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 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.
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