Kurt Godel broke everything. Most famously, he broke mathematics with his incompleteness theorem,
showing that all formal systems contain true statements that can’t be proved within said
system. In studying for his US naturalization interview, he claims to have discovered a legal
means by which the US republic could be transitioned into a fascist dictatorship.
His good friend Albert Einstein along with Oskar Morgenstern insisted on accompanying him to his
interview in the hope of stopping him blurting out his proof - which he very nearly did anyway. Oh,
and speaking of Einstein—once, as a birthday present, Godel gave him a time machine
universe—a solution to the Einstein equations that proved that even general relativity was broken.
energy mugs and shirts are all still available. Link’s in the description. Now, onto the episode.
It’s not an exaggeration to say that Godel and Einstein were best friends. The aging Einstein
once said that the only reason he continued to show up at Princeton was for his walks with
Godel. And so when the occasion of Einstein’s 70th birthday approached, it’s no surprise that
Godel enthusiastically agreed to contribute to a book of tributes to Einstein’s work.
At first Godel thought to write a reflection on the relationship between
Einsteinian spacetime and the ideas of the philosopher Immanuel Kant. As was his wont,
Godel fell down a rabbit hole and in the process of writing this one essay, he came to understand
general relativity as well as anyone in the world. And then he stumbled on something deeply
worrying—an inconsistency at the core of his best friend’s theory. The rabbit hole deepened.
The deadline for the essay came and went—but so important was Godel’s contribution that the
book had to wait. When it was finally done, the reason for Godel’s obsession became clear. He’d
found a solution to the field equation—the core of general relativity—that proves that
Einstein’s theory by itself does not guarantee a clean chain of cause and effect that it was
supposed to. He’d built a spacetime metric—a universe—where any point in space and time
there are return loops that bring you back to the location before you left.
By the time of Godel’s discovery we already knew of solutions to the Einstein equation
that allow time travel. But all of them require something that can be argued is
impossible—negative energy density. So, just add a prohibition of this impossible stuff
and the broken solutions go away. This prohibition is called the weak energy
condition. With the Einstein equation plus the weak energy condition, a deterministic
universe with unambiguous causal ordering seemed guaranteed. That is until Godel. He
found a new way to time travel and to break causal structure—no impossible ingredients required—and
the result is the Godel universe, where time travel isn’t just possible—it’s inevitable.
Let’s take a trip into the Godel universe, first by time traveling back to relativity 101.
In relativity theory, space and time are not cleanly separated—they’re
both part of a 4-dimensional object called spacetime. Depending on motion and gravity,
space and time become mixed, trading into each other. The classic way to depict this is with
a spacetime diagram, with just one dimension of space. Everything moves up through time,
but if you’re also moving through space you move on a sloped line. Shallower is faster,
and 45 degrees represents the speed of light. Because nothing can travel faster than light,
you can only ever influence parts of the future in this cone—your forward lightcone. Meanwhile,
your past lightcone represents the parts of the past that could have had an influence on
you. No possible signal could connect you to regions outside these cones.
So you can travel or can have traveled any path inside these cones. We call
these “time-like paths” because on these, your motion up through time is greater
than your motion sideways through space. Paths into the forbidden zones are called
space-like—more space than time is traversed in these, and that, supposedly, is impossible.
But let’s try anyway. Say we jump in our rocketship and accelerate away from the Earth,
quickly reaching a large fraction of lightspeed. Relativity tells us that
my time axis shifts relative to the direction of the axis back on Earth. From Earth’s perspective,
my time and space coordinates get mixed—my clock slows and my length contracts. If
I actually reach the speed of light, my clock stops, and if I can go faster … well, from the
perspective of some observers my clock actually reverses and I appear to move backwards in time.
This is equivalent to breaking free of your light cone. Once you can do that,
the past is accessible to you. And a long time ago we did some episodes that showed how this works.
The impossibility of superluminal motion is directly connected to the impossibility of time
travel in special relativity—both are baked in via the Lorentz transformation. This is
absolute protection of the causal structure of spacetime in the absence of gravity.
Gravity changes things because it bends the path of light, so it can tip our future light cone. For
example, approaching the black hole light paths are bent, and so our lightcone bends towards
the event horizon until our entire accessible future lies inside the black hole. Below the
event horizon the light cone tips fully sideways, which means the “down” direction becomes your new
time coordinate. Now, if you could turn around and keep your lightcone aligned like this you really
could travel back in time. But in a black hole you can’t turn around your future is only down.
Well, let me actually revise that. In a rotating black hole—a Kerr black hole—there’s a region
deep within where angular motion becomes timelike. Space is spinning so quickly
that a circular path gets you back to where you started at the same time as
you left. We call this a closed timelike curve—a CTC. It’s timelike in this case
because you never violated relativity by leaving your forward light cone.
These regions of the Kerr black hole may not really even exist, and in any case it’s a
pretty useless sort of time travel, because your stuck behind the event horizon anyway.
We can force closed timelike curves into the accessible parts of the universe by bending
spacetime in various exotic ways(wormholes, warp drives etc) but in all cases these require some
sort of negative energy. But then came Godel of course. Even before Roy Kerr came up with
the rotating black hole solution, Godel realized that rotational motion was the key to time travel.
The general effect he relies on is called frame dragging—the twisting of spacetime
due to a rotating mass. And we’ve measured this. Gravity probe B sent a gyroscope in
orbit around the Earth. A gyroscope is supposed to always point in the same direction. In fact,
Godel calls the device an “inertial compass” in his paper. But the gyroscope in Gravity Probe B
slowly swiveled from its original pointing as it orbited due to both the gravitational
curvature and the twisted spacetime—the frame dragging caused by Earth’s rotation.
This angular shift is accompanied by a time shift, but in the case of Earth’s
frame dragging is a far too weak an effect for that component to be measured. But in principle,
if you travel around Earth the right way your clock slows due to the angular motion.
It’s never enough to freeze time altogether though. Even around a rapidly rotating black
hole there are no closed timelike curves on the outside of a black hole—only deep within.
But Godel realised that this frame dragging effect could still break causality if it
happens in a spacetime where the events can add up. Now the key is to describe a
spacetime which has a fundamental twist to it everywhere—all points feel frame dragging—not
just around a single spinning objects. We say it has global vorticity. All spacetime points
rotate relative to neighbors—not like every spot spinning, but rather like it's a vortex
of worldlines in an infinite 4-D spacetime that has no center. There are other requirements—a
negative curvature “hyperbolic” geometry is needed to allow the center-less rotation, and
a balance of smooth positive matter and negative dark energy to keep the universe static in size.
But tuned right, Godel showed that such a universe can be full of closed timelike curves.
Let’s see what it looks like. This saddle shape is a representation of a 2-D hyperbolic space.
We’re reserving the real Z direction for time so that we can stack these planes to represent a 4-D
universe. Zooming in it looks flat locally, but the negative curvature is still there. You don’t
notice anything weird moving through this space, even though the underlying twisting is there.
You notice this when time moves forward, and neighboring particles follow twisted paths.
From any point in this space, as you travel outwards your lightcone tilts relative to
the reference frame of your starting point. And beyond a certain distance—the Godel horizon—it
has tilted far enough to allow travel into the past. Or at least, to claw back some
of the time you spent traveling. But with a carefully planned route, traveling in a loop,
you can steer your future light cone to eventually contain your starting point.
Another way to think about it is that, by mixing time with the angular coordinate,
your motion in a loop takes the place of and even reverses temporal motion. That’s right,
Godel invented the time-turner—turn enough and you travel backwards in time. But with
restrictions. You have to travel out beyond the Godel horizon from your starting point,
which means you can’t just spin on the spot and move into the past.
The time-traveling possibilities of the Godel universe are fun, but they aren’t the real point.
The real point is that this solution to the Einstein equation is a counter-example to the
idea that baseline general relativity—the Einstein equation plus the weak energy condition—are enough
to ensure sensible universes. Before this, any valid GR spacetime could be sliced into layers of
consecutive “nows”, where the exact configuration of everything—the particles, the fields—of each of
those slices can be used to generates the next slice. This perfect global determinism isn’t
true in Godel’s universe, and the past and future become tangled and not clearly definable. Godel
proved that general relativity doesn’t guarantee spacetimes that have a rigid causal ordering. At
least in some cases, like in Godel’s universe, we can’t say whether A caused B or B caused A.
What Godel really showed was that the prohibition against negative energy wasn’t enough to ensure
sensible universes. Others came along and proposed new conditions. “Global Hyperbolicity”
was proposed as an explicit requirement—which has nothing to do with hyperbolic geometries.
It basically states that for any physically reasonable solution to general relativity,
any constant-time slice must fully determine the next constant time slice, and that has to
be true however you do that time slicing. Then came Stephen Hawking’s Chronology Protection
Conjecture, which argues that any spacetime that allows closed timelike curves is unstable—feedback
from time-traveling vacuum energy will cause the ultimate reverb and collapse the universe.
So, that’s really what Godel gave his friend Einstein for his birthday—the
impetus for future generations of physicists to keep working on Einstein’s greatest theory.
Godel didn’t break GR, but he showed the rest of us where some of the few
remaining cracks are. Cracks which may, in time, lead new Einsteins and new Godels to
deeper theories. Really quite the birthday gift, Godel’s time-turning spacetime.
remaining cracks are. Cracks which may, in time, lead new Einsteins and new Godels to
deeper theories. Really quite the birthday gift, Godel’s time-turning spacetime.