Einstein's Theory of Gravity

Here are some useful WWW links (the source for some of what follows)

Spacetime Wrinkles (from NCSA's Expo/Science & Industry)
Strobel on General Relativity

General Relativity

developed from Special Relativity
universal speed limit = speed of light = c = 300,000 km/sec

Example A: Nolan Ryan on a train:

train moves East at v_train= 30 m/sec (~70 mph)
Nolan, who is on the train, throws his fastball at v_ball= 40 m/sec (90 mph)
Nolan sees the ball move at v_ball= 40 m/sec (90 mph)
We see the ball move at v_train+v_ball= 70 m/sec (~160 mph) from the ground

Example B: Nolan switches on a flashlight

Nolan turns on his flashlight pointing East
Nolan sees the light move at c = 300,000 km/sec
Do we see the light move at v_train+c = 300,000.03 km/sec? NO!!
We also see the light move at exactly c = 300,000 km/sec!
Even if the train moved at v_train= 200,000 km/sec, we'd still see the light move at velocity c!

This is the basis for Special Relativity and it has the following other consequences (which we won't cover in detail)

the clocks on a moving train move more slowly
the twin paradox: one twin can go on a long journey moving at high speeds and return younger than the twin who remained on Earth
fast moving objects appear to shrink
E = mc² (you will need to know this!)

The action-at-a-distance feature of Newton's law of Gravity conflicts with Special Relativity's universal speed limit!

suppose we have a star that suddenly explodes
It's gravitational field will change
Newton's theory says that distant observers can detect the change in gravity instantly!
in apparent violation of the universal speed limit
so, Einstein knew that a new theory was needed

Principle of Equivalence (Einstein's Theory of gravity is based on this + Special Relativity:

Newton knew that gravitational and inertial masses were equal

this is why a feather and a bowling ball fall at the same speed (in a vacuum)

Einstein said that this is because these masses are the same
as explained by the elevator example: one at rest on Earth, and one accelerating in space
both observers feel the same thing because the situation is essentially the same

the natural tendency is to be freely falling or moving
in a straight line in a curved spacetime

Einstein's General Theory of Relativity = his Theory of Gravity

uses curved space + time together: curved spacetime: 3 + 1 = 4 dimensions
4 dimensional curved spacetime is a difficult concept,
but we can visualize a curved 2-dimensional surface embedded in 3 dimensions
consider ants on a balloon or rubber sheet.
in flat space the sum of the 3 angles of a triangle = 180^o
in a positively curved space the sum > 180^o
in a negatively curved space the sum < 180^o
Here's a 2-dimensional example of a positively curved space:

This example also includes a negatively curved space.

This diagram shows that the curvature is highest near a large mass, but it leaves out 1 space dimension and the crucial time dimension:

Physicists and Astronomers like General Relativity because it is elegant and simple:

Here is Einstein's equation:

Unfortunately, the form used for actual calculations is slightly more complicated

Tests of General Relativity

Mercury's Orbit

the point of closest approach (the perihelion) moves in time
due to the gravity of other planets
but 43 arc sec = 0.012^o per century is due to the difference between Einstein's and Newton's Theories

Bending of light by the Sun (or another large mass)

first observed by Eddington in 1919

This is the basis of gravitational lensing:

4 images of a distant quasar due to lensing by the gravity of the galaxy in the middle

Here is a diagram seen from the side showing the lightcurve paths:

Multiple images of a few distant blue galaxies lensed by a dense cluster of galaxies (the yellow ones)

Gravitational redshift:

light rays (i.e. photons) lose energy as they climb out of a gravitational field
So, they shift to larger wavelength , lower energy

This has been observed in white dwarf stars like Sirius B:

Gravitational Energy

Energy is conserved - i.e. the total energy does not change but it can be transferred into a different form
consider a baseball in outer space - very far from the Earth - we'll say infinitely far.

let it go from rest
it will reach a high velocity - and gain lots of energy of motion as it falls
energy is conserved - so where did the energy come from?
Gravity - we assign a negative potential energy to an object in a gravitational field
so, the total energy is still the same as before

lots of energy of motion and
a negative gravitational energy that compensates for this to allow energy conservation

Time reversed case: now lets throw a baseball up from the Earth (ignoring air friction)

I throw it at 20 m/sec - it goes up 20 m and falls back to Earth
Nolan Ryan throws it at 40 m/sec - it goes up to 80m at falls back to Earth
shoot it out of a cannon at 10 km/sec - it goes out beyond the communication satellites and then falls back to Earth
shoot it out of a cannon at 11 km/sec - and it goes up and slows down, but never comes back

This is the escape velocity

Suppose the Earth was squeezed down to half its size, but kept the same mass

The escape velocity would be larger - 15 km/sec in this case
the baseball would slow down from 15 km/sec to 11 km/sec by the time it reached the current radius of the Earth

Suppose the Earth was squeezed down to 1 cm

the escape velocity would be = c
any smaller and its a black hole - nothing can escape!