Stationary Elevator with gravity: Ball is accelerated down

Outside of an accelerated elevator: Ball at rest

Inside of accelerated elevator: ball accelerated down

=elevator with gravity!

General relativity Einstein’s fundamental insight: “ Equivalence principle” Gravity accelerates everything ⇒ Gravity must be a property of spacetime Gravity and acceleration are indistinguishable (Galileo) ⇒ Formulate physics in terms of accelerated frames

Equivalence principle

Elevator at rest

Elevator in uniform motion

Inside the moving elevator

Accelerated elevator from outside

Inside the accelerated elevator

= In an elevator in a gravitational field

Light bending: Gravity bends light Recall: light travels on spacetime geodesics ⇒ In spacetime with gravity, geodesics are curved Geodesics are the straightest possible lines ⇒ Gravity curves spacetime

Spacetime curvature

time spac e Spacetime curvature

In curved space Parallel lines don’t stay parallel Triangles don’t add up to 180° The straightest possible lines are “geodesics” The stronger the curvature, the stronger theses effects

In curved spacetime The actual length to a destination is changed (try this yourself!) The circumference of a circle is no longer 2πR (try this yourself!) Sometimes, more than one path is the shortest path (try this yourself!)

What curves spacetime? Gravity curves spacetime We know that mass causes gravity ⇒ Mass curves spacetime

What curves spacetime? Einstein’s most fundamental equation relates the curvature to mass: More mass, more curvature More curvature closer to mass Einstein’s equivalent to Newton’s law of gravity “Field equation”

Is space curved?

Light bending (2): Heavy objects curve spacetime Galaxy clusters are very heavy: 1000 trillion times more massive than the sun They should curve spacetime a lot Light should follow curved path around them

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!

Kepler motion Kepler motion applet

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 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 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

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 R s =2GM/c 2 curvature so strong it bends spacetime inside out Space and time switch roles inside R s : What is our time becomes space Forward in time on our clock means inward in radius for someone inside R s That means: Anything inside must continue to move inward Everything must go inward!

Black holes Make a star smaller than R s =2GM/c 2 curvature so strong it bends spacetime inside out Inside R s 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 R s =2GM/c 2 curvature so strong it bends spacetime inside out Inside R s 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 R s =2GM/c 2 curvature so strong it bends spacetime inside out When does an object become a black hole? Sun: R s = 3km (2 miles) Earth: R s = 1cm (1/3 of an inch) Milkyway:R s = 1/2 lightyear Black holes earth white dwarf stars solar system neutron star galaxies galaxy clusters Black holes Radius Mass

Black holes What happens near Horizon? To us: Clocks stop at R s ⇒ Light emitted at R s has zero frequency To us: Matter “freezes” at R s 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

Light paths Radius Time light cone

Light paths Radius Time

Kepler motion Explore Kepler orbits around Newtonian stars with the following applet: er6.htm er6.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