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Lecture 13 Space Time Diagrams ASTR 340 Fall 2006 Dennis Papadopoulos.

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1 Lecture 13 Space Time Diagrams ASTR 340 Fall 2006 Dennis Papadopoulos

2 Relativity Summary Relativity Postulates –Laws of physics the same in all inertial frames –Speed of light in vacuum constant Corollaries –Space and time form a 4- dim continuum –There are global space- time frames with respect to which non-accelerated objects move in straight lines at constant velocities (inertial frames) Consequences –Simultaneity not preserved for two different observers –Time dilation: proper time t 0 as measured by a clock at rest to the inertial observer Always stretched for the moving observer –Length contraction: proper length l 0 as measured by observer at rest Always contracted for the moving observer

3 Time always runs slower when measured by an observer moving with respect to the clock. The length of an object is always shorter when viewed by an observer who is moving with respect to the object. Everything is slowed/contracted by a factor of: in a frame moving with respect to the observer. Boost Factor

4 Fig. 1-15, p. 21

5 Velocity addition Classical Relativistic V≠v 1 +v 2

6 Relativistic Doppler shift Classical red or blue shift formula for non relativistic speeds v/c<<1 Shift completely due to bunching up (approach) or stretching (recession) of wave crests due to the relative source-observer motion Relativistic shift includes also the effect of time dilation. Frequency of light waves specifies how many times the em field oscillates per second in its rest frame ->     The clock of a moving source runs slow and as a result the emission frequency is reduced as measured by the observer. Time dilation always gives a redshift Relativistic Doppler formula Relativistic Doppler has also a small shift in the perpendicular direction of motion

7 Space-Time Diagram

8 Two dimensional space-time diagram, i.e 3-D


10 Space-time diagrams Because space and time are “mixed up” in relativity, it is often useful to make a diagram of events that includes both their space and time coordinates. This is simplest to do for events that take place along a line in space (one- dimensional space) –Plot as a 2D graph –use two coordinates: x and ct Can be generalized to events taking place in a plane (two-dimensional space) using a 3D graph (volume rendered image): x, y and ct Can also be generalized to events taking place in 3D space using a 4D graph, but this is difficult to visualize x ct light Stationary object Moving objects World lines of events Care should be taken of units if light at 45 degrees

11 Events - Worldlines

12 Lightcones

13 ct Simultaneity

14 ct ct 2 ct 1 xx1x1 x2x2 (x 1,ct 2 ) (x 2,ct 1 ) xx ctct Space-time interval defined as Invariant independent of frame that is measured Physical interpretation Measure time with a clock at rest to the observer  x=0 ->  s=c  t 0 Space-time interval What is the space time interval on a lightcone?

15 Light cone for event “A” “Light Cone”

16 Different kinds of space-time intervals “Light Cone” “time like” “light like” “Space like” Time-like:  s 2 >0 Light-like:  s 2 =0 Space-like:  s 2 <0

17 Past, future and “elsewhere”. “Future of A” (causally- connected) “Past of A” (causally- connected) “Elsewhere” (causally- disconnected)

18 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 x 

19 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.

20 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=  m 0

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