Twin Paradox, The inner product of special relativity

1. TWIN PARADOX
The Inner Product of Special Relativity
by
Hamid Kiabi
Mathematicians develop models to describe various situations. The aim of
mathematician is to develop the mathematical model that most accurately
describe the situation. This model can be used to further our understanding of
the situation. Special Relativity was developed by Albert Einstein is an attempt to
describe the physical world that we live in. At the time Newtonian mechanics was
the theory used to describe the motion of the bodies under forces. However
experiments had led the scientists to believe the motion of the bodies such as
planetary motions, were not accurately described by the Newtonian mechanics.
One of Einstein’s main contribution to science was the development of more
accurate mathematical models. First he developed special relativity, which did
not incorporate gravitation; later on he incorporated gravitation in his theory of
general relativity.
As the title of this short article suggested, it is about special twins. Not just
any twins, the twins are so famous that they were given special name, “TWIN
PARADOX”. The story of our twins starts when they were separated from each
other early in their life. Twin 2 was chosen to travel abroad a spaceship traveling
at 0.8 of the speed of light to the Alpha Centauri which happens to be the nearest
star to earth other than the sun. While Twin 1 remained on earth. Twin2
immediately after arriving at the Alpha Centauri decided to return, and be with
his beloved brother.
When he joins his brother on earth, he finds that his twin on earth is ten years
old (in every sense of the word), while he himself will be only six years old. (These
times would vary according to the speed of Twin 2)! There is experimental
verification of this phenomenon. Physicists have found that certain radioactive
particles moving in the atmosphere decay more slowly than ones at rest on earth.

2. The effect is not likely to be realized on this scale by humans, since the energies
required to produce such high speeds in macroscopic body are prohibitive.
Lets now examine how this phenomenon arises out of Einstein’s model. The
model is one of space-time; hence it involves four coordinates-three space
coordinates,(x1,x2,x3,), and a time coordinate x4. As a result we need to use vector
space notation R4
. There are whole number of operation that can be defined on
a vector space Rn
, for instance the dot product is not the only inner product that
can be defined on Rn
.
For the case of the twins will develop an alternative inner product that will use
for the vector space R4
, where R4
represent the space time, each element of R4
is
an event. Each event has a location in space given by the x1,x2, and x3 and occurs
at a certain time x4. Also there is an additional mathematical structure on R4
, an
inner product. For two arbitrary elements (x1, x2, x3,x4) and (y1, y2, y3, y4) it is
defined as follows:
(x1, x2, x3,x4) . (y1, y2, y3, y4) = x1 y1 - x2 y2 - x3 y3 + x4 y4
Using the inner product, we can now find the norm1
of a vector (x1, x2, x3,x4):
|| (x1, x2, x3,x4) || = [| (x1, x2, x3,x4) |. | (x1, x2, x3,x4) |] 1/2
= [ | (-x1
2
-x2
2
-x3
2-
x4
2
) | ] 1/2
R4
with these structure s is called a Minkowski space. The sweeping innovations in
special relativity were the introduction of the inner product and norm on a four-
dimensional space, implying that space and time are not completely

3. independent, as was assumed in earlier Newtonian model. The physical
interpretation of the norm leads to prediction of an age difference between the
twins. To illustrate, a space-time diagram is shown below( Figure-1), for
convenience, assume that the Alpha Centauri lies in the direction of the x1axis
from the earth. The wins on earth advances in time x4, while the twin in the
rocket advances in time and also moves in the direction of x1. The space-time
diagram shown below in Figure-1.
X4
Q
twin1
R (Alpha Centauri)
twin 2
P
X1
Figure-1
The path of Twin 1 is PQ; he is advancing only in time, x4 . The path of Twin 2 to
Alpha Centauri is PR, advancing in time and in the direction of increasing x1. His
return path to earth is RQ; he rejoins his earth twin at Q. There is no motion in
either the x2 or the x3 coordinate, hence we can suppress these dimensions in the
diagram. We can use the inner product of the special relativity to find certain
norms for vectors; that is, we can calculate lengths of various vectors such as PR.

4. The theory gives a physical interpretation to such lengths. They are the actual
times recorded by the observers moving along these paths. For example, the
length PQ is the time recorded by the Twin 1 in traveling between P and Q. The
length of PR is the time recorded by Twin 2 in traveling between P and R, that is,
from earth to Alpha Centauri; PR + RQ is his age at Q.
Now lets first look at the Twin 1, which stayed on earth with his mother. The
rocket ship travels with 0.8 the speed of the light relative to earth and covers a
distance of 8 light years ( 4 there and 4 back). Thus the round trip, from the point
of view of earth takes 8/0.8 years, that is 10 years. (Time = distance/velocity.) The
age of Twin 1 at Q is 10 years.
We now examine the situation for Twin 2. Let point P be the origin in R4
, (0,0,0,0)
(Figure-2). Q is the point (0,0,0,10) and S is (0,0,0,5).
X4
Q (0,0,0,10)
twin1
S R (4,0,0,5)
twin 2
P (0,0,0,0) X1
Figure -2

5. SR is the spatial distance of Alpha Centauri from earth-four. Thus R is the point
(4,0,0,5) and PR is the vector (4,0,0,5).
|| PR || = [ | (4,0,0,5) . (4,0,0,5) | ]1/2
= [ |-(4)2
- 0 – 0 +(5)2
]1/2
= [ | -16 +25 | ] 1/2
= 3
By the symmetry of the situation, || RQ || = 3.
Then the length PR + RQ is six; therefore the age of Twin 2 when the twins meet
is six!
This model introduces a new kind of geometry in which the straight line is not
necessarily the shortest distance between two points. In Figure -2 the straight
line distance is P and Q is ten, whereas the distance PRQ is six, a smaller distance.
In fact, it turns out that the straight line distance PQ is the longest distance
between P and Q! Thus not only is the physical interpretation of this model
fascinating , but it opens up a new trend in geometrical thinking.
In the general theory of relativity, gravity is taken into account and is represented
mathematically by an inner product. The space becomes curved in nature; curves
leads to extreme distances between points instead of straight lines.

6. If interested consult the very readable article by Alfred Schild entitled “The clock
paradox in Relativity Theory” in the American Mathematical Monthly Vol. 66, No.
1 January 1959. Two books that make very enjoyable light reading on geometries
and dimensions are Flatland, by Edwin Abbott a romance of many dimensions;
and Spherland , by Dionys Burger, Thomas Crowell, a fantasy about curved spaces
and expanding universe.

7. If interested consult the very readable article by Alfred Schild entitled “The clock
paradox in Relativity Theory” in the American Mathematical Monthly Vol. 66, No.
1 January 1959. Two books that make very enjoyable light reading on geometries
and dimensions are Flatland, by Edwin Abbott a romance of many dimensions;
and Spherland , by Dionys Burger, Thomas Crowell, a fantasy about curved spaces
and expanding universe.