Kip Thorne - The Science of Interstellar

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.

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.

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.)

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 Figure 29.7, Cooper has been turned over so he is facing the top of Murph’s head as in Figure 29.6. This suggests that in the wall images 2, 3, 4, and 5, Murph should also be turned over. However, having her upside down in four images and right side up in two would be confusing to a mass movie audience, so the wall images have not been inverted here or in the movie. In fact they are all identical. There is only one bedroom, although to Cooper there appear to be six.

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.

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] In the movie Murph’s bedroom is not a cube; its length, width, and height are 20, 15, and 10 feet, and Cooper’s chamber is three times larger in each dimension: 60, 45, and 30 feet. For simplicity, I idealize the bedrooms and chambers as cubes.