The special theory of relativity is based on the idea that the laws
of physics must be the same for all inertial observers. 

Mach suggested that it should be possible to
reformulate the laws so that they were the same even for observers
in accelerated (noninertial) frames of reference. 

Einstein attempted to
prove that this assumption is valid and the result of his work is
the general theory of relativity (published in 1915). 



The general theory of relativity gave a new
way of looking at gravity. 

It is rather complicated, mathematically, but a few basic ideas
associated with this theory can be introduced here. 



The Principle of Equivalence 

Consider the three situations shown below 





1 
No acceleration. 

Rocket (and mobile laboratory) very far from earth (or
any other similar body). 

Apple released by spaceperson stays near the hand that
released it. 

This situation shows the inertia of the apple. It is an
inertial frame of reference. 


2 
Rocket and laboratory accelerating at 9.8ms^{2}. 

We are now in a noninertial frame. 

Now, if the apple is released it will accelerate
towards the floor. 

The bigger (heavier, more massive) apple will do
exactly the same thing. 


3 
Go back to earth and repeat the observations. 

What do you find? 

The earth’s gravitational field produces exactly the
same effect as the acceleration of the rocket. 




Einstein’s conclusion was that an
accelerated frame of reference must be considered to be
equivalent to a reference frame at rest in a gravitational field. 



The Equivalence
of Gravitational and Inertial Mass 

Newton's second law of motion can be
rearranged to give a definition of inertial mass, m_{i} 



The inertial mass is a measure of how much
force is needed to change the state of the motion of a body. 

Newton's law of universal gravitation also
involved the concept of mass. 

Let's call this mass, gravitational mass, m_{g}
(for obvious reasons!) 

The law leads to the following equation 



In formulating these laws, Newton assumed that
the two masses, m_{i} and m_{g} were equivalent
quantities. 

The fact that all bodies fall with the
same acceleration supports this assumption. 

Einstein’s principle of equivalence
explains why it is reasonable to consider that m_{i}
and m_{g} are the same by stating that acceleration (related
to the idea of inertia) and gravitational field (related to
gravitational mass) are exactly equivalent. 

See also here for more on this. 



Gravity in General Relativity 

In reformulating the laws of physics in
accordance with Mach’s principle, Einstein arrived at a new theory
of gravity. 

He suggested that the presence of a massive
body causes a curvature of spacetime. 

The distortion of spacetime can be
considered to have two effects: 

1. space is curved near a massive
body 

2. time runs more slowly near a
massive body* 



These two effects modify the paths of moving
objects (and of electromagnetic radiations) near massive bodies and
together they form what we call a gravitational field. 



* Don’t confuse this effect with the
timedilation effect. If you are
moving relative to me, I will see your clock as running
slowly and you will see my clock as running slowly.. 
Considering this new effect: if I am very near the
earth and you are far away from the earth, you will measure
my clock to be running slowly and I will
measure your clock to be running fast 




Black Holes 

When a star has "used up" all its nuclear
fuel, it can collapse under the influence of gravity. 

This is thought only to happen to very
massive stars (much more massive than the sun). 

After the collapse, the star becomes very
dense. 

This very dense object is predicted to have
a gravitational field which is so strong that nothing can escape it,
not even light. 



The boundary across which nothing can pass
is called the event horizon and its radius was calculated by Karl
Schwarzschild. 

The Schwarzschild radius is thus a measure
of the size of the black hole. 

It is given by the (perhaps) surprisingly
simple formula 



where M is the mass of the star, c is the
speed of light and G is the universal constant of gravitation. 
