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Lecture 14 Space-time diagrams (cont)

Lecture 14 Space-time diagrams (cont). ASTR 340 Fall 2006 Dennis Papadopoulos. E=mc 2. m o rest mass. Energy due to mass -> rest energy m o c 2 9x10 16 J per kg of mass Energy due to motion Kinetic Energy (1/2) mv 2 Relativistic mass m= g m 0. ct. (x 1 ,ct 2 ). ct 2. c D t. ct 1.

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Lecture 14 Space-time diagrams (cont)

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  1. Lecture 14Space-time diagrams (cont) ASTR 340 Fall 2006 Dennis Papadopoulos

  2. E=mc2 mo rest mass Energy due to mass -> rest energy moc2 9x1016 J per kg of mass Energy due to motion Kinetic Energy (1/2) mv2 Relativistic mass m=gm0

  3. ct (x1,ct2) ct2 cDt ct1 (x2,ct1) Dx x1 x2 x Space-time interval Space-time interval defined as Invariant independent of frame that is measured Physical interpretation Measure time with a clock at rest to the observer Dx=0 -> Ds=cDt0 What is the space time interval on a lightcone?

  4. Spacetime diagrams in different frames • Changing from one reference frame to another… • Affects time coordinate (time-dilation) • Affects space coordinate (length contraction) • Leads to a distortion of the space-time diagram as shown in figure. • Events that are simultaneous in one frame are not simultaneous in another frame ct f q x v/c=tanf=cotq

  5. ct x Space-time interval ct’ B A Space-time interval invariant All inertial observers will agree on the value of Ds, e.g. 2 lightyears, although they will disagree on the value of the time interval and distance interval x’

  6. Reciprocity

  7. “light like” “time like” “Space like” Different kinds of space-time intervals Time-like:s2>0 Light-like: s2=0 Space-like: s2<0 “Light Cone”

  8. Causality B Can I change the time order of events by going to another reference frame? A

  9. Causality • Events A and B… • Cannot change order of A and B by changing frames of reference. • A can also communicate information to B by sending a signal at, or less than, the speed of light. • This means that A and B are causally-connected. • Events A and C… • Can change the order of A and C by changing frame of reference. • If there were any communication between A and C, it would have to happen at a speed faster than the speed of light. • If idea of cause and effect is to have any meaning, we must conclude that no communication can occur at a speed faster than the speed of light.

  10. The twin paradox • Suppose Andy (A) and Betty (B) are twins. • Andy stays on Earth, while Betty leaves Earth, travels (at a large fraction of the speed of light) to visit her aunt on a planet orbiting Alpha Centauri, and returns • When Betty gets home, she finds Andy is greatly aged compared her herself. • Andy attributes this to the time dilation he observes for Betty’s clock during her journey • Is this correct? • What about reciprocity?Doesn’t Betty observe Andy’s clock as dilated, from her point of view? Wouldn’t that mean she would find him much older, when she returns? • Who’s really older??What’s going on???

  11. B (return) A ct B (outbound) x Andy’s point of view • Andy’s world line, in his own frame, is a straight line • Betty’s journey has world line with two segments, one for outbound (towards larger x) and one for return (towards smaller x) • Both of Betty’s segments are at angles 45 to vertical, because she travels at vc • If Andy is older by t years when Betty returns,he expects that due to time dilation she will have aged by t/ years • Since 1/ = (1-v2/c2)1/2 1, Betty will be younger than Andy, and the faster Betty travels, the more difference there will be

  12. B (return) B (return) A A ct ct B (outbound) B (outbound) x x Betty’s point of view • Consider frame moving with Betty’s outbound velocity • Andy on Earth will have straight world line moving towards smaller x • Betty’s return journey world line is not the same as her outbound world line, instead pointing toward smaller x • Both Andy’s world line and Betty’s return world line are at angles  45  to vertical (inside of the light cone) • Betty’s return world line is closer to light cone than Andy’s world line • For frame moving with Betty’s return velocity, situation is similar

  13. Twin Paradox g=1.5 v=.74 c

  14. B (return) A ct B (outbound) x Solution of the paradox • From any perspective, Andy’s world line has a single segment • From any perspective, Betty’s world line has two different segments • There is no single inertial frame for Betty’s trip, so reciprocity of time dilation with Andy cannot apply for whole journey • Betty’s proper time is truly shorter -- she is younger than Andy when she returns

  15. B (return) B (return) A A ct ct B (outbound) B (outbound) x x Different kinds of world lines • Regardless of frame, Betty’s world line does not connect start and end points with a straight line, while Andy’s does • This is because Betty’s journey involves accelerations, while Andy’s does not

  16. ct x More on invariant intervals • Considering all possible world lines joining two points in a space-time diagram, the one with the longest proper time (=invariant interval) is always the straight world line that connects the two points • The light-like world lines (involving reflection) have the shortest proper time -- zero! • Massive bodies can minimize their proper time between events by following a world line near a light-like world line

  17. SR Summary

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