Title: Lecture 17 Special relativity IV: Paradoxes or how twins can or cannot find out who is the older one
1Lecture 17Special relativity IV Paradoxes or
how twins can or cannot find out who is the older
one ?
2World lines slowly moving
3World lines fast moving
4Spacetime interval
?s2 (c??t)2 ?x2
- sign difference between space and time
- ?s2 is invariant under Lorentz transformation
- for particle moving at speed of light?x c??t
? ?s2 0 ? light like (null) distance
5Character of spacetime intervals
- ?s2gt0 ? c??t gt ?x
- spatial distance can be traveled by speed of
light - there exist an inertial frame, in which the two
events happen at the same position - but they never happen simultaneously ? time like
distance - ?s2lt0 ? c??t lt ?x
- spatial distance cannot be traveled by speed of
light - there exist an inertial frame, in which the two
events happen simultaneously - but they never happen at the same place? space
like distance
6Future, past, and elsewhere
Future
?s2gt0
?s2lt0
?s2lt0
Past
?s2gt0
7Principle of causality
- Cause must always precede the effect? A must
not influence D and vice versa? nothing can
movefaster than speed of light
8Faster than speed of light ?
9A comment on superluminal speed
- The key issue is that no information can be
transmitted faster than the speed of light. - This does not exclude or forbid the existence of
apparent superluminal velocities.
10World lines accelerated
11World lines decelerated
12The Twin Paradox
- Ann and Betty are fraternal twins. Betty takes
the chance to participate the first interstellar
space mission traveling to Alpha Centauri and
back nearly at the speed of light.
Ann
Betty
13Anns view (she stays back on Earth)
- Betty is traveling with the Enterprise at
velocities near the speed of light. Her clock
ticks slower. Therefore, upon her return ?
Betty is younger.
14Bettys view (she is traveling to Alpha Centauri)
- Ann is traveling with the Earth at velocities
near the speed of light. Her clock ticks slower.
Therefore, upon her return ? Ann is younger.
15So who is right ?
- Obviously either Ann or Betty must be wrong.
- Lets recall the principle of relativity All
inertial frames of reference are equivalent - Do both stay in an inertial frame of reference?
- Ann (on Earth) yes
- Betty (in the Enterprise) no.
- While she is in an inertial frame during her
travel to and during her travel from Alpha
Centauri, she changes the frame of reference at
least 3 times.
16Anns and Bettys travel in the Minkowski diagram
3
Ann
Betty
2
1
? Betty is younger
17A relativistic murder trial Setup I
- A murdered man is found in the cargo bay of the
star ship Enterprise. - He has two head wounds caused by laser beams.
- The incident is observed from three locations a
space station (at rest), the Enterprise (moving
at 0.5c) and a Klingon ship moving at 0.75c ahead
of the Enterprise
180.5c
0.75c
19A relativistic murder trial Setup II
- All parties agree that the dead man was hit by
the two laser beams simultaneously - The laser was fired by a human at the front of
the Enterprise (i.e. closer to the Klingon space
ship) and by a Klingon in the back (i.e. closer
to the space station). - The human is defended by James T. Kirk who
witnessed the incident from the Space Station,
the Klingon by the captain of the Klingons ship
20Captain Kirks plea
K
H
? The Klingon fired first, his guilt is
the greater one !
21Klingons plea
K
H
? The human fired first, his guilt is the
greater one !
22Witness McCoy
K
H
? Both fired at the same time !
23So what is the verdict ?
- The two suspects (the Klingon and the human) are
separated by a space-like distance (?s2lt0 ). - For space-like distances, the order of events is
relative - But events separated by a space-like distance do
not influence each other. - When the Klingon and the human were firing, they
were not aware that the other is firing, too.
Actually, they only see the other firing when the
victim is already hit by their own laser beam.