An ant is sitting at one end of an
elastic string, which is one metre long.

It starts walking towards the other end at a speed of one centimetre per second.

At the end of every second, the string is instantaneously and uniformly stretched by an additional one metre.

It starts walking towards the other end at a speed of one centimetre per second.

At the end of every second, the string is instantaneously and uniformly stretched by an additional one metre.

Question: Does the ant ever reach the
other end of the string?

At first sight, the answer seems an obvious “No”. After all, the ant moves only a centimetre every second and then its destination recedes by a metre, so clearly not possible.

But here’s the rub – notice the phrase “uniformly stretched”. This changes everything.

For instance, after one second the ant is
1 cm away from the starting point. But when the string is stretched, its length
doubles.

Since the stretching is uniform, it means
the ant gets pulled an

*additional*1 cm, which means it is now*two*centimetres from the starting point.
So, while the destination keeps moving
away, the uniform stretch also pulls the ant somewhat further along each time.

So, which effect wins?

The quickest way to solve this is to work out the

Why ?

Let’s work it out.

At the end of 1 second, the ant has covered 1/100 of the string.

At the end of 2 seconds, it has covered another 1 cm, but the string is now 2 metres long – so the additional fraction is only 1/200. Similarly, at the end of 3 seconds, it covers another 1/300.

The quickest way to solve this is to work out the

*fraction*of the string’s length traversed by the ant at the end of each second.Why ?

*Because uniformly stretching the string leaves this fraction unchanged !**If the fraction ever becomes 1, the ant has reached the other end.*

Let’s work it out.

At the end of 1 second, the ant has covered 1/100 of the string.

At the end of 2 seconds, it has covered another 1 cm, but the string is now 2 metres long – so the additional fraction is only 1/200. Similarly, at the end of 3 seconds, it covers another 1/300.

Thus, the cumulative fraction of covered
by the ant looks like 1/100 + 1/200 + 1/300 +…

Does this ever reach 1 ?

For this we need a little help from the mathematician, who informs us that the
sum of the series (1 + 1/2 + 1/3 +…. )

*diverges*, which means that if you add up the series far enough, it will exceed any number you can think of – a hundred, a million, a googolplex, whatever.
How does this help ? Well, notice that the fraction 1/100 + 1/200
+ 1/300 +… is simply

1/100*(1 + 1/2 + 1/3 +….) and we need
this to reach 1.

But this just means that the term in brackets should eventually reach 100, and our resident mathematician has assured us that it will.

So we are done – the ant

How long does it take ?

But this just means that the term in brackets should eventually reach 100, and our resident mathematician has assured us that it will.

So we are done – the ant

*does*reach the other end.How long does it take ?

This can also be worked out with some
effort and the answer is – about 2.7*10

Hey, I never said this would be quick.

^{43}seconds, which is about 9*10^{35}years or in other words 90 million billion billion times the lifetime of the known universe.Hey, I never said this would be quick.

What if the ant were moving at one

What if the string stretches by a kilometre every second ?

*millimetre*per second ? It still gets there, except that it now takes about 9*10^{350}years, which I won’t even bother to convert to millions and billions.What if the string stretches by a kilometre every second ?

Still the same, just longer. (A

What if the string was a trillion kilometres long at the start ? Same conclusion.

Now let’s go back to the original problem. Since we know the ant does reach the other end, the new question is: What is the total distance the ant has covered ?

A no-brainer, you say.

The ant takes 2.7*10

*lot*longer)What if the string was a trillion kilometres long at the start ? Same conclusion.

Now let’s go back to the original problem. Since we know the ant does reach the other end, the new question is: What is the total distance the ant has covered ?

A no-brainer, you say.

The ant takes 2.7*10

^{43}seconds to reach, it moves 1 cm per second, so a quick multiplication gives 2.7*10^{43}centimetres, right ? Wrong !!As mentioned before, the ant gets an extra shove forward every time the string stretches.

So, the correct approach would be to calculate the

*length of the string*when the ant reaches because we know the ant has traversed this entire distance.

Well, the string stretches 1 metre every second so the correct answer is 2.7*10

^{43}

*metres*– a full one hundred times bigger than the answer we previously got.

In other words, the

*vast majority*of the total distance covered comes not from the ant’s own motion, but the stretching of the string !

Game for more ? Why not ?

Let’s imagine a second ant starts walking exactly one second after the first ant at exactly the same speed – 1 cm per second. In other words, when it starts out, the second ant is 1 cm behind the first.

Question: How far apart are they by the time the first ant reaches the other end ?

You may want to think of this for a bit before seeing the answer…

Solved ? No ? Okay then.

The trick here is to realize that since the ants walk at the same speed, the distance between them

*only*changes when the string is stretched.

By how much ? Easy problem.

Since the stretching is uniform:

*Distance between Ants after Stretch*

= Length of String after Stretch/

**/**Distance between Ants before Stretch= Length of String after Stretch

*Length of String before Stretch*

Applying the same reasoning, if we define
“the end” to be “when the first ant reaches its destination”, we get:

*Distance between Ants at the End*

**/**Distance between Ants at the Start
=

*Length of String at the End*= 2.7*10**/**Length of String at the Start^{43 }In other words, the ants which started out just a centimetre apart will be 2.7*10^{43}centimetres, or about*3 million trillion light years apart*at the end !!
Just one more and we are done.

Consider a string which stretches in a
different manner.

Rather than extending by a fixed amount every second, it

Rather than extending by a fixed amount every second, it

*doubles its length*.
The ant still moves at the same speed.
Now what ?

Revisiting our approach of calculating
the fraction of the string covered by the ant after each second, we now get the
series – 1/100 * (1 + 1/2 + 1/4 + 1/8 + ….)

But now our mathematician friend has new information.

This time the series in brackets

Which means that 1/100 * (1 + 1/2 + 1/4 + 1/8 + ….) is always less than 1/50 – however long it walks, our hapless ant never even covers 2% of the string.

What if the string were longer or shorter ?

But now our mathematician friend has new information.

This time the series in brackets

*converges to 2*– in other words, as you add more and more terms you get ever closer to 2, but never reach or exceed it.Which means that 1/100 * (1 + 1/2 + 1/4 + 1/8 + ….) is always less than 1/50 – however long it walks, our hapless ant never even covers 2% of the string.

What if the string were longer or shorter ?

Unlike what happened before, now it

A quick calculation shows that if the string was 2 cm or longer, the ant never reaches.

If it was any shorter, even by a zillionth of a millimetre, it always does.

Enough about the ants, when do we get to universes ?

Just one more point to clear up first.

**make a difference.***does*A quick calculation shows that if the string was 2 cm or longer, the ant never reaches.

If it was any shorter, even by a zillionth of a millimetre, it always does.

Enough about the ants, when do we get to universes ?

Just one more point to clear up first.

When thinking about this problem, we typically visualize the starting end of the string to be stationary while the other end recedes every time.

Now consider the perspective of an observer at the destination point. From her viewpoint, the ant is always walking towards her, but every time the string stretches,

*the starting point recedes further away and the ant gets pulled back*.

Neither perspective is intrinsically “more correct” than the other, and more importantly, all the conclusions we derived so far remain valid. Now we are ready.

**Cosmic Expansion**

Everybody knows the universe is
expanding. (If you didn’t till now, well, I just told you!)

This is typically presented as a somewhat mysterious phenomenon, frequently with misleading analogies.

But the problem about the ants above offers the best way I know of understanding how it works.

Let’s take the flipped perspective I mentioned at the end.

The observer at the destination end of the string is an astronomer on earth.

The starting end is a distant galaxy.

This is typically presented as a somewhat mysterious phenomenon, frequently with misleading analogies.

But the problem about the ants above offers the best way I know of understanding how it works.

Let’s take the flipped perspective I mentioned at the end.

The observer at the destination end of the string is an astronomer on earth.

The starting end is a distant galaxy.

The ant is light from the galaxy speeding
towards the earth.

The stretching string? That is the space between the earth and galaxy which is expanding.

Many popular articles often have statements like “The galaxies are receding from us because they were thrown apart by the explosion of the Big Bang”.

Nothing of the sort ! The Big Bang wasn’t at all like the explosion of a bomb throwing shrapnel everywhere.

Cosmic expansion happens because

The stretching string? That is the space between the earth and galaxy which is expanding.

*This*is the correct way to understand cosmic expansion.Many popular articles often have statements like “The galaxies are receding from us because they were thrown apart by the explosion of the Big Bang”.

Nothing of the sort ! The Big Bang wasn’t at all like the explosion of a bomb throwing shrapnel everywhere.

Cosmic expansion happens because

*space itself*is stretching, just like our elastic string.
What if you want to look at a different
galaxy?

Just consider a different stretching
string with a different starting length.

Does the starting length matter ? Now we need to get into details….

We started with a string stretching uniformly by a metre every second.

This corresponds to the case where space expands

(Okay, I am cheating a bit. The expansion of the string was “discretized” as an instantaneous stretch at the end of every second, while cosmic expansion is continuous in time.

Does the starting length matter ? Now we need to get into details….

We started with a string stretching uniformly by a metre every second.

This corresponds to the case where space expands

*linearly with time*.(Okay, I am cheating a bit. The expansion of the string was “discretized” as an instantaneous stretch at the end of every second, while cosmic expansion is continuous in time.

However, all the conclusions go through -
we just use calculus rather than summing series.)

Our first result was that the ant always reaches its destination, irrespective of the starting length of the string.

Translated into cosmology this means:

Our first result was that the ant always reaches its destination, irrespective of the starting length of the string.

Translated into cosmology this means:

*If space expands linearly with time (or slower), then the light from***any**galaxy, however distant, will eventually reach the Earth.

In other words, with linear expansion of space, our cosmic horizon grows ever
larger as increasingly distant galaxies come into view.

An interesting corollary is that if the universe as a whole is finite, then we would eventually see all of it – and possibly observe multiple images of the same galaxy.

Now here’s my get-rich scheme: When you read popular articles on astronomy in future, please send me a dollar every time you read something like - “Since the universe is about 13.8 billion years old, the most distant objects we can see are 13.8 billion light years away.”

See what happened there?

This is

This is, in fact, what happens.

Although the Big Bang happened about 13.8 billion years old, the most distant objects we can see in any direction are currently about

In other words, the diameter of the observable universe is currently about 93 billion light years and growing.

What about the third example where two ants get spread far apart by the stretching string ?

This is the analogue of

An interesting corollary is that if the universe as a whole is finite, then we would eventually see all of it – and possibly observe multiple images of the same galaxy.

Now here’s my get-rich scheme: When you read popular articles on astronomy in future, please send me a dollar every time you read something like - “Since the universe is about 13.8 billion years old, the most distant objects we can see are 13.8 billion light years away.”

See what happened there?

This is

*precisely*the error we made when we tried to calculate the length of the string at the end by multiplying the ant’s speed by the time taken – in reality, the string would generally be a*lot*longer.This is, in fact, what happens.

Although the Big Bang happened about 13.8 billion years old, the most distant objects we can see in any direction are currently about

*46.5 billion light years*away.In other words, the diameter of the observable universe is currently about 93 billion light years and growing.

What about the third example where two ants get spread far apart by the stretching string ?

This is the analogue of

*cosmological redshift*.
Light from a distant galaxy is an
electromagnetic wave and the ants in the analogy represent two successive
crests of the wave – the distance between them is the wavelength of light.

The expansion of the universe stretches
out light waves, increasing their wavelength.

Longer wavelength means redder light,
hence the term “redshift”.

This is no mere academic exercise.

Light from the most distant galaxies seen by the Hubble space telescope was emitted about 12 billion years ago when the universe was much smaller and as a result, they appear deep red.

In fact, the most ancient galaxies are invisible to Hubble because their light has been stretched all the way to the infrared spectrum !

The future James Webb telescope, designed with the goal of observing the earliest stars and galaxies is, therefore, equipped to detect infrared radiation.

Light from the most distant galaxies seen by the Hubble space telescope was emitted about 12 billion years ago when the universe was much smaller and as a result, they appear deep red.

In fact, the most ancient galaxies are invisible to Hubble because their light has been stretched all the way to the infrared spectrum !

The future James Webb telescope, designed with the goal of observing the earliest stars and galaxies is, therefore, equipped to detect infrared radiation.

The most ancient light we can detect – the Cosmic Background Radiation – was emitted as visible light when the universe was about 380,000 years old.

This light has been stretched even beyond the infrared spectrum, we see it as microwave radiation – a consequence of the fact that the universe has expanded by a factor of about 1,000 between then and now.

Finally, what about the string which doubles every second ?

This represents a universe where space is
expanding

In this universe, every observer has a

*exponentially*.In this universe, every observer has a

*fixed*cosmic horizon beyond which everything is forever invisible. (This is analogous to the fact that ants 2 cm or more from the end of the string never reach it)
Even worse, since the universe keeps
expanding, points which start off within the horizon get pulled beyond it and
disappear.

In stark contrast to a linearly expanding
universe where ever more distant galaxies keep coming into our view, the skies
in an exponentially universe empty out with time and eventually, every galaxy
or galaxy cluster not tightly bound together by gravity ends up completely
isolated.

So, what kind of universe do we live in ?

Well, right until the end of the 20

Well, right until the end of the 20

^{th}century, all evidence pointed to a universe expanding at a rate even slower than linear.
Then in 1998, the world of cosmology was
overturned when a team led by Saul Perlmutter established that cosmic expansion
was, in fact, “accelerating” – ie, the expansion rate was

*faster*than linear.
Currently, the best theoretical
explanation for this is the presence of a “cosmological constant” which is,
crudely put, a kind of “energy of empty space”. (Nowadays, it’s fashionable to
use the term “dark energy” instead, but given that we know nothing about what it
really is, we might as well call it “The Force”.)

However, the upshot of having a cosmological constant is that space expands exponentially, so we appear to be stuck with a cosmic horizon and emptying skies.

In fact, here is something even more striking.

However, the upshot of having a cosmological constant is that space expands exponentially, so we appear to be stuck with a cosmic horizon and emptying skies.

In fact, here is something even more striking.

Let’s revisit the problem of the doubling
string with an ant starting 1 cm away from the endpoint.

Now this ant reaches its destination in 1 second, but within that time, it starting point is now 2 cm away – which is the horizon.

Now this ant reaches its destination in 1 second, but within that time, it starting point is now 2 cm away – which is the horizon.

Translated into cosmology, this means that
by the time the light from a galaxy reaches the earth, the galaxy itself may
have already crossed the cosmic horizon.

So, while we shall see it shining in our night sky, it is already “gone for good” – beyond the reach of any spaceship or signal we send, even one travelling at the speed of light.

So, while we shall see it shining in our night sky, it is already “gone for good” – beyond the reach of any spaceship or signal we send, even one travelling at the speed of light.

Does this really happen?

Not only does this happen, but it is
estimated that

*97% of the galaxies we see*are already beyond the horizon.
So is this what lies ahead for us?

An inexorable thinning out of the
universe, as the galaxies vanish over the cosmic horizon and the skies grow
black and empty?

Alas, that is indeed our best understanding of the future of the universe, but don’t get too depressed because this is very likely not “the end of everything”.

All evidence so far, not clearly understood yet but fairly consistent, strongly suggests that what I am calling

Alas, that is indeed our best understanding of the future of the universe, but don’t get too depressed because this is very likely not “the end of everything”.

All evidence so far, not clearly understood yet but fairly consistent, strongly suggests that what I am calling

*the*universe is merely*our*universe – an infinitesimally small fragment of “All that is, or ever was, or ever will be”…