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Relativity

The General Theory of Relativity

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 (non-inertial) 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

......

a) No acceleration. Rocket (and mobile laboratory) very far from earth (or any other similar body). Apple released by space-person stays near the hand that released it. This situation shows the inertia of the apple. It is an inertial frame of reference.
b)

Rocket and laboratory accelerating at 9·8ms-2.
Now, if the apple is released it will accelerate towards the floor. The bigger (heavier, more massive) apple will do exactly the same thing. We are now in a non-inertial frame.
c) Go back to earth and repeat all this stuff with apples. 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

When Newton formulated his laws (F = mia and F = Gmg(1)mg(2)/r2) he assumed that gravitational mass, mg, and inertial mass, mi, were the same. Experimental evidence agreed with his theoretical predictions and so supported this assumption.

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

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 space-time. The distortion of space-time can be considered as two effects

i) time runs more slowly near the massive body*
ii) space is curved.

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

* Don’t confuse this effect with the "time-dilation" already discussed. 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 Schwartzchild. The Schwartzchild radius is thus a measure of the size of the black hole. It is given by

where M is the mass of the star and G is the universal constant of gravitation.

 

© David Hoult 2008