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Stretching of time
Stretching of time•Quantum mechanics:
Particles are wave packets with wavelength and frequency
Particle frequency is a “clock”: frequency = ticking rate
Higher energy = higher frequency
Stretching of time•Quantum mechanics:
Particles are wave packets with wavelength and frequency
Particle frequency is a “clock”: frequency = ticking rate
Higher energy = higher frequency
• Drop particle from top of tower
It picks up speed, gains energy
It picks up frequency
Compare to particle at bottom: clock from top ticks faster
Stretching of time• Clock in gravitational
field go slower
Clocks in space go faster than on ground
GPS satellites: extremely accurate clocks
Easily measure gravitational time dilation
Cut the elevator cable
How to make light go straight
g
Then, light will go straight through the elevator
Freely falling objects
• In a freely falling frame, light travels on straight lines
• Light travels on geodesics
⇒ Freely falling frames/objects travel on geodesics as well
This is Einstein’s version of Newton’s first law
Different starting velocity, different geodesic
So, light must travel on very special geodesics
Orbits as free-fall
• Planets orbit the sun, pulled by gravity only
• They are in free fall (no other force)
• Planet orbits are geodesics
• There are many different geodesics/orbits
This astronaut is in free fall!
Spacetime around a star
• A “star” is isotropic (the same in all directions)
Mass
Radius
• Spacetime around a star must be isotropic
What is the curvature of spacetime around a star?
What orbits do planets, particles, photons follow?
What are the geodesics?
Schwarzschild solution
• January 1916 in army hospital
2 months after Einstein invented GR
Died 4 months later
•Solved the field equations
Spacetime structure around spherical stars
Describes how matter and light behave around stars (they follow geodesics)
Far reaching implications...
Karl Schwarzschild
• At large distances:
It reduces to Netwon’s laws
That’s where gravity is weak
Schwarzschild solution
C = 2πR
R
• At large distances:
It reduces to Netwon’s laws
That’s where gravity is weak
• Close to star:
Curvature stretches space: circumference of a circle C < 2πR
Curvature stretches time: clocks go slower
Add more mass: get more curvature
Schwarzschild solution
R
Weak gravity...
Stars...• Stars are big:
Solar radius 430000 miles
Too big for any “extreme” properties to show
⇒ Slight effects only
•Orbits = geodesics
“Almost” ellipses: Not closed (they “precess”)
Light bending: stars behind sun slightly out of position
•Mercury orbit:
Closest to sun: Strongest effect
Observed to precess once every 23000 yrs
Inconsistent with Newton’s laws
Perfectly consistent with General Relativity
Stars...
• Experiment during 1919 eclipse
Eddington detected light deflection
Initial accuracy relatively poor
Confirmed later by radio imaging
Sir Arthur Eddington
Stars...
Relativistic stars
Relativistic stars• What happens when you make a star
smaller and smaller?
Effects become stronger and stronger...
Light should go round and round...
Clocks should go slower and slower...
Relativistic stars• What happens when you make a star
smaller and smaller?
Effects become stronger and stronger...
Light should go round and round...
Clocks should go slower and slower...
Relativistic stars• Make a star smaller than
Rs=2GM/c2
curvature so strong it bends spacetime inside out
• Space and time switch roles inside Rs:
•What is our time becomes space
•Forward in time on our clock means inward in radius for someone inside Rs
•That means: Anything inside must continue to move inward
EverythingEverything must go must go inward!inward!
Black holes• Make a star smaller than
Rs=2GM/c2
curvature so strong it bends spacetime inside out
• Inside Rs everything moves inward
•No information can come back out
⇒“Event horizon”
•Even light must stay inside
•Not light can escape
⇒“black hole”
Black holes• Make a star smaller than
Rs=2GM/c2
curvature so strong it bends spacetime inside out
• Inside Rs everything moves inward
•No information can come back out
⇒“Event horizon”
•Even light must stay inside
•Not light can escape
⇒“black hole”
• Make a star smaller than
Rs=2GM/c2
curvature so strong it bends spacetime inside out
• When does an object become a black hole?
•Sun: Rs = 3km (2 miles)
•Earth: Rs = 1cm (1/3 of an inch)
Milkyway: Rs = 1/2 lightyear
Black holes
earth
whitedwarf
starssolar
systemneutron
star
galaxies
galaxycluster
s
Bla
ck h
oles
RadiusM
ass
Black holes• What happens near Horizon?
To us: Clocks stop at Rs
⇒ Light emitted at Rs has zero frequency
To us: Matter “freezes” at Rs
We never see it fall in
• To the infalling matter:
• Infalling clock ticks infinitely slowly
Infall takes a very short time
• Once inside, the only way is in
Kepler motion• Explore Kepler orbits around Newtonian stars with the
following applet:
• http://galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/kepler6.htm
Tides:•Moon pulls on one side of
earth more strongly
This causes the tides
•This means:
Gravitational acceleration changes from place to place
Curvature changes from place to place
No universal freely falling frame
• Special relativity holds in a tiny, freely falling elevator
• But gravity is not uniform
• Different falling elevators accelerate at different rates
⇒ Spacetime is curved (every observer is different)
Tides:
• Special relativity holds in a tiny, freely falling elevator
• But gravity is not uniform
• Different falling elevators accelerate at different rates
⇒ Spacetime is curved (every observer is different)
Tides:
• Special relativity holds in a tiny, freely falling elevator
• But gravity is not uniform
• Different falling elevators accelerate at different rates
⇒ Spacetime is curved (every observer is different)
That’s why we needed General Relativity in the first place!
Tides:
Tides near a black hole
• Black hole pulls on your feet stronger than on your head
• Your body will follow space-stretching
Very slimming
Very unhealthy