 0 ) ) γ (t 1 -t 0 )(1 /c ) C) γ (t 1 -t 0 )(1 + /c ) D) (t 1 -t 0 ) / γ E) γ(t 1 -t 0 )(1 + '/c )? 6

7 Hint: Use the following frames: t 0 = 0 ' t 1 = 1s Two clocks (one at and one at ) are snchronied. third clock flies past at a elocit. The moment it passes all three clocks show the same time t 0 = 0 (iewed b obserers in and. ee left image.) What time does the third clock show (as seen b an obserer at ) at the moment it passes the clock in? The clock at is showing t 1 = 1s at that moment. à Use Lorent transformation! ) γ (t 1 -t 0 ) ) γ (t 1 -t 0 )(1 /c ) C) γ (t 1 -t 0 )(1 + /c ) D) (t 1 -t 0 ) / γ E) γ(t 1 -t 0 )(1 + '/c ) The moing clock shows the proper time interal!! Δt proper = Δt / γ? Hint: Use the following sstems: t 0 = 0 ' t 1 = 1s The clock traels from to with speed. ssume is at position = 0, then is at position = t, t=(t 1 -t 0 ) Use this to substitute in the Lorent transformation: t = γ (t t ) = γ t(1 c c ) = t / γ We get eal the epression of the time dilation!? 7