Astro 105: The Milky Way

Today we examine the second of Einstein's great achievements - The General Theory of Relativity (GR)

(actually it was his third, he contributed to QM with his theory of the photo-electric effect)

We are also going to focus on the strangest of GR's offspring - Black Holes
 
 

The Death of Action at a Distance

Remember that Newton's great achievement was to unite Heaven and Earth with a single Universal Law of Gravitation.

This law

made gravity into a specific and quantifiable force.

But how did this force reach out across space to act on massive bodies? This almost occult-like property of Newtonian gravity bothered everyone, most of all Newton himself.
 

Einstein built on his special theory of relativity and constructed a way of viewing gravity that did away with the problem of action at a distance.

In special relativity Al considered bodies in Uniform (constant velocity) Motion. In general relativity Einstein considers bodies that accelerate.

Al recognized that if you are in a closed elevator that was accelerating through space you would feel a force down towards the floor just as if you where standing on the Earth.
 
 

Einstein realized that there was an equivalence between an accelerating frame of reference and a stationary one which experiences a gravitational field.

From this Einstein inferred something quite amazing.

                        Gravity was intimately linked to the structure of Space-Time.

Special Relativity had taught him to think of Space-Time and malleable entity. He now explored how Space-Time's elastic properties could be used to embrace the phenomena of gravity.

In his new theory gravity becomes the Geometry of Space-Time: this is why understanding 4-D geometry becomes so important.

General Relativity says:

• Space-Time tells matter how to move (fall).
• Matter tells Space-Time how to bend.

A chunk of matter left on to its own device's (no external forces) will follow the curve of space-time.

But what determines the curvature of Space-Time? Its the amount of matter present.

Actually its Matter-Energy (E=mc² after all).

Now this happens in 4-D which is impossible to visualize. But we can play the 1-2-3 game again.

Imagine a 2-D Space-Time, i.e. the surface of a trampoline.

If you drop a bowling ball onto the trampoline it distorts the elastic fabric creating a depression.
 
 

Shoot a BB across the trampoline's surface and it gets defected, perhaps even caught in the bowling ball's "well".

Just like a orbiting satellite or falling meteor!

Space-Time is an elastic fabric which permeates your being.
 

Since its Space-Time we are talking about, both the space and time parts change character depending on where you are in the well.

For instance, time runs slower the deeper you are in the well.
 
 

Experimental tests of GR.

This is a pretty freaky theory. How do we know its true?

The confirmation of GR came in 1918 during a solar eclipse.

If Space-Time gets bent by matter then even light rays should have their paths altered as they travel near a massive body.

By looking at the relative positions of two stars before and during a solar eclipse astronomers could actually confirm that light paths did indeed get bend by exactly the amount Einstein predicted.

It works like this.
 
 

An extreme form of this is "Gravitational Lensing" where light from a distant object gets bend by a foreground objects' gravity the ways a lens focuses light. This produces multiple images of the distant object.
 
 
Good lensing site: http://spiff.rit.edu/classes/phys240/lectures/grav_lens/grav_lens.html

You can also take two synchronized atomic clocks, put one on a plane and leave the other on the ground.

After the plane flies at high altitude for a while you bring the clocks back together and the
clocks are out of synch by exactly the amount Einstein predicted.

Why is GR so hard?

Einsteins' equations relating space-time curvature and matter-energy can appear quite elegant.
 
 

But if you want to actually work with them you end up with stuff like this....
 
 

Black Holes:

OK so what is a black hole?

Remember the collapse of a massive star.... If the core is too massive nothing can hold it up and it collapses to a point.

There are also super massive black holes at the center of galaxies. They can contain more than 10 ⁶ times the mass of the sun

There are two parts to a black hole -An event horizon which surrounds a singularity.
 
 

The event horizon is the place where the escape velocity from the "star" equals the speed of light.

Photons at the event horizon orbit it forever.
 
 

Nothing, not even light, can escape from inside the event horizon.
 

In terms of the bending of Space-Time, a black hole represents so extreme a curvature that Space-Time "tears"
 

The size of the event horizon is determined from the escape velocity formula. Its radius is called the "Schwarzschild radius" (R_s).
 
 

A 1 solar mass black hole would have a "Schwarzschild radius" of about 3 kilometers across.

At the center of the event horizon is the singularity. This is the geometrical point into which all the mass of the original star has collapsed.

A point has no dimensions! Yet that is where all the mass is! This is an abomination to physics. We just don't understand what happens here. The laws we understand break down.

The event horizon protects our universe from the singularity. Anything that falls past the event horizon can't get out.

Actually the roles of space and time reverse inside a event horizon. The coordinate for distance from the center (r) acts like negative time (-t) and visa versa.

This means that once inside the event horizon you are pulled towards the singularity in the same way you are "pulled" towards the future.

Once inside the event horizon you have to hit the singularity.

In a sense the event horizon protects our universe from the singularity (where all hell could literally break loose)

Can there be singularities without event horizons surrounding them?

Physicists wet their pants in terror just thinking about. They have even invoked something called "cosmic censorship" which states singularities must always have event horizons around them.

What is life like near a black hole.

First off - black holes do not suck the entire universe in. If you are not at the event horizon you can, with enough force escape a black hole. Its just like any other gravitating body until you get near R = R_s.

Photons which do make it out from close to a black hole do get very strongly redshifted because the must fight to escape the enormous pull. They lose energy by becoming weaker (longer wavelength) photons.
 
 

If you fell into a black hole you would get torn apart before you actually hit the event horizon.

This occurs because of "tidal forces" which are the difference in the gravitational force between two points.

Near the event horizon the stretching of space-time is so extreme that your feet feel a much stronger acceleration than your head.

You get pulled into long strand of spaghetti.

These extreme forces pull matter apart down to the subatomic level. The violence of this produces and enormous amount of heat. The radiation from this heat is a signature of black holes.
 
 

A person watching from the outside would watch as you fell into the black hole and would see the sequence of events slow down as you approached the event horizon.

At the event horizon time would appear to stand still.

You would appear, however, forever at the lip of the black hole. Of course the light be would be "redshifted" so strongly that it would be impossible to detect.

If you were indestructible and could survive the tidal forces everything would look OK to you just like in Special Relativity.

Wormholes and White Holes.

The equations of GR have the properties that they are time invariant. You can run them one way or the other. It doesn't matter.

That means that the opposite of a black hole, a time-reversed black hole, a white hole, can mathematically exist.

A black hole can only swallow things. A white hole can only spit things out.

It is hard to see how to make one of these.

On the other hand a charged or rotating black hole has the property that in can join up with a white hole in such a way that you can fall into the black hole and pop out of the white hole.

This is a wormhole.
 
 

The worm hole could lead anywhere, to distant parts of the universe, into the past, even to different universes.

Problem is its hard to make one of these, they don't form from ordinary collapse.

The wormholes are also unstable. Any little jolt and they close up. But who knows.....

Black Hole Evaporation:

Black Holes are not totally black! They can actually emit radiation and in doing so they evaporate!

This can happen because of Quantum Mechanics.

Near the event horizon quantum fluctuations can produce pairs of matter and anti-matter particles

electrons (e-) and positrons (e+).

This would not normally violate the conservation of energy because the two particles would collide and annihilate.

But if one of the particles get swallowed by the event horizon and the other escapes then this is a net lose of energy which must come from the black hole's mass.
 
 

The radiation which is produced this way is called Hawking Radiation .

When a black hole evaporates all the photons stuck at the event horizon get freed. Now you would see all that stuff fall into the black hole.
 
 

Detecting Blackholes

The most important way of detecting balckholes is by watching how fast things rotate around them
   

With the doppler shift you can measure rotation (orbital )speeds of material in accretion disks.
rotation speed depends on the mass of thing being orbited.