Fast forward to the future

Special relativity has thrownup a number of intriguing and bizarre
concepts, chief among which is the idea that time slows down for
fast moving objects. One important aspect of this strange effect
that I have not mentioned so far is that it gives us a way of ‘fast
forwarding’ through time: to travel through time into the future!
So let’s take a closer look at this. Over the years, relativity theory
has provided a rich source of debate and discussion, and not just
among physicists. But by far and away the most puzzling, most
debated, and yet still most misunderstood of all its consequences
is known as the clock paradox, or the paradox of the twins. I
shall give a brief outline of it here and show why there is really no
paradox at all.
Meet the twins Alice and Bob. Alice is the adventurous one
who enjoys travelling around the Galaxy in her high speed rocket while Bob prefers to stay at home. One day, Alice bids her brother
goodbye and heads off in her rocket to the Alpha Centauri system
four lightyears away, travelling at two thirds the speed of light.
Bob monitors her progress and calculates that she should reach
her destination in six years’ time. Once she gets there she will
turn around and head straight back. Taking into account the turnaround
time he expects the round trip to take a little over twelve
years. He is frustrated however by the messages he receives from
her. Not only is there an increasing time delay due to the widening
distance between them, but they are also Doppler shifted towards
longer wavelengths. From the rocket’s speed he works out how
much of a shift there should be and takes it into account. However,
the wavelengths are still too long and he quickly realizes that this is
due to the relativistic effect of time dilation. To him, time on board
her rocket is running a little slower than his and this manifests itself
in a longer wavelength in the signal. Taking this slowing down
of the rocket’s time into account, Bob calculates that according to
Alice the journey should take just nine years, three years less than
the duration of the journey according to Earth time. This would
mean that, on her return, Alice will be three years younger than
her twin brother! This is because time dilation is not something
that affects only moving clocks, but all time on board the rocket,
including Alice’s biological clock.
This is, in fact, not the source of the ‘paradox’ of the title of
the story. Bob has quite correctly used the equations of special
relativity and computed the time difference between his clocks and
his sister’s. No, the paradox, or what at first sight appears to be
a paradox, is that Alice does not believe her brother’s predictions.
She argues that the first principle of relativity is being violated
here. Surely, since all motion is relative, she has just as much right
in claiming that it is not her rocket that is moving away fromEarth,
but the Earth that is moving away from the rocket. It is Bob who is
moving at two thirds light speed and it is his clocks that are running
slower. She therefore claims that, on her return, she should expect
her brother to be the younger of the two. This apparent symmetry
has been the source of much confusion over the years. Both twins
cannot be right, can they?There are many ways of correctly resolving the problem. I
will mention here the simplest one. The answer is that Bob is right
and Alice is not. She will indeed return younger than her brother.
Many books on relativity will state that this is because Alice is the
one who must undergo acceleration and deceleration in the rocket
and it is this that breaks the symmetry between the two twins.
This is true, but saying that their situations are not the same is not
explaining anything. The reason Alice ages less can be explained,
not because of time dilation, but length contraction. To her, the
distance to Alpha Centauri is not four lightyears but only three, and
travelling at two thirds light speed means she can make the trip
there in only four and a half years instead of Bob’s estimate of six.
Areturn journey of a further four and a half years means the whole
trip will take nine years, just as Bob had calculated from the time
dilation of her clocks. The reason whyAlice gets the wrong answer
by appealing to the fact that she sees Bob’s clocks run slower, is
that she is not using the equations of special relativity correctly.
They only apply to observers who do not change their speed or
direction. She does, Bob doesn’t.
A time difference of three years on Alice’s return may not
sound very impressive so let us assume she had been travelling
even faster, at say 99% of the speed of light. She would now return
to Earth (if we ignore turn-around time) after eight years and one
month of Earth time (which is one month longer than it would
take light to complete the trip). But according to Alice, the trip
would take only one year. If she had decided to travel further
afield at this speed on a journey that, for her, would take ten years,
then she would find on her return that eighty years had elapsed
on Earth and that Bob, along with almost everyone she knew, had
already died. She, on the other hand, would be just ten years
older than when she left. This is a clear example of time travel
into the future. If her rocket could have been nudged even closer
to the speed of light she would have returned thousands, or even
millions, of years into the future. So, forget Oil of Ulay. Just hop
on board a fast moving rocket and zip around the solar system for
a while. Friends will be amazed at how young you have kept!
It is sometimes thought that the symmetry between Alice and
Bob’s motion is retrieved if we consider what things look like to a third observer, say a passing space traveller. Wouldn’t he see Alice
and Bob flying apart and back together again? If he is moving in
the same direction as Alice but at half her speed relative to Earth, he
will see the twins moving away from him in opposite directions at
the same speed: a symmetric picture. The problem is that Alice has
to return. If the space traveller continues at the same speed in the
same direction he will see Bob continue to move away from him,
but Alice will eventually turn around and come towards him. She
will pass him on her return journey. Thus the symmetry is broken.
For several years now I have set the twins paradox as a
coursework assignment for my students at Surrey. They are asked
to investigate it using different approaches. So far, the best way
I have seen has been to use the third observer as an adjudicator.
The maths works out very nicely.
I have come across many people who initially think that this
sort of time travel into the future implies that the future must be
already out there, existing alongside our present. This is not the
case here. What is happening is that the future is unfolding on
Earth all the time that Alice is away. It is just that, since less time
elapses for her, she is moving on a different time track to Earth’s.