SPACESHIP RELATIVITY
by
Robert J. Nemiroff
Michigan Technological University
Physics X: About This Course
• Pronounced "Fiziks Ecks"
• Reviews the coolest concepts in physics
• Being taught for credit at Michigan Tech
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Michigan Tech course PH4999
Aimed at upper level physics majors
Light on math, heavy on concepts
Anyone anywhere is welcome
• No textbook required
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Wikipedia, web links, and lectures only
SPECIAL RELATIVITY:
EHRENFEST PARADOX
A rigid ring is rotated about its center. Does it's radius decrease?
Reformulated: Train cars sit on a circular track connected by taut
strings. The train cars all begin to circle the track at once, faster and
faster, eventually reaching relativistic speed. What happens to the
strings?
1. The strings droop.
2. The strings break.
3. The strings remain the same.
4. Did Ehrenfest think of this? Then ask him!
SPECIAL RELATIVITY:
EHRENFEST PARADOX
2. The strings break.
Length contraction makes the length of the train cars plus the strings
contract, so that the strings break. (Assuming circular symmetry.)
Every observer at rest sees the radius as constant. However,
observers rotating with the ring measure a larger circumference than
observers at rest: F' = F / (1 - v2/c2)1/2 . The train cars plus strings
are not long enough to cover this larger circumference.
Einstein himself agreed with this solution.
SPECIAL RELATIVITY:
BELL'S SPACESHIP PARADOX
Two spaceships connected by a string both accelerate. What will
happen to the string?
(Yes, it's the same Bell who created Bell's Inequality.)
SPECIAL RELATIVITY:
BELL'S SPACESHIP PARADOX
1.
2.
3.
The sting will droop. Since length contracts, the string will cover
less length and bunch up.
The string will break. Since length contracts, the string between
the ships contracts but still must cover the same distance as seen
in the rest frame. The string cannot cover this extra distance and
will break.
The string will remain the same. Assume you are in the
spacecrafts frame. A passing observer will not cause your string to
break no matter how fast they pass.
SPECIAL RELATIVITY: PARADOXES
BELL'S SPACESHIP PARADOX
Controversial!
Sets of accelerations should exist that do NOT break the string, in
particular when the lead spaceship should have lower acceleration than
the trailing spaceship.
Simultaneity is important.
Can this be mapped into the Ehrenfest paradox?
If so, then for those cases the string must break.
SPACESHIP RELATIVITY
Is it possible for a human to go anywhere in the visible universe
in a human lifetime without violating the laws of physics?
1. No, one must travel faster than c to do that.
2. Yes, you do not need to travel faster than c.
3. Did you think that Star Trek was real?
SPACESHIP RELATIVITY
2. Yes, you do not need to travel faster than c.
A spaceship traveling at 1 g (comfortable!) can go anywhere in
the visible universe in a normal human life span. This is
essentially the twin paradox, with you being the twin that
leaves. As the ship accelerates, it approaches c in the "home
twin"'s reference frame. The universe "contracts" as well. You
can go anywhere.
See: http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html
SPACESHIP RELATIVITY
Where
How far
Spaceship time at 1 g
(low speed arrival)
Nearest star
4.3 light years
3.6 years
Vega (star)
27 light years
6.6 years
Center of Galaxy
30,000 light years
20 years
M31
2 million light years
28 years
z=1
11 billion light years
(comoving radial)
45 years
Microwave background
45 billion light years
(comoving radial)
48 years
General Formula
n light years
1.94*arccosh(n/1.94+1)
SPACESHIP RELATIVITY:
WARP SPEED
Using Special Relativity, it is possible to "interpret" warp speed not as a
true faster-than light speed but rather as distance in the Earth frame
divided by time in the moving frame.
For example: "warp one" would be the speed a spacecraft must take to
travel one light year, as measured in Earth's frame, by one year's time,
as measured on a starship. This corresponds to a specific velocity
between the spacecraft and Earth.
SPACESHIP RELATIVITY:
WARP SPEED
In other words, how fast must one go to travel one light year away while
experiencing the passing of one year's worth of time? Answer: Warp
one.
How fast must one travel to go ten light years away while experiencing
only one year's worth of time passing?
Answer: Warp 10.
SPECIAL RELATIVITY: A MOVIE
WHAT IT LOOKS LIKE TO MOVE NEAR C