**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 l_{0},
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 *- v ^{2}/c^{2}) ^{˝}*

^{ }

* l = l _{o} *(1

^{ }

^{ }

^{ }*t =
t_{o}*

(1 *- v ^{2}/c^{2}) ^{˝}*

* *

**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
10^{4} ms^{-1}

*(b) *the starship has a velocity of 2 x 10^{8} ms^{-1}

(c) the starship has a velocity of
2.5 x 10^{8} 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?