Theory of relativity

The theory of relativity proposed by Albert Einstein in 1905 deals, as its name suggests, with measurements made in systems that are in motion relative to each other. Such a system is known as a frame of reference. For example, consider a train moving along a railway track. Measurements made in the train are relative to the train and the train is therefore one frame of reference, while measurements made at the side of the track are relative to the ground, the other frame of reference.

While you read this you are probably sitting still - but 'still' relative to what? Even if you are in a building, the building is fixed to the Earth which is itself rotating and also travelling round the Sun. The Sun is moving within our galaxy and the galaxy is moving relative to others in the universe.

There are basically two types of relativity:

    (a)  special relativity, that deals with frames of reference in uniform relative motion, and

    (b) general relativity, that deals with frames of reference in non-uniform relative motion, for example, one frame accelerating relative to the other.

The laws of Newtonian Physics, such as F = ma, hold very well for our everyday lives but at speeds close to that of light considerable divergences occur. Relativity is not considered in detail here, but some of its important facts and consequences are briefly surveyed. Certain basic facts must be assumed:

The two postulates

(a) physical laws are obeyed in all frames of reference,

(b) the velocity of light in a vacuum is constant in all inertial frames of reference.

If we assume these, then we must abandon some of our other more traditional ideas, namely the constancy of mass, length and time. This means that if one object is moving relative to a frame of reference, then the mass and length of the body measured from the frame of reference will be different from those measured with instruments travelling with the body. Even more unusual, time measured by a clock travelling with the body will differ from that measured by a clock at rest in the frame of reference! Length appears to get smaller, mass increases and time appears to pass more slowly in a moving frame of reference when viewed from a stationary frame. Any such differences are very small, however, unless the relative velocities are very large, that is, approaching that of light.

The consequences of special relativity

Imagine for a moment that we live in a world where the velocity of light is small, say 32 m s^-1 (about 70 m.p.h.). Then the predictions of the theory of special relativity would become much more obvious.

If we stood at a street corner in this strange world and watched traffic passing by, then all the cars would appear shortened and even people would appear a little thinner than they were when standing still.

If you tried to pull a trolley along with a rope, not only would it seem to get thinner but you would also find that as you went faster and faster the trolley would seem heavier and heavier and so become more difficult to accelerate.

Imagine that you had gone to the station in the morning to say goodbye to your friends who were going by train to the nearest town (at no more than 70 m.p.h.) and agreed to meet them there in the evening. They would seem to have aged little, but to them you would have looked a lot older - time for a moving frame of reference runs more slowly than for one at rest!

Effects similar to these have been observed in the real world, but because of the very large velocity of light they are much more difficult to see.

The slowing down of time has been noticed in atomic clocks that have been carried in satellites and some high‑energy fundamental particles have been observed at sea level even though knowledge of their half‑life indicates that they should have decayed long before they reached the ground.

The increase of mass becomes a problem in high energy accelerators where as the particles approach the speed of light they become more and more difficult to accelerate further. Even the electrons in our colour television tubes are moving so fast that their actual masses are some 21 per cent heavier than those of electrons at rest.

The equations of special relativity

Consider an object of rest mass m_o and length l₀, moving with velocity v relative to a stationary frame of reference.

The following equations give the mass m, length l and time t as measured from the frame of reference:

m = m_o

(1 - v²/c²) ^½

l = l_o (1 - v²/c²) ^½

t = t_o

(1 -  v²/c²) ^½

Example

An interstellar starship travels through space at high speed.

Find:

the mass of an object with a rest mass of 1kg,

the length of a bar of rest length 1 m and

the new value of the second

when all these quantities are measured relative to a frame of reference at rest outside the starship.

Consider three cases:

(a) the starship has a velocity of 10⁴ ms^-1

(b) the starship has a velocity of 2 x 10⁸ ms^-1

(c) the starship has a velocity of 2.5 x 10⁸ ms^-1

Answer:

Using the above equations we have:

(a) m = 1.00 kg, l = 1.00 m, t = 1.00 s

(b) m = 1.34 kg, l = 0.75 m, t = 1.34 s

(c) m = 4.84 kg, l = 0.20 m, t = 4.84 s

General relativity

The theory of general relativity does not restrict itself to frames of reference that are moving relative to each other at a constant velocity. In the general theory accelerated frames are considered, as are gravitational fields ‑ in fact, it can be shown that the effects of acceleration and gravitational field are equivalent. General relativity theory predicts the following:

(a) that light bends in a gravitational field - light just grazing the surface of the Sun has been observed to be deviated by some 1.75 seconds of arc;

(b) that the perihelion of Mercury, (its nearest point to the Sun), shows precession;

(c) that physical processes such as the vibrations within an atom are slowed down in a high gravitational field, and therefore the light coming from the stars is reddened slightly.

According to the general theory, if a very large triangle were to be surveyed near the Sun or other large astronomical body the angles would not add up to 180% suggesting that space is curved in a gravitational field!

This last idea extends the problem to the whole universe, and so I will end this with some questions ‑ maybe one day one of you who read this will find the answers.

If space is actually curved, which way does it curve?

Has space a positive curvature like the surface of the Earth and therefore a finite size? Or has it a negative curvature like the saddle between two mountain peaks?

Is the universe expanding without limit? Or is it pulsating, so that one day it will collapse back on itself and then expand once more and so on for ever?

If it does this, will another human race like ourselves develop to populate the planet we call Earth ‑ if another Earth ever exists?