Stretching of time Stretching of time Quantum mechanics
- Slides: 32
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
How to make light go straight Cut the elevator cable 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
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) Karl Schwarzschild
• Schwarzschild solution At large distances: It reduces to Netwon’s laws That’s where gravity is weak R C = 2πR
• Schwarzschild solution 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: 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
Stars. . . • 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
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=2 GM/c 2 curvature so strong it bends spacetime inside out • Everything must Space and time switch roles inside go inward! R s: 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
Black holes • Make a star smaller than Rs=2 GM/c 2 curvature so strong it bends spacetime inside out • Inside Rs everything moves inward • No information can come back out ⇒“Event horizon” • Even • Not light must stay inside light can escape ⇒“black hole”
Black holes • Make a star smaller than Rs=2 GM/c 2 curvature so strong it bends spacetime inside out • Inside Rs everything moves inward • No information can come back out ⇒“Event horizon” • Even • Not light must stay inside light can escape ⇒“black hole”
Black holes Make a star smaller than Sun: Rs = 3 km (2 miles) • Earth: Rs = 1 cm (1/3 of an inch) • Milkyway: Rs = 1/2 lightyear galaxies Bla When does an object become a black hole? Mass curvature so strong it bends spacetime inside out ck hol Rs=2 GM/c 2 • galaxy clusters es • neutron star solar system stars white dwarf earth Radius
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/kepl er 6. 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
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)
Tides: • • • Special relativity holds in a tiny, freely falling elevator But gravity is not uniform Different falling elevators accelerate at different rates ⇒ Spacetime is why curved That’s we(every needed General observer. Relativity is different) in the first place!
• • Tides near a black hole Black hole pulls on your feet stronger than on your head Your body will follow spacestretching Very slimming Very unhealthy
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- Wave reflection formula
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