Here’s the story we like to tell about  the beginning of the universe.

Space is   expanding evenly everywhere, but if you rewind  that expansion you find that all of space was   once compacted in an infinitesimal point  of infinite density—the singularity at the   beginning of time.

The expansion of the universe  from this point is called the Big Bang.

We like   to tell this story because it's the correct  conclusion from the description of an expanding   universe that followed Einstein's general theory  of relativity back in the 1910's.

But since then   we've learned so much more.

Does our modern  understanding of the universe still insist   on a point-like Big Bang?

Recent work actually  gives us a way to avoid the beginning of time.

The key to understanding whether the universe had  a beginning is to decide whether there exists what   we call a past singularity—whether all points  in space were at the same point at time t=0.

And from the earliest models of the expanding  universe, the answer seemed yes.

When people like   Alexander Friedmann and Georges Lemaitre solved  the equations of Einstein’s general relativity   back in 19-teens and twenties, they found that,  if you rewind the expansion indeed all points   converged on one point at the start of time.

But these guys made some big assumptions—for   example that the universe is perfectly smooth  everywhere.

It’s not—it’s at least a little   lumpy or we wouldn’t have galaxies.

Could this  non-perfect-smoothness prevent our rewound   universe converging on a single point?

We’ll  come back to that—it’ll be important later.

The fact that the universe is mostly perfectly  smooth hints at something else that may change   our prediction of a beginning of time.

To explain  the general sameness of our modern vast universe,   we needed to add something called cosmic inflation  to stretch out tiny smooth patches in the early   universe into a smooth giant universe.

At first  inflation was described as a period of extreme   exponential expansion that lasted for a tiny  fraction of a second after the Big Bang.

But then   we realized that this inflationary period didn’t  have to stop everywhere—perhaps it stops in spots,   creating bubble universes like ours, in which  case there’d be a greater inflating spacetime   that continues growing everywhere.

But if this  “eternal inflation” lasts forever into the future,   could it last forever into the past?

If  so, perhaps the universe had no beginning.

So, our description of the universe has  gotten a little complicated over the past   100 years.

We’re going to need to carefully  track our path backwards in time to see if   we find a beginning.

So we choose a  coordinate system for space and time,   and we trace those coordinates into the  past they and see if they reach a point   where we cannot trace them any further.

This can  happen if some aspect of the coordinate system   “blows up”--becomes infinite.

We call such  points singularities.

In standard big bang   cosmology there’s a singulartiy in the past  where all paths in space overlap and can’t be   traced further.

This is interpreted as the time  before which there was no space.

But there’s at   least one example that you’ve heard of where a  singularity doesn’t mean the end of spacetime.

Consider the black hole.

The simplest type of  black hole is described by the Schwarzschild   metric.

If we trace the Schwarzschild coordinates  from outside the black hole in, we find they blow   up in two places.

One is expected—at the center  of the black hole we find a proper curvature   singularity—infinite spacetime warping.

But  there’s also a singularity at the event horizon.

There, it appears that the coordinate of time  blows up.

It’s as though we can’t trace time past   that horizon.

We might be tempted to interpret  this as an uncrossable boundary, but we’d be   wrong.

Just because Schwarzschild coordinates  can’t cross this boundary doesn’t mean space   and time end there.

Sorry, you don’t bounce off  the event horizon, you fall straight through it.

We only need a simple-ish coordinate shift  to create a map across the event horizon that   is free from infinities—from  singularities.

For example,   Eddington-Finklestein coordinates combine space  and time in a way that allows this continuous   mapping all the way down to the central  singularity—where we do find an infinity   that does not go away.

The event horizon is what  we call a coordinate singularity—it's similar to   how spherical coordinates have a singularity  at the poles where lines of latitude converge.

We say that spacetime is extendable  beyond the event horizon because a   coordinate shift reveals the space beyond.

In fact, if we switch to a Penrose diagram   we reveal new spaces that look like a mirror  universe and a past white hole—these are the   fully extended coordinates… because we can, in  principle, trace light rays to these places.

But the central singularity of the black  hole remains a physical singularity.

No   light ray can be traced through this  to the other side.

It’s reasonable to   state that spacetime ends at the  center of a black hole.

There’s   no coordinate shift that extends our  map beyond the central singularity.

So what about the beginning of the  universe?

Is there a singularity there,   and if so is it a real physical end-of-spacetime  singularity AKA the beginning of the universe?

Or   is it a coordinate singularity like the event  horizon, with something on the other side.

To figure this out, we need  another tool.

It’s something   we’ve discussed before: geodesic incompleteness.

So, a geodesic is the shortest  path through spacetime in general   relativity.

It describes the curved  path travelled in a gravitational field,   or the straight path traveled out of one.

Light travels something called a null geodesic,   which is a path on which time itself  does not pass for the traveller.

Geodesics are normally thought of as  continuing forever into the past and future,   in that you can keep tracing these spacetime  lines even beyond the segment travelled by a   particle.

But sometimes a geodesic will reach a  dead end where you can’t trace it any further.

These are points of geodesic incompleteness,  and they are interpreted as the literal end of   spacetime—like the end of the map.

The singularity  at the center of a black hole is an example of   such a point.

Geodesic incompleteness is how Roger  Penrose proved that black holes contain physical   singularities with his black hole singularity  theorem—something we have of course covered.

Geodesic incompleteness by itself is pretty  convincing, but we’re always left with a   nagging worry that we plotted our geodesics in  the wrong coordinate system, like with the black   hole event horizon.

Another sure sign of the  end of spacetime in the black hole singularity   is that the curvature blows up— the strength  of the gravitational field becomes infinite.

If we can find such a curvature singularity in a  way that’s independent of the coordinate system,   we can pretty much guarantee that the fabric  of spacetime meets a bad end at that point.

OK, so now that we have some sharper tools  to pry open the beginning of the universe,   let’s see what we can find.

Well, for geodesic incompleteness the answer  appears to be that the universe did have a   beginning.

All geodesics—from the arc followed  by a thrown ball to the Earth’s orbit around   the Sun—can be extended back in time and wind  up in the same infinitesimal point.

And that’s   even true if you include cosmic inflation—even  eternal inflation.

Demonstrating this past   geodesic incompleteness has typically  required certain assumptions.

For example,   many arguments have depended on the weak energy  condition, which is a sort of add-on to the main   equation of general relativity that says  you can’t have negative mass densities.

But things probably get pretty weird  when you approach the past singularity,   so it would be nice if we didn’t even have  to depend on this particular condition.

More recently in 2003, Borde, Guth and Vilenkin  made a stronger argument for geodesic past   incompleteness that didn’t require an energy  condition.

Instead, they only needed to assume   that the average expansion rate was always  positive.

This led to the Borde–Guth–Vilenkin   (BGV) theorem, which states that any universe  that has, on average, been expanding throughout   its history can’t have been expanding  forever—it must have a past boundary.

OK, great.

But can we be sure that this past  boundary is really an end … or a beginning   to spacetime?

What if spacetime is really  extendible, like it was through the black   hole event horizon?

Let’s look at the other  criteria that we used for black holes—whether   curvature becomes infinite.

If so then our  universe surely must contain a physical   singularity in its past.

If it doesn’t contain  curvature singularity then this doesn’t guarantee   that we can extend our spacetime, but we’d  need to do a bit more investigating to be sure.

This is what Geshnizjani, Ling, and  Quintin do in their paper published   last year.

They applied a curvature  test to the beginnings of a range   of different universes to check  the nature of any singularities.

The results depend on the expansion  history of the universe.

Remember   that the BGV theorem tells us that  universes that have, on average,   only expanded over their history must have  a past boundary, which might be interpreted   as a beginning.

This new study finds that  universes with different expansion histories   don’t have to begin with a singularity, and  so, perhaps, don’t have to begin.

For example,   a universe that expanded after a prior phase  of contraction or from a prior static phase   don’t have to have past singularities.

At the  same time these expansion histories also seem   to violate certain energy conditions of general  relativity, and so may not be possible at all.

This isn’t really an entirely new  finding—more a confirmation that   you need to break some aspect  of general relativity to avoid   a beginning to the universe.

Otherwise  they mostly agree with the BGV theorem.

Universe that have only ever  expanded, do have a beginning.

But in the researchers do find one unusual case  in which the BGV theorem may be challenged.

Remember that an inflating universe has  exponential expansion.

A pure exponential   function with no vertical offset looks like  this: it grows rapidly into the future,   but also plateaus approaching but never reaching  zero size in the past.

The BGV theorem tells us   that even this ever-expanding universe should  suffer geodesic incompleteness into the past.

There should be some sort of past boundary that  we’re tempted to identify as the beginning.

That boundary can be depicted  as the lower diagonal edges on   a Penrose diagram.

These represent past boundaries   of the universe if there’s no way to  extend a null geodesic beyond them.

Except, as it turns out there is a way.

This new study found that there’s a type   of spacetime that could extend to this region  beyond the past boundary.

Our universe is   pretty well described with something called the  Friedmann-Lemaitre-Robertson-Walker metric.

It’s   that perfectly smooth, flat, expanding universe  that I mentioned earlier this episode.

If you   add exponential expansion to that—whether as  cosmic inflation or as dark energy—then FLRW   metric can be considered to be a subset of  something called de Sitter space.

In fact,   as these researchers showed,  it’s possible to think of our   universe—our patch of FLRW space—as just  a segment of a larger de Sitter space.

The details of this are a bit much  for now, but long-story-short:   the BGV theorem insists that our universe  has a past boundary.

If there’s no way to   map a null geodesic across that boundary then we  should interpret the boundary as a beginning of   time.

But the new work shows there may be a  place for those null geodesics to travel to,   or have traveled from—the larger de Sitter  space that extends beyond our FLRW universe.

If our universe is extendable in this way,  that might just turn the past boundary of our   universe into a coordinate singularity rather than  a real, physical one with infinite curvature etc.

Now this extended spacetime may  be - probably is - as illusory   as the white hole and mirror universe of  the Penrose diagram.

But it’s intriguing   that we have here a case where it’s not  just blatantly obvious that the universe   rewinds into a non-traversable curvature  singularity at least in this one case.

A word of caution though.

In order to make  this work, our section of the universe—the   FLRW section—has to transition smoothly  into the de Sitter space.

The only way   for it to do that is for the exponentially  accelerating component—often described as   the cosmological constant—to dominate over  any density fluctuations at the beginning of   “our” universe.

If those density fluctuations  are very small then spacetime is extendable   past the past boundary of our universe and  we may not have a physical past boundary.

But any significant density fluctuations  would actually close off the boundary and   should turn this transition into a hard  curvature singularity—eliminating the   possibility that there’s anything “before”  it.

And our universe definitely had   density fluctuations at very, very early  times—otherwise we would not have galaxies   and planets and YouTubers today.

It’s  interesting that the same lumpiness that   Friedman et al glossed over when they predicted  the big bang may make the big bang more likely.

OK, that bodes poorly for an infinite past for  the universe.

There probably was a beginning of   time.

But the resolution of this is buried in the  unknowns of inflationary cosmology.

It also awaits   our theory of quantum gravity, because, even  if the universe approaches a point of infinite   density in the past, our current understanding of  physics breaks down before we reach that point.

The most stunning thing about this sort  of work isn’t even the answer—whether   or not there was a beginning to  the universe.

And by the way,   the current answer is still probably yes.

To me  the most incredible thing is that we can even   hope to answer questions that seem like they  should be far beyond our grasp.

That through   pure reason we can learn one way or another  whether there was a beginning to spacetime.