Outline

•  Black Holes – Schwartzchild radius – River Model of a Black Hole – Light in orbit – Tidal forces

Black Holes

•  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

•  A Black Hole is a collapsed region of space

•  Gravity curves space so much that close enough in light is bent so much it always falls in

•  If you get close enough to a blakc hole, you can never get back out

Black Holes

•  Time flows more slowly near a massive object, space is “stretched” out (circumference < 2πR)

•  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

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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.

gravitational constant

speed of light

critical circumference

mass

Schwarzschild radius

The Schwarzschild Radius

How big does a BH need to be to float?

•  Rs = 3 (M/M¤) km

•  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 information can escape red-shifted

red-shifted into oblivion

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.

Event Horizon

•  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

River Model of a Black Hole

River Model of a Black Hole

River Model of a Black Hole

Tides near a Black Hole

What would happen if you fell into a Black Hole the

mass of the Sun? •  Recall that force of gravity is

F = GMm / R2

What would happen if you fell into a Black Hole the

mass of the Sun? •  Recall that force of gravity is

F = GMm / R2 •  But, if you fall feet first, your head is farther

away than your feet, so F2 = GMm / (R+r)2

What would happen if you fell into a Black Hole the

mass of the Sun? •  Recall that force of gravity is

F = GMm / R2 •  But, if you fall feet first, your head is farther

away than your feet, so F2 = GMm / (R+r)2

•  This force will try to stretch you •  How close can you get before this is a

problem?

What would happen if you fell into a Black Hole the

mass of the Sun? •  How close can you get before you get

pulled apart? •  Difference in force, if r << R

F - F2 = dF = GMm (1/R2 – 1/(R-r)2)

What would happen if you fell into a Black Hole the

mass of the Sun? •  How close can you get before you get

pulled apart? •  Difference in force, if r << R

F - F2 = dF = GMm (1/R2 – 1/(R-r)2)

•  Difference in force, if r << R dF = 2GMm r / R3

What would happen if you fell into a Black Hole the

mass of the Sun? •  How close can you get before you get

pulled apart? •  Difference in force, if r << R

F - F2 = dF = GMm (1/R2 – 1/(R-r)2)

•  Difference in force, if r << R dF = 2GMm r / R3

•  Rearrange, R3 = 2GMm r / dF

What would happen if you fell into a Black Hole the

mass of the Sun? •  R = (2GMm r / dF)1/3

•  M = 2*1030 kg, G = 6.673*10-11

•  r = 1m, m = 40kg

What would happen if you fell into a Black Hole the

mass of the Sun? •  R = (2GMm r / dF)1/3

•  M = 2*1030 kg, G = 6.673*10-11

•  r = 1m, m = 40kg •  How much stretching force to kill you?

What would happen if you fell into a Black Hole the

mass of the Sun? •  R = (2GMm r / dF)1/3

•  M = 2*1030 kg, G = 6.673*10-11

•  r = 1m, m = 40kg •  How much stretching force to kill you? •  dF ~ 1000 kg * 10 m/s2 ~ 10,000

•  Plug in numbers, get Rdeath ~ 800 km

What would happen if you fell into a Black Hole the

mass of the Sun? •  Rdeath ~ 800 km •  Schwarzschild radius

RS = 3km for Solar mass BH •  So you die before you get close •  You also get squeezed from the sides •  Called “spaghetti-fication”

What about a bigger Black Hole?

•  Rdeath = (2GMm r / dF)1/3 Rdeath ~ 800 km * (M/Msun)1/3

•  Schwarzschild radius RS= 2GM/c2

RS = 3km * (M/Msun) •  For Milky way center, M = 4 million Msun

•  Rdeath ~ 130,000 km, RS = 12 million km •  You survive! (sort of)

What would happen if you observed Brian fall

into a black hole ???

sketch

What would happen if you observed Brian fall

into a black hole ???

sketch

•  His infall would appear to slow down due to the high gravity.

•  From your point of view, he would stay at the event horizon forever. •  He would look redder due to gravitational redshift.

•  He would be highly stretched from his perspective due to tidal forces (unpleasant!) •  But for him time will not run slowly, he’ll be dragged into the singularity point quickly.