Lecture 13 Space Time Diagrams

ASTR 340

Fall 2006

Dennis Papadopoulos

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 t0

as measured by a clock at rest to the inertial observer

Always stretched for the

moving observer– Length contraction: proper

length l0 as measured by observer at rest

Always contracted for the moving observer

0tt

/0ll

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 afactor of:

in a frame moving with respect to the observer.

2

21

1

cv

Boost Factor

Fig. 1-15, p. 21

Velocity additionClassical

Relativistic

V≠v1+v2

1 22

1 21

v vV

v v c

Relativistic Doppler shiftClassical 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

0

0

/obsz v c

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

1 /1

1 /

v cz

v c

Relativistic Doppler has also a small shift in the perpendicular direction of motion

Space-Time Diagram

Two dimensional space-time diagram, i.e

3-D

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

ctlight

Stationary object

Moving objects

World lines of events

Care should be taken of units if light at 45 degrees

Events - Worldlines

Lightcones

ct

Simultaneity

ct

ct2

ct1

xx1 x2

(x1,ct2)

(x2,ct1)x

ct2 2 2( ) ( )s c t x

Space-time interval defined as

2 2 2( ) ( )s c t x

Invariant independent of frame that is measured

Physical interpretation

Measure time with a clock at rest to the observer

x=0

-> s=ct0

Space-time interval

What is the space time interval on a lightcone?

0

0

0

s timelike

s lightlike

s spacelike

Light cone for event “A”

“LightCone”

Different kinds of space-time intervals

“LightCone” “time like”“light like”

“Space like”

Time-like: s2>0

Light-like: s2=0

Space-like: s2<0

Past, future and “elsewhere”.

“Future of A” (causally-connected)

“Past of A” (causally-connected)

“Elsewhere”(causally-disconnected)

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

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.

E=mc2

2

2

1

2k

o

E mv

E m c

mo rest mass

n

1/ 2

1/ 2

2 1/ 2 2

2

Binomial theorem

for x 1

(1 x) 1 ...

1(1 ) 1 ..

21

(1 ) 1 ...2

1 1[1 ( / ) ] 1 ( / ) ...

21 ( / )

nx

x x

x x

v c v cv c

2 2 20 0

v/c<<1

1...

2

If

E m c m c mv

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=m0