Transcript Goal: To get to know the ins and outs of relativity

Goal: To get to know the ins and
outs of relativity (relatively
speaking)
Objectives:
1) Black holes vs space-time
2) General Relativity
3) Special Relativity
4) Velocities of relativistic objects
when you are relativistic
Black Hole
• A black hole is an object that is either so
massive or so dense that the escape
velocity on its surface is greater than the
speed of light.
• As Einstein discovered nothing can travel
faster than the speed of light.
• Therefore NOTHING, not even light can
escape from a black hole!
No escape!
• The radius at which the escape velocity is
exactly the speed of light is called the
Schwarzschild radius.
• The Schwarzschild radius is an event
horizon.
• An event horizon is a surface where if
something were to pass through it, it is
gone (event horizon = goodbye forever).
But there’s more!
• Mass warps space. Time is relative to
space. Therefore masses warp time also!
• Tobject = Tuniversal / (1 – rs / r)1/2
• Where rs is the Schwarzschild radius (the
radius of the event horizon of a black hole)
• rs = 1.5 km * Mass of object / Mass of our sun
Black hole astrophysics
• What would happen if we swapped our
sun for a black hole of exact equal mass?
• A) The earth would be sucked into the
black hole
• B) Time on the earth would slow down
• C) The earth would be slingshot out of the
solar system
• D) Nothing would happen to the orbit of
the earth or the clocks on earth.
Black hole astrophysics
• What would happen if we swapped our
sun for a black hole of exact equal mass?
• D) Nothing would happen to the orbit of
the earth or the clocks on earth.
• Black holes are not vacuum cleaners.
They obey gravity just like everything else.
• In fact it is harder to run into a black hole
because it is so frikkn small (diameter of 3
km for one the mass of out sun).
Is that all?
• Nope (but that is all for black holes for
now, sorry).
• Special Relativity
• Clocks progress at a rate RELATIVE to
their position in space.
• Velocity slows the progress of an object’s
clock so that:
• Tobject = Tuniverse / gamma
• Gamma = 1 / (1 – v2/c2)1/2
Example 1
• Tobject = Tuniverse / gamma
• Gamma = 1 / (1 – v2/c2)1/2
• If v = 0.99c then what is the value for
gamma?
Example 1
• Tobject = Tuniverse / gamma
• Gamma = 1 / (1 – v2/c2)1/2
• If v = 0.99c then what is the value for
gamma?
• Gamma = 1 / (1 – (0.99c)2/c2)1/2
• = 7.09
• Note that gamma has no units – it is just a
factor.
Example 2
• A spacecraft flies to the Alpha Centauri star
system at a velocity of 0.9999 c.
• Since Alpha Centauri is 4.3 lightyears away find:
• A) The time that observers on the earth think it
took the spacecraft to get there.
• B) The amount of time that passes by for the
crew of the ship.
• Remember the speed of light is 1 lightyear/year
Example
• A spacecraft flies to the Alpha Centauri star system at a velocity of
0.9999 c.
• Since Alpha Centauri is 4.3 lightyears away find:
• A) The time that observers on the earth think it took the spacecraft to
get there.
• D = VT, T = D/V
• = 4.2 lightyears / (0.9999 lightyears/year)
• = 4.2 years
•
•
•
•
•
B) The amount of time that passes by for the crew of the ship.
Tship = Tuniverse / gamma
Gamma = 1/(1-(0.9999 c / c)2)1/2
Gamma = 70.7
So, T = 4.2 years / 70.7 = 0.059 years = 21.7 days
Lorenz contraction
• Also, the sizes of moving objects are also
RELATIVE to their velocities in space.
• Linmotion = Lrest / gamma
• Gamma = 1 / (1 – v2/c2)1/2
• So (in the direction they are moving) their length
appears to shrink.
• However their other dimensions stay the same.
• A sphere for example would appear as a
saucer…
Lorenz example
• Linmotion(observed) = Lrest / gamma
• Gamma = 1 / (1 – v2/c2)1/2
• A spacecraft which is 700 m long is traveling
directly towards you at a velocity of 0.999 c.
• What is the observed length of the spacecraft by
an observer on earth?
• What is the length of the spacecraft as observed
by the crew of the craft (hint what is the velocity
of the spacecraft from the perspective of the
crew)?
Lorenz example
• Linmotion = Lrest / gamma
• Gamma = 1 / (1 – v2/c2)1/2
• A spacecraft which is 700 m long is traveling directly towards you at
a velocity of 0.999 c.
• What is the observed length of the spacecraft by an observer on
earth?
• Lobserved = Lrest / gamma = 700 m / gamma
• Gamma = 1 / (1 – v2/c2)1/2 = 1 / (1 – (0.999c)2/c2)1/2
• = 22.4
• So, L = 700 m / 22.4 = 31.3 m
• What is the length of the spacecraft as observed by the crew of the
craft (hint what is the velocity of the spacecraft from the perspective
of the crew)?
• 700 m – it appears at rest to any actually on the spaceship
But what happens…
• If you are traveling a fraction of the speed
of light and something flies by you?
• First, a conceptual question, suppose light
goes by you when you are traveling 90%
of the speed of light.
• What velocity does the light appear to be
traveling?
But what happens…
• If you are traveling a fraction of the speed of light
and something flies by you?
• First, a conceptual question, suppose light goes
by you when you are traveling 90% of the speed
of light.
• What velocity does the light appear to be
traveling?
• - the speed of light! Light always appears to go
the speed of light in a vacuum!
• This is why lengths contract and time slows
down.
• If earth was watching though, they would see
light move past you at 0.1 c faster than you…
Okay now for the equation:
• Warning: the book is REALLY confusing about this…
• If you are moving quickly and shot something out at what
appears to you as a velcoity of Vob, then the actual
velocity of the object is:
• Vactual = (Vyours + Vob)
1 + Vyours * Vob / c2
• Vyours is your velocity compared to the rest frame of the
universe
• Vactual is the object’s (the one you shot out) velocity
compared to the rest frame of the universe
Final sample of day
• You are moving at 75% of the speed of
light and you shot out a probe which
moves with a velocity compared to you of
90% of the speed of light.
• What speed is the probe actually moving
at (compared to rest frame of universe)?
Final sample of day
• You are moving at 75% of the speed of light and
a spacecraft flies by you at 90% of the speed of
light.
• What speed does it appear to be moving at?
• Vactual = (Vyours + Vob)
1 + Vyours * Vob / c2
• Vactual = (0.75c + 0.90 c) / (1 + 0.75 * 0.9)
• Vactual = 1.65 c / 1.675 = 0.985 c (or 98.5% the speed of
light)
Some other famous stuff
•
•
•
•
You have probably heard that E = mc2
Too bad it is not completely correct…
This is only the rest energy of matter.
Yes, this means that matter is a form of
energy!
However
• The total energy is E = Gamma mc2
• And the kinetic energy is:
• KE = (Gamma – 1) mc2
• And momentum is:
• p = gamma * mv
• So, most of relativity is multiplying or dividing by
gamma!
Conclusion
• Relativity is strange but cool, and not as
much math as you might think.
• You basically just have to know how to find
gamma, and apply that to everything.