Something Disturbing Happens When You Solve Einstein's Equations This Way

0:00 Kurt Godel broke everything.

0:02 Most famously, he broke mathematics with his incompleteness theorem,

0:07 showing that all formal systems contain true

0:09 statements that can’t be proved within said system.

0:11 In studying for his US naturalization interview,

0:14 he claims to have discovered a legal means by which

0:19 the US republic could be transitioned into a fascist dictatorship.

0:24 His good friend Albert Einstein along with Oskar

0:28 Morgenstern insisted on accompanying him to his interview

0:31 in the hope of stopping him blurting out

0:34 his proof- which he very nearly did anyway.

0:38 Oh, and speaking of Einstein—once, as a birthday present,

0:42 Godel gave him a time machine universe—a solution to the Einstein

0:48 equations that proved that even general relativity was broken.

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1:20 Next up, we’re excited to launch our Universe in a Black Hole T-shit and Hoodie.

1:23 If you believe we’re trapped inside a black hole

1:25 where space and time swapped roles and the Big

1:27 Bang was just the other side of a collapse then this is the shirt for you.

1:31 We’re also still offer up our Black hole light curves desktop and gaming mat

1:34 that shows the ways light is warped by the gravity of a black hole.

1:38 Plus, our UV-glow Kerr rotating black hole hoodies,

1:40 shirt and dark energy mugs and shirts are all still available.

1:44 Link’s in the description.

1:45 Now, onto the episode.

1:46 It’s not an exaggeration to say that Godel and Einstein were best friends.

1:51 The aging Einstein once said that the only reason he continued

1:54 to show up at Princeton was for his walks with Godel.

1:58 And so when the occasion of Einstein’s 70th birthday approached,

2:02 it’s no surprise that Godel enthusiastically agreed to contribute

2:06 to a book of tributes to Einstein’s work.

2:10 At first Godel thought to write a reflection on the relationship

2:14 between Einsteinian spacetime and the ideas of the philosopher Immanuel Kant.

2:18 As was his wont, Godel fell down a rabbit

2:22 hole and in the process of writing this one essay,

2:25 he came to understand general relativity as well as anyone in the world.

2:29 And then he stumbled on something deeply worrying—an

2:32 inconsistency at the core of his best friend’s theory.

2:37 The rabbit hole deepened.

2:39 The deadline for the essay came and went—but so

2:41 important was Godel’s contribution that the book had to wait.

2:45 When it was finally done, the reason for Godel’s obsession became clear.

2:49 He’d found a solution to the field equation—the

2:53 core of general relativity—that proves that Einstein’s theory

2:56 by itself does not guarantee a clean chain

3:00 of cause and effect that it was supposed to.

3:04 He’d built a spacetime metric—a universe—where any point in space and time there

3:11 are return loops that bring you back to the location before you left.

3:17 By the time of Godel’s discovery we already knew

3:20 of solutions to the Einstein equation that allow time travel.

3:23 But all of them require something

3:26 that can be argued is impossible—negative energy density.

3:30 So, just add a prohibition of this impossible

3:35 stuff and the broken solutions go away.

3:38 This prohibition is called the weak energy condition.

3:41 With the Einstein equation plus the weak energy condition,

3:45 a deterministic universe with unambiguous causal ordering seemed guaranteed.

3:51 That is until Godel.

3:53 He found a new way to time travel and to break

3:59 causal structure—no impossible ingredients required—and

4:02 the result is the Godel universe,

4:05 where time travel isn’t just possible—it’s inevitable.

4:09 Let’s take a trip into the Godel universe,

4:13 first by time traveling back to relativity 101.

4:16 In relativity theory, space and time are not cleanly separated—they’re

4:21 both part of a 4-dimensional object called spacetime.

4:25 Depending on motion and gravity,

4:28 space and time become mixed, trading into each other.

4:31 The classic way to depict this is with a spacetime diagram,

4:35 with just one dimension of space.

4:37 Everything moves up through time,

4:39 but if you’re also moving through space you move on a sloped line.

4:43 Shallower is faster, and 45 degrees represents the speed of light.

4:47 Because nothing can travel faster than light, you can only ever influence parts

4:52 of the future in this cone—your forward lightcone.

4:55 Meanwhile, your past lightcone represents the parts of the past

4:58 that could have had an influence on you.

5:01 No possible signal could connect you to regions outside these cones.

5:07 So you can travel or can have traveled any path inside these cones.

5:13 We call these “time-like paths” because on these, your motion

5:16 up through time is greater than your motion sideways through space.

5:21 Paths into the forbidden zones are called space-like—more space than

5:25 time is traversed in these, and that, supposedly, is impossible.

5:29 But let’s try anyway.

5:31 Say we jump in our rocketship and accelerate away from the Earth,

5:34 quickly reaching a large fraction of lightspeed.

5:37 Relativity tells us that my time axis shifts relative

5:41 to the direction of the axis back on Earth.

5:45 From Earth’s perspective,

5:46 my time and space coordinates get mixed—my clock slows and my length contracts.

5:52 If I actually reach the speed of light,

5:54 my clock stops, and if I can go faster… well,

5:57 from the perspective of some observers my clock actually

6:01 reverses and I appear to move backwards in time.

6:04 This is equivalent to breaking free of your light cone.

6:08 Once you can do that, the past is accessible to you.

6:12 And a long time ago we did some episodes that showed how this works.

6:17 The impossibility of superluminal motion

6:19 is directly connected to the impossibility

6:22 of time travel in special relativity—both

6:25 are baked in via the Lorentz transformation.

6:28 This is absolute protection of the causal

6:31 structure of spacetime in the absence of gravity.

6:34 Gravity changes things because it bends the path of light,

6:38 so it can tip our future light cone.

6:41 For example, approaching the black hole light paths are bent,

6:46 and so our lightcone bends towards the event horizon

6:51 until our entire accessible future lies inside the black hole.

6:57 Below the event horizon the light cone tips fully sideways,

7:02 which means the “down” direction becomes your new time coordinate.

7:06 Now, if you could turn around and keep your lightcone

7:10 aligned like this you really could travel back in time.

7:15 But in a black hole you can’t turn around your future is only down.

7:19 Well, let me actually revise that.

7:22 In a rotating black hole—a Kerr black hole—there’s

7:25 a region deep within where angular motion becomes timelike.

7:29 Space is spinning so quickly that a circular path gets you

7:34 back to where you started at the same time as you left.

7:38 We call this a closed timelike curve—a CTC.

7:42 It’s timelike in this case because you never

7:47 violated relativity by leaving your forward light cone.

7:51 These regions of the Kerr black hole may not really even exist,

7:55 and in any case it’s a pretty useless sort of time travel,

7:59 because your stuck behind the event horizon anyway.

8:02 We can force closed timelike curves into the accessible parts

8:06 of the universe by bending spacetime in various exotic ways(wormholes,

8:10 warp drives etc) but in all cases these require some sort of negative energy.

8:16 But then came Godel of course.

8:18 Even before Roy Kerr came up with the rotating black hole solution,

8:24 Godel realized that rotational motion was the key to time travel.

8:29 The general effect he relies on is called frame

8:33 dragging—the twisting of spacetime due to a rotating mass.

8:37 And we’ve measured this.

8:39 Gravity probe B sent a gyroscope in orbit around the Earth.

8:43 A gyroscope is supposed to always point in the same direction.

8:47 In fact, Godel calls the device an “inertial compass” in his paper.

8:53 But the gyroscope in Gravity Probe B slowly

8:57 swiveled from its original pointing as it orbited

9:00 due to both the gravitational curvature and the twisted

9:05 spacetime—the frame dragging caused by Earth’s rotation.

9:09 This angular shift is accompanied by a time shift,

9:13 but in the case of Earth’s frame dragging is a far

9:17 too weak an effect for that component to be measured.

9:21 But in principle, if you travel around Earth the right

9:24 way your clock slows due to the angular motion.

9:28 It’s never enough to freeze time altogether though.

9:31 Even around a rapidly rotating black hole there are no closed

9:35 timelike curves on the outside of a black hole—only deep within.

9:40 But Godel realised that this frame dragging effect could still break causality

9:45 if it happens in a spacetime where the events can add up.

9:49 Now the key is to describe a spacetime which has a fundamental twist

9:55 to it everywhere—all points feel frame

9:59 dragging—not just around a single spinning objects.

10:03 We say it has global vorticity.

10:06 All spacetime points rotate relative to neighbors—not like every spot spinning,

10:11 but rather like it's a vortex of worldlines

10:15 in an infinite 4-D spacetime that has no center.

10:20 There are other requirements—a negative curvature “hyperbolic”

10:24 geometry is needed to allow the center-less rotation,

10:29 and a balance of smooth positive matter and negative

10:33 dark energy to keep the universe static in size.

10:37 But tuned right, Godel showed that such

10:41 a universe can be full of closed timelike curves.

10:45 Let’s see what it looks like.

10:47 This saddle shape is a representation of a 2-D hyperbolic space.

10:51 We’re reserving the real Z direction for time so

10:54 that we can stack these planes to represent a 4-D universe.

10:58 Zooming in it looks flat locally, but the negative curvature is still there.

11:04 You don’t notice anything weird moving through this space,

11:09 even though the underlying twisting is there.

11:12 You notice this when time moves forward,

11:16 and neighboring particles follow twisted paths.

11:19 From any point in this space,

11:22 as you travel outwards your lightcone tilts relative

11:25 to the reference frame of your starting point.

11:29 And beyond a certain distance—the Godel horizon—it has

11:32 tilted far enough to allow travel into the past.

11:36 Or at least, to claw back some of the time you spent traveling.

11:41 But with a carefully planned route, traveling in a loop,

11:45 you can steer your future light cone to eventually contain your starting point.

11:51 Another way to think about it is

11:53 that, by mixing time with the angular coordinate,

11:56 your motion in a loop takes the place of and even reverses temporal motion.

12:03 That’s right, Godel invented the time-turner—turn

12:06 enough and you travel backwards in time.

12:10 But with restrictions.

12:11 You have to travel out beyond the Godel horizon from your starting point,

12:15 which means you can’t just spin on the spot and move into the past.

12:19 The time-traveling possibilities of the Godel universe are fun,

12:23 but they aren’t the real point.

12:26 The real point is that this solution

12:28 to the Einstein equation is a counter-example to the idea

12:32 that baseline general relativity—the Einstein equation plus

12:35 the weak energy condition—are enough to ensure sensible universes.

12:39 Before this, any valid GR spacetime could

12:43 be sliced into layers of consecutive “nows”,

12:47 where the exact configuration of everything—the particles,

12:50 the fields—of each of those slices can be used to generates the next slice.

12:59 This perfect global determinism isn’t true in Godel’s universe,

13:04 and the past and future become tangled and not clearly definable.

13:10 Godel proved that general relativity doesn’t guarantee

13:13 spacetimes that have a rigid causal ordering.

13:17 At least in some cases, like in Godel’s universe,

13:21 we can’t say whether A caused B or B caused A.

13:26 What Godel really showed was that the prohibition

13:29 against negative energy wasn’t enough to ensure sensible universes.

13:34 Others came along and proposed new conditions.

13:39 “Global Hyperbolicity” was proposed as an explicit

13:43 requirement—which has nothing to do with hyperbolic geometries.

13:47 It basically states that for any

13:50 physically reasonable solution to general relativity,

13:52 any constant-time slice must fully determine the next constant time slice,

13:58 and that has to be true however you do that time slicing.

14:04 Then came Stephen Hawking’s Chronology Protection Conjecture,

14:07 which argues that any spacetime

14:11 that allows closed timelike curves is unstable—feedback

14:16 from time-traveling vacuum energy will cause

14:19 the ultimate reverb and collapse the universe.

14:24 So, that’s really what Godel gave his friend Einstein for his birthday—the

14:27 impetus for future generations of physicists

14:30 to keep working on Einstein’s greatest theory.

14:33 Godel didn’t break GR,

14:35 but he showed the rest of us where some of the few remaining cracks are.

14:40 Cracks which may, in time, lead new Einsteins and new Godels to deeper theories.

14:47 Really quite the birthday gift, Godel’s time-turning spacetime.