by Thorne, Kip
Fig. 28.2. An icon representing the Ranger sandwiched between Gargantua’s infalling and outflying singularities. The Ranger is drawn far larger than it should be, so you can see it.
When I explained the two singularities to Chris, he immediately knew which one should hit the Ranger. The outflying singularity. Why? Because Chris had already adopted, for Interstellar, a variant of the laws of physics that prevents physical objects from ever traveling backward in time (Chapter 30). The infalling singularity is produced by stuff that falls into Gargantua long after Cooper falls in (long after, as measured by the external universe’s time; Earth’s time). If Cooper is hit by that singularity and survives, the universe’s far future will be in his past. He will be so far in our future that, even with the help of bulk beings, he won’t be able to return to the solar system until billions of years after he left, if ever. That would prevent him from ever reuniting with his daughter, Murph.
So Chris firmly chose Cooper to be hit by the outflying singularity, not the infalling one—hit by the singularity arising from stuff that fell into Gargantua before the Ranger, not after it.
Chris’s choice, though, presents a bit of a problem for my scientist’s interpretation of the movie. But not a problem so severe as backward time travel. If the Ranger falls directly into Gargantua from the critical orbit, then its infall is slow enough that the infalling singularity will catch up to it and hit it. For the Ranger to hit the outflying singularity instead, as Chris wants, the Ranger must nearly outrun the infalling singularity, which is descending at the speed of light. The Ranger can do so, if it is given a large, inward kick. How? The usual: by a slingshot around a suitable intermediate-mass black hole soon after leaving the Endurance.
What Does Cooper See Inside Gargantua?
Looking up as he falls inward, Cooper sees the external universe. Because his infall has been sped up, he sees time in the external universe flow at roughly the same rate as his own time,51 and he sees the image of the external universe reduced in size,52 from about half of the sky to roughly a quarter.
When I was first shown the movie’s depiction of this, I was pleased to discover that Paul Franklin’s team got it right, and also got right something I had missed: In the movie, the image of the universe above is surrounded by Gargantua’s accretion disk (Figure 28.3). Can you explain why this must be so?
Cooper sees all this above him, but he doesn’t see the infalling singularity. It is moving downward toward him at the speed of light, chasing but not catching the light rays that bring him images of the disk and universe above.
Because we are rather ignorant of what goes on inside black holes, I told Chris and Paul that I’d be comfortable if they used their imaginations in depicting what Cooper sees coming up at him from below, as he falls. I made only one request: “Please don’t depict Satan and the fires of Hades inside the black hole like the Disney Studios did in their Black Hole movie.” Chris and Paul chuckled. They weren’t tempted in the least.
When I saw what they did depict, it made great sense. Looking downward, Cooper should see light from objects that fell into Gargantua before him and are still falling inward. Those objects need not emit light themselves. He can see them in reflected light from the accretion disk above, just as we see the Moon in reflected sunlight. I expect those objects to be mostly interstellar dust, and this could explain the fog he encounters in the movie as he falls.
Fig. 28.3. The universe above, surrounded by the accretion disk, as seen by Cooper inside Gargantua, looking upward across his Ranger’s fuselage. Gargantua’s shadow is the black region on the left. [From Interstellar, used courtesy of Warner Bros. Entertainment Inc.]
Cooper can also overtake stuff that’s infalling more slowly than he. This may explain the white flakes that hit and bounce off his Ranger in the movie.
Rescued by the Tesseract
In my science interpretation, as the Ranger nears the outflying singularity, it encounters mounting tidal forces. Cooper ejects just in the nick of time. Tidal forces tear the Ranger apart. Visually, it splits in two.
At the singularity’s edge the tesseract awaits Cooper—placed there, presumably, by bulk beings (Figure 28.4).
Fig. 28.4. An icon representing Cooper about to be scooped up by the tesseract at the edge of the singularity. The Ranger icon and Cooper icon are drawn far, far larger than they should be, so you can see them, and are drawn two-dimensional, since one space dimension is suppressed from this diagram.
* * *
51 In technical language, signals from above are Doppler shifted to the red by his high speed, which compensates the blue shift produced by the hole’s gravitational pull, so colors look fairly normal.
52 Due to abberration of the starlight.
29
The Tesseract
In Interstellar, the entrance to the tesseract is a white checkerboard pattern. Each white square is the end of a beam. Cooper, entering the tesseract, falls down a channel between beams, dazed and confused, lashing out at what appear to be bricks along the channel wall, but turn out to be books. The channel leads to a large chamber, where he floats and struggles, gradually getting oriented.
The chamber is Christopher Nolan’s unique take on one three-dimensional face of the four-dimensional tesseract, enhanced by Paul Franklin and his visual-effects team. The chamber and its environs are remarkably complex. Seeing them for the first time, I felt as disoriented as Cooper, even though I know what a tesseract is. Chris and Paul had enriched the tesseract so greatly that I only fully understood after talking with them.
Here’s what I know—and what I learned, filtered through my physicist’s eyes. I begin with the standard, simple tesseract, and then I build up to Chris’s complexified tesseract.
From Point to Line to Square to Cube to Tesseract
A standard tesseract is a hypercube, a cube in four space dimensions. In Figures 29.1 and 29.2 I walk you through what this means.
If we take a point (top of Figure 29.1) and move it in one dimension, we get a line. The line has two faces (ends); they are points. The line has one dimension (it extends along one dimension); its faces have one less dimension: zero.
Fig. 29.1. From point to line to square to cube.
If we take a line and move it in a dimension perpendicular to itself (middle of Figure 29.1), we get a square. The square has four faces; they are lines. The square has two dimensions; its faces have one less dimension: one.
If we take a square and move it in a dimension perpendicular to itself (bottom of Figure 29.1), we get a cube. The cube has six faces; they are squares. The cube has three dimensions; its faces have one less dimension: two.
The next step should be obvious, but to visualize it, I need to redraw the cube as you would see it if you were up close to one of the orange faces (top of Figure 29.2). Here the original square (the small, dark orange one), when moved toward you to form the cube, appears to enlarge to become the cube’s front face, the outer square.
Fig. 29.2. From cube to tesseract.
If we take a cube and move it in a dimension perpendicular to itself (bottom of Figure 29.2), we get a tesseract. The picture of the tesseract is analogous to the one above it, of the cube: It looks like two cubes, inside each other. The inner cube has expanded outward, in the picture, to sweep out the four-dimensional volume of the tesseract. The tesseract has eight faces; they are cubes. (Can you identify and count them?) The tesseract has four space dimensions; its faces have one less dimension: three. The tesseract and its faces share one time dimension, not shown in the picture.
The chamber Cooper enters in the film is one of the tesseract’s eight cubical faces, though, as I said earlier, modified in a clever, complex way by Chris and Paul. Before explaining their clever modifications, I use the standard, simple tesseract to describe my interpretation of the movie’s early t
esseract scenes.
Cooper Transported in the Tesseract
Because Cooper is made of atoms held together by electric and nuclear forces, all of which can exist only in three space dimensions and one time, he is confined to reside in one of the tesseract’s three-space-dimensional faces (cubes). He can’t experience the tesseract’s fourth spatial dimension. Figure 29.3 shows him floating in the tesseract’s front face, whose edges I delineated by purple lines.
Fig. 29.3. A Cooper icon in a three-dimensional face of the tesseract.
In my interpretation of the movie, the tesseract ascends from the singularity into the bulk. Being an object with the same number of space dimensions as the bulk (four), it happily inhabits the bulk. And it transports three-dimensional Cooper, lodged in its three-dimensional face, through the bulk.
Now, recall that the distance from Gargantua to Earth is about 10 billion light-years as measured in our brane (our universe, with its three space dimensions). However, as measured in the bulk, that distance is only about 1 AU (the distance from the Sun to the Earth); see Figure 23.7. So, traveling with whatever propulsion system the bulk beings provided, the tesseract, in my interpretation, can quickly carry Cooper across our universe, via the bulk, to Earth.
Figure 29.4 is a snapshot from that trip. One spatial dimension is suppressed from the snapshot, so the tesseract is a three-dimensional cube in a three-dimensional bulk, and Cooper has become a two-dimensional icon of a man, in a two-dimensional face of the cube, traveling parallel to our two-dimensional universe (brane).
To match what is shown in the movie, I imagine this trip is very quick, just a few minutes, while Cooper is still dazed and falling. As he comes to rest, floating in the large chamber, the tesseract docks beside Murph’s bedroom.
Fig. 29.4. The Cooper icon transported through the bulk, above our brane, riding in a face of the tesseract. One space dimension is removed from this picture.
Docking: The View into Murph’s Bedroom
How does this docking work? In my interpretation, arriving in the bulk near Earth the tesseract must penetrate the 3-centimeter-thick AdS layer that encases our brane (Chapter 23) in order to reach Murph’s bedroom. Presumably the bulk beings who built the tesseract equipped it with technology to push the AdS layer to the side, clearing the way for its descent.
Figure 29.5 shows the tesseract, after the clearing, docked alongside Murph’s bedroom in Cooper’s farmhouse. Again, one spatial dimension is suppressed, so the tesseract is depicted as a three-dimensional cube and the farmhouse and bedroom and Murph are two dimensional, as, of course, is Cooper.
Fig. 29.5. The tesseract docked alongside Murph’s bedroom.
The back face of the tesseract coincides with Murph’s bedroom. I’ll explain that more carefully. The back face is a three-dimensional cross section of the tesseract that resides in Murph’s bedroom in the same sense as the circular cross section of a sphere resides in a two-dimensional brane in Figure 22.2, and a spherical cross section of a hypersphere resides in a three-dimensional brane in Figure 22.3. So everything in Murph’s bedroom, including Murph herself, is also inside the tesseract’s back face.
When a light ray traveling out from Murph reaches the common edge of Murph’s bedroom and the tesseract, it has two places to go: The ray can stay in our brane, traveling along route 1 of Figure 29.5 out an open door or into a wall where it is absorbed. Or the ray can stay in the tesseract, traveling along route 2 into and through the next tesseract face, and then onward to Cooper’s eyes. Some of the ray’s photons go along route 1; others go along route 2, bringing Cooper an image of Murph.
Now look at Figure 29.6, in which I restore the suppressed dimension. When Cooper looks through the right wall of his chamber, he sees into Murph’s bedroom through its right wall (right white light ray). Looking through the left wall of his chamber, Cooper sees into Murph’s bedroom through its left wall (left white light ray). Looking through his back wall, he sees into the bedroom through its back wall. Looking through his front wall (orange light ray), he sees into the bedroom through its front wall (though this is not obvious in Figure 29.6; can you explain why it is true?). Looking along the yellow ray, he sees down through her ceiling. Looking along the red ray, he sees up through her floor. To Cooper, as he changes his gaze from one direction to another to another, it seems like he is orbiting Murph’s bedroom. (This is how Chris described it when he first showed me his complexified tesseract.)
Fig. 29.6. The Cooper icon can see into Murph’s bedroom (orange edges) by looking through each of the six walls of his face of the tesseract (purple edges). Here he sees an icon of Murph herself.
In Figure 29.6, all six light rays have to pass through intermediate cubes (tesseract faces) before reaching Murph’s bedroom. In the movie they don’t travel any noticeable distance from chamber to bedroom, so Chris and Paul must have shrunk the tesseract in one dimension; see the gray arrow and notation “make thin” in Figure 29.6.
After that shrinkage, every face of Cooper’s chamber looks directly and immediately into one of the faces (wall or floor or ceiling) of Murph’s bedroom with no intervening space, so to Cooper the situation looks like Figure 29.7. He sees six bedrooms, one bordering each face of his chamber but all identical except for his viewing direction.53 In fact they are all identical. There is only one bedroom, although to Cooper there appear to be six.
Fig. 29.7. The six views of Murph’s bedroom seen by the Cooper icon from his tesseract face. [My own hand sketch.]
Nolan’s Complexified Tesseract
Figure 29.8 is a still, showing Cooper floating in his chamber inside the tesseract. It looks very different from Figure 29.7 because of the complex and rich modifications that Chris conceived, and Paul and his team implemented.
Fig. 29.8. Cooper floating in Nolan’s complexified tesseract. [From Interstellar, used courtesy of Warner Bros. Entertainment Inc.]
The first thing I noticed when I saw Chris’s complexified tesseract was the threefold enlargement of Cooper’s chamber, so the bedroom attached to each chamber face covers only a third of the face. I depict this in Figure 29.9 with all the other tesseract complexities removed and the chamber’s back three faces hidden from view.54
Fig. 29.9. The size of Cooper’s chamber enlarged threefold so the six bedrooms occupy the centers of his chamber’s faces. [My own hand sketch.]
The next thing I noticed were two extrusions extending out of each bedroom along the two directions transverse to Cooper’s chamber (Figures 29.10 and 29.11). As Chris and Paul explained it to me, wherever these extrusions intersect there is a bedroom; for example, bedrooms 7, 8, and 9 as well as the original 1–6.
Fig. 29.10. Extrusions extend out of all the bedrooms, and time flows along them. [My own hand sketch.]
The extrusions extend indefinitely, creating at their intersections a seemingly infinite lattice of bedrooms and of chambers55 like Cooper’s [dashed edges in Fig. 29.10.]. For example, the labeled faces of bedrooms 7, 8, and 9 face into a chamber whose edges are indicated with dots; the back-left-bottom corner of that chamber overlaps the front-right-top corner of Cooper’s chamber.
TARS gives us a clue to the meaning of the extrusions and the latticework of bedrooms and chambers when he tells Cooper, “You’ve seen that time is represented here as a physical dimension.”
Chris and Paul elaborated on that clue for me. The bulk beings, they explained, are displaying time for the blue extrusions as flowing along the blue-arrowed direction in Figure 29.10, and for the green extrusions along the green-arrowed direction, and for the brown extrusions along the brown-arrowed direction.
Fig. 29.11. The lattice of extrusions, drawn by Christopher Nolan in his working notebook when developing the concepts for the complexified tesseract.
To understand this in greater
detail, let’s focus momentarily on the single pair of extrusions that intersect in bedroom 2; see Figure 29.12. Cross sections through the room that are vertical in the picture travel rightward with passing time, along the blue time arrow; and as they travel, they create the blue extrusion. Similarly, cross sections that are horizontal travel upward as time passes, along the green time arrow, creating the green extrusion. Where the two sets of cross sections intersect—where the extrusions intersect—there is a bedroom.
Fig. 29.12. Cross sections of Murph’s bedroom travel along two extrusions. Bedroom 2 resides where the two sets of cross sections intersect. [My own hand sketch.]
The same is true for all other extrusions. At each intersection of two extrusions, the cross sections they carry produce a bedroom.
Because of the cross sections’ finite speed, the various bedrooms are out of time synch with each other. For example, if it takes one second for cross sections to travel along each extrusion from one bedroom to the next, then all the bedrooms in Figure 29.13 are to the future of image 0 by the number of seconds shown in black. In particular, bedroom 2 is one second ahead of bedroom 0, bedroom 9 is two seconds ahead of bedroom 0, and bedroom 8 is four seconds ahead of bedroom 0. Can you explain why?