Principle of Equivalence: Einstein 1907

Box stationary in gravity field

Box falling freely

g

g

g

Box acceleratesin empty space

Box moves through space at constant velocity

Equivalence Principle

• Special relativity: all uniformly moving frames are equivalent, i.e., no acceleration

• Equivalence principle: Gravitational field = acceleration

freely falling frames in GR = uniformly moving frames in SR.

Tides

• Problem:

• Gravity decreases with distance => stretch…

r1

r2

moon

• Tides = gravity changes from place to place

freely falling

not freely falling

not freely falling

?

???

Tides

Light rays and Gravity…• Remember: gravity bends light…

accelerating observer = gravity

Light Rays and Gravity II

• In SR: light rays travel on straight lines

=> in freely falling fame, light travels on straight

lines

• BUT: to stationary observer light travels on curved paths

=> Maybe gravity has something to do with…

curvature of space ?

Curved Spacetime

• Remember: Gravity warps time

slow

fast

BUT: in spacetime, time and space are not separable

=> Both space and time are curved (warped)

This is a bit hard to vizualize (spacetime already 4D…)

GR: Einstein, 1915

• Einstein: mass/energy squeeze/stretch spacetime away

from being “flat”

• Moving objects follow curvature (e.g., satellites, photons)

• The equivalence principle guarantees:

spacetime is “locally” flat

• The more mass/energy there is in a given volume, the

more spacetime is distorted in and around that volume.

GR: Einstein, 1915

• Einstein’s “field equations” correct “action at a

distance” problem:

Gravity information propagates at the speed of light

=> gravitational waves

r?

• Imagine being an ant… living in 2D

• You would understand:

left, right, forward, backward,

but NOT up/down…

• How do you know your world is curved?

Curvature in 2D…

• In a curved space, Euclidean geometry does not apply:

- circumference 2 R

- triangles 180°

- parallel lines don’t stay parallel

<2R

R R

2R

=180

Curvature in 2D…

Curvature in 2D…

Curvature in 2D…

Geodesics

• To do geometry, we need a way to measure distances

=> use ant (let’s call the ant “metric”), count steps it

has to take on its way from P1 to P2 (in spacetime, the

ant-walk is a bit funny looking, but never mind

that)

• Geodesic: shortest line between P1 and P2

(the fewest possible ant steps)

P1 P2

ant

• To the ant, the geodesic is a straight line,

i.e., the ant never has to turn

• In SR and in freely falling frames, objects

move in straight lines (uniform motion)

• In GR, freely falling objects

(freely falling: under the influence of gravity only,

no rocket engines and such; objects: apples, photons,

etc.)

move on geodesics in spacetime.

Geodesics

Experimental Evidence for GR

• If mass is small / at large distances, curvature is weak

=> Newton’s laws are good approximation

• But: Detailed observations confirm GR

1) Orbital deviations for Mercury (perihelion precession)

Newton: Einstein:

2) Deflection of light

Experimental Evidence for GR

Experimental Evidence for GR

• What happens as the star shrinks / its mass increases? How much can spacetime be distorted by a very massive object?

• Remember: in a Newtonian black hole, the escape speed simply exceeds the speed of light

=> Can gravity warp spacetime to the point where even light cannot escape it’s grip?

That, then, would be a black hole.

Black Holes

Black Holes

• Time flows more slowly near a massive object,

space is “stretched” out (circumference < 2R)

• Critical: the ratio of circumference/mass of the object.

If this ratio is small, GR effects are large (i.e., more mass within same region or same mass within smaller region)

Black Holes

???

???

1) massive 2) small

• GR predicts: If mass is contained in a circumference smaller than a certain size

space time within and around that mass concentration qualitatively changes. A far away observer would locate this critical surface at a radius

• Gravitational time dilation becomes infinite as one approaches the critical surface.

kilometersMMc

GMR SunS )/(3

22

2

22

c

GMnceCircumfere

gravitational constant

speed of light

criticalcircumference

mass

Schwarzschild radius

The Schwarzschild Radius

• To a stationary oberserver far away, time flow at the critical surface (at RS) is slowed down infinitely.

• Light emitted close to the critical surface is severely red-shifted (the frequency is lower) and at the critical surface, the redshift is infinite.

From inside this region

no informationcan escape red-shifted

red-shifted intooblivion

Black Holes

• Inside the critical surface, spacetime is so warped that objects cannot move outward at all, not even light.

=> Events inside the critical surface can never affect the region outside the critical surface, since no

information about them can escape gravity.

=> We call this surface the event horizon

because it shields the outside completely from any events on the inside.

Black Holes

• Critical distinction to the Newtonian black hole:

Nothing ever leaves the horizon of a GR black hole.

• Lots of questions…

What happens to matter falling in?

What happens at the center?

Can we observe black holes anyway?

And much, much more…

Newton Einstein

Black Holes