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